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

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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.

Degree is homotopy invariant and multiplicative under composition

Statement

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 $- 1$ .

Facts & Assumptions

Given: The objects and hypotheses in the statement above.

[F1]

Let $n \geq 1$ . Choose a generator $[S^{n}]$ of $H_{n} (S^{n}; Z) ≅ Z$ , using cor-homology-of-spheres. For a continuous self-map $f : S^{n} \to S^{n}$ , its degree is the unique integer satisfying $f_{*} [S^{n}] = \deg (f) [S^{n}] .$ The induced map is furnished by prop-relative-homology-is-functorial-for-maps-of-pairs with empty subspaces. Replacing the same generator in source and target by its negative does not change the integer. For a map between separately oriented copies of $S^{n}$ , use their separately specified generators; reversing just one orientation changes the sign. The unreduced definition here is restricted to $n \geq 1$ . (Degree of a self map of an oriented sphere)

[F2]

If $f, g : X \to Y$ are homotopic continuous maps, then for every $n \geq 0$ and every abelian group $G$ the induced homomorphisms on singular homology agree: $H_{n} (f_{#}) = H_{n} (g_{#}) : H_{n}^{s i n g} (X; G) \to H_{n}^{s i n g} (Y; G) .$ (Homotopic maps induce the same map on singular homology)

Proof

1.1

Homotopic maps induce the same homomorphism on integral $H_{n}$ , so their multiples of the orientation generator agree. This proves homotopy invariance, including constant maps.

F1F2

1.2

Functoriality gives $(g \circ f)_{*} [S^{n}] = g_{*} (\deg (f) [S^{n}]) = \deg (f) \deg (g) [S^{n}]$ , so the integers multiply. The identity has degree $1$ .

F1algebra

2.1

If $u$ is a homotopy inverse of $f$ , then $1 = \deg (u \circ f) = \deg (u) \deg (f)$ . The only units of $Z$ are $\pm 1$ , proving the last assertion without a converse.

step 1.1step 1.2algebra∎

Depends on

Used by

A fixed point free sphere map has antipodal degree Corollary
A group acting freely on a positive even sphere has at most two elements Corollary
Degree of identity constant reflection and antipodal sphere maps Proposition
No nowhere zero tangent vector field on an even sphere Theorem

Dependency tree · two levels

8 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 properties (c),(d), p.134 (standard reference, not scraped)