Let $K$ be a simplicial complex and let $|K|$ be the geometric realization of $K$. Suppose that the vertices ($0$-simplices) of $K$ are $v_0,v_1,\cdots,v_k$, $k$ finite. Let $\Delta^n$ be the standard $n$-simplex and let $$ K_{n}= \text{Simplicial Map}(\Delta^n,K). $$ Then $$ K_n\cong\{f\in \text{Map}(\mathbb{Z}_{n+1},\mathbb{Z}_{k+1})\mid \text{ there exists a simplex } \sigma\in K \text{ such that Im}f \text{ is the set of vertices of } \sigma\}. $$ Here we use $\mathbb{Z}_{n+1}$ to represent the set of all vertices of the standard $n$-simplex $\Delta^n$ and use $\mathbb{Z}_{k+1}$ to represent the set of all vertices of $K$. Analogous to the boundary map in singular homology, there is a boundary map $$ \partial_n: \mathbb{Z} K_n\longrightarrow \mathbb{Z}K_{n-1}. $$ We have that $\{K_n,\partial_n\}_{n\geq 0}$ is a simplicial complex.

Question: Whether is the geometric realization of the simplicial complex $\{K_n,\partial_n\}_{n\geq 0}$ (weak) homotopy equivalent to (or having same homology with) $|K|$?


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