Singular Homology - Homotopy Invariance

Homotopy Invariance

If X and Y are two topological spaces with the same homotopy type, then

for all n ≥ 0. This means homology groups are topological invariants.

In particular, if X is a connected contractible space, then all its homology groups are 0, except .

A proof for the homotopy invariance of singular homology groups can be sketched as follows. A continuous map f: X → Y induces a homomorphism

It can be verified immediately that

i.e. f_# is a chain map, which descends to homomorphisms on homology

We now show that if f and g are homotopically equivalent, then f_* = g_*. From this follows that if f is a homotopy equivalence, then f_* is an isomorphism.

Let F : X × → Y be a homotopy that takes f to g. On the level of chains, define a homomorphism

that, geometrically speaking, takes a basis element σ: Δn → X of C_n(X) to the "prism" P(σ): Δn × I → Y. The boundary of P(σ) can be expressed as

So if α in C_n(X) is an n-cycle, then f_#(α ) and g_#(α) differ by a boundary:

i.e. they are homologous. This proves the claim.