Combinatorial and Geometric Methods in Topology
Carlo Petronio
University of Pisa
April 16, 2007
Abstract: How many different
objects can be obtained by gluing together in pairs the faces of an
octahedron? After deciding what "object" and "different" mean,
this is an apparently very elementary question, but the answer is not
quite so. Before facing it I will go one dimension down and
consider the gluings of the edges of a polygon, discussing surface
topology and showing that there are extremely few surfaces one can get
from a given polygon compared to the number of inequivalent gluing
patterns. Then I will introduce the notions of curvature and
hyperbolic geometry in two and three dimensions, I will discuss
rigidity and I will sketch how this applies to the original question,
yielding the fact that the number of different results is indeed quite
big.