Small simplicial complexes with prescribed torsion in homology

TL;DR

This paper determines the minimal size of simplicial complexes with prescribed torsion in homology, showing it scales with the logarithm of the group order to the power of 1/d, matching known bounds up to constants.

Contribution

It establishes tight bounds on the number of vertices needed for simplicial complexes with specific torsion in homology, generalizing previous results.

Findings

Upper and lower bounds on T_d(G) are proportional to (log |G|)^{1/d}

The bounds match up to a constant factor for all d ≥ 2

Provides a construction for complexes achieving these bounds

Abstract

For $d \geq 2$ and $G$ a finite abelian group, define $T_d(G)$ to be the minimum number of vertices $n$ so that there exists a simplicial complex $X$ on $n$ vertices which has the torsion part of $H_{d - 1}(X)$ isomorphic to $G$. Here we establish an upper bound on $T_d(G)$ which matches the known lower bound up to a constant factor. That is, we prove that for every $d \geq 2$ there exist constants $c_d$ and $C_d$ so that for any finite abelian group $c_d(\log |G|)^{1/d} \leq T_d(G) \leq C_d(\log |G|)^{1/d}.$