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Contractible nonempty spaces have the homology of a point
Statement
If is a nonempty contractible topological space, then for every and every abelian group , where denotes a one-point space.
Facts & Assumptions
Given: A nonempty contractible topological space , an abelian group , and an integer .
For a nonempty space, contractibility is equivalent to the identity map being nullhomotopic (A nonempty space is contractible if and only if its identity map is nullhomotopic).
A map that is homotopic to a constant map is nullhomotopic, and a space is contractible exactly when every map out of it is nullhomotopic (Nullhomotopic maps and contractible spaces).
Homotopy equivalences induce isomorphisms on singular homology (Homotopy equivalences induce isomorphisms on singular homology).
Proof
By [L1], the identity map on is homotopic to the constant map at some point . Let be the unique map and let send the point of to . Then is the constant map at , so , while . Hence is homotopy equivalent to a point.
Apply [L3] to the map . It induces an isomorphism on every singular homology group, which is exactly the displayed conclusion.
Depends on
Used by
- Homotopy spheres of dimension at least five bounding a contractible manifold are standard spheres Corollary
- Suspension isomorphism in reduced singular homology Corollary
- A homology equivalence need not be a homotopy equivalence without simple connectivity Counterexample
- An interval has nonzero Euler characteristic despite being odd-dimensional Counterexample
- An inward radial field violates the outward boundary formula Counterexample
- Ordinary cohomology does not give noncompact Poincaré duality Counterexample
- Absolute and relative Hurewicz homomorphisms Definition
- Alexander duality for the standard equator Example
- Homology of the loop space of an odd sphere Example
- Integral cohomology ring of a closed orientable surface Example
- Mayer–Vietoris computation of sphere homology Example
- Path-loop Serre computation of CP infinity Example
- Poincaré–Lefschetz duality for a disk Example
- The homology of an interval from contractibility Example
- The Lefschetz number of the identity is Euler characteristic Example
- The outward radial field on a disk Example
- A map of nonzero degree between spheres is surjective Lemma
- A relative single cell layer has compatible homotopy and homology bases Lemma
- Finite type and odd-primary acyclicity of K(F₂,q) Lemma
- Integral surface cup pairing from the oriented polygon Lemma
- Local path-fibration and cohomology inputs for mod-two Eilenberg–Mac Lane induction Lemma
- Rational cohomology of K(Z,n) through weak CW fiber comparison Lemma
- Relative homology of the standard handle pair Lemma
- Torsion Eilenberg–Mac Lane spaces are rationally acyclic Lemma
- The outward boundary hypothesis cannot be replaced by nonzero on the boundary Remark
- A transgressive simple fiber system gives a polynomial base Theorem
- Absolute Hurewicz theorem at the first nonzero degree Theorem
- Discrete de Rham cohomology and contractible-mesh exactness Theorem
- No retraction from a disk onto its boundary Theorem
- Rational cohomology of Eilenberg–Mac Lane spaces in one generator 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
- Allen Hatcher, Algebraic Topology (standard reference, not scraped)