$\mathbf {The \ Problem \ is}:$ Let, $(Y,y)$ be a based CW complex with cells only in even degrees; in particular , $r_i$ cells in each of degree $i$ where $i$ is even . Let, $(X,x)$ be a finite based CW complex , then find $H_*(X \wedge Y)$ for all $* \in \mathbb N \cup {0}.$
$\mathbf {My \ approach}:$ I could only try that by cellular homology, $H_i(Y)= \oplus_{r_i} \mathbb Z$ (as all the boundary maps of cellular chain complex of $Y$ are $0$) and let CW dimension of X be $d.$
Then, we are collapsing those cells of $X×Y$ required to construct $X \vee Y$ which in turn homeomorphic to $X×\{y\} \cup \{x\}×Y$ .
Denote cells of $X$ and $Y$ by $e^p$ and $f^q$ respectively . Then all cells of the form $e^{j} ×f^0$ and $e^{0} ×f^j$ getting collapsed .
And, $X \vee Y$ is a subcomplex of $X×Y$ , hence we can apply long exact sequence of relative homology groups .
But, I can't approach further ; we were taught only upto chapter 2 in Hatcher's book and a special case of Künneth formula to find homology of $X×Y$ (in image).
A small help is warmly appreciated . Thanks in advance .