The homotopy type of skeleta of the flag complex over a finite vector space

TL;DR

This paper uses discrete Morse theory to prove that the skeleta of the flag complex over a finite vector space are homotopy equivalent to wedges of spheres, providing explicit formulas for their structure.

Contribution

It offers a Morse theoretic proof of the homotopy type of skeleta of the flag complex and derives explicit formulas for the number of spheres in each wedge.

Findings

Homotopy equivalence of skeleta to wedges of spheres

Explicit formulas for the number of spheres in each wedge

Application of discrete Morse theory to combinatorial topology

Abstract

The aim of this paper is to give a (discrete) Morse theoretic proof of the fact that the $k$-th skeleton of the flag complex $\mathcal{F}$, associated to the lattice of subspaces of a finite dimensional vector space, is homotopy equivalent to a wedge of spheres of dimension $\min\{k,\dim(\mathcal{F})\}$. The tight control provided by Morse theoretic methods allows us to give an explicit formula for the number of spheres appearing in each of these wedge summands.