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On the space of metrics with invertible Dirac operator
KTH, School of Engineering Sciences (SCI), Mathematics (Dept.), Mathematics (Div.).ORCID iD: 0000-0002-9184-1467
2008 (English)In: Commentarii Mathematici Helvetici, ISSN 0010-2571, E-ISSN 1420-8946, Vol. 83, 451-469 p.Article in journal (Refereed) Published
Abstract [en]

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 equivalent to 0, 1, 3, 7 mod 8, n >= 5. As corollaries follow results on the existence of metrics with harmonic spinors by Hitchin and Bar. 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.

Place, publisher, year, edition, pages
2008. Vol. 83, 451-469 p.
Keyword [en]
eigenvalues of the Dirac operator, surgery, positive scalar curvature, harmonic spinors, manifolds, geometry
National Category
URN: urn:nbn:se:kth:diva-47739DOI: 10.4171/CMH/132ISI: 000256732200009ScopusID: 2-s2.0-43649084030OAI: diva2:456095

QC 20111114

Available from: 2011-11-12 Created: 2011-11-12 Last updated: 2012-09-19Bibliographically approved

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Dahl, Mattias
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