I'm trying to understand why $H_0(X) \cong \tilde{H}_0(X) \oplus \mathbb{Z}$. In Hatcher on page 110 he writes
''…Since $\varepsilon \partial_1 = 0$, $\varepsilon$ vanishes on $im \partial_1$ and hence induces a map $H_0(X) \rightarrow \mathbb{Z}$ with kernel $\tilde{H}_0(X)$…''
The bit I don't understand is how exactly it induces this map. I mean, where does $\mathbb{Z}$ come from all of a sudden. It's in the extended diagram for the reduced homology $\tilde{H}_0$ and the $H_0(X)$ is in the regular diagram. So somehow there is a map from normal homology into reduced homology (the exact sequences), but how? Many thanks for your help.
Best Answer
There is a map $\epsilon : C_0(X) \to \mathbb Z$. It is indeed a part of the complex for reduced homology, but it exists independently. Because $\epsilon \partial_1$ is zero, the value of $\epsilon(z)$ doesn't change when you add a boundary so, by definition, it defines a map $\overline \epsilon : H_0(X) = C_0(X)/\mathrm{im} \partial_1 \to \mathbb Z$.
Now, what is the kernel of this map? It's the set of (nonreduced) homology classes represented by an element on which $\epsilon$ vanishes. So that's $\ker \epsilon / \mathrm{im} \partial_1$. That's the very definition of $\widetilde H_0(X)$.
There is really nothing deep going there (and nothing too interesting either). Reduced (co)homology is nothing but a very handy convention that avoids too much boring distinction between zeroth and positive degree for computations you do everyday (looking at the homology of contractible space, looking at the homology of a wedge sum...)