Remarks on a conjecture of Gromov and Lawson

We study obstructions to the existence of Riemannian metrics of positive scalar curvature on closed smooth manifolds arising from torsion classes in the integral homology of their fundamental groups. As an application, we construct new examples of manifolds which do not admit positive scalar curvature metrics, but whose Cartesian products admit such metrics.

In this article we study the space of positive scalar curvature metrics on totally nonspin manifolds with spin boundary. We prove that for such manifolds of certain dimensions, those spaces are not connected and have nontrivial fundamental group. Furthermore we show that a well-known propagation technique for detection results on spaces of positive scalar curvature metrics on spin manifolds ceases to work in the totally nonspin case.

Let $M$ be a smooth closed spin (resp. oriented and totally non-spin) manifold of dimension $n\geq 5$ with fundamental group $\pi$. It is stated, e.g. in [RS95], that $M$ admits a metric of positive scalar curvature (pscm) if its orientation class in $ko_n(B\pi)$ (resp. $H_n(B\pi;\Z)$) lies in the subgroup consisting of elements which contain pscm representatives. This is 2-locally verified loc. cit. and in [Sto94]. After inverting 2 it was announced that a proof would be car...

We introduce Riemannian metrics of positive scalar curvature on manifolds with Baas-Sullivan singularities, prove a corresponding homology invariance principle and discuss admissible products. Using this theory we construct positive scalar curvature metrics on closed smooth manifolds of dimension at least five which have odd order abelian fundamental groups, are nonspin and atoral. This solves the Gromov-Lawson-Rosenberg conjecture for a new class of manifolds with finite fun...

Witten and Yau (hep-th/9910245) have recently considered a generalisation of the AdS/CFT correspondence, and have shown that the relevant manifolds have certain physically desirable properties when the scalar curvature of the boundary is positive. It is natural to ask whether similar results hold when the scalar curvature is zero. With this motivation, we study compact scalar flat manifolds which do not accept a positive scalar curvature metric. We call these manifolds rigidl...

Gromov and Lawson developed a codimension 2 index obstruction to positive scalar curvature for a closed spin manifold M, later refined by Hanke, Pape and Schick. Kubota has shown that also this obstruction can be obtained from the Rosenberg index of the ambient manifold M which takes values in the K-theory of the maximal C*-algebra of the fundamental group of M, using relative index constructions. In this note, we give a slightly simplified account of Kubota's work and rema...

Let $M_\Sigma$ be an $n$-dimensional Thom-Mather stratified space of depth $1$. We denote by $\beta M$ the singular locus and by $L$ the associated link. In this paper we study the problem of when such a space can be endowed with a wedge metric of positive scalar curvature. We relate this problem to recent work on index theory on stratified spaces, giving first an obstruction to the existence of such a metric in terms of a wedge $\alpha$-class $\alpha_w (M_\Sigma)\in KO_n$. I...

We prove the Gromov conjecture on the macroscopic dimension of the universal covering of a closed spin manifold with a positive scalar curvature under the following assumptions on the fundamental group: 1. The Strong Novikov Conjecture holds for $\pi$. 2. The natural map $per:ko_n(B\pi)\to KO_n(B\pi)$ is injective.

K. Grove, L. Verdiani, B. Wilking and W. Ziller gave the first examples of cohomogeneity one manifolds which do not carry invariant metrics with non-negative sectional curvatures. In this paper we generalize their results to a larger family. We also classified all class one representations for a pair (G;H) with G/H some sphere, which are used to construct the examples.

Gromov's Conjecture states that for a closed $n$-manifold $M$ with positive scalar curvature the macroscopic dimension of its universal covering $\tilde M$ satisfies the inequality $\dim_{mc}\tilde M\le n-2$\cite{G2}. We prove this inequality for totally non-spin $n$-manifolds whose fundamental group is a virtual duality group with $vcd\ne n$. In the case of virtually abelian groups we reduce Gromov's Conjecture for totally non-spin manifolds to the vanishing problem whethe...