A ``stable'' version of the Gromov-Lawson conjecture

We discuss a conjecture of Gromov and Lawson, later modified by Rosenberg, concerning the existence of metrics of positive scalar curvature. It says that a closed spin manifold $M$ of dimension $n\ge 5$ has such a metric if and only if the index of a suitable ``Dirac" operator in $KO_n(C^* (\pi_1(M)))$, the real $K$-theory of the group $C^*$-algebra of the fundamental group of $M$, vanishes. It is known that the vanishing of the index is necessary for existence of a positive scalar curvature metric on $M$, but this is known to be a sufficient condition only if $\pi_1(M)$ is the trivial group, $\Bbb Z/2$, an odd order cyclic group, or one of a fairly small class of torsion-free groups. \par We note that the groups $KO_n(C^*(\pi))$ are periodic in $n$ with period $8$, whereas there is no obvious periodicity in the original geometric problem. This leads us to introduce a ``stable'' version of the Gromov-Lawson conjecture, which makes the weaker statement that the product of $M$ with enough copies of the ``Bott manifold" $B$ has a positive scalar curvature metric if and only if the index of the Dirac operator on $M$ vanishes. (Here $B$ is a simply connected $8$-manifold which represents the periodicity element in $KO_8(pt)$.) We prove the stable Gromov-Lawson conjecture for all spin manifolds with finite fundamental group and for many spin manifolds with infinite fundamental group.