Floer homology and covering spaces

Lidman, Tye; Manolescu, Ciprian

doi:10.2140/gt.2018.22.2817

Cited by 15 publications

(13 citation statements)

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“…These inequalities extend the results of Hendricks [10] and Lipshitz-Treumann [14], and are similar if somewhat orthogonal to results for rational homology spheres obtained by Lidman-Manolescu [13] using the Seiberg-Witten-Floer stable homotopy type. Our method essentially follows that of Hendricks, who proved the inequality in the case of knots in Y = S 3 .…”

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confidence: 85%

Equivariant Floer theory and double covers of three-manifolds

Large

2019

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Inspired by Kronheimer and Mrowka's approach to monopole Floer homology, we develop a model for Z/2-equivariant symplectic Floer theory using equivariant almost complex structures, which admits a localization map to a twisted version of Floer cohomology in the invariant set. We then present applications to Steenrod operations on Lagrangian Floer cohomology, and to the Heegaard Floer homology of double covers of three-manifolds.

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confidence: 85%

Equivariant Floer theory and double covers of three-manifolds

Large

2019

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“…In [38,Remark 3.1], the authors define equivariant Seiberg-Witten-Floer homology in the special case that G acts freely on Y . Their construction coincides with ours in such cases.…”

Section: 3mentioning

confidence: 99%

Equivariant Seiberg-Witten-Floer cohomology

Baraglia¹,

Hekmati²

2021

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We develop an equivariant version of Seiberg-Witten-Floer cohomology for finite group actions on rational homology 3-spheres. Our construction is based on an equivariant version of the Seiberg-Witten-Floer stable homotopy type, as constructed by Manolescu. We use these equivariant cohomology groups to define a series of d-invariants d G,c (Y, s) which are indexed by the group cohomology of G. These invariants satisfy a Frøyshov-type inequality under equivariant cobordisms. Lastly we consider a variety of applications of these d-invariants: concordance invariants of knots via branched covers, obstructions to extending group actions over bounding 4-manifolds, Nielsen realisation problems for 4-manifolds with boundary and obstructions to equivariant embeddings of 3-manifolds in 4-manifolds. DAVID BARAGLIA AND PEDRAM HEKMATIHence we obtain a canonical isomorphismThis shows that the Seiberg-Witten-Floer cohomology HSW * (Y, s) does not depend on the choice of metric g. By working in an appropriately defined S 1 -equivariant Spanier-Whitehead category in which suspension by fractional amounts of C is allowed, Manolescu defined the Seiberg-Witten-Floer homotopy type of (Y, s) to beThis is independent of the choice of g by (1.1) and (1.2). Now suppose that a finite group G acts on Y by orientation preserving diffeomorphisms which preserve the isomorphism class of s. Let g be a G-invariant metric on Y . Lifting the action of G to the associated spinor bundle determines an S 1 extensionManolescu's construction of the stable homotopy type SW F (Y, s, g) can be carried out G s -equivariantly, so that SW F (Y, s, g) may be promoted to a G sequivariant stable homotopy type. This is analogous to the construction in [42] of the P in(2)-equivariant Seiberg-Witten-Floer stable homotopy type of (Y, s) where s is a spin-structure on Y . The main difference is that in our construction, the additional symmetries that comprise the group G s come from symmetries of Y rather than internal symmetries of the Seiberg-Witten equations.We define the G-equivariant Seiberg-Witten-Floer cohomology of (Y, s) to beThe right hand side is independent of the choice of metric g by much the same argument as in the S 1 -equivariant case. We make some remarks concerning this construction.(1) In this paper we have chosen to work throughout with cohomology instead of homology. This is simply a matter of preference and we could just as well work with Seiberg-Witten-Floer homology groups. (2) Instead of Borel equivariant cohomology, we could take co-Borel cohomology or Tate cohomology, which correspond to the different versions of Heegaard-Floer cohomology, [39, Corollary 1.2.4]. (3) In a similar fashion we can also define the G-equivariant Seiberg-Witten-Floer K-theory KSW * G (Y, s) = K * +2n(Y,s,g) Gs (SW F (Y, s, g)).More generally we could use any generalised equivariant cohomology theory in which the Thom isomorphism holds. (4) We have not attempted to construct a metric independent G s -equivariant stable homotopy type. To do this one would need to sp...

