Parametrized Constructions and Spaces of Things
by Ryan BUDNEY (UO)
Abstract:
There are many situations in topology where one studies a family
of objects, such as exotic spheres or fiber bundles or embeddings of
spheres in spheres. Typically one introduces a `parametrizing object'
that allows one to get an upper bound on the diversity of such objects.
One constructs exotic spheres by gluing two discs together via a
diffeomorphism of their boundary, one constructs fiber bundles from maps
into a classifying space.
This talk will largely focus on a `parametrizing object' for constructing
embeddings of spheres in spheres, the `long knot spaces'. To understand the
homotopy type of the space of long knots in R3
we use a combination of the known parts of the Thurston
geometrization program for 3-manifolds together with techniques of
Waldhausen and Hatcher. Out of this comes a rather odd fact (from the
point of view of a 3-manifold theorist) that knot spaces admit actions of
certain operads, and these operad actions tell us about the homotopy type
of knot spaces, moreover they are an essential part of the geometrization
of knot complements.