Problem on simplicial complexes.

Question

If $(S_0,P_0)$ and $(S_1,P_1)$ are abstract simplicial complexes, a simplicial map from $(S_0,P_0)$ to $(S_1,P_1)$ is a function $f\colon S_0 \longrightarrow S_1$ such that, if $U\in P_0$, then $f(U)\in P_1$. By $f(U)$ we mean $f$ applied to each element of $U$, i.e., $f(U) = \{f(u) : u \in U\}$ subset $S_1$. Write $C$ for the unit circle in the complex plane, and write $g \colon C \rightarrow C$ for the function $g(z) = z^2$. Find abstract simplicial complexes $(S_1, P_0)$ and $(S_1, P_1)$ and a simplicial map, $f \colon (S_0, P_0 ) \rightarrow(S_1, P_1)$ such that the geometric realization of $(S_0, P_0)$ is homeomorphic to $C$, the geometric realization of $(S_1, P_1)$ is homeomorphic to $C$, and the geometric realization of the map $g$ agrees with the function $f$.

Answer 1

Let $C_k$ be the cycle with $k$ vertices, that is the simplicial complex with vertices $\{0,\ldots,k-1\}$ and maximal faces $\{\{0,1\},\{1,2\},\ldots,\{k-1,0\}\}$ for $k\geq 3$. Its geometric realization is the circle.

Now take a simplicial map $f: C_6\to C_3$ given by $f(i) = i \pmod 3$. It will "wind" the source circle twice onto the second circle, which is what you need.