Higher signature and a codimension two phenomenon

Zhizhang Xie

1:50 pm - 2:50 pm, March 31th, 2017   Science Building A1510

Abstract:

The higher signature index of oriented closed manifolds is invariant under orientation-preserving homotopy equivalences. In this talk, I will talk about a similar invariance statement under the assumption that is slightly weaker than homotopy equivalence. More precisely, let h be an orientation preserving smooth homotopy equivalence between smooth, closed, oriented manifolds N and M. Suppose X is a smooth, closed, oriented submanifold of M of codimension 2, and assume that the map h is transversal to X so that Y = h^{-1}(X) is is a smooth, closed, oriented, codimension 2 submanifold of N. In addition, assume that (a) the fundamental group of X injects into that of M, (b) the second homotopy group of M is trivial, (c) and the normal bundle of X in M is trivial. Then we show that the higher signature index of Y is equal to that of X. The talk is based on joint work with Nigel Higson and Thomas Schick.