  We examine the complexity of the ``Texas Hold'em'' variant of poker from a
topological perspective. We show that there exists a natural simplicial complex
governing the multi-way winning probabilities between various hands, and that
this simplicial complex contains $4$-dimensional spheres as induced
subcomplexes. We deduce that evaluating the strength of a pair of cards in
Texas Hold'em is an intricate problem, and that even the notion of who is
bluffing against whom is ill-defined in some situations.
