7
$\begingroup$

When it comes to induced homomorphisms between homology groups, I have trouble understanding which are surjective or injective. For example, the unique map $p: X \to \{x\}$ where $x$ is a point in $X$ induces a homomorphism $p_*: H_n(X) \to H_n(\{x\})$. Clearly $p$ is surjective, and while it makes sense intuitively, I have trouble showing why $p_*$ is too. It's the same with proving that the map induced by inclusion $i_*: H_n(\{x\}) \to H_n(X)$ is an injection.

In general, is it always the case that if a continuous map of topological spaces $f: X\to Y$ is injective (resp. surjective) then the induced homomorphisms $f_*: H_n(X) \to H_n(Y)$ will be injective (resp. surjective)?

3 Answers 3