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Example 1.1.0.1 (Singular Homology). For any topological space $X$, the singular homology groups $\operatorname{ \mathrm{H} }_{\ast }(X; \operatorname{\mathbf{Z}})$ are defined as the homology groups of a chain complex

\[ \cdots \xrightarrow {\partial } \operatorname{\mathbf{Z}}[ \operatorname{Sing}_{2}(X) ] \xrightarrow { \partial } \operatorname{\mathbf{Z}}[ \operatorname{Sing}_1(X) ] \xrightarrow {\partial } \operatorname{\mathbf{Z}}[ \operatorname{Sing}_0(X) ], \]

where $\operatorname{\mathbf{Z}}[ \operatorname{Sing}_ n(X) ]$ denotes the free abelian group generated by the set $\operatorname{Sing}_ n(X)$ and the differential is given on generators by the formula

\[ \partial (\sigma ) = \sum _{i = 0}^{n} (-1)^{i} d_ i \sigma . \]