On the homotopy type of subcomplexes of Tits buildings (Q1312848)

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.