Hadamard Matrix Torsion

TL;DR

This paper constructs a series of 2-dimensional simplicial complexes with torsion in their first homology group that grows rapidly, using Hadamard matrices, and improves previous constructions in terms of torsion growth rate.

Contribution

The paper introduces an explicit quadratic-time construction of simplicial complexes with torsion growth in ^{n \u221a}, narrowing the gap to the theoretical maximum.

Findings

Torsion growth in ^{n  n} achieved.

Construction is explicit and quadratic in time.

Improves upon previous torsion growth bounds.

Abstract

We construct a series HMT$(n)$ of $2$-dimensional simplicial complexes with torsion $H_1($HMT$(n))=(\mathbb{Z}_2)^{{k}\choose{1}} \times (\mathbb{Z}_4)^{{k}\choose{2}} \times \cdots \times (\mathbb{Z}_{2^k})^{{k}\choose{k}}$, $|H_1($HMT$(n))|=|$det(H$(n))|=n^{n/2} \in \Theta(2^{n \log n})$, where the construction is based on the Hadamard matrices H$(n)$ for $n\geq 2$ a power of $2$, i.e., $n=2^k, \ k \geq 1$. The examples have linearly many vertices, their face vector is $f(HMT(n))=(5n-1,3n^2+9n-6,3n^2+4n-4)$. Our explicit series with torsion growth in $\Theta(2^{n \log n})$ is constructed in quadratic time $\Theta(n^{2})$ and improves a previous construction by Speyer with torsion growth in $\Theta(2^{n})$, narrowing the gap to the highest possible asymptotic torsion growth in $\Theta(2^{n^2})$ proved by Kalai via a probabilistic argument.