Bisimplicial complexes and asphericity

TL;DR

This paper introduces bisimplicial complexes, a new class of regular CW complexes, and presents a Morse-theoretic method to analyze their topological properties, including collapsibility and contractibility, with applications to bipartite graphs and quadric complexes.

Contribution

It defines bisimplices and bisimplicial complexes, and demonstrates their topological properties using discrete Morse theory, advancing understanding of complex structures in geometric group theory.

Findings

Bisimplicial complexes are collapsible when constructed from bi-dismantlable bipartite graphs.

Flag bisimplicial completion of certain complexes is contractible.

Constructs a compact K(G,1) space for torsion-free quadric groups.

Abstract

We present a discrete Morse-theoretic method for proving that a regular CW complex is homeomorphic to a sphere. We use this method to define bisimplices, the cells of a class of regular CW complexes we call bisimplicial complexes. The 1-skeleta of bisimplices are complete bipartite graphs making them suitable in constructing higher dimensional skeleta for bipartite graphs. We show that the flag bisimplicial completion of a finite bipartite bi-dismantlable graph is collapsible. We use this to show that the flag bisimplicial completion of a quadric complex is contractible and to construct a compact K(G,1) for G a torsion-free quadric group.