Surgery and the Spectrum of the Dirac Operator

Details Surgery and the Spectrum of the Dirac Operator Surgery and the Spectrum of the Dirac Operator

2002-01-21

arxiv.org/abs/math/0201195v2

J. reine angew. Math. 552, 53-76 (2002)

Mathematics

Differential Geometry

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

Scientific paper

10.1515/crll.2002.093

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.

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