Maps in homology between one-point-spaces

Question

Given a Homology Theory we know that this uniquelly determinated by the coefficients group $G$, given by $$H_0(pt)\cong G,$$ where $pt$ is the one-point-space. Now, there is only one map $pt\longrightarrow pt$ and this induces the identity map in homology. My problem is the following... what happens if I consider the unique map between different points? Is the induced map in homology still the identity map?

Answer 1

No, and there is a problem (which is also the answer in a way): the homology groups themselves are not in general the same formally speaking in order for a identity map to even make sense. Consider for instance singular homology. If you have $X=\{0\}$, and $Y=\{1\}$, $\Delta_p(X)$ and $\Delta_p(Y)$ are different things, and so are $H_0(X)$ and $H_0(Y)$. In general, what we have is the following diagram

$\require{AMScd}$ \begin{CD} H_0(X) @>f_*>> H_0(Y) \\ @V i_1 V V @VV i_2 V \\ G @>>\xi> G \end{CD}

and, since $f_*$ is an isomorphism by functoriality and $i_1$ and $i_2$ are isomorphisms by the dimension axiom, then $\xi$ is an isomorphism. But $i_1$ and $i_2$ are very arbitrary if we are using only the dimension axiom, and the mere "existential" nature of $i_1,i_2$ do not impose that $\xi$ is the identity. For instance, just revert the usual isomorphism $H_0(pt) \cong \mathbb{Z}$ on singular homology to see that $\xi$ is not necessarily the identity.

OBS: Note the difference from the case that $X=Y$. In this case, the map on the homology is induced by the identity, and hence by functoriality it is the identity on the homology (this doesn't need any consideration of coefficients group and the dimension axiom).