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.

Dranishnikov and his collaborators have recently obtained positive results on this conjecture, or at least on the slightly weaker version which asserts that the macroscopic dimension is \(\le n-1\) rather than \(n-2\). These results, however, depend on suitable Novikov-type conjectures as input, so they don’t give an independent approach to Novikov.  The latter may have been Gromov’s hope, though he did describe his conjecture as “at present completely out of reach”.

The arguments all go by relating the macroscopic dimension to some kind of inessentiality condition, which says that the classifying map for the fundamental group (or some related map) can be retracted into a lower-dimensional skeleton. Here is a result of this kind which Dranishnikov stated in his talk.

Proposition Let \(M\) be a closed \(n\)-manifold. The universal cover \(\tilde{M}\) has macroscopic dimension \(\le n-1\) (i.e. \(M\) is macroscopically small) if and only if the image of the locally finite fundamental class of \(M\) vanishes in integral coarse homology \(HX_n(M;{\mathbb Z})\).

There is an interesting tie-up here with an old paper of Gong and Yu, who called \(M\) macroscopically large if the image of the fundamental class doesn’t vanish in rational coarse homology.  The rationality in their paper is, I think, a side-effect of the fact that I used rational (or real) coefficients exclusively in my original coarse cohomology paper in the Memoirs of the AMS.  But it should be clear that the integral version of the condition is the “right” one.

Anyhow, granted the Proposition above one can prove that if \(M\) is closed and almost spin (i.e. the universal cover is spin) and has positive scalar curvature, and if \(\pi=\pi_1(M)\) satisfies appropriate Rosenberg-Stolz conditions (i.e. an appropriate Novikov statement, here coarse Novikov, plus the rational injectivity of an appropriate map from connective to periodic K-theory), then \(\tilde{M}\) is macroscopically small.  The argument traverses a familiar trajectory: positive curvature implies the index of Dirac vanishes in the K-theory of the Roe algebra, hence in coarse K-homology (NC assumption), hence in connective coarse homology (assumption), hence in ordinary coarse homology (natural transformation of spectra), hence we get macroscopic smallness from the Proposition.

In conversation, Sasha told me he can get down to macroscopic dimension \(n-2\) for certain groups, but I have not understood the details of this.

References

Bolotov, Dmitry, and Alexander Dranishnikov. 2010. “On Gromov’s Scalar Curvature Conjecture.” Proceedings of the American Mathematical Society 138 (4): 1517–1524. doi:10.1090/S0002-9939-09-10199-5.

Dranishnikov, Alexander. 2010. “On Macroscopic Dimension of Rationally Essential Manifolds”. ArXiv e-print 1005.0424. http://arxiv.org/abs/1005.0424.

———. 2013. “On Macroscopic Dimension of Universal Coverings of Closed Manifolds”. ArXiv e-print 1307.1029. http://arxiv.org/abs/1307.1029.

Gong, Guihua, and Guoliang Yu. 2000. “Volume Growth and Positive Scalar Curvature.” Geometric and Functional Analysis 10 (4): 821–828. doi:10.1007/PL00001639.

Gromov, M. 1996. “Positive Curvature, Macroscopic Dimension, Spectral Gaps and Higher Signatures.” In Functional Analysis on the Eve of the 21st Century, Vol.\ II (New Brunswick, NJ, 1993), 132:1–213. Progr. Math. Boston, MA: Birkhäuser Boston. http://www.ams.org/mathscinet-getitem?mr=1389019.

Roe, John. 1993. “Coarse Cohomology and Index Theory on Complete Riemannian Manifolds.” Memoirs of the American Mathematical Society 104 (497): x+90.