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

Thomas Schick
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Abstract

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.

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Thomas Schick. 2004-03-03. A counterexample to the (unstable) Gromov-Lawson-Rosenberg conjecture. https://doi.org/10.1016/s0040-9383(97)00082-7

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