# why is homology isomorphic to reduced homology plus Z?

by redbowlover
Tags: homology, isomorphic, reduced
 P: 16 Working through Hatcher... For any space X, we have an augmented chain complex $...\rightarrow C_1(X) \rightarrow C_0(X)\rightarrow \mathbb{Z}\stackrel{\epsilon}{\rightarrow}0$ Hathcer says that since $\epsilon$ induces a map $H_0(X)\rightarrow \mathbb{Z}$ with kernel $\tilde{H}_0(X)$, we get an isomorphism $H_0(X)\simeq \tilde{H}_0(X)\oplus \mathbb{Z}$ Where is this isomorphism coming from? I understand where the induced map on $H_0(X)$ comes from... Thanks
 Sci Advisor HW Helper PF Gold P: 4,765 There is a short exact sequence 0-->H(reduced)_0-->H_0-->Z-->0, and Z being free, it splits. That is, H_0=H(reduced)_0 x Z.
 P: 16 thanks!

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