Volume 5, issue 2 (2001)

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Positive scalar curvature, diffeomorphisms and the Seiberg–Witten invariants

Daniel Ruberman

Geometry & Topology 5 (2001) 895–924

arXiv: math.DG/0105027


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 Pinc eta invariant.

positive scalar curvature, Seiberg–Witten equations, isotopy
Mathematical Subject Classification 2000
Primary: 57R57
Secondary: 53C21
Forward citations
Received: 1 September 2001
Revised: 2 January 2002
Accepted: 31 December 2001
Published: 3 January 2002
Proposed: Ronald Fintushel
Seconded: Robion Kirby, John Morgan
Daniel Ruberman
Department of Mathematics
Brandeis University
Massachusetts 02454-9110