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

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The first chip identifies the source of the statement or construction; the second identifies the source of its local proof or verification.

Literature-sourced: the exact statement appears in a cited source; only wording and notation differ.
AI-adapted: a semantically identical restatement of literature-sourced material, modulo indexing, notation, and boundary cases adopted by the library.
AI-generated: a genuinely novel statement formulated by AI, with no source for the claim itself.

These labels describe origin, not correctness: citations and verification chips remain separate evidence.

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.

Depends on

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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)