Novikov additivity of the signature as a gluing index theorem

29.04.2016, 11.00  –  Haus 9, Raum 2.22
Arbeitsgruppenseminar Analysis

Juan Camillo Orduz (HU)

Given a 4k-dimensional oriented closed manifold X, its signature sig(X) in Z is a very important topological invariant which can be computed in terms of some combinations of Pontryagin classes. We can understand this relation by means of the Atiyah-Singer index theorem via the so called signature operator, which is a Dirac-type operator. Atiyah, Patodi and Singer found a generalization in the context of manifolds with boundary. The main ingredient of their work was the discovery of appropriate boundary conditions (APS) to obtain a well defined index.

It is natural to ask how does the signature behave when the closed manifold X is decomposed as two manifolds with boundary glued along their common boundary. In this talk we will obtain Novikov’s additivity formula for the signature as a gluing index theorem for manifolds with boundary with APS boundary conditions. The main objective is to illustrate how to use these techniques to find decomposition formulas for the index.