Does smashing always increase the connectivity of a space?
Solution 1:
Yes. In fact, $\operatorname{conn}(X\wedge Y)\ge\operatorname{conn}(X)+\operatorname{conn}(Y)+1$ (if both $X$ and $Y$ are connected).
(Indeed, if $X$ is $n$-connected, it's homotopy equivalent to a CW-complex $X'$ with one $0$-cell and no cells in dimensions $1\le s\le n$. Now note that $S^k\wedge S^l=S^{k+l}$, so $X'\wedge Y'$ is homotopy equivalent to $X\wedge Y$ and doesn't have cells in dimensions $1\le s\le n+m+1$.)