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“…The author's main goal, after arming himself with the cohomotopical invariant, was to study covering spaces. The avoidance of generic perturbations is important in this context, as demonstrated in Lidman & Manolescu (2018), due to the impossibility of producing sufficiently generic equivariant perturbations. The author's cohomotopical contact invariant can also be made G-equivariant for G the group of deck transformations of a finite regular covering.…”

Section: Introductionmentioning

confidence: 99%

Seiberg-Witten Floer Spectra and Contact Structures

Bruno¹

2022

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In this article, the author defines an invariant of rational homology 3-spheres equipped with a contact structure as an element of a cohomotopy set of the Seiberg-Witten Floer spectrum as defined in Manolescu (2003). Furthermore, in light of the equivalence established in Lidman & Manolescu (2016) between the Borel equivariant homology of said spectrum and the Seiberg-Witten Floer homology of , the author shall show that this homotopy theoretic invariant recovers the already well known contact element in the Seiberg-Witten Floer cohomology (vid. e.g. Kronheimer, Mrowka, Ozsváth & Szabó 2007) in a natural fashion. Next, the behaviour of the cohomotopy invariant is considered in the presence of a finite covering. This setting naturally asks for the use of Borel cohomology equivariant with respect to the group of Deck transformations. Hence, a new equivariant contact invariant is defined and its properties studied. The invariant is then computed in concrete examples. The technique leads to new ways to obtain already known results and also opens the possibility of considering scenarios hitherto inaccessible, one of which is studied in detail in this article. AcknowledgementsThe author is greatly indebted to his advisor, S. Sivek, for the invaluable guidance throughout his doctoral studies and the sage commentary during the writing of the present article. The author would also like to thank F. Lin and L. Nicolaescu for very helpful email correspondences. This work was supported by the Engineering and Physical Sciences Research Council. admits a canonical solution, C λ , which, provided a certain parameter r > 0 be made large enough, is nicely behaved in two fundamental ways. It is non-degenerate irrespective of any genericity requirements, and it is not the forward time limit of any Seiberg-Witten trajectory. These two properties shall be instrumental later in the present article. The solution C λ is essentially defined in a manner that make its spinor component bounded away from zero; a feature which is unique to this solution provided r be made large enough.In what follows, agree to fix a metric g on Y with the property that λ ∧ dλ = Vol g . Use ξ := Ker λ. Fix a complex structure J ∈ End(ξ) on the bundle ξ compatible with the symplectic structure on ξ induced by the dλ| ξ . Use R ∈ ΓTY to denote the Reeb vector field field; that is, the vector field satisfying ι R dλ = 0, ι R λ = 1. There is canonical Spin C structure on Y defined via the specification of its spinor representation bundle in the following way.Definition 2.1: Define the spinor bundlewith Clifford multiplication cℓ : TY → End C (S λ ) defined so as to satisfy, for α ∈ Λ 0,q ξ * andRemark 2.2: The notation (X → X * ) : TY ⊗ C → T * Y ⊗ C denotes the C-antilinear isomorphism induced by the metric.Remark 2.3: Note that this fully determines the Clifford action due to the fact that TY = R R ⊕ ξ. This map can be checked to indeed define an irreducible Clifford module; vid. Petit (2005) for a proof and more details about this matter.Definition 2.4: Denote t...

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Floer homology and covering spaces

Cited by 15 publications

References 44 publications

Equivariant Floer theory and double covers of three-manifolds

Equivariant Floer theory and double covers of three-manifolds

Equivariant Seiberg-Witten-Floer cohomology

Seiberg-Witten Floer Spectra and Contact Structures

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