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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.

Depends on

Used by

Cited to discharge well-definedness by Isolated fixed point and local fixed point index.

Dependency tree · two levels

32 results within two dependency steps of this one, each drawn at its shortest distance from it. An arrow runs from a result to what uses it, so the chart reads left to right and ends at this result, which carries a heavier outline. Every node is a link to that result. Click elsewhere on the chart to enlarge it.

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)