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Fixed Point Index and the Lefschetz Theorem

1 · Prerequisites

2 · Summary

This page develops the fixed point index of a smooth self-map and the Lefschetz–Hopf index formula that computes it from rational homology. The starting point is the observation that fixed points of $f$ are exactly the intersections of the graph of $f$ with the diagonal of $M \times M$ , so the local index can be read as a degree of the chart displacement $i d - f$ and, for a nondegenerate fixed point, as the sign of $\det (I - D f_{x})$ . The convention $I - D f_{x}$ is used throughout, and the page records the resulting $(- 1)^{n}$ discrepancy with references that use $D f_{x} - I$ . An isolated fixed point can be split by a small perturbation into finitely many nondegenerate fixed points carrying its index, which reduces all global statements to the transverse case.

The geometric Lefschetz number $I (f)$ is the finite index sum over the fixed points; the algebraic Lefschetz number $L (f)$ is the alternating trace of $f$ on the rational homology of $M$ . For a closed oriented manifold the diagonal-class expansion identifies $L (f)$ with the Poincaré-dual cup pairing of the graph and diagonal classes, and the nondegenerate case of the index formula follows from the sign computation $sign \det (I - D f_{x})$ . Degenerate isolated fixed points are recovered by the splitting lemma, and the identity $L (i d) = χ (M)$ links the theory to the Euler characteristic. The index formula itself is proved for every closed manifold, orientable or not, by pulling the orientation-twisted diagonal class back along the graph; the orientation double cover and its transfer record the same computation on the two-sheeted cover for maps that admit a lift. The Lefschetz fixed point theorem — $L (f) \neq 0$ forces a fixed point — is proved by approximating a continuous fixed-point-free map by a smooth fixed-point-free map and applying the index formula, and the converse is shown to fail by explicit examples with canceling local indices. A final remark recovers Poincaré–Hopf from the index formula by applying it to the normal-projection approximation of the flow of a vector field.

3 · Logical flowchart

4 · Definitions, theorems and proofs

DefinitionDefinition: Literature-sourcedProof: Not applicableOpen item page →

Isolated fixed point and local fixed point index

Definition

Let $M$ be a smooth $n$ -manifold without boundary, $n \geq 1$ (Smooth manifolds and their smooth charts), let $f : M \to M$ be a smooth map ( $C^{r}$ and smooth maps between smooth manifolds) and let $x$ be an isolated fixed point of $f$ , i.e. some neighbourhood of $x$ contains no other fixed point. Choose a smooth chart $(φ, U)$ from the smooth atlas of $M$ (Smooth manifolds and their smooth charts) with $x \in U$ and $φ (x) = 0$ (Manifold charts, coordinate domains, and coordinate functions) and $ε > 0$ such that the closed ball $\overline{B_{ε} (0)}$ lies in $φ (U \cap f^{- 1} (U))$ and $\hat{f} (u) : = φ (f (φ^{- 1} (u))) \neq u$ for $0 < ∣ u ∣ \leq ε$ , and set $g (u) : = u - \hat{f} (u)$ . The local fixed point index of $f$ at $x$ is the degree

${ind}_{x} (f) : = \deg (S^{n - 1} \to S^{n - 1}, v \mapsto \frac{g (ε v)}{∣ g (ε v) ∣}) \in Z$

of Degree of a map between oriented closed manifolds for $n \geq 2$ , both spheres carrying their boundary orientations. For $n = 1$ use the reduced degree of The reduced degree of a map into the 0-sphere: if $h (v) = g (ε v) / ∣ g (ε v) ∣$ , then ${ind}_{x} (f) = (h (+ 1) - h (- 1)) / 2$ . Radius independence follows by radial interpolation in the zero-free punctured ball, using Degree is invariant under proper smooth homotopy for $n \geq 2$ and Reduced degree into the 0-sphere is homotopy invariant and multiplicative for $n = 1$ and of the chart, the ball and the neighbourhood by The local fixed point index is independent of chart, ball and neighbourhood ↗; it uses no orientation of $M$ , because a chart change multiplies source and target orientations by the same sign. The empty sum over a fixed-point-free map is $0$ by convention, and this local-index definition is restricted to $n \geq 1$ .

Remarks

Why a radius can be chosen. Since $x$ is isolated and $φ$ is a homeomorphism with $φ (x) = 0$ , the representative $\hat{f}$ is defined on the open neighbourhood $φ (U \cap f^{- 1} (U))$ of $0$ . Isolation excludes other zeros there. For $ε$ small enough that ${∣ u ∣ \leq ε}$ lies in that neighbourhood and in the representative domain, the continuous function $g$ is nonzero on the compact sphere ${∣ u ∣ = ε}$ , so $∣ g ∣$ attains a positive minimum there (A continuous image of a compact space is compact; a continuous real-valued map on a nonempty compact space attains a maximum and a minimum; and a continuous bijection from a compact space to a Hausdorff space is a homeomorphism, clause 2) and $g (ε v) \neq 0$ on $S^{n - 1}$ ; the displayed map is then defined and smooth. Neither the chart nor $ε$ is part of the value, by the two independence statements cited above.
Convention $I - D f$ , not $D f - I$ . The displacement is $g (u) = u - \hat{f} (u)$ , i.e. $I - D f$ linearized at a fixed point. With the opposite ordering $\hat{f} (u) - u$ the value is multiplied by $(- 1)^{n}$ , the degree of the antipodal map of $S^{n - 1}$ ; Guillemin and Pollack use $d f_{x} - I$ , so their local numbers differ from the ones on this page by $(- 1)^{n}$ . All items on this page use the $I - D f$ convention.
No orientation of $M$ is used. The two spheres in the displayed map are the source and target of a single Euclidean chart expression, both oriented by the standard orientation of $R^{n}$ ; an orientation of $M$ never enters the definition, and none is required for it.

DefinitionDefinition: Literature-sourcedProof: Not applicableOpen item page →

Nondegenerate fixed point

Definition

Let $M$ be a smooth $n$ -manifold and let $f : M \to M$ be a smooth map with a fixed point $x$ ; identify $T_{f (x)} M$ with $T_{x} M$ along $f (x) = x$ , so that the differential is an endomorphism $D f_{x} : T_{x} M \to T_{x} M$ (The differential of a smooth map). The fixed point $x$ is nondegenerate when $I - D f_{x}$ is an isomorphism of $T_{x} M$ (Invertible linear maps, linear isomorphisms, and inverse linear maps), equivalently when $1$ is not an eigenvalue of $D f_{x}$ . A nondegenerate fixed point is isolated: in a chart at $x$ the displacement $u \mapsto u - \hat{f} (u)$ has invertible derivative $I - D \hat{f} (0)$ at $u = 0$ , hence is a local diffeomorphism near $0$ (The smooth inverse function theorem on manifolds) with the unique zero $0$ there, so a neighbourhood of $x$ contains no other fixed point. No choice principle is used, and $n = 0$ is allowed (then $I - D f_{x}$ is an isomorphism of the zero space and the condition is vacuous).

Remarks

Convention. The condition is written $I - D f_{x}$ , not $D f_{x} - I$ ; whether $1$ is an eigenvalue of $D f_{x}$ is insensitive to the order, since $\det (I - D f_{x}) = (- 1)^{n} \det (D f_{x} - I)$ for an $n$ -dimensional $T_{x} M$ and a determinant is nonzero exactly when the endomorphism is invertible. The sign matters for the value of the local index, not for nondegeneracy; see Isolated fixed point and local fixed point index and The index of a nondegenerate fixed point is the sign of det(I-Df).
The two readings agree with graph transversality. The equivalence of the algebraic condition with transversality of the graph of $f$ to the diagonal of $M \times M$ at $(x, x)$ is Graph-diagonal transversality is exactly fixed-point nondegeneracy, and it is what brings nondegenerate fixed points under the intersection theory of the ambient manifold. Neither statement uses an orientation of $M$ .

LemmaStatement: Literature-sourcedProof: Literature-sourcedprecheck passOpen item page →

A closed discrete subset of a compact space is finite

Statement

Let $X$ be a compact topological space (Open cover, subcover, and compact topological space; a compact subset is a subspace that is compact in its own right) and let $S \subseteq X$ be closed and discrete, meaning that for every $s \in S$ there is an open $U_{s} \subseteq X$ with $U_{s} \cap S = {s}$ (Subspace topology: the traces of the open sets, its closed sets and its bases, the continuity of the inclusion, and the characteristic property of a map into a subspace, Topology on a set, open and closed sets, clopen sets, the closed-set axiomatisation, and the coarser/finer comparison). Then $S$ is finite.

Facts & Assumptions

Given: A compact topological space $X$ and a closed subset $S \subseteq X$ such that for every $s \in S$ there is an open set $U \subseteq X$ with $U \cap S = {s}$ .

[F1]

Every open cover of a compact space has a finite subcover, possibly empty when $X = \emptyset$ ; a nonempty finite subcover can be listed as $U_{0}, \dots, U_{n}$ with $X = U_{0} \cup \dots \cup U_{n}$ (Open cover, subcover, and compact topological space; a compact subset is a subspace that is compact in its own right).

[F2]

A subset $S \subseteq X$ is closed exactly when $X ∖ S$ is open (Topology on a set, open and closed sets, clopen sets, the closed-set axiomatisation, and the coarser/finer comparison).

Proof

1.1givenF2

By [F2] the complement $X ∖ S$ is open; form the family $U : = {X ∖ S} \cup {U \subseteq X open : U \cap S is a singleton}$ , a set of open subsets of $X$ defined by comprehension, so forming it selects nothing. It is an open cover of $X$ : a point $x \notin S$ lies in $X ∖ S$ , and a point $s \in S$ lies in some open $U$ with $U \cap S = {s}$ by the hypothesis, and this $U$ is a member of $U$ .

2.1step 1.1F1∎

By [F1] the cover $U$ has a finite subcover. If it is empty, then $X = \emptyset$ and $S = \emptyset$ is finite. Otherwise list it as $U_{0}, \dots, U_{n}$ . Each $U_{i}$ is $X ∖ S$ or open with $U_{i} \cap S = {s_{i}}$ a singleton, and $X ∖ S$ meets $S$ in nothing. Intersecting the covering relation $X = U_{0} \cup \dots \cup U_{n}$ with $S$ gives $S \subseteq {s_{i} : U_{i} \cap S is a singleton}$ , a finite set, so by the listing form of finiteness (Open cover, subcover, and compact topological space; a compact subset is a subspace that is compact in its own right) the set $S$ is finite; if no $U_{i}$ has a singleton trace then $S \subseteq \emptyset$ , so $S = \emptyset$ and $S$ is finite as well.

LemmaStatement: Literature-sourcedProof: Literature-sourcedprecheck passjudge pass (gpt-6.1-sol)Open item page →

Fixed points are exactly the intersections of the graph with the diagonal

Statement

Let $M$ be a smooth manifold and $f : M \to M$ a smooth map ( $C^{r}$ and smooth maps between smooth manifolds) with graph $Γ_{f} = {(x, f (x)) : x \in M} \subseteq M \times M$ (The graph of a smooth map is an embedded submanifold) and diagonal $Δ_{M} = {(x, x) : x \in M}$ (The diagonal $Δ_{X} \subseteq X \times X$ , the diagonal map $δ_{X}$ , and the pairing $⟨ f, g ⟩$ of two maps, The diagonal is an embedded submanifold). Then for every $x \in M$ , $x \in Fix (f) ⟺ (x, f (x)) \in Γ_{f} \cap Δ_{M} ⟺ (x, x) \in Γ_{f},$ so the graph map $γ_{f} : M \to M \times M$ , $γ_{f} (x) = (x, f (x))$ , satisfies $γ_{f}^{- 1} (Δ_{M}) = Fix (f) = {x \in M : f (x) = x}$ : fixed points of $f$ are exactly the intersections of the graph with the diagonal.

Facts & Assumptions

Given: A smooth manifold $M$ , a smooth map $f : M \to M$ , its graph $Γ_{f} = {(x, f (x)) : x \in M}$ and the diagonal $Δ_{M} = {(x, x) : x \in M}$ of the product $M \times M$ ; write $Fix (f) = {x \in M : f (x) = x}$ .

[F1]

The graph $Γ_{f} \subseteq M \times M$ is an embedded submanifold of dimension $\dim M$ , and $(y, z) \in Γ_{f}$ holds exactly when $z = f (y)$ (The graph of a smooth map is an embedded submanifold).

[F2]

$Δ_{M} = {(x, x) : x \in M}$ and the graph map $γ_{f}$ , as the pairing $⟨ {i d}_{M}, f ⟩$ , satisfies $γ_{f}^{- 1} [Δ_{M}] = {x \in M : {i d}_{M} (x) = f (x)}$ (The diagonal $Δ_{X} \subseteq X \times X$ , the diagonal map $δ_{X}$ , and the pairing $⟨ f, g ⟩$ of two maps).

[F3]

The diagonal $Δ_{M}$ is an embedded submanifold of $M \times M$ (The diagonal is an embedded submanifold).

Proof

1.1givenF1F2F3

Let $x \in M$ . By [F2], $(x, f (x)) \in Δ_{M}$ holds exactly when $x = f (x)$ ; by [F1], $(x, x) \in Γ_{f}$ holds exactly when $x = f (x)$ ; and $(x, f (x)) \in Γ_{f}$ always holds by the definition of the graph in [F1], while $(x, x) \in Δ_{M}$ holds exactly when $x = x$ . Hence the three conditions $x \in Fix (f)$ , $(x, f (x)) \in Γ_{f} \cap Δ_{M}$ and $(x, x) \in Γ_{f}$ all say the same equation $f (x) = x$ , so they are equivalent. The intersection is taken inside the product $M \times M$ , in which both factors are embedded submanifolds by [F1] and [F3].

2.1step 1.1F2given∎

The preimage of the diagonal under the graph map is $γ_{f}^{- 1} [Δ_{M}] = {x \in M : γ_{f} (x) \in Δ_{M}} = {x \in M : (x, f (x)) \in Δ_{M}}$ , which by [F2] is ${x \in M : x = f (x)} = Fix (f)$ ; this is the same set whose elements are the points $x$ with $(x, x) \in Γ_{f}$ by step 1.1, so fixed points of $f$ correspond exactly to the intersections $Γ_{f} \cap Δ_{M}$ , through the map $x \mapsto (x, f (x)) = (x, x)$ .

DefinitionDefinition: Literature-sourcedProof: Not applicableOpen item page →

Geometric Lefschetz number (index sum)

Definition

Let $M$ be a closed (compact, boundaryless) smooth $n$ -manifold, $n \geq 1$ , and let $f : M \to M$ be a smooth map all of whose fixed points are isolated (Smooth manifolds and their smooth charts, $C^{r}$ and smooth maps between smooth manifolds, Isolated fixed point and local fixed point index). Then $Fix (f)$ is finite: it is closed, being the preimage under the continuous graph map $x \mapsto (x, f (x))$ of the diagonal, which is closed in the Hausdorff space $M \times M$ (A space is Hausdorff if and only if its diagonal is closed in the square carrying the product topology, Products of smooth manifolds have a canonical product smooth structure, Continuity of a map of topological spaces at a point and globally, Fixed points are exactly the intersections of the graph with the diagonal), and it is discrete by hypothesis, so A closed discrete subset of a compact space is finite gives finiteness. The geometric Lefschetz number of $f$ is

$I (f) : = \sum_{x \in Fix (f)} {ind}_{x} (f) \in Z,$

the finite sum of the local fixed point indices of Isolated fixed point and local fixed point index; for a fixed-point-free $f$ this is the empty sum $I (f) = 0$ . The number is defined without orienting $M$ and without any choice principle. It is not asserted here to be homotopy invariant or to equal a homology trace: those are Lefschetz-Hopf index formula and The Lefschetz number is a homotopy invariant.

Remarks

Isolatedness is a hypothesis, not a conclusion. The definition applies to every smooth self-map of a closed manifold whose fixed points are isolated, degenerate or not; for a nondegenerate fixed point the index is computed by The index of a nondegenerate fixed point is the sign of det(I-Df), but the sum itself does not require nondegeneracy. A map with non-isolated fixed points, such as the identity of a positive-dimensional closed manifold, is outside this definition; its Lefschetz number is defined algebraically in Algebraic Lefschetz number via rational homology traces, and the identity of the two notions on the overlap is Lefschetz-Hopf index formula.
Discreteness from the subspace topology. "Isolated" means that every $x \in Fix (f)$ has a neighbourhood meeting $Fix (f)$ only in $x$ , i.e. that ${x}$ is open in the subspace $Fix (f)$ ; this is the discreteness hypothesis of A closed discrete subset of a compact space is finite, which is what makes the displayed sum finite.

LemmaStatement: Literature-sourcedProof: Literature-sourcedprecheck passjudge pass (gpt-6.1-sol)Open item page →

Graph-diagonal transversality is exactly fixed-point nondegeneracy

Statement

Let $M$ be a smooth $n$ -manifold, $f : M \to M$ smooth, $Γ_{f}$ its graph and $Δ_{M}$ the diagonal (The graph of a smooth map is an embedded submanifold, The diagonal is an embedded submanifold), and let $γ_{f} (x) = (x, f (x))$ . For a fixed point $x$ the following are equivalent:

(i) $Γ_{f}$ and $Δ_{M}$ are transverse at $γ_{f} (x)$ (Transverse embedded submanifolds), i.e. $T_{γ_{f} (x)} Γ_{f} + T_{γ_{f} (x)} Δ_{M} = T_{γ_{f} (x)} (M \times M)$ in the canonical splitting $T_{(x, x)} (M \times M) ≅ T_{x} M \oplus T_{x} M$ of Canonical tangent and cotangent splittings for products;

(ii) $I - D f_{x}$ is invertible (Invertible linear maps, linear isomorphisms, and inverse linear maps);

(iii) $x$ is a nondegenerate fixed point (Nondegenerate fixed point).

