DefinitionDefinition: Literature-sourcedProof: Not applicablePipeline-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.

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Degree of a self map of an oriented sphere

Definition

Let $n \geq 1$ . Choose a generator $[S^{n}]$ of $H_{n} (S^{n}; Z) ≅ Z$ , using 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 Functoriality of relative homology 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 $n \geq 1$ .

Depends on

Used by

A vanishing Lefschetz number with canceling fixed points Counterexample
Degree is not defined by top homology for self maps of s zero Counterexample
Local degree at an isolated preimage Definition
A degenerate isolated fixed point with nonzero local index Example
Degree-d self-maps of a sphere have Lefschetz number 1+(-1)ⁿ d Example
Mapping cone of a degree d circle map Example
Rotations of the two-sphere and their Lefschetz number Example
A degree-zero sphere map extends over the ball without zeros Lemma
A map of nonzero degree between spheres is surjective Lemma
Cellular reduction for a highly connected pair Lemma
Degree is homotopy invariant and multiplicative under composition Proposition
Degree of identity constant reflection and antipodal sphere maps Proposition
Degree-d sphere maps act by multiplication by d in any generalized theory Proposition
Suspension preserves sphere map degree Proposition
Based sphere maps are classified by degree 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, definition p.134 (standard reference, not scraped)