How many triangulations of S^3 are there: exponentially many, or more? This
question was asked among others by Gromov, and it has repercussions on
physics. The count is usually performed with respect to the number N of
facets; two triangulations are considered different if their face posets are
We show that *shellable* triangulations of S^3 are only exponentially many.
In particular, there are exponentially many polytopes with N facets. This is
joint work with Gunter Ziegler (http://www.springerlink.com/content/x8r6mm1155102542/?MUD=MP).
If time permits, we will also sketch some recent progress on this, showing
that under some extra `bounded geometry' assumptions (in the sense of
Cheeger), one can reach exponential bounds for the number of triangulated
d-manifolds with N simplices (for fixed d). Unfortunately, the technical
assumption cannot be removed: Already for d=2, there are more than
exponentially many surfaces with N triangles. This is joint work with Karim