For $x \notin Fix (f)$ one has $γ_{f} (x) \notin Δ_{M}$ and transversality at $γ_{f} (x)$ holds automatically. Consequently the graph map $γ_{f}$ is transverse to $Δ_{M}$ if and only if every fixed point of $f$ is nondegenerate.

Facts & Assumptions

Given: A smooth $n$ -manifold $M$ , a smooth map $f : M \to M$ , a point $x \in M$ , the graph $Γ_{f}$ , the diagonal $Δ_{M}$ and the graph map $γ_{f} = ⟨ {i d}_{M}, f ⟩$ .

[F1]

$Γ_{f}$ and $Δ_{M}$ are embedded submanifolds of $M \times M$ of dimension $n$ ; the first projection restricts to a smooth bijection with smooth inverse $γ_{f}$ on $Γ_{f}$ , and likewise the diagonal map $δ_{M} = ⟨ {i d}_{M}, {i d}_{M} ⟩$ is a smooth bijection onto $Δ_{M}$ with smooth inverse the first projection (The graph of a smooth map is an embedded submanifold, The diagonal is an embedded submanifold, The diagonal $Δ_{X} \subseteq X \times X$ , the diagonal map $δ_{X}$ , and the pairing $⟨ f, g ⟩$ of two maps).

[L1]

The map $γ_{f}$ is smooth with components $π_{0} \circ γ_{f} = {i d}_{M}$ , $π_{1} \circ γ_{f} = f$ , and the canonical splitting identifies $T_{(x, x)} (M \times M)$ with $T_{x} M \oplus T_{x} M$ through the differentials of the two projections; the chain rule computes differentials of composites (Products of smooth manifolds have a canonical product smooth structure, Canonical tangent and cotangent splittings for products, The chain rule for differentials of smooth maps, The differential of a smooth map).

[L2]

$S ⋔ T$ at $p$ means $T_{p} S + T_{p} T = T_{p} M$ (Transverse embedded submanifolds).

[L3]

An endomorphism of a finite-dimensional space is surjective if and only if it is injective, by Rank-nullity: $\dim_{F} V = nullity T + rank T$ ; and $I - D f_{x}$ is invertible exactly when it is bijective, by Invertible linear maps, linear isomorphisms, and inverse linear maps.

Proof

1.1givenF1L1

Tangents of graph and diagonal. Since the first projection restricts to a diffeomorphism $Γ_{f} \to M$ with inverse $γ_{f}$ by [F1], its differential identifies $T_{γ_{f} (x)} Γ_{f}$ with the image of $d (γ_{f})_{x}$ ; by [L1] and the chain rule, the components of $d (γ_{f})_{x} (v)$ are $d π_{0} (d γ_{x} (v)) = d ({i d}_{M})_{x} (v) = v$ and $d π_{1} (d γ_{x} (v)) = d f_{x} (v) = D f_{x} v$ , so $T_{γ_{f} (x)} Γ_{f} = {(v, D f_{x} v) : v \in T_{x} M}$ . The same computation for $δ_{M}$ gives $T_{(x, x)} Δ_{M} = {(v, v) : v \in T_{x} M}$ .

2.1step 1.1L2L3

The sum of the two tangent spaces. Writing vectors of the splitting as pairs, a pair $(a, b)$ lies in $T_{γ_{f} (x)} Γ_{f} + T_{(x, x)} Δ_{M}$ exactly when there are $u, v \in T_{x} M$ with $(a, b) = (v, D f_{x} v) + (u, u) = (u + v, u + D f_{x} v)$ , i.e. exactly when $b - a = (D f_{x} - I) v$ is in the image of $D f_{x} - I$ . Hence the sum is all of $T_{x} M \oplus T_{x} M$ if and only if $D f_{x} - I$ is surjective; as an endomorphism of the finite-dimensional space $T_{x} M$ this holds if and only if $D f_{x} - I$ is invertible by [L3], and $D f_{x} - I$ is invertible if and only if $I - D f_{x}$ is.

3.1step 2.1given∎

Conclusion. For a fixed point $x$ , clause (i) holds if and only if the sum of step 2.1 is the whole tangent space, i.e. if and only if $I - D f_{x}$ is invertible, which is clause (ii), and this is the definition of nondegeneracy, clause (iii). If $x \notin Fix (f)$ , then $γ_{f} (x) = (x, f (x)) \notin Δ_{M}$ because $f (x) \neq x$ , so there is no point of $Γ_{f} \cap Δ_{M}$ over $x$ and the transversality condition at $γ_{f} (x)$ is vacuous; the equivalence for the graph map therefore reduces to the fixed points, giving the stated global criterion. No orientation of $M$ , no metric and no choice principle is used.

LemmaStatement: Literature-sourcedProof: Literature-sourcedprecheck passOpen item page →

The local fixed point index is invariant under conjugation by a local diffeomorphism

Statement

Let $M$ be a smooth $n$ -manifold without boundary, $n \geq 1$ , let $f : M \to M$ be smooth with an isolated fixed point $x$ , let $h : N \to M$ be a diffeomorphism from a neighbourhood of $y \in N$ onto a neighbourhood of $x$ with $h (y) = x$ , and let $f^{'} : = h^{- 1} \circ f \circ h$ (defined near $y$ ) have the isolated fixed point $y$ . Then

${ind}_{y} (f^{'}) = {ind}_{x} (f)$

(Isolated fixed point and local fixed point index). In particular, if $π : \tilde{M} \to M$ is a smooth covering map that is a local diffeomorphism, and $\tilde{f}$ is a smooth lift of $f$ ( $π \circ \tilde{f} = f \circ π$ ) and $\tilde{x}$ is a fixed point of $\tilde{f}$ with $π (\tilde{x}) = x$ , then ${ind}_{\tilde{x}} (\tilde{f}) = {ind}_{x} (f)$ whenever $x$ is an isolated fixed point of $f$ .

Facts & Assumptions

Given: Smooth manifolds $M, N$ without boundary, a smooth map $f : M \to M$ with isolated fixed point $x$ , a local diffeomorphism $h$ as above, and $f^{'} = h^{- 1} f h$ with isolated fixed point $y$ .

[F1]

For an isolated fixed point $z$ of a smooth self-map $F$ of an $n$ -manifold, a chart $(χ, W)$ with $χ (z) = 0$ and an admissible radius $ε$ give ${ind}_{z} (F)$ as the degree of $v \mapsto (u - \hat{F} (u)) (ε v) / ∣ (u - \hat{F} (u)) (ε v) ∣$ on $S^{n - 1}$ , and the value does not depend on the admissible radius (Isolated fixed point and local fixed point index): for two admissible radii the straight-line homotopy through $u - \hat{F}$ along the annulus is nowhere zero, so Degree is invariant under proper smooth homotopy gives equal degrees.

[L1]

Degree is multiplicative under composition (Degree is multiplicative under composition), the radial self-map $w \mapsto A w / ∣ A w ∣$ of $S^{n - 1}$ determined by a linear isomorphism $A$ is a diffeomorphism of degree $sign \det A$ (Degree of an orientation-preserving or reversing diffeomorphism, Regular-value formula for degree), and degree is invariant under smooth homotopy of maps of spheres (Degree is invariant under proper smooth homotopy). For $n = 1$ , use reduced degree: homotopy invariance and multiplicativity are supplied by Reduced degree into the 0-sphere is homotopy invariant and multiplicative, and $ρ_{A} (v) = sign (A) v$ has reduced degree $sign (A)$ .

[L2]

A smooth map $Φ$ of an open set of $R^{n}$ satisfies $Φ (x + h) = Φ (x) + d Φ_{x} (h) + O (∣ h ∣^{2})$ as $h \to 0$ , uniformly on compact sets where the second derivatives are bounded (the Lagrange remainder of Multivariable Taylor formula with $o (∥ h ∥^{k})$ remainder is controlled by the continuity of the second derivatives on a compact neighbourhood).

Proof

1.1givenF1

Charts and setup. Choose charts $φ$ at $x$ and $ψ$ at $y$ with $φ (x) = ψ (y) = 0$ , admissible for $f$ and $f^{'}$ respectively, and put $k : = φ \circ h \circ ψ^{- 1}$ , a smooth local diffeomorphism near $0$ with $k (0) = 0$ and $A : = D k_{0}$ invertible; on a neighbourhood of $0$ the identity $\hat{f^{'}} = k^{- 1} \circ \hat{f} \circ k$ holds, and the displacement maps $g (u) : = u - \hat{f} (u)$ , $g^{'} (u) : = u - \hat{f^{'}} (u)$ vanish only at $u = 0$ in some ball. By [F1] the values ${ind}_{x} (f)$ and ${ind}_{y} (f^{'})$ are computed by the normalized sphere maps of $g$ and $g^{'}$ at any admissible radii, so it suffices to produce one common degree for such normalized maps.

1.2F1L1L2

Degree of the linearized comparison map. Choose $R > 0$ so that $g$ is defined and nonzero on $0 < ∣ w ∣ \leq R$ . Choose $r > 0$ so that $k$ is defined and injective on $∣ u ∣ \leq r$ and $k ({\overline{B}}_{r}) \subseteq B_{R}$ . Fix $0 < ε \leq r$ and an admissible radius $0 < δ \leq R$ for $g$ , so that $Φ (v) : = g (δ v) / ∣ g (δ v) ∣$ has degree ${ind}_{x} (f)$ by [F1]. For $v \in S^{n - 1}$ the point $k (ε v)$ is nonzero and satisfies $∣ k (ε v) ∣ < R$ , and $k (ε v) = ∣ k (ε v) ∣ σ (v)$ with $σ (v) : = k (ε v) / ∣ k (ε v) ∣$ . For fixed $v$ the points $k (ε v)$ and $δ σ (v)$ lie on one ray through $0$ and have moduli in $(0, R]$ , so the radial interpolation $H_{t} (v) : = g (((1 - t) ∣ k (ε v) ∣ + t δ) σ (v)) / ∣ g (((1 - t) ∣ k (ε v) ∣ + t δ) σ (v)) ∣$ , $t \in [0, 1]$ , is a homotopy of nowhere-zero maps of $S^{n - 1}$ from $v \mapsto g (k (ε v)) / ∣ g (k (ε v)) ∣$ to $Φ \circ σ$ ; hence $\deg (g (k (ε \cdot)) / ∣ g (k (ε \cdot)) ∣) = \deg Φ \cdot \deg σ$ by [L1]. By [L2], $k (u) = A u + O (∣ u ∣^{2})$ with $A$ invertible, so $t \mapsto k (t v) / ∣ k (t v) ∣$ on $0 < t \leq ε$ , extended by $ρ_{A} (v) : = A v / ∣ A v ∣$ at $t = 0$ , is a smooth homotopy (the quotient $k (t v) / t = \int_{0}^{1} D k_{s t v} v d s$ extends smoothly to $t = 0$ ) from $σ$ to $ρ_{A}$ through nowhere-zero maps, whence $\deg σ = \deg ρ_{A} = sign \det A$ by [L1]. Finally the normalized map of $u \mapsto A^{- 1} g (k (u))$ at radius $ε$ is $ρ_{A^{- 1}} (g (k (ε \cdot)) / ∣ g (k (ε \cdot)) ∣)$ with $ρ_{A^{- 1}} (w) = A^{- 1} w / ∣ A^{- 1} w ∣$ , so by [L1] its degree is $\deg ρ_{A^{- 1}} \cdot sign \det A \cdot {ind}_{x} (f) = {ind}_{x} (f)$ , because $\deg ρ_{A^{- 1}} = sign \det A^{- 1} = sign \det A$ .

2.1step 1.1step 1.2L2

Second-order comparison. Write $\hat{f} (u) = u - g (u)$ and compute, using [L2] for $k^{- 1}$ at the point $k (u)$ with increment $- g (k (u))$ , $g^{'} (u) = u - k^{- 1} (k (u) - g (k (u))) = d (k^{- 1})_{k (u)} (g (k (u))) + O (∣ g (k (u)) ∣^{2})$ uniformly near $0$ . Since $u \mapsto d (k^{- 1})_{k (u)}$ is continuous and equals the invertible $A^{- 1}$ at $u = 0$ , on a ball of radius $ε_{0}$ the estimates $∣ d (k^{- 1})_{k (u)} v ∣ ≍ ∣ v ∣$ hold uniformly, and $∣ g (k (u)) ∣ \leq C ∣ u ∣$ ; hence $∣ g^{'} (u) - A^{- 1} g (k (u)) ∣ = o (∣ g (k (u)) ∣)$ as $u \to 0$ , uniformly, and in particular, after shrinking the radius $ε$ fixed in step 1.2 if necessary, $∣ g^{'} (u) - A^{- 1} g (k (u)) ∣ \leq \frac{1}{2} ∣ A^{- 1} g (k (u)) ∣$ for $0 < ∣ u ∣ \leq ε$ .

3.1step 2.1F1L1

Consequence: equal degrees. For $0 < ∣ u ∣ \leq ε$ the vector $A^{- 1} g (k (u))$ is nonzero because $g (k (u)) \neq 0$ and $A$ is invertible, and by step 2.1 every point of the segment from $A^{- 1} g (k (u))$ to $g^{'} (u)$ lies within $∣ g^{'} (u) - A^{- 1} g (k (u)) ∣ \leq \frac{1}{2} ∣ A^{- 1} g (k (u)) ∣$ of $A^{- 1} g (k (u))$ , hence is nonzero. So $(t, u) \mapsto (1 - t) A^{- 1} g (k (u)) + t g^{'} (u)$ is a smooth homotopy of nowhere-zero maps on $S^{n - 1}$ , the normalized maps of $A^{- 1} g (k (ε \cdot))$ and of $g^{'} (ε \cdot)$ have the same degree by [L1], and that degree is ${ind}_{y} (f^{'})$ by [F1].

4.1step 3.1step 1.2given∎

Conclusion. Combining steps 3.1 and 1.2, the normalized maps computing ${ind}_{y} (f^{'})$ and ${ind}_{x} (f)$ have the same degree, so ${ind}_{y} (f^{'}) = {ind}_{x} (f)$ . For the covering clause, $π$ is a local diffeomorphism, so it restricts to a diffeomorphism from an open neighbourhood of $\tilde{x}$ onto an open neighbourhood of $x$ , and $π \circ \tilde{f} = f \circ π$ gives $\tilde{f} = π^{- 1} \circ f \circ π$ there; the first clause applies with $h = π$ and $f^{'} = \tilde{f}$ . No orientation of $M$ or $N$ is used and no choice principle is used.

LemmaStatement: Literature-sourcedProof: Literature-sourcedprecheck passOpen item page →

The local fixed point index is independent of chart, ball and neighbourhood

Statement

Let $M$ be a smooth $n$ -manifold without boundary, $n \geq 1$ , and let $f : M \to M$ be smooth with an isolated fixed point $x$ . Then the choices entering Isolated fixed point and local fixed point index do not affect the value: any two admissible charts at $x$ produce the same degree, and any two admissible radii in one chart produce homotopic normalized sphere maps. For $n = 1$ the degree is reduced degree; maps from different charts need not be homotopic (for $f (u) = u - u^{2}$ on $R$ , the charts $u$ and $- u$ give the two different constant maps $S^{0} \to S^{0}$ , each of reduced degree $0$ ). Consequently ${ind}_{x} (f)$ is a well-defined integer depending only on the germ of $f$ at $x$ , and it is computed by the displayed formula in every admissible smooth chart and every sufficiently small ball around $x$ .

Facts & Assumptions

Given: A smooth $n$ -manifold $M$ without boundary, $n \geq 1$ , a smooth map $f : M \to M$ and an isolated fixed point $x$ .

[F1]

The index ${ind}_{x} (f)$ is the degree of the normalized displacement $v \mapsto g (ε v) / ∣ g (ε v) ∣$ on $S^{n - 1}$ for a chart $(φ, U)$ with $φ (x) = 0$ , $g = u - \hat{f} (u)$ and an admissible $ε > 0$ (Isolated fixed point and local fixed point index).

[L1]

For a smooth self-map $f : M \to M$ with isolated fixed point $x$ and a local diffeomorphism $h$ taking $y$ to $x$ , the proof of The local fixed point index is invariant under conjugation by a local diffeomorphism, steps 1.1–4.1, compares the normalized displacement degrees in arbitrary admissible charts at $x$ for $f$ and at $y$ for the local conjugate $h^{- 1} f h$ . Its sphere-map comparison includes reduced degree when $n = 1$ and uses only that the representatives are defined near $0$ , not that they preserve their Euclidean domains.

[L2]

Degree is invariant under smooth homotopy of maps of spheres (Degree is invariant under proper smooth homotopy); degree is multiplicative under composition and the radial map of a linear isomorphism has degree its determinant sign (Degree is multiplicative under composition, Degree of an orientation-preserving or reversing diffeomorphism, Regular-value formula for degree); a diffeomorphism's differential is an isomorphism (The differential of a diffeomorphism is an isomorphism). For $n = 1$ the homotopy and composition assertions use Reduced degree into the 0-sphere is homotopy invariant and multiplicative.

Proof

1.1givenF1L2

Radius independence. Let $0 < ε^{'} < ε$ be admissible radii in a chart $(φ, U)$ , so $g$ is defined on a neighbourhood of the closed ball and $g \neq 0$ on $0 < ∣ u ∣ \leq ε$ . The family $(t, v) \mapsto g (t ε v) / ∣ g (t ε v) ∣$ , $t \in [ε^{'} / ε, 1]$ , is a smooth homotopy of maps $S^{n - 1} \to S^{n - 1}$ taking the values nonzero throughout, hence the two normalized maps have the same degree by [L2]; this is precisely the independence of the displayed degree from the admissible radius, and it also compares a large admissible ball with any smaller admissible ball inside it.

