Geometry and Topology Seminar

Homeomorphism groups of manifolds and Erdős space

Jan Dijkstra, Free University Amsterdam

Details
Wednesday January 16, 2008  
  809 Physical Sciences Center
  3:45 pm  

Abstract Consider the Hilbert space l2 consisting of all square summable sequences of real numbers. In 1940 Paul Erdős introduced the subspace of l2 consisting of sequences of rational numbers and showed that this space (which is now called Erdős space) is totally disconnected but one-dimensional. Since Erdős space is like Hilbert space topologically equivalent to its own square the space has the counterintuitive property that when squared the dimension does not become 2 but remains 1. This fact makes Erdős space a classic and important example in Dimension Theory. The Erdős Project is a collaborative effort of Dijkstra and van Mill at the Vrije Universiteit Amsterdam. We will discuss the primary result of the Erdős Project thus far - the topological characterization of Erdős space and its main application, a complete topological classification of the groups of homeomorphisms H(M;D) of topological manifolds, Hilbert cube manifolds, and Menger manifolds M that leave a countable dense subset D invariant.