Homology computations for complex braid groups

TL;DR

This paper develops methods to compute the homology of complex braid groups, providing explicit results like Poincaré polynomials and second cohomology groups, which help distinguish these groups.

Contribution

It introduces new computational techniques for homology of complex braid groups and provides explicit homological invariants for a large class of these groups.

Findings

Poincaré polynomial with finite field coefficients computed for many groups

Second integral cohomology groups determined for all studied groups

Non-isomorphism results established for different complex braid groups

Abstract

Complex braid groups are the natural generalizations of braid groups associated to arbitrary (finite) complex reflection groups. We investigate several methods for computing the homology of these groups. In particular, we get the Poincar\'e polynomial with coefficients in a finite field for one large series of such groups, and compute the second integral cohomology group for all of them. As a consequence we get non-isomorphism results for these groups.