2.1givenstep 1.1F1L1

Chart independence. Let $(φ, U)$ and $(ψ, V)$ be two admissible charts at $x$ . Apply the sphere-map comparison in [L1] to the given self-map $f : M \to M$ , with $N = M$ , $y = x$ , $h = {id}_{M}$ and $f^{'} = f$ , choosing $φ$ and $ψ$ as its two charts. Its transition is $k = φ \circ ψ^{- 1}$ , and ${\hat{f}}_{ψ} = k^{- 1} \circ {\hat{f}}_{φ} \circ k$ holds near $0$ : continuity at the fixed point permits shrinking the source so that both the source and its image lie in $U \cap V$ . The cited proof compares these two displacement sphere maps directly and gives equal degrees; its self-map hypothesis is satisfied by $f$ on $M$ . By [F1] these are exactly the displayed degrees in the two charts. Combined with step 1.1, this proves independence of every admissible chart, ball and radius.

3.1step 2.1given∎

Dependence on the germ only. If $f_{0}, f_{1} : M \to M$ agree on a neighbourhood $W$ of $x$ and have there the isolated fixed point $x$ , choose an admissible chart and radius inside $W$ ; the displacement representatives coincide, so the two displayed degrees coincide and, by step 2.1 applied to each, ${ind}_{x} (f_{0}) = {ind}_{x} (f_{1})$ : the index depends only on the germ of $f$ at $x$ . The neighbourhood-independence clause is the case $f_{0} = f_{1} = f$ with two admissible neighbourhoods.

TheoremStatement: Literature-sourcedProof: AI-adaptedprecheck passOpen item page →

The index of a nondegenerate fixed point is the sign of det(I-Df)

Statement

Let $M$ be a smooth $n$ -manifold without boundary, $n \geq 1$ , and let $x$ be a nondegenerate fixed point of a smooth map $f : M \to M$ (Nondegenerate fixed point). Then ${ind}_{x} (f) = sign \det (I - D f_{x} : T_{x} M \to T_{x} M) \in {+ 1, - 1},$ so every nondegenerate fixed point has index $+ 1$ or $- 1$ . The convention is $I - D f_{x}$ , not $D f_{x} - I$ ; in the other ordering the value is multiplied by $(- 1)^{n}$ , which is the source of sign discrepancies between references.

Facts & Assumptions

Given: A smooth $n$ -manifold without boundary, $n \geq 1$ , and a nondegenerate fixed point $x$ of the smooth self-map $f$ .

[F1]

The index is the degree of the normalized chart displacement, with reduced degree for $n = 1$ , independent of chart and admissible radius (Isolated fixed point and local fixed point index, The local fixed point index is independent of chart, ball and neighbourhood).

[F2]

Nondegeneracy means $I - D f_{x}$ is invertible (Nondegenerate fixed point). In a chart at $x$ , the chain rule identifies the derivative of $g (u) = u - \hat{f} (u)$ with the conjugate $A = I - D {\hat{f}}_{0}$ of $I - D f_{x}$ (The differential of a smooth map, The chain rule for differentials of smooth maps), and differentiability gives $g (u) = A u + o (∣ u ∣)$ (Multivariable Taylor formula with $o (∥ h ∥^{k})$ remainder, $k = 1$ , componentwise).

[F3]

Degree is invariant under smooth homotopies of connected spheres (Degree is invariant under proper smooth homotopy) and an orientation-preserving or reversing sphere diffeomorphism has degree $+ 1$ or $- 1$ (Degree of an orientation-preserving or reversing diffeomorphism). For $n = 1$ use reduced-degree homotopy invariance (Reduced degree into the 0-sphere is homotopy invariant and multiplicative).

Proof

1.1givenF2

Choose a chart centered at $x$ and a closed ball on which the chart displacement $g$ is defined. By [F2], $g (u) = A u + R (u)$ with $A$ invertible and $R (u) = o (∣ u ∣)$ . Let $c = ∥ A^{- 1} ∥^{- 1} > 0$ and shrink the ball until $∣ R (u) ∣ \leq c ∣ u ∣ / 2$ there. Then $∣ A u ∣ \geq c ∣ u ∣$ , so $g$ has no zero in the punctured ball and a positive radius $ε$ in it is admissible.

2.1step 1.1F1F3

For $v \in S^{n - 1}$ and $t \in [0, 1]$ , the vector $A ε v + t R (ε v)$ has norm at least $c ε / 2 > 0$ . Its normalization is a smooth homotopy from $L_{A} (v) = A v / ∣ A v ∣$ to the sphere map defining the fixed-point index. Therefore [F1] and [F3] identify ${ind}_{x} (f)$ with $\deg L_{A}$ , using reduced degree for $n = 1$ .

3.1step 2.1F1F2F3∎

The inverse of $L_{A}$ is $L_{A^{- 1}}$ . For $n \geq 2$ , radial normalization subtracts only an outward-normal component from $A w$ on tangent vectors $w$ , then rescales by a positive scalar. Thus, in outward-normal-first sphere orientations, the orientation sign of $d L_{A}$ is $sign \det A$ : the ambient ordered frame $(v, w_{1}, \dots, w_{n - 1})$ is sent to $(A v, A w_{1}, \dots, A w_{n - 1})$ , and deleting normal components and positive rescaling leave its determinant sign unchanged. By [F3], $\deg L_{A} = sign \det A$ . For $n = 1$ , $L_{A} (v) = sign (A) v$ , of reduced degree $sign (A)$ directly from [F1]. Finally $A$ is conjugate to $I - D f_{x}$ , so their determinant signs agree. Hence ${ind}_{x} (f) = sign \det (I - D f_{x}) \in {+ 1, - 1}$ . This local-coordinate proof uses no orientation of $M$ and no choice principle.

Remarks

Sign convention. With the opposite ordering, $\det (D f_{x} - I) = (- 1)^{n} \det (I - D f_{x})$ by multilinearity of the determinant in the columns of an $n \times n$ matrix, so a reference that uses $d f_{x} - I$ reports $(- 1)^{n}$ times the index defined here. Guillemin and Pollack use that ordering; the displacement convention here is fixed throughout the proof.
Isolatedness is not enough. The formula needs the invertibility of $I - D f_{x}$ ; for a degenerate isolated fixed point the index is still defined, but it is not determined by the first derivative. See Isolated fixed points need not be nondegenerate.

LemmaStatement: Literature-sourcedProof: AI-adaptedOpen item page →

An isolated fixed point splits under perturbation, preserving its index

Statement

Assume countable choice. Let $M$ be a closed smooth $n$ -manifold, $n \geq 1$ , and $f : M \to M$ smooth with all fixed points isolated. For every open neighbourhood $V$ of a fixed point $x$ , there is a smooth $g$ , arbitrarily close to $f$ , homotopic to it through a homotopy supported in a compact subset of $V$ , such that every fixed point of $g$ in $V$ is nondegenerate. Choose disjoint small closed chart balls $B_{z} ⋐ V$ around the finitely many points $z \in Fix (f) \cap V$ . The construction keeps $g = f$ outside their interiors, and each local replacement satisfies $\sum_{y \in Fix (g) \cap B_{z}} {ind}_{y} (g) = {ind}_{z} (f) .$ In particular if $V \cap Fix (f) = {x}$ , its new fixed-point index sum is ${ind}_{x} (f)$ ; for general $V$ the global index sum is unchanged. All sums are finite. Thus one isolated point can be split in an isolating neighbourhood, or all isolated points in a prescribed neighbourhood can be split simultaneously.

Facts & Assumptions

Given: Countable choice and a closed smooth $n$ -manifold $M$ , $n \geq 1$ , a smooth $f : M \to M$ with all fixed points isolated, a fixed point $x$ of $f$ and a neighbourhood $V$ of $x$ .

[F1]

The index ${ind}_{x} (f)$ is the degree of the normalized displacement $v \mapsto d (ε v) / ∣ d (ε v) ∣$ in an admissible chart, where $d (u) = u - \hat{f} (u)$ ; it is independent of the chart and radius (Isolated fixed point and local fixed point index, The local fixed point index is independent of chart, ball and neighbourhood).

[F2]

For a smooth field on a closed Euclidean ball, nonzero on its boundary, its finite isolated-zero index sum equals the boundary degree (reduced degree for $n = 1$ ), by The index sum of an outward field is the Gauss degree. Use only the chart-induced trivialization here; the corresponding restricted case is also The local index is additive under a transverse perturbation. Two fields agreeing on the boundary therefore have the same index sum.

[F3]

For a smooth map $G$ with isolated fixed point $y$ , nondegeneracy of $y$ is the invertibility of $I - D G_{y}$ (Nondegenerate fixed point), and then ${ind}_{y} (G) = sign \det (I - D G_{y})$ (The index of a nondegenerate fixed point is the sign of det(I-Df)); the chart displacement $u \mapsto u - \hat{G} (u)$ is a smooth vector field whose zeros are the fixed points of $\hat{G}$ , with nondegenerate zeros corresponding to nondegenerate fixed points and with the same index (Isolated zero and local index of a vector field, Nondegenerate zero of a vector field).

[F4]

Regular values of a smooth map are dense and their complement is null (Morse-Sard for smooth manifolds, Regular values have null complement and are dense); for a compact set $K$ inside an open set $U$ there is a smooth bump equal to $1$ near $K$ and supported in $U$ (A manifold bump for a compact set inside an open set).

[F5]

A closed discrete subset of a compact space is finite (A closed discrete subset of a compact space is finite). Continuous images of compact sets are compact, and a continuous real-valued function on a nonempty compact set attains its minimum (A continuous image of a compact space is compact; a continuous real-valued map on a nonempty compact space attains a maximum and a minimum; and a continuous bijection from a compact space to a Hausdorff space is a homeomorphism, clauses 1–2).

Proof

1.1givenF1F4F5

Choose a target chart $(φ, U)$ at $x$ with $φ (x) = 0$ and restrict the representative to $W = φ (U \cap f^{- 1} (U) \cap V)$ . It contains $0$ . Choose $R > 0$ with ${\overline{B}}_{R} \subset W$ and $d (u) : = u - \hat{f} (u) \neq 0$ on $0 < ∣ u ∣ \leq R$ . Fix $0 < r < R$ and use [F4] to choose $ρ : R^{n} \to [0, 1]$ equal to one near ${\overline{B}}_{r}$ and supported in $B_{R}$ . The compact annulus $K = {r \leq ∣ u ∣ \leq R}$ is nonempty, and [F5] gives $m = \min_{K} ∣ d ∣ > 0$ . The compact image $\hat{f} ({\overline{B}}_{R})$ lies in the open target chart image; a finite cover by balls with doubled radii inside that image supplies $η > 0$ such that adding a vector of norm less than $η$ stays in it.

2.1step 1.1F4construct

By [F4] choose a regular value $a$ of $d$ with $∣ a ∣ < \min (m / 2, η)$ . Define $g_{t} (p) = φ^{- 1} (\hat{f} (u) + t ρ (u) a)$ for $u = φ (p) \in {\overline{B}}_{R}$ , and $g_{t} = f$ elsewhere, for $0 \leq t \leq 1$ . These definitions agree on an open collar of the boundary because $supp ρ ⋐ B_{R}$ . They give a smooth homotopy, supported in the compact set $φ^{- 1} (supp ρ) \subset V$ , with $g_{0} = f$ ; put $g = g_{1}$ . The vector $a$ can be arbitrarily small.

3.1step 1.1step 2.1F3F5

On $K$ , the displacement $d - ρ a$ has norm at least $m - ∣ a ∣ > 0$ . Inside $B_{r}$ one has $ρ = 1$ on a neighbourhood, so the fixed points of $g$ are exactly the preimages of the regular value $a$ under $d$ . Their displacement derivative is $D d$ , which is invertible there. Hence they are nondegenerate by [F3]. The zero set is closed in ${\overline{B}}_{R}$ and discrete, so it is finite by [F5].

4.1step 1.1step 3.1F1F2F3

The fields $d$ and $d - ρ a$ have identical nonzero boundary values on $\partial B_{R}$ . By [F2] their index sums agree. The first field has only the zero $0$ , of index ${ind}_{x} (f)$ by [F1]; each zero of the second has the corresponding fixed-point index by [F3]. This proves the local sum identity. Outside the support, $g = f$ on a neighbourhood of each old fixed point, so the germ clause of The local fixed point index is independent of chart, ball and neighbourhood preserves its index. Countable choice is inherited from Sard.

5.1step 1.1step 2.1step 3.1step 4.1F5∎

For an arbitrary $V$ , the fixed points of $f$ lying in it form a finite set. Choose mutually disjoint balls $B_{z} ⋐ V$ around all of them, each isolating its centre and satisfying step 1.1. Perform steps 1.1–4.1 in each ball. The supports are disjoint and the formulas equal $f$ near every ball boundary, so they glue to one smooth map and one smooth supported homotopy. No new fixed point occurs outside the balls, and every old fixed point inside $V$ was included; hence every new fixed point in $V$ is nondegenerate. Summing the local identities gives global index preservation. Since there are finitely many bumps and all perturbation vectors may be chosen arbitrarily small, any prescribed smooth-neighbourhood bound is met by taking their finitely many vectors small enough.

LemmaStatement: Literature-sourcedProof: Literature-sourcedprecheck passOpen item page →

The local intersection sign of graph against diagonal is sign det(I-Df)

Statement

Let $M$ be a closed oriented smooth $n$ -manifold, $n \geq 1$ , let $f : M \to M$ be smooth with all fixed points nondegenerate, let $Γ_{f} \subseteq M \times M$ be the graph oriented by the diffeomorphism $x \mapsto (x, f (x))$ onto $M$ and let $Δ_{M}$ be the diagonal oriented by $x \mapsto (x, x)$ , with $M \times M$ carrying the product orientation (Product orientations). Then at each fixed point $x$ , for the ordered pair of factors $(Γ_{f}, Δ_{M})$ , the local oriented intersection sign of The local oriented intersection sign is $ε_{(Γ_{f}, Δ_{M})} (x) = sign \det (I - D f_{x}) = {ind}_{x} (f) .$ Consequently the oriented intersection numbers of The oriented intersection number satisfy $I (γ_{f}, Δ_{M}) = I (Γ_{f}, Δ_{M}) = \sum_{x \in Fix (f)} {ind}_{x} (f) = I (f),$ the graph map being $γ_{f} (x) = (x, f (x))$ (The graph of a smooth map is an embedded submanifold), and the sum is finite because $Γ_{f}$ and $Δ_{M}$ are closed complementary submanifolds of the closed oriented $M \times M$ . The factor order matters: in the order $(Δ_{M}, Γ_{f})$ every local sign is multiplied by $(- 1)^{n}$ (Intersection number under factor interchange).

Facts & Assumptions

Given: A closed oriented smooth $n$ -manifold $M$ , a smooth $f : M \to M$ with all fixed points nondegenerate, the graph $Γ_{f}$ and diagonal $Δ_{M}$ in the closed oriented manifold $M \times M$ .

[F1]

At a coincidence point of transverse oriented maps the local sign compares the ordered direct sum $T_{a} X \oplus T_{b} Z \to T_{y} M$ of the two tangent spaces, with the product orientation, to the ambient orientation (The local oriented intersection sign, Product orientations).

[F2]

The graph map $γ_{f}$ and the diagonal map $δ_{M} (x) = (x, x)$ are diffeomorphisms onto the embedded submanifolds $Γ_{f}, Δ_{M}$ of dimension $n$ ; in the splitting $T_{(x, x)} (M \times M) ≅ T_{x} M \oplus T_{x} M$ the tangent space of the graph is ${(v, D f_{x} v)}$ and that of the diagonal ${(v, v)}$ (The graph of a smooth map is an embedded submanifold, The diagonal is an embedded submanifold, Canonical tangent and cotangent splittings for products, The differential of a smooth map).

[F3]

$Γ_{f}$ and $Δ_{M}$ meet transversely at $γ_{f} (x)$ exactly when $x$ is a nondegenerate fixed point (Graph-diagonal transversality is exactly fixed-point nondegeneracy), and then ${ind}_{x} (f) = sign \det (I - D f_{x})$ (The index of a nondegenerate fixed point is the sign of det(I-Df)).

[L1]

The oriented intersection number of a map transverse to a closed oriented submanifold is the finite sum of the local signs over the preimages (The oriented intersection number), and exchanging the two ordered factors multiplies every local sign by $(- 1)^{a b}$ for dimensions $a, b$ (Intersection number under factor interchange). Countable Choice is inherited by the intersection number, not by its displayed finite sums (The Axiom of Countable Choice ( ${A C}_{ω}$ )).

Proof

1.1givenF1F2F3

The determinant at a fixed point. Let $x$ be a fixed point and let $(e_{1}, \dots, e_{n})$ be a positively oriented basis of $T_{x} M$ . By [F2] the tuples $(e_{i}, D f_{x} e_{i})_{i \leq n}$ and $(e_{j}, e_{j})_{j \leq n}$ are positively oriented bases of $T_{γ_{f} (x)} Γ_{f}$ and $T_{(x, x)} Δ_{M}$ ; concatenated in the order $(Γ_{f}, Δ_{M})$ and expressed in the ambient basis $((e_{1}, 0), \dots, (e_{n}, 0), (0, e_{1}), \dots, (0, e_{n}))$ of $T_{(x, x)} (M \times M)$ they form the columns of the block matrix $(\begin{array}{cc} I & I \\ D f_{x} & I \end{array})$ . Subtracting the $i$ -th column from the $(n + i)$ -th column for each $i$ (a column shear of determinant $1$ , The determinant is alternating and multilinear in the rows as well as in the columns applied to the transpose, using For every square matrix over a commutative ring, $\det (A^{T}) = \det (A)$ ) gives the block lower triangular matrix $(\begin{array}{cc} I & 0 \\ D f_{x} & I - D f_{x} \end{array})$ , whose determinant is $\det (I - D f_{x})$ . The orientation comparison of [F1] is therefore $sign \det (I - D f_{x})$ , and by [F3] this is ${ind}_{x} (f)$ .

