Hadamard Matrix Torsion

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.