Polyfolds and Fredholm Theory

Hofer, Helmut

doi:10.1093/oso/9780198784913.003.0004

Cited by 11 publications

(15 citation statements)

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“…To make sense of virtual fundamental classes of generalized DT 4 moduli spaces as homology classes in M c 's, one would in general rely on Joyce's D-manifolds theory [29] or Fukaya-Oh-OhtaOno's theory of Kuranishi spaces [21] or Hofer's polyfolds theory [24] (see [57] for a comparison between them). Assuming this part, which is claimed by Borisov-Joyce, we have Theorem 2.6.…”

Section: Dt4 Cmentioning

confidence: 99%

Relative Donaldson-Thomas theory for Calabi-Yau 4-folds

Cao

Leung

2017

Trans. Amer. Math. Soc.

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Section: Dt4 Cmentioning

confidence: 99%

Relative Donaldson-Thomas theory for Calabi-Yau 4-folds

Cao

Leung

2017

Trans. Amer. Math. Soc.

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“…3 Remark 1.3.8. (The polyfold approach) If X is the zero set of a Fredholm section s of a polyfold bundle E → S of index d, then one can use the fact that the realization |S| supports partitions of unity to give a very simple construction for a weighted branched manifold M and section S whose corresponding relative Euler class agrees with that of s : S → E. (In the applications of interest to us S is a category 14 whose realization is a space of stable maps with the Gromov topology: see [H,HWZ2].) Here is a very brief outline of the construction: for full details see [MW4].…”

Section: Reductions and Zero Setsmentioning

confidence: 99%

“…There are many possible approaches to this problem, e.g. [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%

“…Further, if one works with polyfolds, then the proof can be radically simplified. Indeed, it is not difficult to define a smooth Kuranishi atlas on any space X that appears as the (compact) zero set of a polyfold bundle [HWZ1,H,Y,MW4]. Because the polyfolds of Gromov-Witten theory support sc-smooth partitions of unity, if the isotropy is trivial, one can even define such an atlas with just one chart.…”

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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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“…In full generality one needs an alternative approach. One possibility is the Polyfold Theory due to Hofer, Wysocki and Zehnder [Ho,HWZ1,HWZ2,HWZ3], which will provide the analytic background for such constructions. Our goal is to implement a finite-dimensional Morse homological approach to local contact homology at the chain level.…”

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Transversality for local Morse homology with symmetries and applications

2019

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We prove the transversality result necessary for defining local Morse chain complexes with finite cyclic group symmetry. Our arguments use special regularized distance functions constructed using classical covering lemmas, and an inductive perturbation process indexed by the strata of the isotropy set. A global existence theorem for symmetric Morse-Smale pairs is also proved. Regarding applications, we focus on Hamiltonian dynamics and rigorously establish a local contact homology package based on discrete action functionals. We prove a persistence theorem, analogous to the classical shifting lemma for geodesics, asserting that the iteration map is an isomorphism for good and admissible iterations. We also consider a Chas-Sullivan product on non-invariant local Morse homology, which plays the role of pair-of-pants product, and study its relationship to symplectically degenerate maxima. Finally, we explore how our invariants can be used to study bifurcation of critical points (and periodic points) under additional symmetries. Contents1 arXiv:1702.04609v3 [math.SG] 16 Sep 2018 1.2. Main results. In Section 1.2.1 we state our transversality results, which are used in Section 1.2.2 to define local invariant Morse homology. Applications to subharmonic invariants of isolated periodic points are described in Section 1.2.3,where the comparison to local contact homology is explained. Our Persistence Theorem is stated in Section 1.2.4, along with its non-symmetric version. Chas-Sullivan products and symplectically degenerate maxima are discussed in Section 1.2.5. In Section 1.2.6 we compute our invariants in three bidimensional bifurcation scenarios under Z 2 -symmetry (according to [DX], generically these are the only three cases in two dimensions). In Sections 1.2.7 and 1.2.8 we discuss basic global examples. and allow for the definition non-invariant local homology groupswhich are discrete versions of local Floer homology groups.Remark 1.7. The fact that we have the term N`2 in the symmetric inflation map I Z k N and N`1 in the non-symmetric one (and consequently the shift in the degree is 2nk for I Z k N and nk for I N ) is due to an orientation issue; see Remark 2.12 for details.Remark 1.8 (Gradings). The Conley-Zehnder index of a path M : r0, T s Ñ Spp2nq satisfying M p0q " I, det M pT q´I ‰ 0 is defined in a standard way, see for instance [SZ,RS1]. Its extension to general paths is not standard. Here we use its maximal lower semicontinuous extension. Namely, if det M pT q´I " 0 then its Conley-Zehnder index is defined as the lim inf in C 0 asM Ñ M of the Conley-Zehnder indices of pathsM : r0, T s Ñ Spp2nq satisfying detpM pT q´Iq ‰ 0. With these conventions, the Conley-Zehnder index of t P r0, ks Þ Ñ dϕ t H p0q will be denoted by CZpH, kq. This extension to degenerate paths is smaller than or equal to the one defined in [RS1], and in general disagrees with it; for instance the constant path equal to the identity matrix will have index´n according to our conventions, but will have index zero according to the conventi...

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Polyfolds and Fredholm Theory

Cited by 11 publications

References 45 publications

Relative Donaldson-Thomas theory for Calabi-Yau 4-folds

Relative Donaldson-Thomas theory for Calabi-Yau 4-folds

Constructing the virtual fundamental class of a Kuranishi atlas

Transversality for local Morse homology with symmetries and applications

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