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.

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 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

Dependency tree · 2 levels

Sources

Hatcher, Algebraic Topology, Degree, definition p.134