TheoremStatement: Literature-sourcedProof: AI-adaptedPipeline-generated

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.

The Hopf mod-two degree theorem for nonorientable domains

Statement

Assume ${A C}_{ω}$ . Let $M$ be a closed connected nonorientable smooth $m$ -manifold, $m \geq 1$ . (i) Two smooth maps $f, g : M \to S^{m}$ are smoothly homotopic if and only if $\deg_{2} (f) = \deg_{2} (g)$ . (ii) Both elements of $Z / 2$ are realized: constant maps have mod-two degree $0$ , and the pinch map of a closed coordinate ball has mod-two degree $1$ . (iii) Consequently $\deg_{2}$ induces a bijection $[M, S^{m}] \to Z / 2$ , so two continuous maps $M \to S^{m}$ are homotopic if and only if their mod-two degrees agree.

Facts & Assumptions

Given: A closed connected nonorientable smooth $m$ -manifold $M$ with $m \geq 1$ , smooth maps $f, g : M \to S^{m}$ , and the mod-two degree of The mod-two degree of a map to a sphere (Orientable manifolds, Regular and critical points and values, The Axiom of Countable Choice ( ${A C}_{ω}$ )).

[F1]

The mod-two degree is well defined, is invariant under smooth homotopy, and descends to free homotopy classes of continuous maps (The mod-two degree is well defined and homotopy invariant).

[F2]

For a smooth map $f : M \to S^{m}$ , a regular value $y$ with positive basis $b$ and the framed regular preimage $(f^{- 1} (y), f_{*} b)$ , framed cobordism classes of closed framed $0$ -manifolds in $M$ are classified by the parity of the cardinality: two such framed preimages are framed cobordant exactly when their parities agree, and null-cobordism is exactly even cardinality (Framed zero-dimensional bordism in a nonorientable manifold is mod two, Framed regular preimages of a map to a sphere).

[F3]

If two smooth maps have framed cobordant regular preimages at regular values with positive bases, they are smoothly homotopic (A framed cobordism of regular preimages produces a homotopy).

[F4]

The explicit smooth pinch model of the realization lemma supplies a smooth map $M \to S^{m}$ with a regular value whose preimage has exactly one point, obtained by reading the model in a coordinate ball and extending by the base point; its mod-two degree is therefore $1$ , while a constant map has an empty regular fibre over any value different from the constant and hence mod-two degree $0$ (Every integer is realized by a map to the sphere, Regular and critical points and values).

[F5]

Regular values exist by Sard's theorem; every continuous map is homotopic to a smooth map and continuously homotopic smooth maps are smoothly homotopic (Morse-Sard for smooth manifolds, Every continuous map between smooth manifolds is homotopic to a smooth map, Continuously homotopic smooth maps are smoothly homotopic, Homotopies of continuous maps, homotopies relative to a subspace, and path homotopies relative to the endpoints).

Proof

technique · direct

1.1F1givenalgebra

(Forward direction.) If $f$ and $g$ are smoothly homotopic then $\deg_{2} (f) = \deg_{2} (g)$ by [F1].

1.2F2F3F5

(Converse.) Suppose $\deg_{2} (f) = \deg_{2} (g)$ . Choose regular values $y$ of $f$ and $y^{'}$ of $g$ and positive bases by [F5]; the parities of the framed preimages are the mod-two degrees, hence equal, so by [F2] the two framed preimages are framed cobordant, and [F3] makes $f$ and $g$ smoothly homotopic.

1.3F4F1

(Realization of both values.) Constant maps have mod-two degree $0$ and the pinch map of [F4] has mod-two degree $1$ , so both elements of $Z / 2$ occur.

2.1F1F5step 1.1step 1.2step 1.3∎

(Bijection on free homotopy classes.) Steps 1.1 and 1.2 classify smooth maps by $\deg_{2}$ , and [F5] lets every continuous map be replaced by a homotopic smooth one and every continuous homotopy by a smooth one, so $\deg_{2}$ is a well-defined bijection $[M, S^{m}] \to Z / 2$ with the two values realized in step 1.3; no orientation of $M$ is used anywhere.

Depends on

Used by

Dependency tree · two levels

80 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

Daniel S. Freed, Bordism: Old and New (lecture notes, UT Austin, Fall 2012) (standard reference, not scraped)
John Milnor, Topology from the Differentiable Viewpoint (standard reference, not scraped)
Victor Guillemin and Alan Pollack, Differential Topology (standard reference, not scraped)