all: This should not be too hard, but I am stuck. $S_g$ is the orientable, genus-g surface, and $H_1(S_g,\mathbb{Z})$ is the first homology with coeffs. in $\mathbb{Z}$. I am trying to avoid using the universal coefficient theorem, if possible.
I have been trying to find an answer using chains and mod2 reduction, i.e., using the fact that we can embed non-trivial $\mathbb{Z}/2$ $1$-chains $c$ into the group of $\mathbb{Z}$ $1$-chains, as just chains c', with odd coefficients. Then $f$ will send the embedded c' into a homologous ( in $H_1(S_g,\mathbb{Z})$ ) $\mathbb{Z}$ $1$-chain c''. But I don't know if mod 2 reduction will preserve homology, i.e., if c'\sim c'' , does it follow that c'\pmod{2}\sim c''\pmod{2} ( i.e., reducing coefficients term-by-term)?
I know some properties of mod 2-reduction, but this is from $\mathrm{SL}(2,\mathbb{Z})$ to $\mathrm{SL}(2, \mathbb{Z}/n)$; kernel has finite index in $\mathrm{SL}(2,\mathbb{Z})$; it is finitely-generated by elements of finite order, but this does not seem to lead anywhere useful.
Maybe showing naturality of the map would help. Thanks.