JournalscmhVol. 83 , No. 2DOI 10.4171/cmh/132

On the space of metrics with invertible Dirac operator

  • Mattias Dahl

    Royal Institute of Technology, Stockholm, Sweden
On the space of metrics with invertible Dirac operator cover

Abstract

On a compact spin manifold we study the space of Riemannian metrics for which the Dirac operator is invertible. The first main result is a surgery theorem stating that such a metric can be extended over the trace of a surgery of codimension at least three. We then prove that, if non-empty, the space of metrics with invertible Dirac operators is disconnected in dimensions n ≡ 0,1,3,7 mod 8, n ≥ 5. As corollaries follow results on the existence of metrics with harmonic spinors by Hitchin and Bär. Finally we use computations of the eta invariant by Botvinnik and Gilkey to find metrics with harmonic spinors on simply connected manifolds with a cyclic group action. In particular this applies to spheres of all dimensions n ≥ 5.