2.1step 1.1F3L1

The global intersection number. By [F3] the graph map is transverse to the diagonal precisely because all fixed points are nondegenerate, and transversality plus closedness of the complementary-dimensional submanifolds in the compact $M \times M$ makes the intersection finite; by [L1] the oriented intersection number $I (γ_{f}, Δ_{M})$ is the sum of the local signs over the fixed points, which by step 1.1 is $\sum_{x} {ind}_{x} (f) = I (f)$ , the geometric Lefschetz number of Geometric Lefschetz number (index sum). Replacing the graph map by the inclusion of the graph changes nothing: the two are identified by the diffeomorphism $x \mapsto (x, f (x))$ , which is orientation-preserving and conjugates the local data, so $I (Γ_{f}, Δ_{M}) = I (γ_{f}, Δ_{M})$ .

3.1step 2.1L1∎

Factor order. In the opposite order $(Δ_{M}, Γ_{f})$ the ambient tangent space is presented with the two $n$ -dimensional factors exchanged, and [L1] applies with $a = b = n$ , giving $I (Δ_{M}, Γ_{f}) = (- 1)^{n^{2}} I (Γ_{f}, Δ_{M}) = (- 1)^{n} I (Γ_{f}, Δ_{M})$ ; the same factor appears pointwise because the local sign of [F1] is computed from the ordered sum. No metric is used, and the only choice principle involved is the Countable Choice recorded in [L1] for the intersection number of non-transverse representatives, which the transverse case of this lemma does not use.

DefinitionDefinition: Literature-sourcedProof: Not applicableOpen item page →

Algebraic Lefschetz number via rational homology traces

Definition

Assume the Axiom of Choice (The Axiom of Choice). Let $M$ be a closed smooth $n$ -manifold and let $f : M \to M$ be continuous. Its Lefschetz number is $L (f) : = \sum_{i = 0}^{n} (- 1)^{i} tr (f_{*} : H_{i} (M; Q) \to H_{i} (M; Q)) \in Q,$ the alternating sum of the traces of the induced rational homology endomorphisms (The singular chain complex and singular homology, The basis-independent trace of an endomorphism of a finite-dimensional vector space, The rationals as equivalence classes of pairs of integers). The sum is finite and every trace is of a finite-dimensional operator because $\dim_{Q} H_{i} (M; Q) < \infty$ and $H_{i} (M; Q) = 0$ for $i > n$ (Finiteness and additivity of the Euler characteristic, clause (i), which is where the Axiom of Choice enters, through the existence of an excellent Morse function (Every compact smooth manifold admits an excellent Morse function)). When a finite CW model whose cellular chains compute $H_{*} (M; Q)$ is available (The handle chain complex computes singular homology, Cellular homology computes singular homology), this number agrees with the published finite-CW Lefschetz number Lefschetz number of a finite CW self-map of the induced self-map transported along the homotopy equivalence (Homotopy equivalences induce isomorphisms on singular homology); the comparison is a consistency statement, not part of the definition. Homotopic maps have the same $L$ because they induce the same maps on homology (Homotopic maps induce the same map on singular homology). No orientation, no smoothness of $f$ and no field other than $Q$ is used.

Remarks

Why the sum stops at $n$ . A closed $n$ -manifold has $H_{i} (M; Q) = 0$ for $i > n$ , by the finiteness proposition cited above, which is proved from the existence of an excellent Morse function on the double of $M$ and the in-run handle chain complex. This is the only place where the Axiom of Choice is used in the definition; the trace is a basis-independent invariant of an endomorphism of a finite-dimensional $Q$ -vector space (The basis-independent trace of an endomorphism of a finite-dimensional vector space).
Value field. Each rational homology trace, and hence $L (f)$ , is an integer. Choose the finite CW model from the Morse handle presentation and A handle decomposition gives a relative CW complex, and transport $f$ using a homotopy inverse. Cellular approximation (Cellular approximation for maps of CW pairs) gives a homotopic cellular map with integral matrices on the finite free cellular chains (Relative homology of consecutive CW skeleta, Cellular maps induce cellular chain maps). Its cycles are finite free (A submodule of a free module of finite rank over a PID is free of no larger rank), so integral homology is finitely generated. Quotienting torsion gives a finite free lattice (Every finitely generated torsion-free module over a PID is free). Clearing denominators in rational cycles shows that this lattice spans rational homology; clearing denominators in a rational bounding chain shows that its kernel before quotienting is exactly torsion. Thus rational homology is the rationalization of the lattice, and the induced map has an integral lattice matrix and integer trace. Cellular comparison and homology conjugation give the same trace for $f_{*}$ . Alternatively Hopf trace formula computes $L (f)$ as the alternating integral cellular-chain trace.
Not the same as the geometric number. This definition applies to every continuous self-map, including maps with non-isolated fixed points such as the identity; the equality with the finite index sum $I (f) = \sum_{x} {ind}_{x} (f)$ for smooth self-maps of closed smooth $n$ -manifolds with $n \geq 1$ and isolated fixed points is Lefschetz-Hopf index formula, and the corresponding geometric number is Geometric Lefschetz number (index sum).

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The diagonal and graph classes contract to the alternating trace

Statement

Assume AC (The Axiom of Choice). Let $M$ be a closed oriented smooth $n$ -manifold and $f : M \to M$ smooth. For a homogeneous basis $α_{p, j}$ of $H^{p} (M; Q)$ , choose the dual basis $β_{p, j} \in H^{n - p} (M; Q)$ with $⟨ β_{p, j} ⌣ α_{p, k}, [M] ⟩ = δ_{j k}$ . In the cohomology-first cap convention, $P D [Δ_{M}] = \sum_{p, j} (- 1)^{p} β_{p, j} \times α_{p, j} .$ Writing $γ_{f} (x) = (x, f (x))$ for the graph map, $⟨ γ_{f}^{*} P D [Δ_{M}], [M] ⟩ = ⟨ P D [Γ_{f}] ⌣ P D [Δ_{M}], [M \times M] ⟩ = L (f) .$ The graph Poincare dual is characterized by $⟨ P D [Γ_{f}] ⌣ φ, [M \times M] ⟩ = ⟨ γ_{f}^{*} φ, [M] ⟩$ for every degree- $n$ cohomology class $φ$ . When $n \geq 1$ and the fixed points are nondegenerate, this value is $I (f)$ , the graph-diagonal intersection number in the local displacement convention $u - v$ (Geometric Lefschetz number (index sum), The geometric intersection pairing on a closed oriented manifold).

Facts & Assumptions

Given: $M, f$ , the bases, and AC as in the statement.

[F1]

The orientation-twisted diagonal realizes the Lefschetz trace supplies the normalized diagonal class, the signed dual-basis expansion, graph-pullback trace, and nondegenerate local evaluation. The chosen orientation trivializes its orientation coefficient system. Manifold components are open by local path-connectedness (Topological manifolds are locally compact and locally path connected, A connected, locally path-connected space is path-connected, because its path components are open), hence compactness gives finitely many, and homology splits over them (The singular homology of a disjoint union is the direct sum).

[F2]

Poincare duality is the inverse of cohomology-first cap with the fundamental class (The cap-duality map of an oriented manifold, Poincaré duality for oriented topological manifolds). Cup and cap satisfy the composition and naturality formulas (Cap naturality and projection formula, Kronecker evaluation pairing).

[F3]

The local displacement convention $u - v$ identifies the graph-diagonal local signs with fixed-point indices (The local intersection sign of graph against diagonal is sign det(I-Df), The geometric intersection pairing on a closed oriented manifold).

Proof

1.1givenF1F2

For disconnected $M$ , apply [F1] on each component: the diagonal has support only in $C \times C$ , and the product components $C \times D$ with $C \neq D$ have zero diagonal class. In component-adapted bases its expansion is the sum of the component expansions; it is independent of the basis since a basis change and its inverse dual change cancel in the tensor sum. Components mapped to a different component have zero diagonal trace block and no diagonal intersection. Thus the graph-pullback trace identity also sums over the components, including the empty case. Trivialize the orientation system by the given orientation of $M$ . The cap-normalized diagonal class of [F1] becomes $P D [Δ_{M}]$ by [F2]. Its expansion is exactly the displayed formula, and [F1]'s graph pullback gives $⟨ γ_{f}^{*} P D [Δ_{M}], [M] ⟩ = L (f)$ .

2.1F2step 1.1

Since the graph is oriented by $γ_{f}$ , its fundamental homology class is $(γ_{f})_{*} [M]$ . For every degree- $n$ class $φ$ , [F2] gives $⟨ P D [Γ_{f}] ⌣ φ, [M \times M] ⟩ = ⟨ φ, P D [Γ_{f}] \cap [M \times M] ⟩ = ⟨ φ, (γ_{f})_{*} [M] ⟩ = ⟨ γ_{f}^{*} φ, [M] ⟩$ . Taking $φ = P D [Δ_{M}]$ proves the cup contraction identity from step 1.1.

3.1F1F3step 1.1step 2.1∎

If $n \geq 1$ and the fixed points are nondegenerate, [F1] evaluates this graph pullback as the sum of the local signs $sign \det (I - D f_{x})$ . By [F3] this is both $I (f)$ and the stated graph-diagonal intersection number. AC is inherited from [F1] and Poincare duality.

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Lefschetz-Hopf index formula for nondegenerate fixed points (orientable case)

Statement

Assume AC (The Axiom of Choice). Let $M$ be a closed oriented smooth $n$ -manifold, $n \geq 1$ , and let $f : M \to M$ be smooth with every fixed point nondegenerate. Then $Fix (f)$ is finite, the geometric Lefschetz number is defined (Geometric Lefschetz number (index sum)), and $I (f) = \sum_{x \in Fix (f)} {ind}_{x} (f) = L (f),$ where $L (f)$ is the algebraic Lefschetz number of Algebraic Lefschetz number via rational homology traces.

Facts & Assumptions

Given: A closed oriented smooth $n$ -manifold $M$ and a smooth self-map $f$ whose fixed points are all nondegenerate; assume AC.

[F1]

The orientation-twisted diagonal realizes the Lefschetz trace proves the graph-pullback trace and its nondegenerate fixed-point evaluation, without an orientation restriction.

[F2]

Geometric Lefschetz number (index sum) defines the finite index sum, and Algebraic Lefschetz number via rational homology traces defines the rational homology alternating trace.

Proof

1.1givenF1F2

On each component carried into itself, [F1] gives finiteness and identifies the finite local index sum with its Lefschetz trace. A component carried into a different component has no fixed points and a zero source-to-source diagonal block in homology, hence contributes zero to both quantities. Compactness gives finitely many components, so these finite sums exhaust both quantities of [F2].

2.1F1step 1.1∎

Summing the component identities gives $I (f) = L (f)$ . The supplied orientation is compatible with [F1]'s twisted proof by trivializing its orientation system, and AC is inherited from that supplier.

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The orientation double cover is canonically oriented and preserves closedness

Statement

Let $M$ be a connected smooth $n$ -manifold. The deck-group and component assertions below assume $M \neq \emptyset$ ; for the empty base, use the empty cover and its trivial deck group. Then there is a smooth $n$ -manifold $\tilde{M}$ , the orientation double cover, together with a smooth two-sheeted covering map $π : \tilde{M} \to M$ (Covering maps, evenly covered neighbourhoods, fibres, sheets, and trivial coverings) and a smooth involution $τ : \tilde{M} \to \tilde{M}$ with $τ^{2} = {i d}_{\tilde{M}}$ and $π \circ τ = π$ (Deck transformations and the deck-transformation group of a covering), such that $\tilde{M}$ is orientable (Orientable manifolds) and indeed canonically oriented (Oriented smooth manifolds and oriented charts). Its deck group is $Z / 2 = {i d, τ}$ , acting freely and transitively on every fibre; when $M$ is nonorientable, $\tilde{M}$ is connected and the covering is regular (Regular coverings), while when $M$ is orientable, $\tilde{M} ≅ M \times Z / 2$ with $τ$ exchanging the two components. If $M$ is closed then $\tilde{M}$ is closed. The underlying set is $\tilde{M} = {(x, o_{x}) : x \in M, o_{x} a ray in \det T_{x} M}, π (x, o_{x}) = x, τ (x, o_{x}) = (x, - o_{x}),$ the tangent-space model of the orientation double cover.

Facts & Assumptions

Given: A connected smooth $n$ -manifold $M$ ; assume it nonempty until the empty case at the end.

[F1]

A covering map is a continuous surjection whose base points have evenly covered neighbourhoods, over which the total space splits into sheets mapped homeomorphically onto the neighbourhood; a deck transformation is a homeomorphism over the base, and a covering with path-connected total space is regular when its deck group acts transitively on every fibre (Covering maps, evenly covered neighbourhoods, fibres, sheets, and trivial coverings, Deck transformations and the deck-transformation group of a covering, Regular coverings).

[F2]

A smooth manifold is a Hausdorff, second-countable, locally Euclidean space with a maximal smooth atlas; a cover of a smooth manifold whose total space is connected carries a unique smooth structure of the same dimension for which the covering map is a smooth local diffeomorphism (Smooth manifolds and their smooth charts, Smooth atlases, Connected covers of smooth manifolds have a canonical smooth structure).

[F3]

A topological manifold is locally path-connected, and local path connectedness lifts and descends along covering maps; a connected locally path-connected space is path-connected (Topological manifolds are locally compact and locally path connected, Local path-connectedness lifts and descends along covering maps, A connected, locally path-connected space is path-connected, because its path components are open).

[F4]

For a finite-sheeted covering, the total space is compact exactly when the base is compact (For a finite-sheeted covering, the total space is compact exactly when the base is compact).

[F5]

An orientation of $M$ is a smooth choice of a ray in $\det T_{x} M$ for every $x$ ; $M$ is orientable when such a choice exists (Oriented smooth manifolds and oriented charts, Orientable manifolds).

Proof

1.1givenF5

The model. Let $\tilde{M}$ be the set of pairs $(x, o_{x})$ with $x \in M$ and $o_{x}$ a ray in $\det T_{x} M$ , with $π (x, o_{x}) = x$ and $τ (x, o_{x}) = (x, - o_{x})$ . For a chart $(U, φ)$ of $M$ write $s_{U, φ}^{+} (x) : = d φ_{x}^{- 1} (the positive ray of R^{n})$ for the ray in $\det T_{x} M$ pulled back from the standard ray, and $s_{U, φ}^{-} : = - s_{U, φ}^{+}$ ; for open $W \subseteq U$ put $S_{W, φ}^{\pm} : = {(x, s_{U, φ}^{\pm} (x)) : x \in W}$ . These sets cover $\tilde{M}$ and are closed under finite intersections: for a second chart $(V, ψ)$ and open $W^{'} \subseteq V$ , the intersection $S_{W, φ}^{ϵ} \cap S_{W^{'}, ψ}^{δ}$ equals $S_{Ω, φ}^{ϵ}$ where $Ω = {x \in W \cap W^{'} : ϵ s_{U, φ}^{+} (x) = δ s_{V, ψ}^{+} (x)}$ , and $Ω$ is open in $W \cap W^{'}$ because the comparison of the two chart rays is governed by the sign of the nowhere-zero continuous function $x \mapsto \det d (ψ φ^{- 1})_{x}$ , which is locally constant; if no point of $W \cap W^{'}$ satisfies the comparison the intersection is empty. So these sets form a basis of a topology on $\tilde{M}$ , and by construction $x \mapsto (x, s_{U, φ}^{\pm} (x))$ is a homeomorphism of $W$ onto $S_{W, φ}^{\pm}$ .

2.1step 1.1F1F2

The basis description gives the covering and the involution. Every point of $\tilde{M}$ lies in some basis element $S_{W, φ}^{\pm}$ , which is homeomorphic to the open set $W \subseteq R^{n}$ , so $\tilde{M}$ is locally Euclidean of dimension $n$ . Every chart domain $U$ of $M$ satisfies $π^{- 1} (U) = S_{U, φ}^{+} ⊔ S_{U, φ}^{-}$ , a disjoint union of two open sets on each of which $π$ restricts to a homeomorphism onto $U$ ; since the chart domains cover $M$ , $π$ is continuous, open, surjective and a two-sheeted covering map. The map $τ$ is continuous with $τ^{2} = i d$ and $π \circ τ = π$ , because in the basis it exchanges $S_{W, φ}^{+}$ with $S_{W, φ}^{-}$ . The space $\tilde{M}$ is Hausdorff: points with distinct images are separated by the inverse images of disjoint open neighbourhoods in the Hausdorff space $M$ , and the two points of a fibre lie in the two disjoint sheets over any chart domain containing the image.

3.1step 2.1F1F3

Components are covering spaces. Suppose $M$ is nonempty and connected. By [F3] both $M$ and its cover are locally path-connected, their components are open path components, and $M$ is path-connected. Fix $a$ in a component $C$ and let $y \in M$ . A path from $π (a)$ to $y$ lifts from $a$ by Existence and uniqueness of path lifts through a covering map; its image stays in $C$ , so $C$ meets the fibre over every $y$ . Hence there are at most two components. If there are two, each meets every two-point fibre exactly once; if there is one, it contains both fibre points. Over a connected evenly covered neighbourhood, each sheet is connected and therefore lies in one component. Thus $π ∣_{C} : C \to M$ is a covering.

4.1step 3.1F2

Smooth structure. By [F2] each component $C$ of $\tilde{M}$ carries a unique smooth $n$ -manifold structure for which $π ∣_{C}$ is a smooth local diffeomorphism; on a nonempty connected base the components are at most two disjoint open sets, so these structures combine into a smooth $n$ -manifold structure on $\tilde{M}$ for which $π$ is a smooth local diffeomorphism and a two-sheeted covering map. In particular $\tilde{M}$ is a topological $n$ -manifold and $π$ is a local diffeomorphism, so a chart of $M$ pulls back along $π$ on each sheet to a chart of $\tilde{M}$ .

