Physically Motivated Questions in Topology: Manifold Pairings
by
Michael Freedman
Microsoft Research

Since Milnor built the exotic 7-spheres in 1956 by gluing together two copies of S³ ×D⁴, gluing has played a key role in manifold topology. In this talk, I will borrow and idea from quantum mechanics and consider gluing superpositions of manifolds. It turns out that there is a dramatic difference according to dimension. The natural pairings induced by gluing have null vectors when the manifold dimension is 4 or higher, in contrast the pairings are positive when the dimension is 3 or lower. This difference has a profound implication for what topological features can be captured within the physics of a d-dimensional quantum mechanical system; it gives another perspective on the fact that the known topological phases of matter are 2+1 dimensional.

Date received: October 30, 2007