CorollaryStatement: Literature-sourcedProof: AI-adaptedPipeline-generatedjudge pass (gpt-5.6-terra)audited 2026-09-07

How statement and proof provenance work

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.

A fixed point free sphere map has antipodal degree

Statement

For $n \geq 1$ , any continuous fixed-point-free map $f : S^{n} \to S^{n}$ has degree $(- 1)^{n + 1}$ . Consequently, a self-map of any other degree has a fixed point.

Facts & Assumptions

Given: The objects and hypotheses in the statement above.

[F1]

On $S^{n} \subset R^{n + 1}$ for $n \geq 1$ , the identity, a constant map, any single coordinate reflection, and the antipodal map have degrees $1$ , $0$ , $- 1$ , and $(- 1)^{n + 1}$ respectively. (Degree of identity constant reflection and antipodal sphere maps)

[F2]

For $n \geq 1$ and continuous sphere self-maps $f, g$ , homotopic maps have the same degree and $\deg (g \circ f) = \deg (g) \deg (f)$ . Every homotopy equivalence $S^{n} \to S^{n}$ has degree $1$ or $- 1$ . (Degree is homotopy invariant and multiplicative under composition)

Proof

1.1

Let $u (x, t) = (1 - t) f (x) - t x$ . Its endpoint values are unit vectors. If it vanished for $0 < t < 1$ , equality of norms would imply $1 - t = t$ and hence $f (x) = x$ , contrary to the hypothesis. Thus $H (x, t) = u (x, t) / ∣ u (x, t) ∣$ is a continuous sphere homotopy.

givenalgebra

2.1

Its endpoints are $f$ and the antipodal map. F2 and F1 give $\deg (f) = (- 1)^{n + 1}$ . If a map of any other degree lacked fixed points, the just-proved equality would contradict its degree, proving the consequence.

F1F2step 1.1∎

Depends on

Used by

A group acting freely on a positive even sphere has at most two elements Corollary

Dependency tree · two levels

6 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

Hatcher, Algebraic Topology, Degree property (g), p.134 (standard reference, not scraped)