Surgery and the Spectrum of the Dirac Operator

Abstract

We show that for generic Riemannian metrics on a simply-connected closed spin manifold of dimension at least 5 the dimension of the space of harmonic spinors is no larger than it must be by the index theorem. The same result holds for periodic fundamental groups of odd order. The proof is based on a surgery theorem for the Dirac spectrum which says that if one performs surgery of codimension at least 3 on a closed Riemannian spin manifold, then the Dirac spectrum changes arbitrarily little provided the metric on the manifold after surgery is chosen properly.

23 pages, 4 figures, to appear in J. Reine Angew. Math

Cited by in corpus (19)

• Topological mirror symmetry with fluxes
• Surgery and Harmonic Spinors
• Dirac-harmonic maps from index theory
• Prescribing eigenvalues of the Dirac operator
• Complex/Symplectic Mirrors
• The first conformal Dirac eigenvalue on 2-dimensional tori
• Dirac operators on manifolds with periodic ends
• On the space of metrics with invertible Dirac operator
• Dirac eigenvalues for generic metrics on three-manifolds
• Dirac eigenspinors for generic metrics
• Harmonic spinors and local deformations of the metric
• Mass endomorphism, surgery and perturbations
• Manifolds with small Dirac eigenvalues are nilmanifolds
• The spinorial τ-invariant and 0-dimensional surgery
• Dirac Eigenvalues of higher Multiplicity
• Weyl laws on open manifolds
• Existence of Dirac Eigenvalues of higher Multiplicity
• Homotopy equivalence of spaces of metrics with invertible Dirac operator
• Highly connected manifolds of positive $p$-curvature