PropositionStatement: Literature-sourcedProof: AI-adaptedPipeline-generatedaudited 2026-09-07

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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

1.1F2

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.

2.1F1step 1.1algebra∎

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.

Depends on

Used by

The zero stable stem is the integers Example
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)