= Solution
A <simplicial approximation> to $f:|K|\to|L|$ is a simplicial map $g:K\to L$ satisfying the open-star condition
$$
f(\operatorname{st}_K(v))\subseteq\operatorname{st}_L(g(v))
$$
for every vertex $v$. Equivalently, whenever $f(x)$ lies in the interior of a simplex of $L$, $|g|(x)$ lies in that simplex. In particular $f$ and $|g|$ are joined by the simplexwise straight-line <homotopy>.
The <Simplicial approximation theorem> says that, for finite $K$, every continuous $f$ has such an approximation after sufficiently many <barycentric subdivisions> of $K$. For general complexes one uses a subdivision adapted to the open-star cover; a single uniform number of barycentric subdivisions is the compact-domain version.
Given triangulations $X\cong|K|$, $Y\cong|L|$, choose a simplicial approximation $g$ on a suitable subdivision $K'$. Its <chain map> sends an oriented simplex $[v_0,\ldots,v_r]$ to $[g(v_0),\ldots,g(v_r)]$, with repeated-vertex simplices sent to zero. It commutes with the boundary operator and induces $g_*$. Compose this with the subdivision isomorphism $H_r(K)\to H_r(K')$ and the triangulation identifications to define $f_*:H_r(X)\to H_r(Y)$.
For $z\mapsto z^n$, a positively oriented fundamental loop winds $n$ times around the target circle. With integer coefficients,
$$
\boxed{h_*:H_0(S^1)\to H_0(S^1)\ \hbox{is the identity},\qquad h_*:H_1(S^1)\to H_1(S^1)\ \hbox{is multiplication by }n.}
$$
All higher homology groups vanish. The $H_1$ result can also be read from a subdivision into $n$ copies of a target fundamental cycle.
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