Alphabeta MathHow 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.
Homotopy equivalences, homotopy inverses and spaces of the same homotopy type
Definition
Let be topological spaces. A continuous map is a homotopy equivalence if there is a continuous map such that
in the sense of Homotopies of continuous maps, homotopies relative to a subspace, and path homotopies relative to the endpoints. Such a is a homotopy inverse of .
The spaces and have the same homotopy type, or are homotopy equivalent, written , when a homotopy equivalence exists.
The equations required of an ordinary inverse have been weakened to homotopies. Neither composite need equal the corresponding identity map, and a homotopy equivalence need not be bijective.
Depends on
Used by
- A continuous map homotopic to a homotopy equivalence is itself a homotopy equivalence Corollary
- A homotopy equivalence induces a bijection between path components Corollary
- A weakly contractible CW complex is contractible Corollary
- Homotopy spheres of dimension at least five bounding a contractible manifold are standard spheres Corollary
- Sphere self-maps of degree ±1 are exactly the homotopy equivalences Corollary
- A homology cobordism need not be an h-cobordism Counterexample
- Whitehead theorem fails without CW type Counterexample
- Fiber and fiber homotopy equivalence Definition
- h-Cobordism Definition
- Simple homotopy equivalence Definition
- Whitehead torsion of a finite CW homotopy equivalence Definition
- A one-point space and ℝ are homotopy equivalent but not homeomorphic Example
- Simply connected h-cobordisms have zero Whitehead obstruction Example
- FALSE: homotopy-equivalent spaces must be homeomorphic False statement
- A contractible relative group-ring complex with a pi-one isomorphism detects a homotopy equivalence Lemma
- A finitely punctured open disk has the homotopy type of a finite wedge of circles Lemma
- A lifted finite CW equivalence has a contractible group-ring mapping cone Lemma
- A product collar deformation retracts onto its face Lemma
- An elementary CW expansion has zero Whitehead torsion Lemma
- Relative homology of an h-cobordism vanishes at both ends Lemma
- The handle complex of an h-cobordism is contractible over the group ring, with an explicit contraction Lemma
- The interior-disc and closed-disc configuration spaces are homotopy equivalent Lemma
- The target of a finite cellular mapping cylinder is a simple subcomplex Lemma
- The first Stiefel–Whitney class classifies orientability Proposition
- De rham cohomology is smooth homotopy invariant Theorem
- Having the same homotopy type is an equivalence relation on topological spaces Theorem
- Homotopy equivalences induce isomorphisms on singular homology Theorem
- Simple homotopy equivalences have zero torsion Theorem
- The inclusion of a deformation retract is a homotopy equivalence with the retraction as homotopy inverse Theorem
- Whitehead theorem Theorem
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
- A. Hatcher, Algebraic Topology, Section 0 (standard reference, not scraped)