Bonn Topology Group - Abstracts

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Talk

Thomas Schick (Goettingen): Mapping the surgery exact sequence to analysis using higher index theory (03/12/2013)

Abstract

The surgery exact sequence is a tool which combines algebra (L-theory) and generalized homology to get information about the possible different manifolds which have the same homotopy type.

Much of the information can be read off using the signature of an oriented manifold and variants of this.

The signature can also be computed as the index of the signature operator. It has "higher" variants which take values in the K-theory of C*-algebras and give further information.

Indeed, Higson and Roe introduced a C*-algebraic counterpart to the surgery exact sequence, which is closely related to the Baum-Connes conjecture, and constructed a transformation from the topologists surgery exact sequence to their C*-algebraic counterpart. Here, information can often be read off in a more direct way. Higson and Roe's construction is using quadratic forms and is in parts quite indirect.

In the talk, we will introduce in the analysis of higher Dirac operators (using also methods from large-scale/coarse geometry). Using this, we will derive a very direct and explicit transformation from the surgery exact sequence to the Higson-Stolz sequence. Along the way, we have to prove a secondary index theorem for manifolds with boundary which is of independent interest.

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