CorollaryStatement: Literature-sourcedProof: AI-adaptedPipeline-generatedjudge pass (gpt-6.1-sol)

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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.

Depends on

Used by

The Lefschetz number of the identity is the Euler characteristic Corollary
A vanishing Lefschetz number with canceling fixed points Counterexample
A torus translation has zero Lefschetz number and no fixed points Example
The Lefschetz index formula recovers Poincare-Hopf Remark
Lefschetz fixed point theorem Theorem

Dependency tree · two levels

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Sources

Victor Guillemin and Alan Pollack, Differential Topology (Prentice-Hall 1974; complete 236-page PDF) (standard reference, not scraped)
Peter Wong, Lectures on Fixed Point Theory, Mini-Course XV Encontro Brasileiro de Topologia, Rio Claro 2006 (complete notes) (standard reference, not scraped)