# Definition of Reduced Homology

I'm having some trouble with Hatcher's introduction of reduced homology on p. 110 of his Algebraic Topology:

...This is done by defining the reduced homology groups $$\tilde{H}_n(X)$$ to be the homology groups of the augmented chain complex $$\cdots \to C_2(X) \overset{\partial_2}{\to} C_1(X) \overset{\partial_1}{\to} C_0 \overset{\epsilon}{\to} \mathbb{Z} \to 0$$ [where $$\epsilon(\sigma) = 1$$ for all singular 0-simplices $$\sigma$$]...Since $$\epsilon\partial_1 = 0$$, $$\epsilon$$ vanishes on $$\operatorname{Im}{\partial_1}$$ and hence induces a map $$H_0(X) \to \mathbb{Z}$$ with kernel $$\tilde{H}(X)$$, so $$H_0(X) \cong \tilde{H}_0(X) \oplus \mathbb{Z}$$.

I understand everything except the last claim that $$H_0$$ is a direct sum. All I see from the rest of the discussion is that we have an exact sequence $$0 \to \tilde{H_0} \to H_0 \to \mathbb{Z} \to 0$$, but I can't figure out why this sequence splits.

• Does this answer your question? math.stackexchange.com/questions/55399/reduced-homology?rq=1
– User
Sep 18 '20 at 18:29
• There is always some argumentation when the issue of reduced homology comes up. While Hatcher is a good book, I recommend you not take his definition of reduced homology seriously. When you are a little more familiar with homology I recommend you look for an explanation about the point of reduced homology, the main being that it is basically only useful when you have a basepoint. Sep 18 '20 at 18:37
• @0-thUser I saw that post, but it seemed like all the responses where just explaining the quotient relation $H_0/\tilde{H}_0 \cong \mathbb{Z}$ and not the direct sum. Sep 19 '20 at 14:52

Actually, since $$\mathbb{Z}$$ is a projective abelian group, any exact sequence of the form $$0 \to A \to B \to \mathbb{Z} \to 0$$ splits, although not canonically. If you haven't seen this argument before, just note that for any $$b \in B$$ mapping to $$1$$, we can define a map $$\mathbb{Z} \to B$$ splitting the sequence by sending $$n$$ to $$nb$$.