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Are there any relationship between the scalar curvature and the simplicial volume?

The simplicial volume is zero (positive) on Torus (Hyperbolic manifold) and those manifolds does not admit a Riemannian metric with positive scalar curvature. What do we know about the simplicial volume of a Riemannian manifold with positive scalar curvature?

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  • $\begingroup$ If scalar curvature is bounded from below by a positive number, then $\pi_1$ is finite and so simplicial volume is zero $\endgroup$ – Grisha Papayanov Oct 22 '19 at 9:09
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    $\begingroup$ @Grisha, any closed manifold times with 2-sphere admits metrics with positive scalar curvature. $\endgroup$ – Jialong Deng Oct 22 '19 at 9:18
  • $\begingroup$ I am dumb and read "sectional" instead of "scalar". Should get more sleep. $\endgroup$ – Grisha Papayanov Oct 22 '19 at 9:23
  • $\begingroup$ Does anyone knows the simplicial volume of the connected sum of two copies of $\mathbb{S}^{n-1}\times\mathbb{S}^1$ ? It does have metric with positive scalar curvature thanks to the construction of Gromow and Lawson but its $\pi_1$ has exponential growth, that would be a good test case, as are product $M\times\mathbb{S}^2$. $\endgroup$ – Thomas Richard Oct 22 '19 at 9:36
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    $\begingroup$ @ThomasRichard Product of anything with a sphere have vanishing simplicial volume (since such a space admits a self-map of positive degree), and in dimensions more than two simplicial volume is additive with respect to connected sum, so for these examples volume is zero. $\endgroup$ – Grisha Papayanov Oct 22 '19 at 9:39
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In a preliminary version of what would become Gromov's "A Dozen Problems, Questions and Conjectures about Positive Scalar Curvature", he writes on page 88:

Neither is one able to prove (or disprove) that manifolds with positive scalar curvatures have zero simplicial volumes. Possibly, these conjectures need significant modifications to become realistic.

Simplicial volume didn't make it into the published version of these notes, maybe because he thought his conjectural relationship between scalar curvature and simplicial volume was too hopeless to be worth mentioning, but at any rate this is reasonable evidence that this question was an open problem in 2017, and I haven't seen any evidence of progress since then.

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By theorems of Wilking and Milnor (see On fundamental groups of manifolds of nonnegative curvature by Wilking), the fundamental group of a compact manifold with nonnegative sectional (not scalar, I've misread the original question) curvature is of polynomial growth. By Gromov's theorem, it is virtually nilpotent, hence amenable. Compact manifolds with amenable fundamental group have vanishing simplicial volume.

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  • $\begingroup$ I think you're right, I'll edit the answer $\endgroup$ – Grisha Papayanov Oct 22 '19 at 9:24

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