I am currently doing Hatcher's problem 2.1.13:
Verify that $f \simeq g$ implies $f_* = g_*$ for induced homomorphisms of reduced homology groups.
By inspected of the regular and reduced chains, it is evident that $H_n(X) = \tilde{H}_n(X)$ for $n > 0$, so all that is left to prove is the case $n = 0$. Since $H_0(X) = \frac{\ker \partial_0}{\text{im } \partial _1}$ and $\tilde{H}_0(X) = \frac{\ker \epsilon}{\text{im } \partial_1}$, is it enough to show that $\ker\epsilon \cong \ker \partial_0$ to prove the statement? If so, how would you go about starting it?