Title: Small simplicial complexes with prescribed torsion in homology
Speaker: Andrew Newman (Ohio State University)
Abstract: By Kalai's generalization of Cayley's formula, it is known that for every dimension $d \geq 2$, there exists a constant $k_d > 0$ so that for every $n$, there is a simplicial complex $X$ on $n$ vertices so that the torsion subgroup of $H_{d - 1}(X)$ has order at least $e^{k_d n^d}$. In this talk, I use the probabilistic method to affirmatively answer an inverse question. That is, I prove that for any $d \geq 2$, there is a constant $C_d$ so that for \emph{any} abelian group $G$ there is a simplicial complex $X$ on at most $C_d \sqrt[d] {\log |G|}$ vertices so that the torsion subgroup of $H_{d - 1}(X)$ is isomorphic to $G$.
Seminar URL: https://u.osu.edu/probability/
