Surgery groups of the fundamental groups of hyperplane arrangement complements

TL;DR

This paper proves the Farrell-Jones Fibered Isomorphism conjecture for certain groups including Artin braid groups and computes their surgery groups, extending to fundamental groups of hyperplane arrangement complements.

Contribution

It establishes the conjecture for groups with finite index strongly poly-free subgroups and explicitly computes surgery groups for these groups.

Findings

Farrell-Jones conjecture holds for groups with strongly poly-free subgroups

Explicit surgery group computations for Artin pure braid groups

Extension of results to fundamental groups of hyperplane arrangement complements

Abstract

Using a recent result of Bartels and Lueck (arXiv:0901.0442) we deduce that the Farrell-Jones Fibered Isomorphism conjecture in L-theory is true for any group which contains a finite index strongly poly-free normal subgroup, in particular, for the Artin full braid groups. As a consequence we explicitly compute the surgery groups of the Artin pure braid groups. This is obtained as a corollary to a computation of the surgery groups of a more general class of groups, namely for the fundamental group of the complement of any fiber-type hyperplane arrangement in the complex n-space.