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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.

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Sources

Victor Guillemin and Alan Pollack, Differential Topology (Prentice-Hall 1974; complete 236-page PDF) (standard reference, not scraped)
Eleny Ionel, notes by Andrew Lin, Stanford Math 215B Differential Topology, Winter 2023 (complete 63-page lecture notes) (standard reference, not scraped)