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)?
Best Answer
1) The map $p$ has one-sided inverse $i\colon\{x\}\to X$. So $p_*\circ i_*=(p\circ i)_*=id_*=id$, so $i_*$ is an injection and $p_*$ is a surjection.
2) It's not always the case. For example, any reasonable $X$ can be embedded in $\mathbb R^N$ (for $N$ large enough) and $H_n(\mathbb R^N)=0$ for $n>0$, so, usually, the induced map on homology is not injective (perhaps, the easiest example is $S^1\to D^2$).
But if $f\colon X\to Y$ is not just injective but has a (right) inverse, then by the argument from 1) the statement is true.