The fundamental class of smooth Kuranishi atlases with trivial isotropy

McDuff, Dusa; Wehrheim, Katrin

doi:10.1142/s1793525318500048

Cited by 37 publications

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“…In previous literature [1,3,59], it was stated that when transversality could not be achieved by perturbing the almost complex structure that the difficulty could still be resolved via a delicate virtual cycle technique involving multivalued perturbations. However, full details were never given and recent literature by [11,19,38,44,52] suggests that this procedure is even more delicate than previously indicated.…”

Section: Motivation and Resultsmentioning

confidence: 99%

Automatic transversality in contact homology I: regularity

Nelson

2015

Abh. Math. Semin. Univ. Hambg.

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This paper helps to clarify the status of cylindrical contact homology, a conjectured contact invariant introduced by Eliashberg, Givental, and Hofer in 2000. We explain how heuristic arguments fail to yield a well-defined homological invariant in the presence of multiply covered curves. We then introduce a large subclass of dynamically convex contact forms in dimension 3, termed dynamically separated, and demonstrate automatic transversality holds, thereby allowing us to define the desired chain complex. The Reeb orbits of dynamically separated contact forms satisfy a uniform growth condition on their ConleyZehnder index under iteration, typically up to large action; see Definition 1.15. These contact forms arise naturally as perturbations of Morse-Bott contact forms such as those associated to S 1 -bundles. In subsequent work, we give a direct proof of invariance for this subclass and, when further proportionality holds between the index and action, powerful geometric computations in a wide variety of examples.

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Section: Motivation and Resultsmentioning

confidence: 99%

Automatic transversality in contact homology I: regularity

Nelson

2015

Abh. Math. Semin. Univ. Hambg.

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“…3)) while M IJ will simply be defined as the image α IJ (M IJ ). As in [MW2], we use the notation φ IJ := ρ −1 IJ : V IJ → V IJ for the inverse of the atlas structural map ρ IJ .…”

Section: The Main Argumentsmentioning

confidence: 99%

“…if x ∈ V IJ ∩ V HJ for I H J then The last condition means that the composition rule holds directly, without having to introduce analogs of the paths P(e, x). Of course, the choice of the g I , ε I requires some attention to detail as in the proof of Lemma 3.1.11 below; see also the construction of the perturbation section in [MW2,§7.3]. Thus one begins with a family of shrinkings V κ < · · · < V 1 < V 0 of an initial reduction V 0 , where κ := max{|J| | J ∈ I K } and then chooses metrics g J on V |J| J , starting with J of length |J| = 1, that satisfy the above conditions for the submanifolds V |J| IJ of V |J| J for some constant ε I > 0.…”

Section: The Main Argumentsmentioning

confidence: 99%

“…[FO,FF,HWZ1,H]. In this note we develop the work of McDuff-Wehrheim [MW1,MW2,MW3] and Pardon [P] that uses atlases, in an attempt to clarify the passage from atlas to virtual fundamental class.…”

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confidence: 99%

“…The notion of topological atlas introduced in [MW1] is somewhat different; in particular the domains there need not be manifolds. For more details on all topics mentioned in this section, see the original papers [MW1,MW2,MW3] or [M2]. 5 For simplicity, we assume that Ei is even dimensional so that the orientation of a product of the Ei does not depend on their order.…”

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confidence: 99%

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Constructing the virtual fundamental class of a Kuranishi atlas

McDuff

2019

Algebr. Geom. Topol.

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Consider a space X, such as a compact space of J-holomorphic stable maps, that is the zero set of a Kuranishi atlas. This note explains how to define the virtual fundamental class of X by representing X via the zero set of a map SM : M → E, where E is a finite dimensional vector space and the domain M is an oriented, weighted branched topological manifold. Moreover, SM is equivariant under the action of the global isotropy group Γ on M and E. This tuple (M, E, Γ, SM ) together with a homeomorphism from S −1 M (0)/Γ to X forms a single finite dimensional model (or chart) for X. The construction assumes only that the atlas satisfies a topological version of the index condition that can be obtained from a standard, rather than a smooth, gluing theorem. However if X is presented as the zero set of an sc-Fredholm operator on a strong polyfold bundle, we outline a much more direct construction of the branched manifold M that uses an sc-smooth partition of unity.1 See [C1, C2] for a weak form of these requirements.CONSTRUCTING THE VIRTUAL FUNDAMENTAL CLASS OF A KURANISHI ATLAS 3 axiom. Instead of gluing the chart domains together to form a topological space |K|, Pardon works with K-homotopy sheaves of (co)chain complexes defined on homotopy colimits of spaces that are obtained from the chart domains. This gives a flexible way of assembling local homological information into a global object. Though this approach may be useful in many contexts, it is hard for a nonexpert in sheaf theory to understand where the technical difficulties are, and what actually has to be checked to ensure that the method works in any particular case. This becomes an issue if one wants to extend the method to cases (such as Hamiltonian Floer theory, or symplectic field theory) in which one must deal with a family of related moduli spaces and so should work on the chain level. The current paper was prompted by the desire to develop a different approach, that would replace Pardon's sophisticated sheaf theory by more elementary arguments that yet do not require smoothness.This note only considers the simplest case, appropriate to Gromov-Witten theory, in which the aim is to construct a homology class [X] vir K ∈Ȟ d (X; Q). Working with Pardon's submersion axiom, we define a consistent thickening of the domains of the atlas charts to make them all have the same dimension. In the case with trivial isotropy, one thereby constructs an oriented topological manifold M of dimension D := d + dim E A , together with a map S M : M → E A whose zero set can be identified with X. If the isotropy is nontrivial, M is a branched manifold with a weighting function Λ and a global action of the total isotropy group Γ A , and there is a homeomorphismis the union of two circles, each of weight 1 2 , identified along a closed subarc A, so that the points x ∈ A have weight Λ(x) = 1, while the others all have weight Λ(x) = 1 2 . See also §1.4.) Here is the first main result. (See Theorem 1.3.4 for a more precise statement.)Theorem A: Let K be a d-dimensional Kuranishi atl...

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Floer Field Philosophy

Wehrheim

2016

Association for Women in Mathematics Series

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Floer field theory is a construction principle for e.g. 3-manifold invariants via decomposition in a bordism category and a functor to the symplectic category, and is conjectured to have natural 4-dimensional extensions. This survey provides an introduction to the categorical language for the construction and extension principles and provides the basic intuition for two gauge theoretic examples which conceptually frame Atiyah-Floer type conjectures in Donaldson theory as well as the relations of Heegaard Floer homology to Seiberg-Witten theory.Contents 1 A smooth four manifold can be thought of as a curved 4-dimensional space-time. ASD (anti-selfdual) instantons in this space-time satisfy a reduction of Maxwell's equations for the electro-magnetic potential in vacuum, which has an infinite dimensional gauge symmetry.

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The fundamental class of smooth Kuranishi atlases with trivial isotropy

Cited by 37 publications

References 39 publications

Automatic transversality in contact homology I: regularity

Automatic transversality in contact homology I: regularity

Constructing the virtual fundamental class of a Kuranishi atlas

Floer Field Philosophy

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