Non-vanishing of homotopy groups of Manin--Schechtman arrangements

TL;DR

This paper investigates Manin--Schechtman arrangements, showing their complements have non-zero higher homotopy groups, thus they are not $K(pi,1)$ spaces in many cases.

Contribution

It proves that the complements of these arrangements have non-vanishing higher homotopy groups, providing new insights into their topological properties.

Findings

Manin--Schechtman arrangements have non-zero higher homotopy groups.

Their complements are not $K(pi,1)$ spaces in many cases.

Abstract

One of the central problems in the topology of hyperplane arrangements is determining whether the complement is a $K(\pi,1)$-space. In this paper, we study Manin--Schechtman arrangements, introduced as higher-dimensional analogs of the braid arrangement, and prove that their complements have non-vanishing higher homotopy groups. Consequently, these arrangements fail to be $K(\pi,1)$ in a broad range of cases.