5.1step 4.1F5

The canonical orientation. At a point $p = (x, o_{x})$ the differential $d π_{p} : T_{p} \tilde{M} \to T_{x} M$ is an isomorphism, so $d π_{p}^{- 1} (o_{x})$ is a ray in $\det T_{p} \tilde{M}$ . On the sheet $S_{W, φ}^{+}$ the pulled-back chart $Φ : = φ \circ (π ∣_{S_{W, φ}^{+}})$ has differential $d Φ_{p} = d φ_{x} \circ d π_{p}$ , so it carries this ray to the ray $d φ_{x} (o_{x})$ , which is the standard ray of $R^{n}$ because $o_{x} = s_{U, φ}^{+} (x)$ ; on the sheet $S_{W, φ}^{-}$ the same computation gives the opposite standard ray. The ray assignment is therefore constant in these charts, hence a smooth choice of rays, so it is an orientation of $\tilde{M}$ by [F5]. It is canonical: it is defined from $π$ and the points $(x, o_{x})$ themselves, with no chart or orientation of $M$ chosen.

5.2step 4.1step 2.1F1F5

The deck group. Let $h : \tilde{M} \to \tilde{M}$ be a deck transformation. Since $π \circ h = π$ , $h$ maps each fibre into itself, so $h (x, o_{x})$ is $(x, o_{x})$ or $(x, - o_{x})$ , and because $h$ is injective on the two-point fibre it acts by a well-defined sign $h (x, o_{x}) = (x, ϵ (x) o_{x})$ with $ϵ : M \to {\pm 1}$ . Continuity of $h$ makes $ϵ$ locally constant: over a chart domain $U$ the two sheets $S_{U, φ}^{\pm}$ are disjoint open sets, and a connected neighbourhood of $x$ maps into one of them, so $ϵ$ is constant near $x$ . Hence $ϵ$ is continuous into the discrete group ${\pm 1}$ and therefore constant because $M$ is connected; so $h = i d$ if $ϵ = + 1$ and $h = τ$ if $ϵ = - 1$ . Moreover $τ$ is smooth, being locally the sheet exchange between the charts of $\tilde{M}$ , and $π \circ τ = π$ , so $τ$ is a deck transformation; thus $Deck (π) = {i d, τ} ≅ Z / 2$ , and it acts freely (only $i d$ has a fixed point) and transitively on every two-point fibre.

6.1step 4.1step 5.2F5

Orientable case. Suppose $M$ is orientable and let $x \mapsto o_{x}$ be an orientation of $M$ by [F5]. Then $Φ : \tilde{M} \to M \times Z / 2$ , $Φ (x, o_{x}^{'}) = (x, ϵ)$ where $o_{x}^{'} = ϵ o_{x}$ , is a bijection over $M$ , and in the charts of step 5.1 the map $Φ$ and its inverse change only the locally constant sign of the second coordinate, so $Φ$ is a diffeomorphism for the product smooth structure on $M \times Z / 2$ (Products of smooth manifolds have a canonical product smooth structure); it carries $τ$ to the map exchanging the two components $M \times {+ 1}$ and $M \times {- 1}$ .

6.2step 5.2step 3.1F1F3F5

Nonorientable case. Suppose $M$ admits no orientation. Then $\tilde{M}$ is connected: if $\tilde{M} = C_{1} ⊔ C_{2}$ with two components, step 3.1 makes each $C_{i}$ a covering of $M$ meeting each fibre exactly once, so $π ∣_{C_{1}}$ is a bijective local homeomorphism, i.e. a homeomorphism, and its inverse $s : M \to \tilde{M}$ is a continuous section; writing $s (x) = (x, o_{x})$ , the assignment $x \mapsto o_{x}$ is in the pulled-back charts of step 4.1 locally constant, hence a smooth choice of rays and an orientation of $M$ by [F5], a contradiction. So $\tilde{M}$ is connected when $M$ is nonorientable, hence path-connected by [F3] since $M$ is locally path-connected as a manifold and local path connectedness ascends to the cover; the deck group acts transitively on every fibre by step 5.2, so the covering is regular by [F1].

7.1step 4.1F4∎

Closedness. If $M$ is closed, i.e. compact and boundaryless, then $\tilde{M}$ is compact by [F4], and it is boundaryless because $π$ is a local diffeomorphism onto a boundaryless manifold; hence $\tilde{M}$ is closed. For the empty base $M = \emptyset$ , $\tilde{M} = \emptyset$ , the projection is a covering map vacuously, the deck group is trivial, and the empty ray choice gives its canonical orientation; the two-element deck-group assertion was restricted to a nonempty connected base.

Remarks

The canonical orientation is reversed by the deck transformation. In the charts of step 5.1 the ray at $(x, o_{x})$ is the pullback of $o_{x}$ , and $τ (x, o_{x}) = (x, - o_{x})$ has the opposite ray, so $τ$ is orientation- reversing for the canonical orientation. The local fixed-point index is nevertheless unchanged by this deck transformation as a conjugation, since both chart orientations reverse.
Relation to the homological orientation cover. The library's Orientation local system and orientation cover builds a two-sheeted covering from the local homology fibres $H_{n} (M, M ∖ {x}; Z)$ ; the construction above is the tangent-space model of the same cover, obtained by reading a chart-induced ray in $\det T_{x} M$ as the corresponding local homology generator. This item proves all covering, smoothness, orientability and connectedness properties for the model it defines, by Smooth orientation sign is the local integral homology multiplier: chart changes act on both models by the same determinant sign, including the signed-point convention in dimension zero. Sending each chart ray to its chart-induced local generator therefore defines a fibrewise bijection that respects local sheet charts and path transport. This identifies the tangent model with the orientation local system used in the twisted diagonal argument.

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A smooth local diffeomorphism lifts canonically to the orientation double cover

Statement

Let $M$ be a connected smooth $n$ -manifold, $π : \tilde{M} \to M$ its orientation double cover with deck transformation $τ$ (The orientation double cover is canonically oriented and preserves closedness) and let $f : M \to M$ be a local diffeomorphism (Diffeomorphisms and local diffeomorphisms of manifolds), so that $D f_{x} : T_{x} M \to T_{f (x)} M$ is an isomorphism for every $x$ (The differential of a smooth map, The smooth inverse function theorem on manifolds). Then $\tilde{f} (x, o_{x}) : = (f (x), D f_{x} (o_{x})), {\tilde{f}}^{'} : = τ \circ \tilde{f},$ define smooth maps $\tilde{M} \to \tilde{M}$ with $π \circ \tilde{f} = π \circ {\tilde{f}}^{'} = f \circ π$ , $τ \circ \tilde{f} = \tilde{f} \circ τ$ and $τ \circ {\tilde{f}}^{'} = {\tilde{f}}^{'} \circ τ$ ; when $M$ is nonorientable these are exactly the two lifts of $f \circ π$ through $π$ (the total space $\tilde{M}$ is then connected). If $f$ is a diffeomorphism, so are $\tilde{f}$ and ${\tilde{f}}^{'}$ . The construction is canonical, i.e. it involves no choices.

Facts & Assumptions

Given: A connected smooth $n$ -manifold $M$ , its orientation double cover $(\tilde{M}, π, τ)$ and a local diffeomorphism $f : M \to M$ .

[F1]

$\tilde{M} = {(x, o_{x})}$ is the set of rays $o_{x}$ in $\det T_{x} M$ , with $π (x, o_{x}) = x$ , $τ (x, o_{x}) = (x, - o_{x})$ ; over a chart $(U, φ)$ of $M$ the two sheets $S_{U, φ}^{\pm} = {(x, \pm s_{U, φ}^{+} (x)) : x \in U}$ are charts with $φ \circ π$ as coordinate map, and $π$ is a two-sheeted covering map (The orientation double cover is canonically oriented and preserves closedness).

[F2]

A local diffeomorphism is a smooth map that is a diffeomorphism from a neighbourhood of each point onto an open set, equivalently a smooth immersion of the same dimension; its differential is everywhere invertible, and an invertible linear map carries rays in the determinant line to rays (Diffeomorphisms and local diffeomorphisms of manifolds, The smooth inverse function theorem on manifolds, The differential of a smooth map, Oriented smooth manifolds and oriented charts).

Proof

1.1givenF1F2

The formula defines maps and the covering and commutation identities. For $(x, o_{x}) \in \tilde{M}$ the differential $D f_{x}$ is an isomorphism by [F2], so $D f_{x} (o_{x})$ is a ray in $\det T_{f (x)} M$ and $\tilde{f} (x, o_{x}) : = (f (x), D f_{x} (o_{x}))$ is a point of $\tilde{M}$ ; the inverse linear map sends the opposite ray to the opposite ray, so $D f_{x} (- o_{x}) = - D f_{x} (o_{x})$ and hence $τ \tilde{f} = \tilde{f} τ$ and $τ {\tilde{f}}^{'} = {\tilde{f}}^{'} τ$ from $τ^{2} = i d$ . The identities $π \circ \tilde{f} = π \circ {\tilde{f}}^{'} = f \circ π$ are the definitions, using [F1].

2.1step 1.1F1F2

Smoothness in the sheet charts. Let $(U, φ)$ be a chart of $M$ and $(V, ψ)$ a chart of $M$ with $f (U) \subseteq V$ ; write $\hat{F} : = ψ \circ f \circ φ^{- 1}$ on $φ (U)$ , so $\det D {\hat{F}}_{u} \neq 0$ for all $u$ by [F2] and $u \mapsto \det D {\hat{F}}_{u}$ is continuous with locally constant sign. In the sheet charts of [F1] the point $(x, s_{U, φ}^{+} (x))$ is carried by $\tilde{f}$ to the point whose ray is $D f_{x} (s_{U, φ}^{+} (x)) = d ψ_{f (x)}^{- 1} (D {\hat{F}}_{φ (x)} (standard ray))$ , which equals $sign \det D {\hat{F}}_{φ (x)} \cdot s_{V, ψ}^{+} (f (x))$ ; hence the coordinate expression of $\tilde{f}$ is $(u, ϵ) \mapsto (\hat{F} (u), sign \det D {\hat{F}}_{u} \cdot ϵ)$ , smooth because $\hat{F}$ is smooth and the sign is locally constant on $φ (U)$ . The expression for $τ \tilde{f}$ differs only by the locally constant factor $- 1$ on the second coordinate, so ${\tilde{f}}^{'}$ is smooth too.

2.2step 1.1F1

Uniqueness of the two lifts. Suppose $M$ is nonorientable, so that $\tilde{M}$ is connected by [F1]; then $π$ is a two-sheeted covering with connected total space and deck group ${i d, τ}$ . Let $g : \tilde{M} \to \tilde{M}$ satisfy $π \circ g = f \circ π$ . Then $g$ and $\tilde{f}$ both lift the map $f \circ π$ through $π$ , so their difference is measured by a deck transformation: at each point, $g (p) = \tilde{f} (p)$ or $g (p) = τ \tilde{f} (p)$ , and continuity on the connected $\tilde{M}$ makes the choice constant; hence $g = \tilde{f}$ or $g = τ \tilde{f} = {\tilde{f}}^{'}$ . The two are distinct because $τ$ has no fixed point on $\tilde{M}$ , while ${\tilde{f}}^{'} = \tilde{f}$ would force $τ$ to fix every point of the nonempty set $\tilde{f} (\tilde{M})$ .

3.1step 2.1step 1.1given∎

Diffeomorphisms lift to diffeomorphisms. If $f$ is a diffeomorphism with inverse $f^{- 1}$ , form $\tilde{f^{- 1}}$ by the same construction, using the invertible differentials $D (f^{- 1})_{f (x)} = (D f_{x})^{- 1}$ that follow from the chain rule for $f^{- 1} \circ f = {i d}_{M}$ (The chain rule for differentials of smooth maps); then $\tilde{f^{- 1}} \circ \tilde{f} (x, o_{x}) = (x, (D f_{x})^{- 1} (D f_{x} (o_{x}))) = (x, o_{x})$ and likewise in the other order, so $\tilde{f}$ is a bijection with smooth inverse $\tilde{f^{- 1}}$ by step 2.1, hence a diffeomorphism; so is ${\tilde{f}}^{'} = τ \circ \tilde{f}$ . The construction uses only the given map, its differential and the cover, so it is canonical.

Remarks

Local diffeomorphism is exactly the hypothesis under which the formula is defined. If $D f_{x}$ is singular then $D f_{x} (o_{x})$ is the zero element of $\det T_{f (x)} M$ , not a ray, so $(f (x), D f_{x} (o_{x}))$ is not a point of $\tilde{M}$ . Moreover a general smooth self-map of a nonorientable closed manifold need not lift to the orientation double cover at all: for $M = R P^{2} \times S^{1}$ and $f$ collapsing the first factor to a point while wrapping the second factor once around the projective line, the induced map on $π_{1}$ sends the kernel of the orientation character outside that kernel, so the lifting criterion (Lifting criterion for maps from path-connected locally path-connected spaces) gives no lift. The transfer items on this page therefore carry the existence of a lift as an explicit hypothesis.
The orientable case. If $M$ is nonempty and orientable, $\tilde{M} = M \times Z / 2$ is disconnected and the formula produces only two of the four continuous lifts of $f \circ π$ ; the mixed lifts that act by $\tilde{f}$ on one component and $τ \tilde{f}$ on the other are never used, and the nonorientable case of the Lefschetz–Hopf formula is the only place where uniqueness of the two lifts is invoked. For $M = \emptyset$ , the orientation cover is empty and $f \circ π$ has exactly one lift; the two displayed formulas coincide with the unique empty map.

LemmaStatement: Literature-sourcedProof: Literature-sourcedprecheck passOpen item page →

The two lifts of a self-map carry twice the fixed point index sum

Statement

Let $M$ be a connected closed smooth $n$ -manifold, $n \geq 1$ , $π : \tilde{M} \to M$ its orientation double cover with deck transformation $τ$ (A smooth local diffeomorphism lifts canonically to the orientation double cover), let $f : M \to M$ be smooth with isolated fixed points, and let $\tilde{f} : \tilde{M} \to \tilde{M}$ be a smooth lift of $f$ commuting with $τ$ ( $π \circ \tilde{f} = f \circ π$ and $τ \circ \tilde{f} = \tilde{f} \circ τ$ ); set ${\tilde{f}}^{'} : = τ \circ \tilde{f}$ . (Such a lift exists when $f$ is a local diffeomorphism, by the derivative lift; it need not exist for general smooth $f$ .) Then $\tilde{f}$ and ${\tilde{f}}^{'}$ have isolated fixed points and finite fixed point sets, and $\sum_{\tilde{x} \in Fix (\tilde{f})} {ind}_{\tilde{x}} (\tilde{f}) + \sum_{\tilde{x} \in Fix ({\tilde{f}}^{'})} {ind}_{\tilde{x}} ({\tilde{f}}^{'}) = 2 \sum_{x \in Fix (f)} {ind}_{x} (f) .$ More precisely, over each fixed point $x$ of $f$ exactly one of the two lifts has fixed points, it fixes both points of the fibre $π^{- 1} (x)$ , and each of those two fixed points has local index ${ind}_{x} (f)$ .

Facts & Assumptions

Given: The connected closed smooth $n$ -manifold $M$ , its orientation double cover $(\tilde{M}, π, τ)$ , a smooth $f : M \to M$ with isolated fixed points and a $τ$ -commuting lift $\tilde{f}$ .

[F1]

$π$ is a smooth two-sheeted covering map with deck transformation $τ$ , $τ^{2} = i d$ and $π \circ τ = π$ ; each fibre is ${a, τ a}$ with $a \neq τ a$ ; $\tilde{M}$ is closed when $M$ is (A smooth local diffeomorphism lifts canonically to the orientation double cover).

[F2]

For an isolated fixed point $z$ of a smooth self-map of a boundaryless $n$ -manifold the local index is defined and unchanged under conjugation by a local diffeomorphism of a neighbourhood of the point (Isolated fixed point and local fixed point index, The local fixed point index is invariant under conjugation by a local diffeomorphism).

[L1]

Fixed points of a self-map are the points whose graph meets the diagonal, and the fixed point set of a smooth self-map of a manifold is closed; a closed discrete subset of a compact space is finite (Fixed points are exactly the intersections of the graph with the diagonal, A closed discrete subset of a compact space is finite).

Proof

1.1givenF1

Fixed points over a fixed point. Let $x \in Fix (f)$ and $π^{- 1} (x) = {a, τ a}$ . Since $π (\tilde{f} (a)) = f (π (a)) = f (x) = x$ , the point $\tilde{f} (a)$ lies in ${a, τ a}$ . If $\tilde{f} (a) = a$ , then $\tilde{f} (τ a) = τ \tilde{f} (a) = τ a$ by the commutation, so both fibre points are fixed by $\tilde{f}$ , while ${\tilde{f}}^{'} (a) = τ (a) = τ a \neq a$ and ${\tilde{f}}^{'} (τ a) = a \neq τ a$ ; if $\tilde{f} (a) = τ a$ , then $\tilde{f} (τ a) = τ \tilde{f} (a) = τ^{2} a = a$ , so both fibre points are fixed by ${\tilde{f}}^{'} = τ \tilde{f}$ and neither by $\tilde{f}$ . In both cases exactly two of the four pairs $(\tilde{g}, \tilde{x}) \in {\tilde{f}, {\tilde{f}}^{'}} \times π^{- 1} (x)$ satisfy $\tilde{g} (\tilde{x}) = \tilde{x}$ , namely one lift fixing both points of the fibre. Conversely, a fixed point of $\tilde{f}$ or of ${\tilde{f}}^{'}$ projects to a fixed point of $f$ , because $π \circ \tilde{f} = f \circ π$ .

2.1step 1.1F1L1

Isolation and finiteness. Let $\tilde{x}$ be a fixed point of $\tilde{f}$ and $x = π (\tilde{x})$ . Choose a neighbourhood $W$ of $x$ containing no fixed point of $f$ other than $x$ . Since $π$ is a local homeomorphism and $\tilde{M}$ is Hausdorff, choose a neighbourhood $\tilde{W}$ of $\tilde{x}$ with $π (\tilde{W}) \subseteq W$ and $τ \tilde{x} \notin \tilde{W}$ . Any fixed point of $\tilde{f}$ in $\tilde{W}$ projects into $W$ , hence lies over $x$ , hence is $\tilde{x}$ or $τ \tilde{x}$ ; the second is excluded by $τ \tilde{x} \notin \tilde{W}$ . So the fixed points of $\tilde{f}$ are isolated, and the same argument applies to ${\tilde{f}}^{'}$ . Their fixed sets are closed by [L1] and discrete, and $\tilde{M}$ is compact by [F1], so both fixed sets are finite by [L1].

