Showing Homology Sequence breaks up into short sequences that split

Question

Let $X$ and $Y$ be polyhedra.

Show that if $p \in Y$, the homology exact sequence of $(X \times Y, X \times p)$ breaks up into short exact sequences that split.

Definitely am a bit lost on this one... what is the homology sequence of this going to look like exactly? Doesn't it depend on what subsets we use? Is their a canonical choice? Once we have this sequence, I might be able to see why it splits myself.

Answer 1

Presumably this is referring to the long exact sequence of homology for the relative homology of the pair $(X\times Y,X\times p)$. That is, it's the sequence $$\dots\to H_n(X\times p)\to H_n(X\times Y)\to H_n(X\times Y,X\times p)\to H_{n-1}(X\times p)\to \cdots$$

A hint for how to get this to split is hidden below.

Try splitting the map $H_n(X\times p)\to H_n(X\times Y)$.