  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.