3.1step 1.1step 2.1F2given∎

Local indices. Let $\tilde{x}$ be a fixed point of $\tilde{f}$ with $π (\tilde{x}) = x$ . Since $π$ is a local diffeomorphism, it restricts to a diffeomorphism from an open neighbourhood of $\tilde{x}$ onto an open neighbourhood of $x$ , and from $π \circ \tilde{f} = f \circ π$ we get $\tilde{f} = π^{- 1} \circ f \circ π$ on that neighbourhood; the conjugation lemma [F2] therefore gives ${ind}_{\tilde{x}} (\tilde{f}) = {ind}_{x} (f)$ . The same computation applies to every fixed point of ${\tilde{f}}^{'}$ , which also satisfies $π \circ {\tilde{f}}^{'} = f \circ π$ . Step 1.1 says that over each fixed point $x$ of $f$ exactly one of the two lifts has fixed points, and it fixes both points of the fibre, so the total of the local indices of $\tilde{f}$ and ${\tilde{f}}^{'}$ over $x$ is $2 {ind}_{x} (f)$ . Summing over the finite set $Fix (f)$ — the geometric Lefschetz number of $f$ is the finite index sum of Geometric Lefschetz number (index sum) — gives the displayed identity.

LemmaStatement: Literature-sourcedProof: Literature-sourcedprecheck passjudge pass (gpt-6.1-sol)Open item page →

The Lefschetz numbers of the two lifts sum to twice the base Lefschetz number

Statement

Assume AC (The Axiom of Choice). Let $M$ be a connected closed smooth $n$ -manifold, $π : \tilde{M} \to M$ its orientation double cover with deck transformation $τ$ (The orientation double cover is canonically oriented and preserves closedness), and let $\tilde{f} : \tilde{M} \to \tilde{M}$ be a continuous lift of a continuous $f : M \to M$ commuting with $τ$ ( $π \circ \tilde{f} = f \circ π$ , $τ \circ \tilde{f} = \tilde{f} \circ τ$ ; such a lift exists when $f$ is a local diffeomorphism, by A smooth local diffeomorphism lifts canonically to the orientation double cover, and need not exist otherwise). Then $L (\tilde{f}) + L (τ \circ \tilde{f}) = 2 L (f),$ where $L$ is the algebraic Lefschetz number of Algebraic Lefschetz number via rational homology traces. Equivalently, if $H_{*} (\tilde{M}; Q) = V_{+} \oplus V_{-}$ is the eigenspace decomposition of $τ_{*}$ with eigenvalues $+ 1$ and $- 1$ , then $π_{*}$ restricts to an isomorphism $V_{+} ≅ H_{*} (M; Q)$ that conjugates ${\tilde{f}}_{*} ∣_{V_{+}}$ to $f_{*}$ , and $L (\tilde{f}) + L (τ \circ \tilde{f}) = 2 \sum_{i} (- 1)^{i} tr ({\tilde{f}}_{*} ∣ V_{+}^{i}) = 2 L (f) .$

Facts & Assumptions

Given: A connected closed smooth $n$ -manifold $M$ , its orientation double cover $(\tilde{M}, π, τ)$ , a continuous $f : M \to M$ and a $τ$ -commuting lift $\tilde{f}$ .

[F1]

$π$ is a two-sheeted covering with deck transformation $τ$ , $τ^{2} = i d$ and $π \circ τ = π$ (Covering maps, evenly covered neighbourhoods, fibres, sheets, and trivial coverings, Deck transformations and the deck-transformation group of a covering, The orientation double cover is canonically oriented and preserves closedness).

[F2]

For a singular simplex $σ : Δ^{k} \to M$ the set of lifts through $π$ has exactly two elements: the standard simplex is connected and simply connected, so the lifting criterion gives a lift once the image of a vertex is chosen, and lifts from a connected space are unique (Lifting criterion for maps from path-connected locally path-connected spaces, Two lifts from a connected space that agree at one point agree everywhere, A connected covering of a locally path-connected simply connected space is one-sheeted and trivial, The standard topological simplex and its affine face maps, Singular simplices and singular chain groups with coefficients).

[F3]

Singular chains and homology are covariantly functorial, and singular cohomology is contravariantly functorial (Singular chains and singular homology are covariantly functorial, Singular cohomology is contravariantly functorial); $L$ is the alternating trace sum over rational homology, well defined because the rational homology of a closed manifold is finite-dimensional and vanishes above degree $n$ (Algebraic Lefschetz number via rational homology traces).

[L1]

Over a field, cohomology is dual to homology, and the trace of the dual endomorphism equals the trace of the original; the alternating trace may be computed in either (Cohomology over a field is dual to homology over that field, The basis-independent trace of an endomorphism of a finite-dimensional vector space).

Proof

1.1givenF2F3

The transfer. For a singular simplex $σ$ let $T_{#} (σ)$ be the sum of its two lifts through $π$ ; the sum over the full lift set is independent of any selection, so it defines a rational-linear map $T_{#} : C_{*} (M; Q) \to C_{*} (\tilde{M}; Q)$ . It is a chain map: restriction to a face bijects the lift set of $σ$ with the lift set of its faces, since a lift of a face extends uniquely along the inclusion of the connected, simply connected simplex, so boundaries commute with $T_{#}$ . By [F1] and [F2], $π_{#} T_{#} = 2 i d$ and $T_{#} π_{#} = i d + τ_{#}$ on chains, hence on homology $π_{*} T_{*} = 2 i d$ and $T_{*} π_{*} = i d + τ_{*}$ . Also $τ_{*} T_{*} = T_{*}$ because the deck involution exchanges the two summands of every transfer.

2.1step 1.1

The invariant decomposition. Since $τ^{2} = i d$ , the involution $τ_{*}$ of $H_{*} (\tilde{M}; Q)$ has eigenvalues $\pm 1$ and splits the space as $V_{+} \oplus V_{-}$ . From step 1.1, $T_{*} (y) / 2 \in V_{+}$ and $π_{*} (T_{*} (y) / 2) = y$ for every $y$ , so $π_{*} ∣_{V_{+}}$ is surjective; and $π_{*} ∣_{V_{+}}$ is injective, because $π_{*} x = 0$ with $x \in V_{+}$ gives $0 = T_{*} π_{*} x = x + τ_{*} x = 2 x$ ; hence $π_{*}$ restricts to an isomorphism $V_{+} \to H_{*} (M; Q)$ . Naturality $π_{*} {\tilde{f}}_{*} = f_{*} π_{*}$ conjugates ${\tilde{f}}_{*} ∣_{V_{+}}$ to $f_{*}$ , and because $\tilde{f}$ commutes with $τ$ the map ${\tilde{f}}_{*}$ preserves each $V_{\pm}$ .

3.1step 2.1F3L1∎

The trace identities. By step 2.1, $tr ({\tilde{f}}_{*} ∣ V_{+}^{i}) = tr (f_{*} ∣ H_{i} (M; Q))$ , and $(τ \tilde{f})_{*} = τ_{*} {\tilde{f}}_{*}$ acts as $+ {\tilde{f}}_{*}$ on $V_{+}$ and $- {\tilde{f}}_{*}$ on $V_{-}$ ; hence the alternating sums satisfy $L (\tilde{f}) = \sum_{i} (- 1)^{i} [tr ({\tilde{f}}_{*} ∣ V_{+}^{i}) + tr ({\tilde{f}}_{*} ∣ V_{-}^{i})]$ and $L (τ \tilde{f}) = \sum_{i} (- 1)^{i} [tr ({\tilde{f}}_{*} ∣ V_{+}^{i}) - tr ({\tilde{f}}_{*} ∣ V_{-}^{i})]$ , whose sum is $2 \sum_{i} (- 1)^{i} tr (f_{*} ∣ H_{i} (M; Q)) = 2 L (f)$ . The same computation may be read in cohomology by [L1], which is how the source states the transfer. AC enters only through the finiteness in Algebraic Lefschetz number via rational homology traces; the transfer itself is canonical and the two-lift sums involve no selection.

LemmaStatement: Literature-sourcedProof: AI-adaptedjudge pass (gpt-6.1-sol)Open item page →

Orientation coefficients are deck eigenspaces, with product and duality pairings

Statement

Assume AC. Let $M$ be a connected closed smooth $n$ -manifold, $O = O_{M}^{Q}$ its rational orientation system, and $π : \tilde{M} \to M$ its orientation cover with involution $τ$ . Cohomology with constant coefficients and with $O$ identifies respectively with the $+ 1$ and $- 1$ eigenspaces of $τ^{*}$ on $H^{*} (\tilde{M}; Q)$ . With $p_{1}, p_{2} : M \times M \to M$ , cross products give $H^{r} (M \times M; p_{1}^{*} O) ≅ ⨁_{i + j = r} H^{i} (M; O) \otimes H^{j} (M; Q),$ and the analogous formula with $p_{2}^{*} O$ has the two coefficient systems reversed. The pairing $H^{n - p} (M; O) \otimes H^{p} (M; Q) ⟶ Q, (β, α) ⟼ ⟨ β ⌣ α, [M]^{t w} ⟩$ is perfect. These products obey the usual Koszul rule, including graded commutativity with the coefficient factors interchanged, and evaluation on the product twisted fundamental class is the product of the two factor evaluations. The orientation system of $M \times M$ is $p_{1}^{*} O \otimes p_{2}^{*} O$ , with factor order first then second.

Facts & Assumptions

Given: The objects and AC in the statement.

[F1]

Homology and cohomology with local coefficients.

[F2]

Cup and cap products with local coefficients.

[F3]

The orientation double cover is canonically oriented and preserves closedness.

[F4]

Lifting criterion for maps from path-connected locally path-connected spaces.

[F5]

Two lifts from a connected space that agree at one point agree everywhere.

[F6]

Cohomological Kunneth isomorphism under finite free hypotheses.

[F7]

Poincare duality with the orientation local system.

[F8]

Canonical twisted fundamental classes over compact subsets.

[F9]

Cohomology over a field is dual to homology over that field.

[F10]

Singular cohomology is graded commutative.

[F11]

Algebraic Lefschetz number via rational homology traces.

[F12]

The Axiom of Choice.

Proof

1.1givenF1F3F4F5

On the cover, the tautological local orientation at $(x, o)$ trivializes $π^{*} O$ : a fiber coefficient is written $c o$ with $c \in Q$ . A local cochain $φ$ therefore assigns to a lifted simplex $\tilde{σ}$ the scalar obtained by expressing $φ (π \tilde{σ})$ in the tautological orientation at its initial vertex. Replacing the lift by $τ \tilde{σ}$ negates this scalar. Conversely an anti-invariant scalar cochain defines a fiber value independent of the lift, since both scalar and orientation negate. For constant coefficients the same construction has no sign and gives invariant cochains. Each simplex lifts after its initial vertex is specified, and lift uniqueness makes restriction to faces agree with coefficient transport; thus these are inverse cochain maps, also for relative pairs.

2.1step 1.1algebra

For a cochain complex with an involution $t$ commuting with its differential, $P_{\pm} = (1 \pm t) / 2$ are complementary cochain projections. An invariant or anti-invariant cohomology class has a cocycle representative in the same subcomplex by applying the corresponding projection. If a cocycle in that subcomplex bounds in the full complex, applying the projection to a primitive makes it bound there. Thus cohomology of the subcomplex equals the corresponding cohomology eigenspace. This proves the first assertion using step 1.1. For chains the identical proof uses the weighted sum of the two lifts: changing a chosen orientation negates both its coefficient and the lift difference. This identifies local chains with the anti-invariant chains, up to the harmless normalization factor two, and respects boundaries.

3.1F6F11step 1.1step 2.1

Apply step 1.1 on the four-sheeted cover $q = π \times π$ . A $p_{1}^{*} O$ cochain is exactly a cochain anti-invariant under $(τ, 1)$ and invariant under $(1, τ)$ ; a $p_{2}^{*} O$ cochain has the reversed parities. The commuting projections $(1 \pm (τ, 1)^{*}) / 2$ and $(1 \pm (1, τ)^{*}) / 2$ show, by step 2.1, that these identifications also hold in cohomology. The ordinary rational Kunneth isomorphism on $\tilde{M} \times \tilde{M}$ is natural for both involutions. Restricting it to the $(-, +)$ and $(+, -)$ summands gives exactly the two claimed cross-product isomorphisms; each degree has finitely many summands and finite-dimensional factors by closed-manifold finiteness.

4.1F2F10step 1.1step 3.1

The local cup formula lifts to the ordinary scalar cup formula because transport of the tautological orientation along a lifted simplex is precisely its orientation-system transport. Consequently its cross-product and cup signs are the ordinary ones on $q$ . Pullback to the relevant parity subcomplex is injective on cohomology by step 2.1, so graded commutativity and the Koszul rule upstairs prove these identities downstairs. On every product chart the ordered tangent splitting identifies the product orientation system with $p_{1}^{*} O \otimes p_{2}^{*} O$ . This local identification is independent of the two orientation choices since a reversal negates the corresponding factor on each side. Hence it is a global identification.

5.1F2F8step 2.1step 4.1

The canonical twisted fundamental chain on $M$ pulls up by the weighted lift construction to the ordinary fundamental class of the canonically oriented cover; pairing a lifted orientation-coefficient cocycle with that class is twice its downstairs evaluation, since each simplex has two lifts with equal signed evaluations. For $q$ the factor is four. Ordinary product evaluation on $\tilde{M} \times \tilde{M}$ , divided by four, is therefore the product of the two downstairs evaluations (each divided by two). These statements also follow simplexwise from the lift sums, so do not depend on a triangulation. The local characterization of the twisted fundamental classes supplies the classes used here.

6.1F2F7F9F11F12step 5.1∎

Twisted Poincare duality sends $β \in H^{n - p} (M; O)$ to $β \cap [M]^{t w} \in H_{p} (M; Q)$ using the canonical pairing $O \otimes O \to \underline{Q}$ , $(c o) \otimes (d o) \mapsto c d$ ; this pairing is independent of $o$ since both factors negate. The cohomology-first cap identity gives $⟨ β ⌣ α, [M]^{t w} ⟩ = ⟨ α, β \cap [M]^{t w} ⟩$ . Duality is an isomorphism and rational cohomology is the full dual of finite-dimensional rational homology, so this pairing is perfect. AC is inherited from the duality, Kunneth and finiteness suppliers. No map $f$ , and no lift of a map $f$ , has been assumed or constructed.

LemmaStatement: Literature-sourcedProof: AI-adaptedOpen item page →

The orientation-twisted diagonal realizes the Lefschetz trace

Statement

Assume AC. Let $M$ be a connected closed smooth $n$ -manifold and $O = O_{M}^{Q}$ . Orient the normal coordinate to its diagonal by first minus second: $(u, v) \mapsto u - v$ . There is a normalized supported diagonal class $u_{Δ} \in H^{n} (M \times M, (M \times M) ∖ Δ; p_{1}^{*} O)$ ; write $U$ for its absolute image. With $δ (x) = (x, x)$ , $U \cap [M \times M]^{t w} = δ_{*} [M]^{t w} in H_{n} (M \times M; p_{2}^{*} O) .$ Choose a basis $α_{p, j}$ of $H^{p} (M; Q)$ and the uniquely dual basis $β_{p, j} \in H^{n - p} (M; O)$ satisfying $⟨ β_{p, j} ⌣ α_{p, k}, [M]^{t w} ⟩ = δ_{j k}$ . Then $U = \sum_{p, j} (- 1)^{p} β_{p, j} \times α_{p, j} .$ For every smooth map $f : M \to M$ , its graph map $γ_{f} (x) = (x, f (x))$ pulls the coefficient system $p_{1}^{*} O$ back to $O$ , and $⟨ γ_{f}^{*} U, [M]^{t w} ⟩ = \sum_{p} (- 1)^{p} tr (f^{*} : H^{p} (M; Q) \to H^{p} (M; Q)) = L (f) .$ If $n \geq 1$ and all fixed points of $f$ are nondegenerate, the left side is $\sum_{x} sign \det (I - D f_{x}) = I (f)$ . No orientability or lifting hypothesis on $f$ is needed.

Facts & Assumptions

Given: The objects and AC in the statement.

[F1]

Orientation coefficients are deck eigenspaces, with product and duality pairings.

[F2]

Poincare duality with the orientation local system.

[F3]

Canonical twisted fundamental classes over compact subsets.

[F4]

Cup and cap products with local coefficients.

[F5]

The tubular neighbourhood theorem in a smooth ambient manifold.

[F6]

Excision and Mayer–Vietoris with local coefficients.

[F7]

Thom isomorphism for oriented vector bundles.

[F8]

Naturality and uniqueness of Thom classes.

[F9]

The normal Thom class realizes the Poincare dual of a closed submanifold.

[F10]

The orientation double cover is canonically oriented and preserves closedness.

[F11]

Algebraic Lefschetz number via rational homology traces.

[F12]

The index of a nondegenerate fixed point is the sign of det(I-Df).

[F13]

Graph-diagonal transversality is exactly fixed-point nondegeneracy.

[F14]

Cohomology over a field is dual to homology over that field.

[F15]

The basis-independent trace of an endomorphism of a finite-dimensional vector space.

[F16]

The Axiom of Choice.

