Simplicial Complexes and Simplicial Homology

due by Monday, Feb 7, 2022

1. Fix $n\geq 1$ and consider the partially ordered set $$S = \{k\in \mathbb{N}: k\vert n\},$$ where $k\vert n$ means $k$ divides $n$, with the partial order given by $k\leq m$ if $k\vert m$. Let $X$ be the Poset complex associated to S. Prove or disprove the following.

   • (a) $X$ has a $2$-simplex if and only if $n$ is composite.
   • (b) $X$ has a $1$-simplex if and only if $n > 1$.
   • (c) $X$ is the standard $k$-simplex for some $k$ (i.e., the set of simplices is a power set) if and only if $n$ is the power of a prime.

2. Let $\Sigma$ be the simplicial complex $$\Sigma=\{\{0\}, \{1\}, \{2\}, \{0, 1\}, \{0, 2\}\}.$$ Then prove or disprove that the geometric realization $X =\vert\Sigma\vert$ is homeomorphic to a closed interval.

3. Let $\Sigma$ be the simplicial complex $$\Sigma=\{\{0\}, \{1\}, \{2\}, \{0, 1\}, \{0, 2\}\}.$$ Prove or disprove the following.

   • (a) $Z_1$ is trivial.
   • (b) $C_1$ is isomorphic to $\Z$.