Category Archives: My Publications

Positive scalar curvature partial vanishing theorems and coarse indices

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This paper, http://arxiv.org/abs/1210.6100, has been accepted by the Proceedings of the Edinburgh Mathematical Society.  I just sent off the copyright transfer form this evening, so everything is now set, I hope.

The paper is mostly paying an expository debt.  In my CBMS lecture notes I said that if one has the Dirac operator on a complete spin manifold \(M\), and if there is some subset \(N\subseteq M\) such that \(D\) has uniformly positive scalar curvature outside \(N\), then the index of \(D\) belongs to the K-theory of the ideal \(I_N \triangleleft C^*(M) \) associated to the subset \(N\).  A very special case of this is the observation of Gromov-Lawson that \(D\) is Fredholm if we have uniformly positive scalar curvature outside a compact set.  There are of course analogous results using thepositivity of the Weitzenbock curvature  term for other generalized Dirac operators.

Until now, I had not written up the proof of this assertion, but I felt last year that it was (past) time to do so.  This paper contains the proof and also that of the associated general form of the Gromov-Lawson relative index theorem which also appears in my CBMS notes. The latter proof uses some results from my paper with Paul Siegel on sheaf theory and Pashcke duality.

The submission to PEMS is in honor of a very pleasant sabbatical spent in Edinburgh in fall 2004.

 

The real form of the coarse Baum-Connes conjecture

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Following up on my post a few days back about Dranishnikov’s talk… After the talk, Sasha asked me if I knew a reference where some “standard” facts about the real version of the coarse Baum-Connes conjecture were stated (as, for example, that the real coarse index of the Dirac operator vanishes for positive scalar curvature manifolds, or that the complex form of the coarse Baum-Connes conjecture implies the real form.

I was sure that these “well known to experts” results must be written down somewhere. Maybe they are, but I couldn’t find a clean reference.  So I thought it might be helpful to put together a little note summarizing some of these standard facts.  I’ve now posted this on the arXiv and it is available here.  If you need the real version of CBC for something, this might be useful.

Originally, Nigel and I were going to cover the real version of everything in Analytic K-Homology.  But at some point we got fed up with Clifford algebras and retreated to the complex world.  I think that was the only way to get the book finished, but it has left a few loose ends!

Ghostbusting and Property A

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Let \(X\) be a bounded geometry discrete metric space.  Guoliang Yu defined a ghost to be an element of the Roe algebra \( C^*_u(X) \) that is given by a matrix \(T_{xy}\) whose entries tend to zero as \(x,y\to\infty\).

The original counterexamples of Higson to the coarse Baum-Connes conjecture were noncompact ghost projections on box spaces derived from property T groups.  On the other hand, all ghost operators on a property A space are compact.

In Ghostbusting and Property A, Rufus Willett and I show that all ghosts on \(X\) are compact if and only if \(X\) has property A.  (Appropriately enough, on a space without property A we construct ghosts using the spectral theorem.) The paper will appear in the Journal of Functional Analysis.

Question: To what characterization of ordinary amenability does this correspond?