Reduced Homology

Define the reduced homology groups \widetilde{H}_n(X) to be the homology groups of the augmented chain complex \displaystyle \dots\to C_2(X)\xrightarrow{\partial_2}C_1(X)\xrightarrow{\partial_1}C_0(X)\xrightarrow{\epsilon}\mathbb{Z}\to 0 where \epsilon(\sum_i n_i\sigma_i)=\sum_in_i. We require X to be nonempty, to avoid having a nontrivial homology group in dimension -1.

Relation between H_n and \widetilde{H}_n
Since \epsilon\partial_1=0, \epsilon vanishes on \text{Im}\,\partial_1 and hence induces a map \tilde{\epsilon}:H_0(X)\to\mathbb{Z} with \ker\tilde{\epsilon}=\ker\epsilon/\text{Im}\,\partial_1=\widetilde{H}_0(X). So H_0(X)\cong\widetilde{H}_0(X)\oplus\mathbb{Z}. Clearly, H_n(X)\cong\widetilde{H}_n(X) for n>0.

Author: mathtuition88

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