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THE PROBABILITY THEORY OF 3-MANIFOLDS. (3 of 4)
What conclusions can we draw from the empirical densities? At first, one
probably expects
statistical quantities
associated to such random walks to reflect very little of the full
topological structure of
the manifold. It turns out, however, that in certain special
situations, the mean
commute time carries enough information to allow one to determine the
manifold completely. We
prove for example:
THM. Suppose an
unknown manifold M is triangulated with 27 tetrahedra. If the maximal mean
commute time associated
to the combinatorial random walk on this triangulation is
≥
88,
then M is the 3-sphere.
In general, we are
interested in calculating, for a given number of tetrahedra n and a
given threshold time t,
the a posteriori probabilities Pr(M|C
≤
t), that is, the
probability that the
manifold one is walking in is M after having observed that the
commute time C is
≤
t.
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