On the $K(\pi, 1)$-problem for restrictions of complex reflection arrangements

TL;DR

This paper investigates the $K(ta,1)$ property of certain hyperplane arrangement complements associated with complex reflection groups, proving it for monomial groups and narrowing down remaining cases.

Contribution

It proves the $K(ta,1)$ property for arrangements from monomial groups and identifies only eight unresolved cases among all irreducible complex reflection groups.

Findings

Proved the $K(ta,1)$ property for arrangements from monomial groups $G(r,p,ll)$.

Reduced the problem to only eight unresolved arrangements among all irreducible complex reflection groups.

Identified three remaining irreducible complex reflection groups with unresolved $K(ta,1)$ status.

Abstract

Let $W\subset GL(V)$ be a complex reflection group, and ${\mathscr A}(W)$ the set of the mirrors of the complex reflections in $W$. It is known that the complement $X({\mathscr A}(W))$ of the reflection arrangement ${\mathscr A}(W)$ is a $K(\pi,1)$ space. For $Y$ an intersection of hyperplanes in $\mathscr A(W)$, let $X(\mathscr A(W)^Y)$ be the complement in $Y$ of the hyperplanes in $\mathscr A(W)$ not containing $Y$. We hope that $X(\mathscr A(W)^Y)$ is always a $K(\pi,1)$. We prove it in case of the monomial groups $W = G(r,p,\ell)$. Using known results, we then show that there remain only three irreducible complex reflection groups, leading to just eight such induced arrangements for which this $K(\pi,1)$ property remains to be proved.