Another one that has been bugging me:
Say $X$ is a finite CW complex. What is the simplest CW structure on $S^n \times X$
So I assume that $E$ is the family of cells in $X$ and $\Phi = \{ \Phi_e:e \in E \}$ is the family of attaching maps (technically I guess $\Phi_e | S^{k-1}$ is the attaching map of a $k$-cell).
I think that if I take the usual CW structure on the $n$-sphere (1 0-cell, $e^0_s$ and 1 $n$-cell $e^n_s$) that the family of cells E' of $S^n \times X$ is just E'=\{ e^0_s \times e, e^n_s \times e: e \in E \}
But I am unsure how to attach it? I guess we only really need to worry in the instances we are attaching a 0-cell and an $n$-cell, else we can just use the usual maps.
Writing down the cellular chain complex is not too bad - it will just have an extra copy of $\mathbb{Z}$ in the $0$-th and $n$-th position.
Can we then calculate $H_k(S^n \times X)$ in terms of $H_k(X)$? (Again, the boundary formulas will only change in the $0$-th and $n$-th case
Edit: And can we do it without the Kunneth formula?