Spherical subcomplexes of spherical buildings

TL;DR

This paper investigates the structure of spherical subcomplexes within spherical buildings, showing that certain convex subsets lead to non-contractible spherical subcomplexes in the building's complement.

Contribution

It characterizes the maximal subcomplexes supported by the complement of specific convex subsets in spherical buildings, extending understanding of their topological properties.

Findings

Maximal subcomplexes are spherical and non-contractible when the convex subset is open or a closed ball of radius pi/2.

The results apply to thick spherical buildings with the CAT(1) metric.

Provides conditions under which the complement's subcomplexes have specific topological features.

Abstract

Let B be a thick spherical building equipped with its natural CAT(1) metric and let M be a proper, convex subset of B. If M is open or if M is a closed ball of radius pi/2, then the maximal subcomplex supported by the complement of M is spherical and non contractible.