Unfolded Seiberg–Witten Floer spectra, I :
Definition and invariance

Khandhawit, Tirasan; Lin, Jianfeng; Sasahira, Hirofumi

doi:10.2140/gt.2018.22.2027

Cited by 15 publications

(32 citation statements)

References 32 publications

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“…Because of that (and in applications concerning the Seiberg-Witten stable homotopy type, cf. [KLS18]), it is preferable to take the Coulomb slice as the model for the quotient by the identity component of the gauge group. Indeed, for Y a rational homology sphere the Hodge decomposition gives the L 2 -orthogonal decomposition…”

Section: Coulomb Slice On 3-manifoldsmentioning

confidence: 99%

“…Finally, we discuss the gauge slice used by Lipyanskiy [Lip08] and Khandhawit [Kha,KLS18]. They require that a ∈ Ω 1 CC (X) and that for each component Y i ⊂ ∂X we have Y i ι * (⋆a) = 0.…”

Section: The Other Direction Follows By Chasing Arrows In the Diagram...mentioning

confidence: 99%

“…Firstly, we choose a gauge twisting τ (Definition 3.24) and define the (S 1 -equivariant) twisted restriction map R τ : M • s (X, ŝ) → C cC (∂X, s), taking values in the configurations on ∂X in the Coulomb slice (Definition 3.2). The gauge splittings and gauge twistings we introduce generalize the double Coulomb slice used in [Lip08,Kha] and twistings utilized in [KLS18]. We prove regularity, denseness and "semi-infinite-dimensionality" of the Seiberg-Witten moduli spaces:…”

Section: Introductionmentioning

confidence: 99%

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Low-regularity Seiberg-Witten moduli spaces on manifolds with boundary

Suwara¹

2021

Preprint

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For a compact spin c manifold X with boundary b 1 (∂X) = 0, we consider moduli spaces of solutions to the Seiberg-Witten equations in a generalized double Coulomb slice in W 1,2 Sobolev regularity. We prove they are Hilbert manifolds, prove denseness and "semi-infinite-dimensionality" properties of the restriction to ∂X, and establish a gluing theorem.To achieve these, we prove a general regularity theorem and a strong unique continuation principle for Dirac operators, and smoothness of a restriction map to configurations of higher regularity on the interior, all of which are of independent interest.

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Section: Coulomb Slice On 3-manifoldsmentioning

confidence: 99%

Section: The Other Direction Follows By Chasing Arrows In the Diagram...mentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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Low-regularity Seiberg-Witten moduli spaces on manifolds with boundary

Suwara¹

2021

Preprint

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“…In [6], Cohen, Jones, and Segal proposed the problem of lifting Floer homology to a Floer spectrum or pro-spectrum, in the sense of stable homotopy theory. Since then, stable homotopy refinements of Floer homology have been constructed in Seiberg-Witten theory [19,12,36] and symplectic geometry [7,15,1]. In a similar vein, there is a lift of Khovanov homology to a stable homotopy type [18,17].…”

Section: Introductionmentioning

confidence: 99%

A knot Floer stable homotopy type

Manolescu,

Sarkar

2021

Preprint

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Given a grid diagram for a knot or link K in S 3 , we construct a spectrum whose homology is the knot Floer homology of K. We conjecture that the homotopy type of the spectrum is an invariant of K. Our construction does not use holomorphic geometry, but rather builds on the combinatorial definition of grid homology. We inductively define models for the moduli spaces of pseudo-holomorphic strips and disk bubbles, and patch them together into a framed flow category. The inductive step relies on the vanishing of an obstruction class that takes values in a complex of positive domains with partitions. Contents1. Introduction 2. Background 3. The complex of positive domains 4. The complex of positive domains with partitions 5. n -manifolds 6. Stratified spaces 7. Local models 8. Moduli spaces 9. The stratification 10. Embeddings and framings 11. The embedded framed cobordism group 12. Constructing the moduli spaces 13. Embedding and framing the permutohedra 14. The Cohen-Jones-Segal construction References

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“…Steps towards describing Floer homology in terms of a polarized manifold have been taken by Lipyanskiy [52].) Although the Cohen-Jones-Segal approach has been stymied by analytic difficulties, it has inspired other constructions of stable homotopy refinements of various Floer homologies and related invariants [1,6,7,12,13,16,24,[33][34][35][40][41][42]56,74].…”

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confidence: 99%

Spatial Refinements and Khovanov Homology

Lipshitz

Sarkar

2019

Proceedings of the International Congress of Mathematicians (ICM 2018)

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We review the construction and context of a stable homotopy refinement of Khovanov homology. IntroductionIn the late 1920s, in his study of critical points and geodesics [61][62][63], Morse introduced what is now called Morse theory-using functions for which the second derivative test does not fail (Morse functions) to decompose manifolds into simpler pieces. The finite-dimensional case was further developed by many authors (see [9] for a survey of the history), and an infinite-dimensional analogue introduced by Palais-Smale [65,66,80]. In both cases, a Morse function f on M leads to a chain complex C * (f ) generated by the critical points of f . This chain complex satisfies the fundamental theorem of Morse homology: its homology H * (f ) is isomorphic to the singular homology of M . This is both a feature and a drawback: it means that one can use information about the topology of M to deduce the existence of critical points of f , but also implies that C * (f ) does not see the smooth topology of M . (See [59,60] for an elegant account of the subject's foundations and some of its applications.)In the 1980s, Floer introduced several new examples of infinite-dimensional, Morse-like theories [21][22][23]. Unlike Palais-Smale's Morse theory, in which the descending manifolds of critical points are finite-dimensional, in Floer's setting both ascending and descending manifolds are infinite-dimensional. Also unlike Palais-Smale's setting, Floer's homology groups are not isomorphic to singular homology of the ambient space (though the singular homology acts on them). Indeed, most Floer (co)homology theories seem to have no intrinsic cup product operation, and so are unlikely to be the homology of any natural space.In the 1990s, Cohen-Jones-Segal [14] proposed that although Floer homology is not the homology of a space, it could be the homology of some associated 2010 Mathematics subject classification. 57M25,55P42

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Unfolded Seiberg–Witten Floer spectra, I : Definition and invariance

Abstract: Let Y be a closed and oriented 3-manifold. We define different versions of unfolded Seiberg-Witten Floer spectra for Y . These invariants generalize Manolescu's Seiberg-Witten Floer spectrum for rational homology 3-spheres. We also compute some examples when Y is a Seifert space.

Cited by 15 publications

References 32 publications

Low-regularity Seiberg-Witten moduli spaces on manifolds with boundary

Low-regularity Seiberg-Witten moduli spaces on manifolds with boundary

A knot Floer stable homotopy type

Spatial Refinements and Khovanov Homology

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