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

Lin, Jianfeng; Ruberman, Daniel; Saveliev, Nikolai

doi:10.2140/gt.2018.22.2865

Cited by 23 publications

(36 citation statements)

References 49 publications

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“…In light of Manolescu's result, it is also natural to ask this in the more restricted setting of having Rokhlin invariant one, in which case there is no 2-torsion. For further results in this direction, see work of Saveliev [24] and Lin-Ruberman-Saveliev [14].…”

Section: Overview and Motivationmentioning

confidence: 96%

On homology cobordism and local equivalence between plumbed manifolds

Dai

Stoffregen

2019

Geom. Topol.

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We establish a structural understanding of the involutive Heegaard Floer homology for all linear combinations of almost-rational (AR) plumbed three-manifolds. We use this to show that the Neumann-Siebenmann invariant is a homology cobordism invariant for all linear combinations of AR plumbed homology spheres. As a corollary, we prove that if Y is a linear combination of AR plumbed homology spheres with µpY q " 1, then Y is not torsion in the homology cobordism group. A general computation of the involutive Heegaard Floer correction terms for these spaces is also included.2010 Mathematics Subject Classification. 57R58, 57M27.

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Section: Overview and Motivationmentioning

confidence: 96%

On homology cobordism and local equivalence between plumbed manifolds

Dai

Stoffregen

2019

Geom. Topol.

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“…Thus all eigenvalues are real and non-negative. The proof of [18,Proposition 7.1] then proceed to show that spinors corresponding to small eigenvalues λ 2 of D − z (X T )D + z (X T ) are glued by spinors in the kernels of (D…”

Section: )mentioning

confidence: 99%

“…Following the construction in [1, p. 533], we get a family of metrics g ρ , ρ ∈ (0, r 0 ) for some small r 0 > 0, on the surgered manifold M ∞ by modifying g| N near a neighborhood of the surgery spheres. As explained in the proof of [18,Theorem 10.3], the proof of [1, Theorem 1.2] can be modified to this non-compact case so that for each fixed z 0 ∈ S 1 one can find ρ z0 > 0 such that for all ρ < ρ z0 we have…”

Section: )mentioning

confidence: 99%

“…The formula (1.7) is proved by Lin-Ruberman-Saveliev [19] in the case when Y n is a rational homology sphere. Combining with their splitting formula [18], one is able to obtain the Lefschetz number on the reduced monopole Floer homology HM red (Y n ) when Y n is a rational homology sphere. On the other hand, when Y n is not a rational homology sphere, (1.7) provides us with the computation for λ SW with X admitting generating hypersurfaces of non-vanishing first Betti number, which are the first known examples of this sort.…”

Section: Introductionmentioning

confidence: 99%

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A surgery formula for the Casson–Seiberg–Witten invariant of integral homology S1×S3

2021

Journal of Topology

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We prove a surgery formula of the Casson-Seiberg-Witten invariant of integral homology S 1 × S 3 along an embedded torus, which could either be regarded as an extension of the product formula for Seiberg-Witten invariants or a manifestation of the surgery exact triangle in fourdimensional Seiberg-Witten theory of homology S 1 × S 3 . As an application, we compute this invariant for mapping tori of 3-manifolds under diffeomorphisms of finite-order and fixed-point set being a simple closed curve. This computation generalizes the result of Lin-Ruberman-Saveliev in [On the monopole lefschetz number of finite order diffeomorphisms, Preprint, 2020].

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“…Note that the number − ind C D q − sign(M ) 8 always belongs to m(Y, s). Moreover, as we varies {c n }, this number changes by the spectral flow of / D q ,B0 .…”

Section: Consider the Operatormentioning

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

Cited by 23 publications

References 49 publications

On homology cobordism and local equivalence between plumbed manifolds

On homology cobordism and local equivalence between plumbed manifolds

A surgery formula for the Casson–Seiberg–Witten invariant of integral homology S1×S3

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

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