Tag Archives: index theorem

Positive scalar curvature partial vanishing theorems and coarse indices

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.

 

Macroscopic dimension and PSC, after Dranishnikov

Sasha Dranishnikov gave a talk describing some of his results about Gromov’s conjecture relating positive scalar curvature and macroscopic dimension.

Definition (Gromov) Let \(X\) be a metric space.  We say that \(X\) has macroscopic dimension \(\le n\) if there exists a continuous, uniformly cobounded \(f\colon X\to K\), where \(K\) is an \(n\)-dimensional simplicial complex.  We recall that uniformly cobounded means that there is an upper bound on the diameters of inverse images of simplices.

This is a metric notion, but it is quite different from the familiar asymptotic dimension.  One way of defining the latter says that \(X\) has asymptotic dimension \(\le n\) if, for each \(\epsilon>0\), there is an \(\epsilon\)-Lipschitz uniformly cobounded map to an \(n\)-dimensional simplicial complex (here, we agree to metrize \(K\) as a subset of the standard simplex in infinite-dimensional Euclidean space).  From this definition it is apparent that the macroscopic dimension is less than or equal to the asymptotic dimension.  On the other hand, it is also clear that the macroscopic dimension is less than or equal to the ordinary topological dimension.

Gromov famously conjectured that the universal cover of a compact \(n\)-manifold that admits a metric of positive scalar curvature should have macroscopic dimension \(\le n-2\).  The motivating example for this conjecture is a manifold  \(M^n = N^{n-2}\times S^2 \) – this clearly admits positive scalar curvature, and its universal cover has macroscopic dimension at most \(n-2\).  Gromov’s conjecture suggests that this geometric phenomenon is “responsible” for all positive scalar curvature metrics. Continue reading

Lin Shan’s index theory

OK, I am going to try to revive this blog in the hope that it will encourage me to read and keep up with the mathematical literature. We shall see…

Anyhow, I just wrote a review for Mathematical Reviews of the paper “Equivariant higher index theory and nonpositively curved manifolds” by Lin Shan (JFA 255(2008), 1480-1496. This paper defines and studies an analytic assembly map that includes both the coarse assembly map and the Baum-Connes assembly map as special cases, and it proves a Novikov conjecture type statement.