[F17]

The closed smooth bases $Z_{i} ≅ \tilde{M}$ are paracompact Hausdorff (Topological manifolds are metrizable and paracompact) and have finite CW type: choose an excellent Morse function (Every compact smooth manifold admits an excellent Morse function), its finite handle presentation (The handle chain complex computes singular homology), and the finite CW model of A handle decomposition gives a relative CW complex. A finite trivializing cover admits a subordinate smooth partition by taking finitely many compactly supported chart bumps whose positive sets cover the compact base and dividing by their positive sum (A manifold bump for a compact set inside an open set). This supplies numerability for [F7].

Proof

1.1givenF1F5F6F7F8F10F17

Construct the supported class without an unoriented Thom theorem. Pull a tubular neighborhood of $Δ$ back to $\tilde{M} \times \tilde{M}$ . Its zero set is the disjoint union $Z_{0} = {(a, a)}$ and $Z_{1} = {(a, τ a)}$ . On either normal bundle, $d π_{a} (v) - d π_{b} (w)$ identifies the normal quotient with $T_{π (a)} M$ ; use the tautological orientation at $a$ to orient it. Each is an oriented rank- $n$ bundle on a base satisfying [F17], so [F7] supplies its unique fiber-normalized rational Thom class. Their sum, extended from disjoint tubes, is a relative class on the four-sheeted cover. The first deck involution reverses the specified normal orientation, and the second preserves it; uniqueness of the Thom classes therefore makes the sum anti-invariant under the first and invariant under the second. For relative descent, apply the cochain and projection construction in the proof of [F1] to $X = M \times M$ , $A = X ∖ Δ$ and $q = π \times π$ . Expressing a local cochain value in the first tautological orientation identifies it with a scalar cochain anti-invariant under $t_{1} = (τ, 1)$ and invariant under $t_{2} = (1, τ)$ ; restrictions to faces agree with coefficient transport. Vanishing on simplices in $A$ corresponds exactly to vanishing on their lifts in $q^{- 1} A$ , so the identification restricts to the relative complexes. The cochain projection $P = (1 - t_{1}^{*}) (1 + t_{2}^{*}) / 4$ preserves that relative complex and commutes with its differential. Applying $P$ to cocycle representatives of $(-, +)$ classes, and to primitives of exact $(-, +)$ cocycles, proves both surjectivity and injectivity onto the relative cohomology eigenspace. Thus the Thom sum descends uniquely to $u_{Δ}$ with coefficient $p_{1}^{*} O$ . On a common base chart its fiber normalization is the generator for $u - v$ , with coefficient the orientation of that chart.

2.1F2F3F4F8F9step 1.1

Check the cap normalization. On the oriented four-sheeted ambient manifold, let $Z_{i}$ have the orientation of its first factor. The ordered tangent-then-normal basis with normal coordinate $u - v$ has determinant $(- 1)^{n}$ relative to ambient first-then-second coordinates: in equal base coordinates its block matrix is $(\begin{matrix} I & I \\ I & 0 \end{matrix})$ . If the second lift has the opposite base orientation there is the additional sign from that orientation reversal. Apply [F9] first with integral coefficients on the oriented cover and then send its normalized Thom and fundamental classes to rational coefficients; the singular cup, cap and inclusion formulas commute with this coefficient map, and uniqueness in [F8] identifies the rational Thom class. Thus the cohomology-first normal-cap formula supplies the further shuffle sign $(- 1)^{n^{2}} = (- 1)^{n}$ . Thus the two $(- 1)^{n}$ factors cancel. After descent the second-factor orientation discrepancy is precisely carried by the coefficient $p_{2}^{*} O$ , and the result is the diagonal's canonical twisted fundamental class with that coefficient. For completeness this is a global equality, not merely a local sign test: in the tube, cap has support in the zero section after the fiber retraction; its pushforward to that section is a top twisted class, and its restrictions at every point are the just-computed canonical local generators. Uniqueness of the top twisted fundamental class gives that class, and the natural cap formula and open-tube inclusion give the asserted ambient equality. These formulas hold with local coefficients because the face transports in [F4] are exactly the scalar formulas in every lifted chart, and step 1.1's relative descent is injective.

3.1F1F4step 2.1

Characterize $U$ by testing. For every $φ \in H^{n} (M \times M; p_{2}^{*} O)$ , the cap identity and step 2.1 give $⟨ U ⌣ φ, [M \times M]^{t w} ⟩ = ⟨ δ^{*} φ, [M]^{t w} ⟩$ . The coefficient contraction is $p_{1}^{*} O \otimes (p_{1}^{*} O \otimes p_{2}^{*} O) \to p_{2}^{*} O$ , so the typing is exact. This pairing separates $H^{n} (M \times M; p_{1}^{*} O)$ : [F1]'s Kunneth decompositions and perfect factor pairings make its matrix a blockwise tensor product of invertible matrices, with unit Koszul signs.

4.1F1step 3.1algebra

Test the proposed expansion on $φ = α_{p, k} \times β_{p, l}$ . A term $β_{r, j} \times α_{r, j}$ can pair nontrivially only for $r = p$ . Its product evaluation is $(- 1)^{p} ⟨ β_{p, j} ⌣ α_{p, k}, [M]^{t w} ⟩ ⟨ α_{p, j} ⌣ β_{p, l}, [M]^{t w} ⟩ = (- 1)^{p + p (n - p)} δ_{j k} δ_{j l}$ , where the first $(- 1)^{p}$ is the cross-product Koszul sign $(- 1)^{p^{2}}$ . Including the proposed coefficient $(- 1)^{p}$ leaves $(- 1)^{p (n - p)} δ_{k l}$ . The diagonal evaluation is exactly $⟨ α_{p, k} ⌣ β_{p, l}, [M]^{t w} ⟩ = (- 1)^{p (n - p)} δ_{k l}$ . These tests span by [F1], and step 3.1 separates classes, proving the expansion with the stated sign.

5.1F1F4F11F14F15step 4.1

Pull back along the graph. Since $p_{1} γ_{f} = {i d}_{M}$ , no coefficient comparison involving $f^{*} O$ is required. Write $f^{*} α_{p, j} = \sum_{k} a_{k j} α_{p, k}$ . Naturality of cup and cross products in [F4] gives $γ_{f}^{*} U = \sum_{p, j} (- 1)^{p} β_{p, j} ⌣ f^{*} α_{p, j}$ , whose evaluation is $\sum_{p, j} (- 1)^{p} a_{j j}$ by the chosen duality. This is the alternating cohomology trace, and it equals the homology trace by field duality. This uses the graph pullback directly; it never asserts that $f \times i d$ is a diffeomorphism.

6.1F3F6F8F12F13step 1.1step 5.1

If every fixed point is nondegenerate, graph transversality gives a finite preimage of the diagonal. Pull the relative supported class back along $γ_{f}$ and excise disjoint coordinate balls around these points. In such a ball its normal coordinate is $u - \hat{f} (u)$ , and its derivative at the point is $I - D f_{x}$ . Pullback of the oriented normal Thom generator evaluates on the local twisted fundamental class by the degree of this map; for its invertible derivative this degree is $sign \det (I - D f_{x})$ by [F12]. Excision and the finite decomposition of the relative fundamental class add these evaluations. Consequently the absolute evaluation in step 5.1 is the sum of these local signs, namely $I (f)$ . Changing chart orientation reverses both the normal generator and the twisted fundamental coefficient, so the integer local value is unchanged.

7.1F1F2F5F7F11F16step 5.1step 6.1∎

Empty fixed set gives a relative pullback through an empty support, hence zero and the empty index sum. The construction also covers orientable $M$ (its orientation cover has two components when $M$ is nonempty); an orientation trivializes $O$ and gives the ordinary diagonal and graph-pullback formula. AC enters through the stated duality, Kunneth, tubular and Thom suppliers and the finite-dimensional trace definition.

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Lefschetz-Hopf index formula

Statement

Assume AC (The Axiom of Choice). Let $M$ be a closed smooth $n$ -manifold, $n \geq 1$ , possibly disconnected or nonorientable, and let $f : M \to M$ be smooth with only isolated fixed points. Then $Fix (f)$ is finite, its geometric index sum $I (f) = \sum_{x} {ind}_{x} (f)$ is defined (Geometric Lefschetz number (index sum)), and $I (f) = L (f)$ , where $L (f)$ is the algebraic Lefschetz number of Algebraic Lefschetz number via rational homology traces.

Facts & Assumptions

Given: $M, n, f$ and AC as in the statement.

[F1]

A continuous self-map of a Hausdorff space has a closed fixed set, since the diagonal is closed; a closed discrete subset of a compact space is finite (A space is Hausdorff if and only if its diagonal is closed in the square carrying the product topology, A closed discrete subset of a compact space is finite).

[F2]

An isolated fixed point splits under perturbation, preserving its index permits a smooth homotopy supported in a small ball isolating a fixed point which replaces that point by finitely many nondegenerate fixed points with the same total index.

[F3]

The orientation-twisted diagonal realizes the Lefschetz trace proves $I (g) = L (g)$ for every smooth self-map of a connected closed manifold whose fixed points are nondegenerate, without an orientation or lifting hypothesis.

[F4]

Manifold components are open; compactness gives finitely many. Their rational homology groups decompose as a finite direct sum, and the trace is the sum of the diagonal component-block traces (Connected components, quasicomponents, and totally disconnected spaces, The singular homology of a disjoint union is the direct sum, Algebraic Lefschetz number via rational homology traces). Homotopic maps induce the same homology maps (Homotopic maps induce the same map on singular homology).

Proof

1.1givenF1F2

By [F1] isolation and compactness make the fixed set finite. Choose pairwise disjoint admissible balls isolating its points. Applying [F2] successively in these balls yields a smooth map $g$ homotopic to $f$ , unchanged outside the balls, with every fixed point nondegenerate and $I (g) = I (f)$ . There are no additional fixed points outside the balls because $g = f$ there, and the finite index sum is defined by Geometric Lefschetz number (index sum).

2.1F3F4step 1.1

Each connected component $C$ is carried by $g$ into a single component, since its image is connected. If that component is $C$ , [F3] applies to the self-map $g ∣_{C}$ and gives $I (g ∣_{C}) = L (g ∣_{C})$ . If it is a different component, $C$ has no fixed points, and the source-to- $C$ diagonal block of $g_{*}$ on the homology direct sum in [F4] is zero. Thus that component contributes zero both to the index sum and to the trace. Summing the finitely many diagonal-block identities gives $I (g) = L (g)$ .

3.1F4step 1.1step 2.1∎

Since $f$ and $g$ are homotopic, [F4] gives $L (f) = L (g)$ . Combining with the preceding steps gives $I (f) = I (g) = L (g) = L (f)$ . AC is inherited from the finite-dimensional Lefschetz-number and twisted diagonal suppliers; the perturbations and the finite component decomposition require no global orientation or lift of either map.

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The Lefschetz number is a homotopy invariant

Statement

Assume AC (The Axiom of Choice). Let $M$ be a closed smooth manifold and let $f, g : M \to M$ be homotopic continuous maps. Then $L (f) = L (g)$ (Algebraic Lefschetz number via rational homology traces). If $\dim M \geq 1$ and $f$ and $g$ are smooth with isolated fixed points, then their geometric index sums satisfy $I (f) = I (g)$ (Geometric Lefschetz number (index sum)).

Facts & Assumptions

Given: $M, f, g$ and AC as in the statement.

[F1]

The Lefschetz number is the alternating rational homology trace, and homotopic maps induce the same homology maps (Algebraic Lefschetz number via rational homology traces, Homotopic maps induce the same map on singular homology).

[F2]

For smooth maps with isolated fixed points, Lefschetz-Hopf index formula identifies the geometric index sum of Geometric Lefschetz number (index sum) with the algebraic Lefschetz number.

Proof

1.1givenF1

A homotopy from $f$ to $g$ gives $f_{*} = g_{*}$ on every rational homology group by [F1]. The finite alternating sums of their traces therefore agree: $L (f) = L (g)$ .

2.1F2step 1.1∎

If $\dim M \geq 1$ and both maps are smooth with isolated fixed points, [F2] applies to each map without any orientability or lifting restriction. Hence $I (f) = L (f) = L (g) = I (g)$ . AC is inherited from the Lefschetz-number and index-formula suppliers.

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The Lefschetz number of the identity is the Euler characteristic

Statement

Assume AC (The Axiom of Choice). Let $M$ be a closed smooth $n$ -manifold. Then $L ({i d}_{M}) = χ (M),$ the Euler characteristic of Euler characteristic of a compact manifold. Consequently every smooth self-map of a closed smooth manifold homotopic to the identity has Lefschetz number $χ (M)$ .

Facts & Assumptions

Given: A closed smooth $n$ -manifold $M$ .

[F1]

$L (f) = \sum_{i = 0}^{n} (- 1)^{i} tr (f_{*} : H_{i} (M; Q) \to H_{i} (M; Q))$ , and the identity map induces the identity on homology (Algebraic Lefschetz number via rational homology traces).

[F2]

$χ (M) = \sum_{i = 0}^{n} (- 1)^{i} \dim_{Q} H_{i} (M; Q)$ , a finite sum because the rational homology is finite-dimensional and vanishes above degree $n$ (Euler characteristic of a compact manifold, Finiteness and additivity of the Euler characteristic clause (i)).

[L1]

Homotopic maps have equal Lefschetz numbers (The Lefschetz number is a homotopy invariant), and on the scope of Lefschetz-Hopf index formula the Lefschetz number equals the geometric index sum of a smooth map with isolated fixed points.

Proof

1.1givenF1F2

The identity's traces. By [F1] the induced map $({i d}_{M})_{*}$ is the identity of $H_{i} (M; Q)$ for each $i$ , so $tr (({i d}_{M})_{*}) = \dim_{Q} H_{i} (M; Q)$ ; therefore $L ({i d}_{M}) = \sum_{i} (- 1)^{i} \dim_{Q} H_{i} (M; Q)$ , the same finite alternating sum that defines $χ (M)$ in [F2]. Hence $L ({i d}_{M}) = χ (M)$ .

2.1step 1.1L1∎

Maps homotopic to the identity. If $f$ is smooth and homotopic to ${i d}_{M}$ , then $L (f) = L ({i d}_{M}) = χ (M)$ by [L1] and step 1.1. When in addition $\dim M \geq 1$ and $f$ has isolated fixed points, the same number is the geometric index sum $I (f)$ , which is how the identity's Lefschetz number is recovered geometrically by a small perturbation of the identity. For nonempty $M$ with $n \geq 1$ , every point is fixed by ${i d}_{M}$ and no point is isolated, so its geometric index sum is not defined directly. If $M = \emptyset$ and $n \geq 1$ , the identity has no fixed points and $I ({i d}_{M}) = 0 = L ({i d}_{M}) = χ (M)$ .

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Lefschetz fixed point theorem

Statement

Assume AC (The Axiom of Choice). Let $M$ be a closed smooth manifold and let $f : M \to M$ be continuous. If $L (f) \neq 0$ (Algebraic Lefschetz number via rational homology traces), then $f$ has a fixed point. No converse is asserted: $L (f) = 0$ does not force $f$ to be fixed-point-free, as the companion counterexample shows.

Facts & Assumptions

Given: AC, a closed smooth manifold $M$ and a continuous self-map $f$ .

[F1]

Under countable choice, $M$ admits a proper smooth Euclidean embedding (The weak Whitney proper embedding theorem) and its closed image $S$ has an open neighbourhood $U$ with smooth retraction $r : U \to S$ (A closed Euclidean submanifold has a smooth neighborhood retraction). AC implies the required countable choice (The Axiom of Choice, The Axiom of Countable Choice ( ${A C}_{ω}$ )).

[F2]

A continuous Euclidean-valued map has a smooth approximation within any positive continuous error bound (Whitney approximation for Euclidean-valued maps).

[F3]

Continuous images of compact spaces are compact, and continuous real-valued functions on nonempty compact spaces attain extrema (A continuous image of a compact space is compact; a continuous real-valued map on a nonempty compact space attains a maximum and a minimum; and a continuous bijection from a compact space to a Hausdorff space is a homeomorphism, clauses 1–2). Closed bounded Euclidean subsets are compact (Heine-Borel in $R^{n}$ : with the Euclidean metric a subset of $R^{n}$ is compact if and only if it is closed and bounded, and the proof by bisection uses no choice principle; the same holds on the real line).

[F4]

Homotopic self-maps have the same Lefschetz number (The Lefschetz number is a homotopy invariant). A smooth fixed-point-free self-map in positive dimension has $L = I = 0$ , by Lefschetz-Hopf index formula and the empty-sum convention of Geometric Lefschetz number (index sum). The trace definition is Algebraic Lefschetz number via rational homology traces.

[F5]

Singular chains are finite formal sums of continuous simplices (Singular simplices and singular chain groups with coefficients).

Proof

1.1givenF4F5cases

Prove the contrapositive, assuming $f$ has no fixed point. If $M$ is empty, its homology groups and $L (f)$ are zero. If $\dim M = 0$ , compactness makes the discrete manifold finite. Every singular simplex is constant; in each point summand its boundary is multiplication by $\sum_{j = 0}^{k} (- 1)^{j}$ , which is one for positive even $k$ and zero for odd $k$ . Thus homology is zero in positive degrees and $H_{0}$ has the point basis. The matrix of $f_{*}$ on this basis has a diagonal one exactly at a fixed point, so its trace is zero. Hence $L (f) = 0$ . Assume now $M \neq \emptyset$ and $\dim M \geq 1$ .

1.2givenF1F3

Embed $M$ by $e$ as $S \subset R^{N}$ and take $r : U \to S$ from [F1]. Write $F = e \circ f$ . The function $x \mapsto ∥ e (x) - F (x) ∥$ is continuous and positive, so [F3] gives a minimum $d > 0$ . Choose $b > 0$ such that the closed $b$ -neighbourhood $K$ of $S$ lies in $U$ : finitely many open balls whose doubled balls lie in $U$ cover compact $S$ , and the minimum of their radii supplies such a $b$ after shrinking. The set $K$ is compact by Euclidean closedness and boundedness. Continuity of $r$ on $K$ supplies $η > 0$ , with $η < b$ , such that $∥ y - z ∥ < η$ for $y, z \in K$ implies $∥ r (y) - r (z) ∥ < d / 2$ : cover $K$ by neighbourhood balls on which oscillation is less than $d / 4$ , take a finite cover by their half-sized balls, and use the minimum half-radius.

