# Does positive scalar curvature imply vanishing of the simplicial volume on a closed Riemannian manifold?

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?

• If scalar curvature is bounded from below by a positive number, then $\pi_1$ is finite and so simplicial volume is zero – Grisha Papayanov Oct 22 '19 at 9:09
• @Grisha, any closed manifold times with 2-sphere admits metrics with positive scalar curvature. – Jialong Deng Oct 22 '19 at 9:18
• I am dumb and read "sectional" instead of "scalar". Should get more sleep. – Grisha Papayanov Oct 22 '19 at 9:23
• 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 $M\times\mathbb{S}^2$. – Thomas Richard Oct 22 '19 at 9:36
• @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. – Grisha Papayanov Oct 22 '19 at 9:39