Topology and combinatorics of the complex of flags

Let V be an n-dimensional space over a fixed finite field. The complex of flags X(V) is the simplicial complex whose vertices are the non-trivial linear subspaces of V, and whose simplices are ascending chains of subspaces. This complex, also known as the spherical building associated to the linear group GL(V), appears in a number of different mathematical areas, including topology, combinatorics and representation theory. After recalling the classical homological properties of X(V), we will discuss some more recent results including:

1. Minimal weight cocycles in the Lusztig-Dupont homology.

2. Coding theoretic aspects of X(V) and the existence of homological codes.

2. Coboundary expansion of X(V) and its applications.