Scalar Curvature , Diffeomorphisms , and the Seiberg - Witten Invariants

One of the striking initial applications of the Seiberg-Witten invariants was to give new obstructions to the existence of Riemannian metrics of positive scalar curvature on 4– manifolds. The vanishing of the Seiberg–Witten invariants of a manifold admitting such a metric may be viewed as a non-linear generalization of the classic conditions [16, 15, 21] derived from the Dirac operator. If a manifold Y has a metric of positive scalar curvature, it is natural to investigate the topology of the space PSC(Y ) of all such metrics. Perhaps the simplest question which one can ask is whether PSC(Y ) is connected; examples of manifolds for which it is disconnected were previously known in all dimensions greater than 4. This phenomenon is detected via the index theory of the Dirac operator, often in conjunction with the Atiyah–Patodi–Singer index theorem [2]. In the first part of this paper, we use a variation of the 1–parameter Seiberg-Witten invariant introduced in [22] to prove that on a simply–connected 4–manifold Y , PSC(Y ) can be disconnected. Our examples cannot be detected by index theory alone, ie without the intervention of the Seiberg–Witten equations. These examples also yield a negative answer, in dimension 4, to the question of whether metrics of positive scalar curvature which are concordant are necessarily isotopic. Apparently (cf. the discussion in [21, §3 and §6]) this is the first result of this sort in any dimension other than 2. An a priori more difficult problem than showing that PSC is not necessarily connected is to find manifolds for which the “moduli space” PSC /Diff is disconnected. (The action of the diffeomorphism group on the space of metrics is by pull–back, and preserves the subset of positive scalar curvature metrics.) The metrics lying in different components of PSC(Y ) constructed in the first part of the paper are obtained by pulling back a positive scalar curvature metric via one of the diffeomorphisms introduced in [22], and hence give no information about PSC /Diff . Building on constructions of Gilkey [8] we give explicit examples of non–orientable 4–manifolds for which the moduli space is disconnected. These examples are detected, as in [8, 4], by an η–invariant associated to a Pinc Dirac operator.