Positive scalar curvature, diffeomorphisms and the Seiberg–Witten invariants

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

Mathematical Subject Classification 2000

Primary: 57R57

Secondary: 53C21

References

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Publication

Received: 1 September 2001

Revised: 2 January 2002

Accepted: 31 December 2001

Published: 3 January 2002

Proposed: Ronald Fintushel

Seconded: Robion Kirby, John Morgan

Authors

Daniel Ruberman
Department of Mathematics Brandeis University Waltham Massachusetts 02454-9110 USA

Volume 5, issue 2 (2001)

Daniel Ruberman