Diffeomorphism equivariance and the scanning map

Diffeomorphism equivariance and the scanning map

Thu, 31/05/2012
12:00 		Richard Manthorpe Junior Geometry and Topology Seminar Add to calendar L3
Diffeomorphism equivariance and the scanning map
Given a manifold $ M $ and a basepointed labelling space $ X $ the space of unordered finite configurations in $ M $ with labels in $ X $, $ C(M;X) $ is the space of finite unordered tuples of points in $ M $, each point with an associated point in $ X $. The space is topologised so that particles cannot collide. Given a compact submanifold $ M_0\subset M $ we define $ C(M,M_0;X) $ to be the space of unordered finite configuration in which points `vanish' in $ M_0 $. The scanning map is a homotopy equivalence between the configuration space and a section space of a certain $ \Sigma^nX $-bundle over $ M $. Throughout the 70s and 80s this map has been given several unsatisfactory and convoluted definitions. A natural question to ask is whether the map is equivariant under the diffeomorphism group of the underlying manifold. However, any description of the map relies heavily on `little round $ \varepsilon $-balls' and so only actions by isometry have any chance at equivariance. The goal of this talk is to give a more natural definition of the scanning map and show that diffeomorphism equivariance is an easy consequence.