Homology computations for complex braid groups II

TL;DR

This paper completes the integral homology computation for complex braid groups associated with exceptional complex reflection groups, using recursive differential calculations and parallel computing, and also derives rational homology of related Milnor fibers.

Contribution

It provides the full integral homology of these complex braid groups and the rational homology of their Milnor fibers, advancing understanding of their topological properties.

Findings

Complete integral homology for complex braid groups of exceptional type

Explicit differential computations using parallel computing

Rational homology of Milnor fibers derived from the calculations

Abstract

We complete the computation of the integral homology of the generalized braid group $B$ associated to an arbitrary irreducible complex reflection group $W$ of exceptional type. In order to do this we explicitely computed the recursively-defined differential of a resolution of $\mathbf{Z}$ as a $\mathbf{Z} B$-module, using parallel computing. We also deduce from this general computation the rational homology of the Milnor fiber of the singularity attached to most of these reflection groups.