On the homotopy type of subcomplexes of Tits buildings
type: Item
instance of: scholarly article
title: On the homotopy type of subcomplexes of Tits buildings
zbMATH Open document ID: 0789.51013
author: Herbert Abels
author: Peter Abramenko
published in: Advances in Mathematics
publication date: 1994-02-10T00:00:00Z
review text: This paper studies the following question. Let \({\mathcal E}\) be a finite subset of subspaces of a building of type \(A_ n\), viewed as a simplicial complex \(X\) arising from the vector space \(V\). Let \(T_{\mathcal E} (V)\) be the subcomplex of \(X\) obtained from \(X\) by considering only those subspaces \(D\) of \(V\) for which \(D\) and \(E\) generate \(V\), for all \(E \in {\mathcal E}\), whenever \(D\) and \(E\) meet nontrivially. Under which condition(s) is \(T_{\mathcal E} (V)\) spherical, i.e. has homotopy type of a bouquet of \((n-2)\)-spheres (where \(n\) is the dimension of \(V)\)? This question is answered in the paper under review and the condition expresses that the dimensions of the members of \({\mathcal E}\) cannot be too big compared with the order \(q\) of the underlying field. Examples and counter-examples are also mentioned in the paper.
Mathematics Subject Classification ID: 51E24
zbMATH DE Number: 495497
zbMATH Keywords: projective space
zbMATH Keywords: Solomon-Tits theorem
zbMATH Keywords: homotopy type
reviewed by: Hendrik Van Maldeghem
MaRDI profile type: Publication
full work available at URL: aima.1993.1042
OpenAlex ID: W2123925179
DOI: 10.1006/AIMA.1993.1042
description: scientific article; zbMATH DE number 495497