  We consider the effect of compactification of extra dimensions on the onset
of classical chaotic "Mixmaster" behavior during cosmic contraction. Assuming a
universe that is well-approximated as a four-dimensional
Friedmann-Robertson--Walker model (with negligible Kaluza-Klein excitations)
when the contraction phase begins, we identify compactifications that allow a
smooth contraction and delay the onset of chaos until arbitrarily close the big
crunch. These compactifications are defined by the de Rham cohomology (Betti
numbers) and Killing vectors of the compactification manifold. We find
compactifications that control chaos in vacuum Einstein gravity, as well as in
string theories with N = 1 supersymmetry and M-theory. In models where chaos is
controlled in this way, the universe can remain homogeneous and flat until it
enters the quantum gravity regime. At this point, the classical equations
leading to chaotic behavior can no longer be trusted, and quantum effects may
allow a smooth approach to the big crunch and transition into a subsequent
expanding phase. Our results may be useful for constructing cosmological models
with contracting phases, such as the ekpyrotic/cyclic and pre-big bang models.
