A counterexample to the (unstable) Gromov–Lawson–Rosenberg conjecture

Doing surgery on the 5-torus, we construct a five-dimensional closed spin-manifold M with π1(M)≅Z4×Z/3, so that the index invariant in the KO-theory of the reduced C∗-algebra of π1(M) is zero. Then we use the theory of minimal surfaces of Schoen/Yau to show that this manifold cannot carry a metric of positive scalar curvature. The existence of such a metric is predicted by the (unstable) Gromov–Lawson–Rosenberg conjecture.