Seiberg-Witten Equations, End-Periodic Dirac Operators, and a Lift of Rohlin's Invariant

Mrowka, Tomasz; Ruberman, Daniel; Saveliev, Nikolai

doi:10.4310/jdg/1320067650

Cited by 35 publications

(121 citation statements)

References 47 publications

(97 reference statements)

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“…Similar moduli spaces and Casson-type invariants have been defined by Ruberman and Saveliev [25] for Z[Z]-homology Hopf surfaces, going back to work of Furuta and Ohta [16], and for Z[Z]-homology 4-tori [24]. The first of these is also related in a non-trivial way to Seiberg-Witten theory [21] by work of Mrowka, Ruberman and Saveliev. In particular, these invariants are in general non-vanishing.…”

Section: Introductionmentioning

confidence: 61%

“…Furthermore, the holonomy sections Hol qi (A) on B i continue to span gl(E) in a neighbourhood of x, because l was chosen so that we have a Sobolev inclusion L 2 l → C 0 , and so the holonmy sections are continuous. Multiplying the finite number of holonomy sections Hol qi (A) with convenient one-forms β i ∈ Ω 1 (B i ; C) supported in small enough neighbourhoods of x, we obtain a perturbation β = (β i ) ∈ V , where all but these finite number of one-forms are zero, so that equation (21) implies that Σ is zero in a neighbourhood U of x. As we also have D − A,α Σ = 0, the unique continuation principle [1] for solutions to the perturbed Dirac equation implies that Σ = 0 on the whole of X.…”

Section: Transversality In the Main Stratummentioning

confidence: 99%

“…The first of these is also related in a non-trivial way to Seiberg-Witten theory [22] by work of Mrowka, Ruberman and Saveliev. In particular, these invariants are in general non-vanishing.…”

Section: Introductionmentioning

confidence: 99%

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A vanishing result for a Casson-type instanton invariant

Zentner

2014

Proceedings of the London Mathematical Society

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Abstract. Casson-type invariants emerging from Donaldson theory over certain negative definite 4-manifolds were recently suggested by Andrei Teleman. These are defined by an algebraic count of points in a zero-dimensional moduli space of flat instantons. Motivated by the cobordism program of proving Witten's conjecture, we use a moduli space of P U (2) Seiberg-Witten monopoles to exhibit an oriented one-dimensional cobordism of the instanton moduli space to the empty space. The Casson-type invariant must therefore vanish.

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Section: Introductionmentioning

confidence: 61%

Section: Transversality In the Main Stratummentioning

confidence: 99%

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A vanishing result for a Casson-type instanton invariant

Zentner

2014

Proceedings of the London Mathematical Society

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“…• the 4-dimensional Casson-type invariant λ SW (X), defined by Mrowka-Ruberman-Saveliev [10]; • the Frøyshov invariant h(Y, s), defined by Frøyshov [2], where s is the unique spin structure on Y that can be extended to a spin structure on X. (It was proved in [2] that this invariant does not depend on the choice of Y .…”

Section: Introductionmentioning

confidence: 99%

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

Lin

2019

Journal of Topology

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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, ...

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“…(We call Y a cross-section of X.) This invariant has interpretations in terms of Seiberg-Witten theory [21] and conjecturally in terms of Yang-Mills theory [34,35,37,36]. More recently, Frøyshov [5] observed that if X has a cross-section Y that is a rational homology 3-sphere, the invariant h(Y, s X ) ∈ Q associated to the unique Spin c structure s X on Y induced from X, which is defined using monopole Floer homology, is also a smooth invariant of X. Frøyshov's argument uses only the rational homology cobordism invariance property of h(Y, s X ), so it applies verbatim to the version of h(Y, s X ) defined by Kronheimer and Mrowka [14, §39.1] (presumed, but not known, to be equal to Frøyshov's) and the similarly defined Heegaard Floer correction term d(Y, s X ) [25].…”

Section: Introductionmentioning

confidence: 99%

Heegaard Floer invariants in codimension one

Levine

Ruberman

2018

Trans. Amer. Math. Soc.

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Abstract. Using Heegaard Floer homology, we construct a numerical invariant for any smooth, oriented 4-manifold X with the homology of S 1 × S 3 . Specifically, we show that for any smoothly embedded 3-manifold Y representing a generator of H 3 (X), a suitable version of the Heegaard Floer d invariant of Y , defined using twisted coefficients, is a diffeomorphism invariant of X. We show how this invariant can be used to obstruct embeddings of certain types of 3-manifolds, including those obtained as a connected sum of a rational homology 3-sphere and any number of copies of S 1 × S 2 . We also give similar obstructions to embeddings in certain open 4-manifolds, including exotic R 4 s.

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Seiberg-Witten Equations, End-Periodic Dirac Operators, and a Lift of Rohlin's Invariant

Cited by 35 publications

References 47 publications

A vanishing result for a Casson-type instanton invariant

A vanishing result for a Casson-type instanton invariant

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

Heegaard Floer invariants in codimension one

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