Some Homology Definitions

Chain Complex
A sequence of homomorphisms of abelian groups \displaystyle \dots\to C_{n+1}\xrightarrow{\partial_{n+1}}C_n\xrightarrow{\partial_n}C_{n-1}\to\dots\to C_1\xrightarrow{\partial_1}C_0\xrightarrow{\partial_0}0 with \partial_n\partial_{n+1}=0 for each n.

nth Homology Group
H_n=\text{Ker}\,\partial_n/\text{Im}\,\partial_{n+1}

\Delta_n(X)
\Delta_n(X) is the free abelian group with basis the open n-simplices e_\alpha^n of X.

n-chains
Elements of \Delta_n(X), called n-chains, can be written as finite formal sums \sum_\alpha n_\alpha e_\alpha^n with coefficients n_\alpha\in\mathbb{Z}.

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