Binomial rings, and integral homology of complements of compact toric arrangements

TL;DR

This paper provides an explicit chain complex for computing the homology of complements of finite collections of affine subtori in a compact torus, using binomial models and analyzing spectral sequence collapse.

Contribution

It introduces a new explicit chain complex for integral homology of toric arrangement complements and demonstrates spectral sequence collapse under certain conditions.

Findings

Explicit chain complex for homology computation

Spectral sequence collapses at second page rationally

Integral collapse under specific arrangement conditions

Abstract

An \emph{affine subtorus} of the compact torus $T=(S^1)^n$ is a translated copy of a Lie subgroup. Given a finite collection $T_1,\ldots, T_k$ of such subtori, and a prime $p$, we describe an explicit chain complex that calculates the group $H_*(T-\bigcup_{i=1}^k T_i,\mathbb{Z}_{(p)})$. %The complex is determined by the integral homology maps induced by the inclusions $T_J\subset T_I$ where $I\subset J\subset\{1,\ldots, k\}$ and $T_I$ denotes $\bigcap_{i\in I} T_i$. Our main tool is the binomial models for spaces constructed by T.~Ekedahl. We use these results to express the groups $H_*(T-\bigcup_{i=1}^k T_i,\mathbb{Z})$. We also show that the Mayer-Vietoris spectral sequence that converges to the homology of $T-\bigcup_{i=1}^k T_i$ collapses at the second page rationally, and also integrally under some assumptions on the arrangement $T_1,\ldots, T_k$, with all extension problems being trivial in the latter case.