PropositionStatement: Literature-sourcedProof: AI-adaptedPipeline-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.

Every integer occurs as the degree of a sphere map

Statement

For every $d \in Z$ and $n \geq 1$ there exists a continuous self-map of $S^{n}$ of degree $d$ .

Facts & Assumptions

Given: The objects and hypotheses in the statement above.

[F1]

Let $f : S^{n} \to S^{n}$ be continuous, $n \geq 1$ , with source and target orientations fixed. If $f^{- 1} (y)$ is finite, then $\deg (f) = \sum_{x \in f^{- 1} (y)} \deg_{x} f .$ The sum over an empty fibre is $0$ . (Global sphere degree is the sum of local degrees)

[F2]

On $S^{n} \subset R^{n + 1}$ for $n \geq 1$ , the identity, a constant map, any single coordinate reflection, and the antipodal map have degrees $1$ , $0$ , $- 1$ , and $(- 1)^{n + 1}$ respectively. (Degree of identity constant reflection and antipodal sphere maps)

Proof

1.1

For $d = 0$ take a constant map, which has degree zero by F2.

1.2

For $k = ∣ d ∣ > 0$ , choose $k$ disjoint closed coordinate balls with nonempty interiors. Collapse their boundaries and the complement of their interiors to a single point. The quotient is a wedge of $k$ copies of $D^{n} / \partial D^{n} ≅ S^{n}$ : a homeomorphism of the ball interior with $R^{n}$ , followed by inverse stereographic projection, extends to this quotient by sending the boundary to the omitted pole. Fold these copies to a common target sphere. Use an orientation-preserving homeomorphism on each copy if $d > 0$ , and compose each with a coordinate reflection if $d < 0$ . The maps agree at the collapsed point, so the quotient construction is continuous.

givenF2

2.1

A point distinct from the common pole has exactly one preimage in each ball. At each such point the local degree is $+ 1$ or $- 1$ as chosen: an orientation-preserving chart restricts the local generator unchanged, and reflection reverses it. F1 gives degree $k sgn (d) = d$ . This works for $k = 1$ and for one-dimensional balls as well.

F1F2step 1.2∎

Depends on

Used by

Degree zero does not imply a sphere map is constant Counterexample

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, Example 2.31, p.136 (standard reference, not scraped)