Relations between the homology group of quotient space and the relative homology group

Question

Let $X$ topological space with subspace $A$.Under what conditions,$H_{n}(X,A)$ is isomorphic to $H_{n}(X/A)$

Answer 1

If $ X $ is compact Hausdorff and $ A $ is a closed subset of $ X $ which is a strong deformation retract of a compact neighbourhood of $ A $ in $ X $, then $$ H_n(X,A) \cong \tilde H_n(X/A), $$ $ \tilde H_n(X/A) $ being the $ n $-th reduced homology group of $ X/A $ (Vick, Homology Theory (2nd ed), Cor. 2.15). Now $ \tilde H_n(X/A) \cong H_n(X/A) $ for all $ n > 0 $, whereas $ \tilde H_0(X/A) $ is free abelian with one fewer basis element than $ H_0(X/A) $ (Vick, ibid., p. 48).

Does that help?

Answer 2

Hatcher defines a good pair $(X,A)$ to consist of any space $X$ and a nonempty closed subspace $A$ such that $A$ is a strong deformation retract of some neighborhood in $X$ (page 114). He then proves that for such good pairs we have $H_n(X,A)\cong\widetilde H_n(X/A)$ (page 124).