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Positive scalar curvature, diffeomorphisms and the Seiberg-Witten invariants. (English) Zbl 1002.57064

This paper studies the space of positive scalar curvature (psc) metrics on a 4-manifold, giving examples of simply connected 4-manifolds for which it is disconnected. These examples imply that concordance of psc metrics does not imply isotopy of such metrics.
The method of proof is to use a modification of a Seiberg-Witten invariant for 1-parameter families introduced by D.Ruberman in [Math. Res. Lett. 5, 743-758 (1998; Zbl 0946.57025)]. The former invariant counted, for a spinc structure whose Seiberg-Witten moduli space has virtual dimension \(-1\), the number of solutions to the Seiberg-Witten equations in a generic 1-parameter family of metrics. The modified invariant counts, for a diffeomorphism \(f\) of the 4-manifold, the number of solutions to a path of metrics joining an initial metric \(h_{0}\) with its pull-back by \(f\) and the successive iterates.
Therefore these examples give psc metrics \(h_{0}\) and \(f^{*}h_{0}\) which are not connected by psc metrics. The invariant also shows that the diffeomorphism group of a 4-manifold can be disconnected.
The paper also deals with the moduli space of psc metrics modulo diffeomorphism, giving examples of non-orientable 4-manifolds for which this space is disconneted. The components in the moduli space are distinguished by a \(Pin^{c}\) eta invariant introduced by P.B.Gilkey [Adv. in Math. 58, 243-248 (1985; Zbl 0602.58041)].

MSC:

57R57 Applications of global analysis to structures on manifolds
53C21 Methods of global Riemannian geometry, including PDE methods; curvature restrictions
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