Geometry & Topology, Vol. 5 (2001)
Paper no. 28, pages 895--924.
Positive scalar curvature, diffeomorphisms and the Seiberg-Witten invariants
Daniel Ruberman
Abstract.
We study the space of positive scalar curvature (psc) metrics on a
4-manifold, and give examples of simply connected manifolds for which
it is disconnected. These examples imply that concordance of psc
metrics does not imply isotopy of such metrics. This is demonstrated
using a modification of the 1-parameter Seiberg-Witten invariants
which we introduced in earlier work. The invariant shows that the
diffeomorphism group of the underlying 4-manifold is disconnected. We
also study the moduli space of positive scalar curvature metrics
modulo diffeomorphism, and give examples to show that this space can
be disconnected. The (non-orientable) 4--manifolds in this case are
explicitly described, and the components in the moduli space are
distinguished by a Pin^c eta invariant.
Keywords.
Positive scalar curvature, Seiberg-Witten equations, isotopy
AMS subject classification.
Primary: 57R57.
Secondary: 53C21.
DOI: 10.2140/gt.2001.5.895
E-print: arXiv:math.DG/0105027
Submitted to GT on 1 September 2001.
(Revised 2 January 2002.)
Paper accepted 31 December 2001.
Paper published 3 January 2002.
Notes on file formats
Daniel Ruberman
Department of Mathematics, MS 050
Brandeis University
Waltham, MA 02454-9110, USA
Email: ruberman@brandeis.edu
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