  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.
