On aspherical configuration Lie groupoids

TL;DR

This paper investigates the asphericity of complements of hyperplanes in products of aspherical 2-manifolds and orbifolds, extending the understanding to Lie groupoids and proposing a general conjecture.

Contribution

It proves asphericity for complements in certain 2-dimensional orbifolds and generalizes the problem to the framework of Lie groupoids, suggesting broader applicability.

Findings

Complement of hyperplanes in aspherical 2-manifold products is aspherical.

Proved asphericity for a class of 2-dimensional orbifolds.

Extended the problem to Lie groupoids, proposing a general conjecture.

Abstract

The complement of the hyperplanes $\{x_i=x_j\}$, for all $i\neq j$ in $M^n$, for $M$ an aspherical $2$-manifold, is known to be aspherical. Here we consider the situation, when $M$ is a $2$-dimensional orbifold. We prove this complement to be aspherical for a class of aspherical $2$-dimensional orbifolds, and predict that it should be true in general also. We generalize this question in the category of Lie groupoids, as orbifolds can be identified with a certain kind of Lie groupoids.