The homotopy type of toric arrangements

TL;DR

This paper constructs a CW-complex homotopy equivalent to the complement of a toric arrangement, providing combinatorial and algebraic descriptions, with applications in robotics.

Contribution

It introduces a new CW-complex model for toric arrangement complements and offers algebraic tools for cohomology calculations, extending known methods to this setting.

Findings

Constructed a CW-complex homotopy equivalent to the arrangement complement.

Provided a combinatorial description similar to the Salvetti complex.

Developed an algebraic description for arrangements defined by Weyl groups.

Abstract

A toric arrangement is a finite set of hypersurfaces in a complex torus, every hypersurface being the kernel of a character. In the present paper we build a CW-complex homotopy equivalent to the arrangement complement, with a combinatorial description similar to that of the well-known Salvetti complex. If the toric arrangement is defined by a Weyl group we also provide an algebraic description, very handy for cohomology computations. In the last part we give a description in terms of tableaux for a toric arrangement appearing in robotics.