Let $K$ consist of n-simplex and it's faces, let $K_0$ be the set of proper faces of K. Then why does the group $C_p(K)/C_p(K_0)$ vanish for $p<n$? Is it because there could be some elements in $K_0$ which are not faces of $K$?
2026-04-02 07:09:52.1775113792
Why does the relative homology group vanish modulo the set of proper faces
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$K_0$ contains all faces of $K$ of dimension $< n$. Hence for $p < n$ the complexes $K$ and $K_0$ have the same (oriented) $p$-simplices.