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The homotopy type of elliptic arrangements

Emanuele Delucchi and Roberto Pagaria

Algebraic & Geometric Topology 21 (2021) 2037–2063

We give combinatorial models for the homotopy type of complements of elliptic arrangements, ie 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.

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elliptic arrangements, Salvetti complexes, polyhedral complexes, CW complexes, acyclic categories, braid group, configuration spaces
Mathematical Subject Classification 2010
Primary: 55U05
Secondary: 05E45, 20F36, 55R80
Received: 18 March 2020
Revised: 23 June 2020
Accepted: 9 July 2020
Published: 18 August 2021
Emanuele Delucchi
University of Applied Arts and Sciences of Southern Switzerland
Roberto Pagaria
Dipartimento di Matematica
Università di Bologna