Given a topological space $X$, $X$ is $k$-connected if any $-1\le \ell \le k$ and continuous map $f:S^{\ell}\to X$ can be extended to $\bar{f}:B^{\ell+1}\to X$, where $S^\ell$ is viewed as the boundary of $B^{\ell+1}$. The connectivity $c(X)$ of $X$ is the maximum of such $k$.
The join $X*Y$ of topological spaces $X,Y$ is $X\times Y\times [0,1]/\sim$, where the equivalence relation $\sim$ is defined by $(x,y,0)\sim (x',y,0)$ for all $x,x'\in X, y\in Y$ and $(x,y,1)\sim(x,y',1)$ for all $x\in X, y,y'\in Y$.
It is known that $c(X*Y)\ge c(X)+c(Y)+2$. I am wondering is there any example that the strict inequality holds, i.e., $c(X*Y)> c(X)+c(Y)+2$? (An example of simplicial complexes will be more appreciated.)