PropositionStatement: Literature-sourcedProof: AI-adaptedPipeline-generatedaudited 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.

Suspension preserves sphere map degree

Statement

For $n \geq 1$ and $f : S^{n} \to S^{n}$ , let $Σ f$ be its two-cone (unreduced) suspension. Orient $Σ S^{n} ≅ S^{n + 1}$ by the natural suspension isomorphism. Then $\deg (Σ f) = \deg (f)$ .

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]

Let $G$ be an abelian group. For a based well-pointed space $X$ —meaning that the basepoint inclusion ${x_{0}} ↪ X$ is a cofibration—reduced singular homology has natural isomorphisms ${\tilde{H}}_{n + 1} (Σ X; G) ≅ {\tilde{H}}_{n} (X; G)$ for all integers $n$ . Here $Σ X$ is the suspension with two distinct apices, as in def-adjunction-cone-suspension. (Suspension isomorphism in reduced singular homology)

Proof

1.1

Choose a source basepoint $x_{0}$ and the target basepoint $f (x_{0})$ . Each sphere is well-pointed at its chosen point: rotate a CW structure with a vertex to that point and use the vertex cofibration. Thus $f$ is a based map between these choices; the underlying two-cone suspension is unchanged. Thus F2 gives a natural isomorphism $s : {\tilde{H}}_{n + 1} (Σ S^{n}; Z) \to {\tilde{H}}_{n} (S^{n}; Z)$ . Choose the upstairs generator mapping to the downstairs generator.

2.1

Naturality gives $s (Σ f)_{*} = f_{*} s$ . Applied to the upstairs generator, the right side is $\deg (f)$ times the downstairs generator. Since $s$ is injective, the upstairs multiple is also $\deg (f)$ . For $n \geq 1$ these reduced groups equal the top unreduced groups defining degree.

F1step 1.1algebra∎

Depends on

Used by

Degree of identity constant reflection and antipodal sphere maps Proposition

Dependency tree · two levels

9 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, Proposition 2.33, p.137 (standard reference, not scraped)