Binomial rings, and integral homology of complements of compact toric arrangements
Alexey G. Gorinov, Alexander V. Zakharov
TL;DR
The paper addresses the problem of computing the integral and rational homology of complements of finite arrangements of affine subtori in a compact torus $T=(S^1)^n$. It introduces binomial-models in the sense of Ekedahl to build explicit cosimplicial chain complexes and uses a Mayer–Vietoris double complex $\mathrm{MV}({C_I^*})$ to compute $H^*(T,\bigcup T_i,R)$, thereby obtaining $H_{\dim T-*}(T-\bigcup T_i,R)$. In the rational setting, the authors show $\mathrm{MV}({H^*(T_I,R)})$ provides a minimal model with $E_2$-degeneration, and under certain integral hypotheses (e.g., intersections path connected and $H_1(-,\mathbb{Z})$ lifting to free groups) the first page suffices for integral coefficients as well. They further develop finite-rank, computable models via binomial filtrations and establish torsion-bounds through the MV framework; they also construct diagrams of classifying spaces to realize finite polyhedral models that underpin the integral results. The results have potential applications in topological combinatorics and quasi-crystal theory, offering explicit, computable descriptions of the homology of toric-arrangement complements.
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.