Gromov-Lawson-Rosenberg conjecture

Doing surgery on the 5-torus, we construct a 5-dimensional closed spin-manifold M with π1(M) ∼= Z ×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 manifolds cannot carry a metric of positive scalar curvature. The existence of such a metric is predicted by the (unstable) Gromov-Lawson-Rosenberg conjecture. 1 Obstructions to positive scalar curvature 1.1 Definition. A manifold M which admits a metric of positive scalar curvature is called a pscm-manifold. We start with a discussion of the index obstruction for spin manifolds to be pscm, constructed by Lichnerowicz [6], Hitchin [4] and in the following refined version due to Rosenberg [9]. 1.2 Theorem. Let M be a closed spin-manifold, π := π1(M). One can construct a homomorphism, called index, from the singular spin bordism Ω ∗ (Bπ) to the (real) KO-theory of the reduced real C-algebra of π: ind : Ω ∗ (Bπ) → KO∗(C ∗ redπ) Let u : M → Bπ be the classifying map for the universal covering of M . If M is pscm, then ind([u : M → Bπ]) = 0 ∈ KOm(C ∗ red) ∗e-mail: [email protected] www: http://www.uni-math.gwdg.de/schick/