Anders Björner characterized which finite, graded posets
are closure posets of regular CW complexes, and he also observed that
a regular CW complex is homeomorphic to the order complex of its
closure poset. This suggests that one might use combinatorics to
determine topological structure; however, it is possible for two
different CW complexes with very different topological structure to
have the same closure poset if one of them is not regular. I will talk
about a new criterion for determining whether a particular finite CW
complex is regular; this will involve a mixture of combinatorics and
topology. Along the way, I will review the notions from topology we
will need as well as the history of Björner's work and related work in
this area. To the extent that time permits, I will also discuss an
application in which Bruhat order is the closure poset.