The homotopy type of elliptic arrangements

TL;DR

This paper develops combinatorial models to understand the homotopy type of complements of elliptic arrangements, providing new insights into their fundamental groups and configuration spaces.

Contribution

It introduces finite polyhedral CW complex models for elliptic arrangement complements and characterizes face categories of such complexes.

Findings

Provides a presentation of the fundamental group of elliptic arrangement complements.

Models the homotopy type using acyclic categories and polyhedral CW complexes.

Analyzes ordered configuration spaces of elliptic curves.

Abstract

We give combinatorial models for the homotopy type of complements of elliptic arrangements (i.e., certain sets of abelian subvarieties in a product of elliptic curves). We give a presentation of the fundamental group of such spaces and, as an application, we treat the case of ordered configuration spaces of elliptic curves. Our models are finite polyhedral CW complexes, and our combinatorial tools of choice are acyclic categories (small categories without loops). As a stepping stone, we give a characterization of which acyclic categories arise as face categories of polyhedral CW complexes.