The Seiberg–Witten equations on end‐periodic manifolds and an obstruction to positive scalar curvature metrics

Abstract

By studying the Seiberg-Witten equations on end-periodic manifolds, we give an obstruction on the existence of positive scalar curvature metric on compact 4-manifolds with the same homology as S 1 × S 3 . This obstruction is given in terms of the relation between the Frøyshov invariant of the generator of H3(X; Z) with the 4-dimensional Casson invariant λSW (X) defined in [Mrowka, Ruberman and Saveliev, J. Differential Geom. 88 (2011) 333-377]. Along the way, we develop a framework that can be useful in further study of the Seiberg-Witten theory on general end-periodic manifolds.Since it was proved in [10] that the mod-2 reduction of λ SW (X) is always ρ(Y, s) (the Rohlin invariant of (Y, s)), we have the following corollary:Then, X admits no Riemannian metric with positive scalar curvature.This corollary gives a large family of interesting examples of 4-manifolds (with b 2 = 0) admitting no positive scalar curvature metric.Example 1.4. Let X be obtained by furling up any homology cobordism from Y = Σ(2, 3, 7) (the Brieskorn sphere) to itself. Then X admits no Riemannian metric with positive scalar curvature because ρ(Y ) = 1 and h(Y ) = 0.We summarize the idea in the proof of Theorem 1.2 as follows: Let W be the cobordism from Y to itself obtained by cutting X along Y . We consider the manifoldThis non-compact manifold has two ends: one is cylindrical and the other one is periodic. (The word 'periodic' indicates the fact that we are gluing together infinitely many copies of the same manifold W . See [18] for the precise definition.) For a Riemannian metric g X on X, we can construct, using a cutoff function, a metric on Z + that equals the lift of g X over the periodic-end and restricts to the product metric on the cylindrical end. Now consider the (suitably perturbed) Seiberg-Witten equations on Z + . More specifically, let [b] be a critical point of the Chern-Simons-Dirac functional with certain absolute grading. We consider the moduli space M ([b], Z + ) of gauge equivalent classes of solutions that approaches [b] on the cylindrical end and has exponential decay on the periodic end. By adding end points to the moduli space M ([b], Z + ), which correspond to 'broken solutions' on Z + , we get the moduli space M + ([b], Z + ), which is a 1-manifold with boundary. Now we use the assumption that g X has positive scalar curvature. Under this assumption, we can prove that M + ([b], Z + ) is compact. Therefore, the number of points in ∂M + ([b], Z + ), counted with sign, should be 0. This actually implies that a certain reducible critical point [a 0 ] cannot be 'killed by the boundary map' and hence survives in the monopole Floer homology. By this argument, we show that + A : L 2 1 (X; S + ) → L 2 (X; S − ), associated to the connection A = A 0 + ia · f * (dθ) for any a ∈ R, has trivial kernel. We call such metric 'admissible metric'.Assumption 2.1. Throughout this paper, we fix a choice of admissible metric g X .Remark 2. By the Weitzenböck formula, any metric with positive scalar curvature is admissible. However, ...

A splitting theorem for the Seiberg-Witten invariant of a homology S1× S3

“…We add to this body of knowledge the following theorem, which was originally proved by the first-named author [33, Theorem 1.2] using different techniques. It was conjectured in [33,Remark 1] that there should exist a proof along the lines of this paper.…”

“…It was shown in [41] that λ SW pXq reduces modulo 2 to the Rohlin invariant ρpXq. As in [33], this fact leads to the corollary that, if X admits a metric of positive scalar curvature, any rational homology sphere Y carrying the generator of H 3 pX; Zq must satisfy the relation hpY, sq " ρpXq pmod 2q with respect to the induces spin structure s. For example, if a generator of H 3 pX; Zq is carried by the Brieskorn homology sphere Σp2, 3, 7q, then X cannot admit a positive scalar curvature metric.…”

“…To obtain estimate (32), simply combine the estimate C ¨t´1{2 ¨e´pu´vq 2 {4t on the former kernel with the estimate respectively. To obtain estimate (33), write the operator D z expp´tD ź D z q in the form du " pB{Bu ´ln z ¨f 1 q exp pt ¨pB{Bu ´ln z ¨f 1 q 2 q ‰ ¨expp´tD 2 q ´du ¨exp pt ¨pB{Bu ´ln z ¨f 1 q 2 q ¨"D expp´tD 2 q ‰ (34)…”

“…Let ǫ 0 ą 0 be any small number such that DpY, hq has no eigenvalues in the interval r´ǫ 0 , ǫ 0 s (the existence of ǫ 0 follows from Assumption 1). Then one can prove as in [33,Proposition 2.8] that there exist perturbations q satisfying the following assumption. Assumption 2.…”

Instantons for 4-manifolds with periodic ends and an obstruction to embeddings of 3-manifolds

“…There are two typical studies of gauge theory for 4-manifolds with periodic end by C.H.Taubes [13] and J.Lin [11]. They gave a sufficient condition to exist a natural compactification of the instanton and the Seiberg-Witten moduli spaces for such non-compact 4-manifolds.…”