Index Theory and the Baum-Connes conjecture

TL;DR

This paper explores how operator algebras, especially $C^*$-algebras, are used to study manifolds through index theorems, focusing on the Baum-Connes conjecture and its implications for topology.

Contribution

It provides an overview of index theory, $C^*$-algebras, and the Baum-Connes conjecture, highlighting their roles in connecting analysis, topology, and geometry of manifolds.

Findings

Index theorems relate analysis to topology via $K$-theory.

The Baum-Connes conjecture links operator algebras to manifold topology.

Implications include progress on the Novikov conjecture.

Abstract

These notes are based on lectures on index theory, topology, and operator algebras at the "School on High Dimensional Manifold Theory" at the ICTP in Trieste, and at the Seminari di Geometria 2002 in Bologna. We describe how techniques coming from the theory of operator algebras, in particular $C^*$-algebras, can be used to study manifolds. Operator algebras are extensively studied in their own right. We will focus on the basic definitions and properties, and on their relevance to the geometry and topology of manifolds. The link between topology and analysis is provided by index theorems. Starting with the classical Atiyah-Singer index theorem, we will explain several index theorems in detail. Our point of view will be in particular, that an index lives in a canonical way in the K-theory of a certain $C^*$-algebra. The geometrical context will determine, which $C^*$-algebra to use. A central pillar of work in the theory of $C^*$-algebras is the Baum-Connes conjecture. Nevertheless, it has important direct applications to the topology of manifolds, it impliese.g. the Novikov conjecture. We will explain the Baum-Connes conjecture and put it into our context. Several people contributed to these notes by reading preliminary parts and suggesting improvements, in particular Marc Johnson, Roman Sauer, Marco Varisco und Guido Mislin. I am very indebted to all of them. This is an elaboration of the first chapter of the author's contribution to the proceedings of the above mentioned "School on High Dimensional Manifold Theory" 2001 at the ICTP in Trieste.