Thomas Schick: A counterexample to the (unstable) Gromov-Lawson-Rosenberg conjecture
Doing surgery on the 5-torus, we construct a 5-dimensional closed
spin-manifold M with $\pi_1(M) = Z^4times Z/3$, so that the index
invariant in the KO-theory of the reduced $C^*$-algebra of $\pi_1(M)$
is zero. Then we use the theory of minimal surfaces of Schoen/Yau to
show that this manifolds cannot carry a metric of positive scalar curvature.
The existence of such a metric is predicted by the (unstable)
Gromov-Lawson-Rosenberg
conjecture.
Thomas Schick