A flexible construction of equivariant Floer homology and applications

Hendricks, Kristen; Lipshitz, Robert; Sarkar, Santanu

doi:10.1112/jtopol/jtw022

Cited by 46 publications

(87 citation statements)

References 76 publications

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“…In the case that G = Z/2Z, Floer homology coupled with Morse homology on EG is used in [SS10] by Seidel and Smith to define equivariant Lagrangian Floer homology . More recently, Hendricks, Lipshitz and Sarkar employed homotopy theoretic methods to define Lagrangian Floer homology in the presence of the action of a Lie group [HLS16b,HLS16a]. There are also various other equivariant theories for other Floer homologies (see, for example, [Don02, KM07, AB96]).…”

Section: Equivariant Lagrangian Floer Homologymentioning

confidence: 99%

Atiyah-Floer conjecture: A formulation, a strategy of proof and generalizations

Daemi¹,

Fukaya²

2018

Proceedings of Symposia in Pure Mathematics

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Around 1988, Floer introduced two important theories: instanton Floer homology as invariants of 3-manifolds and Lagrangian Floer homology as invariants of pairs of Lagrangians in symplectic manifolds. Soon after that, Atiyah conjectured that the two theories should be related to each other and Lagrangian Floer homology of certain Lagrangians in the moduli space of flat connections on Riemann surfaces should recover instanton Floer homology. However, the space of flat connections on a Riemann surface is singular and the first step to address this conjecture is to make sense of Lagrangian Floer homology on this space. In this note, we formulate a possible approach to resolve this issue. A strategy to construct the desired isomorphism in the Atiyah-Floer conjecture is also sketched. We also use the language of A∞categories to state generalizations of the Atiyah-Floer conjecture.

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Section: Equivariant Lagrangian Floer Homologymentioning

confidence: 99%

Atiyah-Floer conjecture: A formulation, a strategy of proof and generalizations

Daemi¹,

Fukaya²

2018

Proceedings of Symposia in Pure Mathematics

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“…At several points in Section 7 of our paper we assert that an almost complex structure

j

on the algebraic surface

S

used to define symplectic Khovanov homology induces an almost complex structure

{prefixHilb}^{n} false(j false)

on the Hilbert scheme (or Douady space, following )

{prefixHilb}^{n} false(S false)

of length

n

subschemes of

S

. If

j

is a complex structure, this is true, but there is no known extension of the Hilbert scheme of points in a complex manifold to the almost‐complex case.…”

Section: The Mistakementioning

confidence: 99%

“…This (false) principle is used in a ‘cylindrical’ formulation of symplectic Khovanov homology in [, Lemma 7.10], which is then used in the proof of stabilization invariance for symplectic Khovanov homology in [, Section 7.4.1], equivariant symplectic Khovanov homology in [, Theorem 1.26], and reduced symplectic Khovanov homology in [, Theorem 7.25]. (See also Abouzaid‐Smith's paper for a more careful cylindrical reformulation of the curves in symplectic Khovanov homology in certain cases, and Mak‐Smith's recent paper for a more general cylindrical reformulation.…”

Section: The Mistakementioning

confidence: 99%

Corrigendum: A flexible construction of equivariant Floer homology and applications

Hendricks

Lipshitz

Sarkar

2020

Journal of Topology

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We correct a mistake regarding almost complex structures on Hilbert schemes of points in surfaces in Hendricks, Lipshitz and Sarkar (J. Topol. 9 (2016) 1153–1236). The error does not affect the main results of the paper, and only affects the proofs of invariance of equivariant symplectic Khovanov homology and reduced symplectic Khovanov homology. We give an alternate proof of the invariance of equivariant symplectic Khovanov homology.

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“…8. The (symplectic) Khovanov complex admits, in some sense, an O(2)-action [25,57,73,78]. Does the Khovanov stable homotopy type?…”

Section: Speculationmentioning

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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A flexible construction of equivariant Floer homology and applications

Cited by 46 publications

References 76 publications

Atiyah-Floer conjecture: A formulation, a strategy of proof and generalizations

Atiyah-Floer conjecture: A formulation, a strategy of proof and generalizations

Corrigendum: A flexible construction of equivariant Floer homology and applications

Spatial Refinements and Khovanov Homology

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