On Page 108 of the book Algebraic Topology, Allen Hatcher, the singular homology of a topological space $X$ is defined to be the homology of the chain complex by setting the $n$-chains $C_n(X)$ as the free abelian group with basis $\text{Map}(\Delta^n,X)$, where $\Delta^n$ is the standard $n$-simplex, and the boundary map $\partial_n$ as $$ \partial _n (\sigma)=\sum_i (-1)^i\sigma\mid_{[v_0,\cdots,\hat v_i,\cdots,v_n]}, $$ where $\sigma$ is a map from $\Delta^n$ to $X$ and $\sigma\mid $ is the restriction of $\sigma$.
Let $K$ be a simplicial complex and let $|K|$ be the geometric realization of $K$. Suppose the vertices ($0$-simplices) of $K$ are $v_0,v_1,\cdots,v_k$, $k$ finite. 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 $\Delta^n$ and use $\mathbb{Z}_{k+1}$ to represent the set of all vertices of $K$. Let $C_n(K)$ be the free abelian group with basis $K_n$. Analogous to the boundary map in singular homology, there is a boundary map $$ \partial_n: C_n(K)\longrightarrow C_{n-1}(K). $$
Question: Is the singular homology of $|K|$ same or different with the homology of the chain complex $$ \cdots C_n(K)\overset{\partial_n}{\longrightarrow} C_{n-1}(K)\overset{\partial_{n-1}}{\longrightarrow}\cdots\overset{\partial_1}{\longrightarrow} C_0(K) ? $$