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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2 thoughts on “Reduced Homology”

Intuitively what does reduced homology mean ?

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mathtuition88 says:

April 24, 2017 at 7:01 am

Same as homology, except for degree 0. For a point, reduced homology is zero for all degree.

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