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

An \emph{affine subtorus} of the compact torus $T=(S^1{)}^n$ is a translated copy of a Lie subgroup. Given a finite collection $T_1,\dots ,T_k$ of such subtori, and a prime $p$, we describe an explicit chain complex that calculates the group $H_{\ast }(T-{\bigcup }_{i=1}^k{T}_i,{\mathbb{Z}}_{(p)})$. %The complex is determined by the integral homology maps induced by the inclusions $T_J\subset T_I$ where $I\subset J\subset \{1,\dots ,k\}$ 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 $H_{\ast }(T-{\bigcup }_{i=1}^k{T}_i,\mathbb{Z})$. 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 $T_1,\dots ,T_k$, with all extension problems being trivial in the latter case.