I was down in Princeton at the Institute yesterday and heard two interesting talks. The first was the beginning of a series of four lectures by Paul Baum about the Baum-Connes conjecture, the second was by Chris Woodward about “equivariant localization”. I’ll write a bit about Baum-Connes here, perhaps something about the topic of Woodward’s talk in a second posting.
The Baum-Connes conjecture was first formulated in 1982 by Paul Baum and Alain Connes in an unpublished paper.Roughly it says that the K-theory of the reduced C* algebra of a group G is identical with the equivariant K-homology of a certain sort of classifying space for the group. Equivariant K-homology classes can be represented by certain generalizations of the Dirac operator, and the map to the K-theory of the C* algebra is given by taking the index of the operator.
There’s a huge literature about this by now and a few years ago Nigel Higson put together a detailed bibliography. Some recent expository articles about the conjecture include an ICM talk by Higson, a survey talk by Wolfgang Lueck, and a book by Alain Valette.
The conjecture remains unproved for discrete groups in general, and Baum said that he suspects it is not true in full generality, invoking what he called “Gromov’s principle”. According to Baum, this principle states that “No statement about all finitely presented groups is both non-trivial and true.” While the conjecture has been proved for some classes of discrete groups, there are many for which it is expected to be true but remains unproved (e.g. SL(3,Z)). For a while it was thought that the conjecture applied also to groupoids, but counterexamples for groupoids have been found.
I’ve always been fascinated by part of the philosophy behind the Baum-Connes conjecture, which is to use equivariant K-homology, classifying spaces and Dirac operators to get information about representation theory of groups in cases where little is known about this representation theory. This is in some very vague sense related to what seems to me to be going on in QFT, where Dirac operators and path integrals over the classifying space of the gauge group (the space of connections) are somehow related to the representation theory of the gauge group. The Baum-Connes conjecture itself just involves locally compact groups and thus doesn’t say anything about gauge groups, so it is not directly relevant to the case that may be of interest in QFT.
Gromov’s principle can be used as a jumping off point for some very interesting discussions. Let G be a discrete group. BG is defined as any CW-complex which is connected, and has G for its fundamental group, and has all higher homotopy groups zero. We can then take the homology or cohomology H*(BG). On the other hand in the realm of pure algebra we can write down a complex whose homology is H*(BG). It is a theorem that these two are the same — and this is valid for any discrete group. Of course many mathematicians would declare this to be trivial. But it took the best mathematicians of the nineteen-thirties (e.g. Heinz Hopf) much thought and much work to figure this out. So a lot of the content of Gromov’s principle hinges on exactly what we mean by “non-trivial”. Another issue with GP is what “about” means. In the last of the four lectures at IAS I’ll give two statements which are relevant to Baum=Connes for
discrete groups and which are true for all discrete groups. In the sense that Gromov had in mind these are undoubtedly trivial statements.But it took my co-workers and me quite a few years to figure this out and to come up with the proofs. So in the colloquial sense that mathematicians use the term “non-trivial” these statements are (it seems to me) non-trivial. So exactly what does “non-trivial” mean in Gromov’s principle. Perhaps it means a statement which tells us something really deep and really meaningful about all discrete groups. Gromov is probably right about such statements.
Peter Woit’s comments about QFT suggest that someday we might have something like Baum-Connes for gauge groups. I hope this happens. It might be really fascinating.