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If $X$ is a $n$-dim polyhedran, then using geometric techniques, we construct groups $H_p(X{)}_{\Delta }$ and $H^p(X{)}_{\Delta }$ such that there are natural isomorphisms $H^p(X{)}_{\Delta }\simeq H_{n-p}\left(X\right)$ and $H_p(X{)}_{\Delta }\simeq H^{n-p}\left(X\right)$ which induce an intersection pairing. These groups give a geometric interpretation of two spectral sequences studied by Zeeman and allow us to prove a conjecture of Zeeman about them.