Hadamard Matrix Torsion

Abstract

We construct a series HMT$(n)$ of $2$-dimensional simplicial complexes with torsion $H_1($HMT$(n))=({\mathbb{Z}}_2{)}^{\left(\genfrac{}{}{0ex}{}{k}{1}\right)}\times ({\mathbb{Z}}_4{)}^{\left(\genfrac{}{}{0ex}{}{k}{2}\right)}\times \cdots \times ({\mathbb{Z}}_{2^k}{)}^{\left(\genfrac{}{}{0ex}{}{k}{k}\right)}$, $|H_1($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 $n\ge 2$ a power of $2$, i.e., $n=2^k,k\ge 1$. 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.

16 pages, 5 figures