Bisimplicial complexes and asphericity

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.