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 induce isomorphisms on singular homology
Statement
If is a homotopy equivalence, then for every and every abelian group the induced map is an isomorphism.
Facts & Assumptions
Given: A homotopy equivalence , an abelian group , and an integer .
A homotopy equivalence has a homotopy inverse (Homotopy equivalences, homotopy inverses and spaces of the same homotopy type).
Singular homology is functorial for identities and composites (Singular chains and singular homology are covariantly functorial).
Homotopic maps induce the same map on singular homology (Homotopic maps induce the same map on singular homology).
Proof
By [L1], choose a homotopy inverse with and . Applying [L3] gives
By [L2], Combining this with step 1.1 shows that and are two-sided inverses. Hence is an isomorphism.
Depends on
Used by
- Contractible nonempty spaces have the homology of a point Corollary
- Homotopy spheres of dimension at least five bounding a contractible manifold are standard spheres Corollary
- Morse Euler characteristic identity Corollary
- Suspension isomorphism in reduced singular homology Corollary
- Algebraic Lefschetz number via rational homology traces Definition
- Fiber transport and monodromy action Definition
- Weighted Lagrangian ribbon bordisms Definition
- A created cancelling pair contributes a (1+t)tᵏ term Example
- Mobius band as an interval bundle with monodromy Example
- Proper Cech descent does not require acyclic fibers Example
- A product collar deformation retracts onto its face Lemma
- Cyclic p-fold power construction Lemma
- Finite type and odd-primary acyclicity of K(F₂,q) Lemma
- Finite-cellular cyclic squares agree with singular cup-i squares Lemma
- Rational homotopy comparison with one endpoint surjection implies homology comparison Lemma
- Rational sphere homotopy below the first unstable degree Lemma
- Relative homology of an h-cobordism vanishes at both ends Lemma
- The Alexander module of a link complement is finitely presented Lemma
- The cyclic cover retracts onto the lifted flower and has a deck-equivariant spine model Lemma
- The rational homology of a closed smooth manifold is finite-dimensional in each degree Lemma
- Torsion Eilenberg–Mac Lane spaces are rationally acyclic Lemma
- Finiteness and additivity of the Euler characteristic Proposition
- Singular homology is invariant under deformation retracts Proposition
- The unreduced module fits an exact sequence with the reduced module Proposition
- Poincare-Hopf with outward-pointing boundary Theorem
Dependency tree · two levels
12 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
- Allen Hatcher, Algebraic Topology (standard reference, not scraped)