Let $X$ be both a simplicial set and a closed $n$-dimensional manifold. We have a duality isomorphism $H^k(X)\to H_{n-k}(X)$. Furthermore, let $Y,Y'\subseteq X$ be two subcomplexes and closed submanifolds of dimension $i$ resp. $j$. We have simplicial fundamental classes $[Y]\in H_i(X)$ and $[Y']\in H_j(X)$.
I want to find a simplicial description of the intersection product $[Y]\bullet[Y']\in H_{i+j-n}(X)$
My idea was to consider the simplicial cross product $[Y\times Y']\in H_{i+j}(X\times X)$ and the diagonal map $d:X\to X\times X$ ($X\times X$ can be described as a bisimplicial set) and describe $d_![Y\times Y']$. Now two questions arise:
Is there a simplicial description $d$? I assume, it is just $d(\Sigma)=\Sigma\otimes\Sigma$ for each simplex.
Is there a simplicial description of the homology transfer?