Manifolds of positive scalar curvature

The talk is a report about particl results concerning the following
Question. Let $M$ be a smooth, compact manifold without boundary. Does $M$ admit a Riemannian metric of positive scalar curvature?

It follows from results proved independently by Gromov-Lawson [GL] and Schoen-Yau [SY] that the answer depends only on the bordism calss of $M$ in a suitable bordism group. For simply connected non-spin manifolds, the relevant bordism group is oriented bordism, which was calculated by Wall. Using his results, Gromov-Lawson show that every simply connected non-spin manifold of dimension $n\geq 5$ admits a positive scalar curvature metric. This is not the case for spin manifolds, since the existence of a positive scalar curvature metric on an $n$-dimensional spin manifold $M$ implies the vanishing of a topoloical invariant $\alpha(M)$ (cf. [L], [H], [R]). Analytically, this invariant is the index of a suitable "Dirac" opertor and lives in ${\rm KO}_n(C^*\pi)$, the real $K$-theory of the group $C^*$-algebra of the fundamental group $\pi=\pi_1(M)$.
Conjecture (Gromov-Lawson-Rosenberg). A spin manifold of dimension $n\geq 5$ has a positive scalar curvature metric if and only if $\alpha(M)$ vanishes.

Using homotopy theoretic methods to analyze the relevant bordism group, the conjecture has been verified for manifolds with trivial fundamental group [S] or fundamental group of order 2 [RS1]. We note that the groups ${\rm KO}_n(C^*\pi)$ are periodic 8, and this periodicity is given by multiplication by the "Bott element" $b\in {\rm KO}_8$. Geometrically, $b=\alpha(B)$ for some simply connected 8-dimensional spin manifold $B$, and so the conjecture above implies
Cancellation Conjecture. A spin manifold of dimension $n\geq 5$ has a positive scalar curvature metric if and only if $M\times B$ does.

In fact, the Gromov-Lawson-Rosenberg conjecture is equivalent to the Cancellation Conjecture plus the following
Stable Conjecture. Let $M$ be a spin manifold of dimension $n\geq 5$. The product of $M$ with sufficiently many copies of $B$ has a positive scalar curvature metric if and only if $\alpha(M)$ vanishes.

The stable conjecture has been verified for finite groups [RS2], and for groups, for which the "assembly map" is injective. This map plays a central rôle in the Novikov Conjecture, and is known to be injective e.g. for torsionfree discrete subgroups of Lie groups [Ka, Thm. 6.7].

Bibliography:

1. [GL] M. Gromov a nd H. B. Lawson, Jr., The classificatio nof simply connected manifolds of positive scalar curvature, Ann. of Math. 111 (1980), 423--434.
2. [H] N. Hitchin, Harmonic spinors, Adv. Math. 14 (1974), 1--55.
3. [K] G. G. Kasparov, Equivariant $KK$-theory and the Novikov Conjecture, Invent. Math. 91 (1988), 147--201.
4. [L] A. Lichnerowicz, Spineurs harmoniques, C. R. Acad. Sci. Paris, Sér A-B 257 (1963), 7--9.
5. [R] J. Rosenberg, $C^*$-algebras, positive scalar curvature, and the Novikov Conjecture, III, Topology 25 (1986), 319--336.
6. [RS1] J. Rosenberg and S. Stolz, Manifolds of positive scalar curvature, Algebraic Topology and its Applications (G. Carlsson, R. Cohen, W.-C. Hsiang, and J. D. S. Jones, eds.), M. S. R. I. Publications, vol. 27, Springer, New York, 1994, pp. 241--267.
7. [RS2] J. Rosenberg and S. Stolz, A "stable" version of the Gromov-Lawson conjecture, to apper in the Proceedings of the Cech Centennial Conference.
8. [S] S. Stolz, Simply connected manifolds of positive scalar curvature, Annals of Math. 136 (1992), 511--549.
9. [SY] R. Schoen and S.-T. Yau, On the structure of ma nifolds with positive scalar curvature, Manuscripta Math. 28 (1979), 159--183.