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 ${\pi }_1(M)=Z^{4}timesZ\mathrm{/}3$, so that the index invariant in the KO-theory of the reduced $C^{\ast }$-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.