Tag Archives: asymptotic dimension

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

Property A and large scale paracompactness

Jerzy Dydak and his collaborators have published a very interesting series of papers recently, whose overall theme is that property A is the large scale analog of paracompactness.  The papers that I have seen on the arXiv are these:

Cencelj, M., J. Dydak, and A. Vavpetič. “Asymptotic Dimension, Property A, and Lipschitz Maps.” Revista Matemática Complutense 26, no. 2 (July 1, 2013): 561–571. doi:10.1007/s13163-012-0102-2.

Cencelj, M., J. Dydak, and A. Vavpetič. Coarse Amenability Versus Paracompactness. ArXiv e-print, August 13, 2012. http://arxiv.org/abs/1208.2864.

Dydak, Jerzy. Coarse Amenability and Discreteness. ArXiv e-print, July 15, 2013. http://arxiv.org/abs/1307.3943.

Dydak, Jerzy, and Atish Mitra. “Large Scale Absolute Extensors.” arXiv:1304.5987 (April 22, 2013). http://arxiv.org/abs/1304.5987.

Recall that a Hausdorff topological space \(X\) is paracompact if every open cover of \(X\) has a refinement to a locally finite open cover.  (It is metacompact, or weakly paracompact, if every open cover has a refinement to a pointwise finite open cover.)  Most often though one applies paracompactness via the existence of partitions of unity: \(X\) is paracompact if and only if there exists a (locally finite) partition of unity subordinate to any open cover.  The papers above elucidate what should be the “coarse version” of paracompactness both in the “covering” and the “partition of unity” interpretations, and in both cases they relate it to property A. Continue reading

Permanence properties in coarse geometry

I hate to think how long it has been since I last posted here. My apologies – it has been a difficult summer for various non-mathematical reasons. Anyhow, trying to get back on track let me mention a survey article that Erik Guentner sent me called “Permanence properties in coarse geometry”. What Erik means by “permanence properties” is statements like “the property of having finite asymptotic dimension is closed under group extensions”. Many statements of this kind, for a variety of coarse properties (asymptotic dimension, embeddability in Hilbert space, property A/exactness, etc) have by now been proved and this is a very nice survey bringing together general techniques for obtaining such results with specific applications.