Operator algebras and topology

TL;DR

This paper surveys the interplay between operator algebras and topology, focusing on the Baum-Connes conjecture, its implications for scalar curvature, and the role of $L^2$-invariants in high-dimensional manifold theory.

Contribution

It introduces operator algebra techniques to topology, explains the Baum-Connes conjecture and its applications, and discusses $L^2$-invariants in the context of manifold theory.

Findings

The Baum-Connes conjecture implies the Novikov and stable Gromov-Lawson-Rosenberg conjectures.

Counterexamples to the unstable Gromov-Lawson-Rosenberg conjecture are presented.

An introduction to $L^2$-cohomology and $L^2$-Betti numbers is provided.

Abstract

These notes cover the contents of three survey lectures held at the ICTP Trieste Summer school on High dimensional manifold theory 2001. They introduce techniques coming from the theory of operator algebras. We will focus on the basic definitions and properties, and on their relevance to the geometry and topology of manifolds. An central pillar of work in the theory of C*-algebras is the Baum-Connes conjecture. It implies the Novikov conjecture. In the first talk, the Baum-Connes conjecture will be explained and put into our context. One application of the Baum-Connes conjecture is to the positive scalar curvature question. It implies the so called ``stable Gromov-Lawson-Rosenberg conjecture''. The unstable version of this conjecture said that, given a closed spin manifolds M, a certain obstruction, living in a certain (topological) K-theory group, vanishes if and only M admits a Riemannian metric with positive scalar curvature. It turns out that this is wrong, and counterexamples will be presented in the second talk. The third talk introduces $L^2$-cohomology, $L^2$-Betti numbers and other $L^2$-invariants. These invariants, their basic properties, and the central questions about them are introduced.