2.1step 1.2F1F2construct

By [F2], choose smooth $A : M \to R^{N}$ with $∥ A (x) - F (x) ∥ < η$ for every $x$ . The segments $(1 - t) F (x) + t A (x)$ lie in $K \subset U$ , so $H (t, x) = e^{- 1} r ((1 - t) F (x) + t A (x))$ is a continuous homotopy from $f$ to the smooth self-map $g = e^{- 1} r A$ . Moreover $∥ e (g (x)) - F (x) ∥ = ∥ r (A (x)) - r (F (x)) ∥ < d / 2$ since $r (F (x)) = F (x)$ . Thus $∥ e (x) - e (g (x)) ∥ > d / 2$ , and $g$ is fixed-point-free.

3.1step 1.1step 2.1F4∎

By [F4], $L (g) = 0$ and $L (f) = L (g)$ , proving the contrapositive and hence the fixed-point theorem for every closed smooth manifold. AC supplies the approximation and embedding hypotheses as well as those of the index formula.

Remarks

The converse fails: the identity of a positive-dimensional closed manifold with Euler characteristic zero has $L = 0$ and fixes every point. The companion counterexample has two isolated fixed points with canceling indices.

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Small-time flow fixed point indices and vector field zero indices

Statement

Assume countable choice (The Axiom of Countable Choice ( ${A C}_{ω}$ )) as in the vector-field index suppliers. Let $M$ be a smooth $n$ -manifold without boundary, $n \geq 1$ , and let $X$ be a smooth vector field on $M$ with an isolated zero at $p$ (Isolated zero and local index of a vector field).

(i) Tangent families. Let $t \mapsto f_{t}$ be a smooth family of maps defined on a neighbourhood of $p$ with $f_{0} = i d$ , tangent to $X$ at time zero, i.e. $\frac{d}{d t} ∣_{t = 0} f_{t} (x) = X (x)$ for every $x$ , and suppose that there is a neighbourhood of $p$ in which, for every sufficiently small $t \neq 0$ , the point $p$ is the only fixed point of $f_{t}$ . Then ${ind}_{p} (f_{t}) = (- 1)^{n} {ind}_{p} X (t > 0), {ind}_{p} (f_{- t}) = {ind}_{p} X (t > 0) .$

(ii) The flow at a nondegenerate zero. The local flow $φ$ of $X$ (Local and global flows generated by a vector field, Integral curves of a vector field) is such a family; if the zero $p$ is nondegenerate (Nondegenerate zero of a vector field), then the isolation hypothesis of (i) holds for every sufficiently small $t \neq 0$ , so the two displayed identities hold for $φ_{t}$ : the small-time flow at a nondegenerate zero satisfies ${ind}_{p} (φ_{t}) = (- 1)^{n} {ind}_{p} X$ for $t > 0$ and ${ind}_{p} (φ_{- t}) = {ind}_{p} X$ for $t > 0$ .

The sign $(- 1)^{n}$ is the consistent short-time sign: the displacement $i d - f_{t}$ is asymptotic to $- t X$ near a zero, so the two local indices differ by the sign of $- i d$ on $R^{n}$ .

Facts & Assumptions

Given: A smooth $n$ -manifold $M$ without boundary, $n \geq 1$ , a smooth vector field $X$ with isolated zero $p$ ; in (i) a tangent family $f_{t}$ as in the statement, in (ii) the local flow $φ_{t}$ of $X$ .

[F1]

On a smooth chart ball $B$ around $p$ on which $X$ vanishes only at $p$ , the flow satisfies the integral identity $φ_{t} (x) = x + \int_{0}^{t} X (φ_{s} (x)) d s$ and depends smoothly on $(t, x)$ ; hence ${\hat{φ}}_{t} (u) = u + t \tilde{X} (u) + t^{2} r (t, u)$ with $r$ smooth near $(0, 0)$ , where ${\hat{φ}}_{t}$ and $\tilde{X}$ are the chart representatives. More generally, a smooth family $t \mapsto f_{t}$ with $f_{0} = i d$ and $\frac{d}{d t} ∣_{t = 0} f_{t} (x) = X (x)$ satisfies ${\hat{f}}_{t} (u) = u + t \tilde{X} (u) + t^{2} r (t, u)$ with $r$ smooth, and then ${\hat{f}}_{- t} (u) = u - t \tilde{X} (u) + t^{2} r_{-} (t, u)$ with $r_{-}$ smooth, because $\frac{d}{d t} ∣_{t = 0} f_{- t} (x) = - X (x)$ ; this is the fundamental theorem of calculus applied twice to each coordinate (Local existence, uniqueness, and smooth dependence for manifold integral curves, Local and global flows generated by a vector field).

[F2]

The local fixed point index ${ind}_{p} (φ_{t})$ is the degree of the normalized chart displacement $v \mapsto (u - {\hat{φ}}_{t} (u)) (ε v) / ∣ \cdot ∣$ on $S^{n - 1}$ , independent of admissible smooth chart and radius (Isolated fixed point and local fixed point index, The local fixed point index is independent of chart, ball and neighbourhood); the local index of a smooth vector field with isolated zero is the same degree of its normalized chart representative, independent of smooth chart, ball and admissible trivialization (Isolated zero and local index of a vector field, The local index is independent of chart, ball and trivialization). Degree, and in dimension zero the reduced degree, is invariant under homotopies of maps of spheres (Degree is invariant under proper smooth homotopy, Reduced degree into the 0-sphere is homotopy invariant and multiplicative).

[F3]

Negation multiplies the local index of a vector field by $(- 1)^{n}$ (Negation scales the local index by $(- 1)^{n}$ ), and for a nondegenerate zero the index is $sign \det D X_{p}$ (Nondegenerate zero of a vector field, The index of a nondegenerate vector-field zero).

[F4]

The inverse function theorem: a smooth map of Euclidean open sets with invertible differential at a point is a local diffeomorphism there, and the manifold form applies in charts (The smooth inverse function theorem on manifolds).

Proof

1.1givenF1F2

The expansion. Work in a smooth chart $(φ, U)$ at $p$ with $φ (p) = 0$ , write $\tilde{X}$ for the chart representative of $X$ and ${\hat{f}}_{t}$ for the chart representative of a family as in (i) or of the flow. By [F1], ${\hat{f}}_{t} (u) = u + t \tilde{X} (u) + t^{2} r (t, u)$ and ${\hat{f}}_{- t} (u) = u - t \tilde{X} (u) + t^{2} r_{-} (t, u)$ with $r, r_{-}$ smooth near $(0, 0)$ ; for the flow, ${\hat{φ}}_{t} (0) = 0$ for all $t$ , so the expansion gives $t^{2} r (t, 0) = 0$ and hence $r (t, 0) = 0$ near $t = 0$ . Choose $ε > 0$ with $\tilde{X} \neq 0$ on $0 < ∣ u ∣ \leq ε$ ; then $c : = \min_{∣ u ∣ = ε} ∣ \tilde{X} (u) ∣ > 0$ .

2.1step 1.1F2F3

The index identities for a tangent family. Let the family of (i) satisfy its isolation hypothesis on a neighbourhood containing the closed ball $∣ u ∣ \leq ε$ . For small $t > 0$ the normalized displacement $v \mapsto (u - {\hat{f}}_{t} (u)) (ε v) / ∣ (u - {\hat{f}}_{t} (u)) (ε v) ∣$ is defined, and by step 1.1 it equals $(- \tilde{X} (ε v) - t r (t, ε v)) / ∣ - \tilde{X} (ε v) - t r (t, ε v) ∣$ . Since $∣ \tilde{X} ∣ \geq c$ on the sphere and $r$ is bounded there, for $∣ t ∣$ small every vector $- \tilde{X} (ε v) - s t r (t, ε v)$ , $s \in [0, 1]$ , has norm at least $c / 2 > 0$ ; hence the straight-line homotopy in $s$ is one of nowhere-zero maps of $S^{n - 1}$ , and [F2] gives ${ind}_{p} (f_{t}) = \deg (v \mapsto - \tilde{X} (ε v) / ∣ \tilde{X} (ε v) ∣) = (- 1)^{n} {ind}_{p} X$ by [F3]. The same computation with ${\hat{f}}_{- t}$ gives ${ind}_{p} (f_{- t}) = \deg (\tilde{X} / ∣ \tilde{X} ∣) = {ind}_{p} X$ , again by [F2] and [F3].

3.1step 1.1step 2.1F1F2F4∎

The flow at a nondegenerate zero. The flow is tangent to $X$ at time zero and fixes $p$ , so it satisfies all hypotheses of (i) except possibly the isolation one. Suppose $p$ is nondegenerate, so that $D {\tilde{X}}_{0}$ is invertible, and consider $G (t, u) : = (t, \tilde{X} (u) + t r (t, u))$ near $(0, 0)$ ; its differential at $(0, 0)$ is block triangular with diagonal blocks $1$ and $D {\tilde{X}}_{0}$ , hence invertible. By [F4] $G$ is a local diffeomorphism at $(0, 0)$ , so there are $α, β > 0$ such that for $∣ t ∣ < α$ every solution of $\tilde{X} (u) + t r (t, u) = 0$ with $∣ u ∣ < β$ is unique; the fixed points of $φ_{t}$ in the chart are exactly these solutions, and $u = 0$ is one of them because $r (t, 0) = 0$ by step 1.1. Therefore for every $0 < ∣ t ∣ < α$ the point $p$ is the only fixed point of $φ_{t}$ in the ball $∣ u ∣ < β$ , the isolation hypothesis of (i) holds, and step 2.1 applied to $φ_{t}$ gives ${ind}_{p} (φ_{t}) = (- 1)^{n} {ind}_{p} X$ and ${ind}_{p} (φ_{- t}) = {ind}_{p} X$ . No orientation of $M$ is used, and no further choice is used after the vector-field index suppliers.

The common isolating neighbourhood in (i) must be checked for a tangent family. For $X (u) = u^{3}$ on $R$ and $f_{t} (u) = u + t u^{3} - t^{2} u$ , one has $f_{0} = i d$ and $\partial_{t} f_{t} ∣_{t = 0} = X$ , but for $t > 0$ the fixed points are $0, \sqrt{t}, - \sqrt{t}$ . They approach the isolated zero $0$ , so no common isolating neighbourhood works for all small positive $t$ . Part (ii) establishes the required common neighbourhood for the stated nondegenerate flow case. The normal-projection family in the Poincare–Hopf remark supplies it directly for arbitrary isolated zeros.

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Isolated fixed points need not be nondegenerate

Remark

Isolatedness of a fixed point is strictly weaker than nondegeneracy. A fixed point is nondegenerate when $I - D f_{x}$ is invertible (Nondegenerate fixed point), whereas the local fixed point index of Isolated fixed point and local fixed point index is defined for every isolated fixed point, including degenerate ones. The equivalence of nondegeneracy with transversality of the graph to the diagonal is Graph-diagonal transversality is exactly fixed-point nondegeneracy, and it is exactly this transversality that fails at a degenerate isolated point.

The standard example. Take the local model $f (z) = z + z^{2}$ on $C$ , a smooth self-map of the plane ( $C^{r}$ and smooth maps between smooth manifolds). Its fixed point equation is $z + z^{2} = z$ , i.e. $z^{2} = 0$ , so $0$ is the only fixed point near the origin and it is isolated. Its differential is $D f_{0} = I$ and $I - D f_{0} = 0$ is not invertible (Invertible linear maps, linear isomorphisms, and inverse linear maps), so $0$ is isolated but degenerate, and the determinant formula of The index of a nondegenerate fixed point is the sign of det(I-Df) does not apply.

Its index is nevertheless defined and equals $2$ . With the convention $I - D f$ of this page the displacement is $i d - f = - z^{2}$ , whose representative in real coordinates on the circle $∣ z ∣ = ε$ is $v \mapsto - ε^{2} e^{2 i θ}$ with $θ$ the polar angle; after normalization this is the self-map $e^{i θ} \mapsto e^{i (2 θ + π)}$ of $S^{1}$ , of degree $2$ . Hence ${ind}_{0} (f) = 2$ : an isolated degenerate fixed point can carry a nonzero index, and its value is not controlled by $I - D f_{0}$ .

The two theorems on this page that survive. The definition of the local index applies verbatim, and under countable choice (The Axiom of Countable Choice ( ${A C}_{ω}$ )), An isolated fixed point splits under perturbation, preserving its index splits the degenerate point into nondegenerate ones with the same total index. For the explicit quadratic perturbation $f_{a} (z) = z + z^{2} - a$ , $a \in C ∖ {0}$ small, the two fixed points satisfy $z^{2} = a$ . At either point the displacement derivative is multiplication by $- 2 z$ , of real determinant $4 ∣ z ∣^{2} > 0$ , so each index is $+ 1$ . General smooth perturbations can have more fixed points; the splitting theorem preserves the total index, not their number. The polynomial extends to a smooth self-map of the Riemann sphere $S^{2} = C \cup {\infty}$ whose only fixed points are $0$ and $\infty$ : in the chart $w = 1 / z$ the map is $w \mapsto w^{2} / (w + 1)$ and the fixed point equation $w^{2} / (w + 1) = w$ has the unique solution $w = 0$ , with displacement $w - w^{2} / (w + 1) = w / (w + 1)$ , whose linear part at $0$ is the identity, so ${ind}_{\infty} (f) = + 1$ . This is the standard example showing that the converse direction of the Lefschetz theory needs the index and not merely the first derivative; the companion examples page computes both indices.

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The Lefschetz index formula recovers Poincare-Hopf

Remark

The derivation. Assume AC (The Axiom of Choice). Let $M$ be a closed smooth $n$ -manifold, $n \geq 1$ , and let $X$ be a smooth vector field with isolated zeros (Isolated zero and local index of a vector field). Embed $M$ as a closed smooth submanifold of a Euclidean space (The weak Whitney proper embedding theorem), let $N$ be an open tubular neighbourhood with its normal-fibre retraction $r : N \to M$ (The Euclidean tubular neighbourhood theorem, A closed Euclidean submanifold has a smooth neighborhood retraction), and for small $t > 0$ define $f_{t} (x) : = r (x - t X (x)) .$ Compactness gives a uniform tube margin around $M$ (take a finite cover by balls whose doubled balls lie in $N$ ), while the Euclidean norm of $X$ is bounded by A continuous image of a compact space is compact; a continuous real-valued map on a nonempty compact space attains a maximum and a minimum; and a continuous bijection from a compact space to a Hausdorff space is a homeomorphism, clause 2. Hence $f_{t}$ and the same formula for every $s \in [0, t]$ are defined for a common small $t > 0$ . This is Guillemin and Pollack's normal-projection approximation to the flow of $- X$ , and it has the following properties.

The fixed points of $f_{t}$ are exactly the zeros of $X$ . If $r (x - t X (x)) = x$ , put $z : = x - t X (x)$ ; then $z - x = - t X (x)$ is perpendicular to $T_{x} M$ because $r$ is the normal-fibre projection, while $X (x) \in T_{x} M$ , so $t X (x) = 0$ and $X (x) = 0$ ; conversely $X (x) = 0$ gives $f_{t} (x) = r (x) = x$ . Hence the fixed points of $f_{t}$ are the isolated zeros of $X$ , for every sufficiently small $t > 0$ for which the family is defined.
$f_{t}$ is homotopic to the identity. The formula $(s, x) \mapsto r (x - s X (x))$ , $s \in [0, t]$ , is a homotopy from $f_{0} = {i d}_{M}$ to $f_{t}$ , so $L (f_{t}) = L ({i d}_{M}) = χ (M)$ by The Lefschetz number is a homotopy invariant and The Lefschetz number of the identity is the Euler characteristic.
The index identification. Since $r$ restricts to the identity on $M$ with identity differential along $T_{x} M$ , the family satisfies $f_{t} (x) = x - t X (x) + O (t^{2})$ : it is tangent to $- X$ at time zero, and its fixed points are isolated. The tangent-family part of Small-time flow fixed point indices and vector field zero indices, applied to the field $- X$ , gives ${ind}_{p} (f_{t}) = (- 1)^{n} {ind}_{p} (- X)$ , and the negation law ${ind}_{p} (- X) = (- 1)^{n} {ind}_{p} X$ (Negation scales the local index by $(- 1)^{n}$ ) leaves ${ind}_{p} (f_{t}) = {ind}_{p} X$ at every zero $p$ of $X$ , for all sufficiently small $t > 0$ .

The Lefschetz–Hopf index formula Lefschetz-Hopf index formula applies to the smooth map $f_{t}$ , whose fixed points are exactly the isolated zeros of $X$ , and combines the three properties into $\sum_{p : X (p) = 0} {ind}_{p} X = \sum_{p \in Fix (f_{t})} {ind}_{p} (f_{t}) = L (f_{t}) = χ (M),$ which is precisely the Poincaré–Hopf theorem Poincare-Hopf for closed manifolds — recovered here as a corollary of the Lefschetz–Hopf index formula. The normal-projection family gives the fixed-set description directly, including at degenerate zeros; it does not require a periodic-orbit analysis of the actual flow. Compactness makes the isolated zero set finite (it is closed and discrete), by A closed discrete subset of a compact space is finite: locally, continuity makes the nonzero locus open. Thus the finitely many local small-time bounds have a common positive bound. For odd $n$ the conclusion is also consistent with the vanishing of $χ$ recorded in Closed odd-dimensional manifolds have zero Euler characteristic.

What is used. The argument uses a proper embedding of $M$ , a tubular neighbourhood with its normal-fibre retraction, the tangent-family index computation, the Lefschetz–Hopf index formula and the homotopy invariance of $L$ ; no countability or orientation hypothesis on $M$ is added beyond the ones already carried by those suppliers.

5 · Examples, counterexamples and false statements

None yet.