  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.

Differential Geometry;Differential Geometry