Quotient theorems in polyfold theory and S1‐equivariant transversality

Abstract

We introduce group actions on polyfolds and polyfold bundles. We prove quotient theorems for polyfolds, when the group action has finite isotropy. We prove that the sc-Fredholm property is preserved under quotient if the base polyfold is infinite dimensional. The quotient construction is the main technical tool in the construction of equivariant fundamental class in our future work. We also analyze the equivariant transversality near the fixed locus in the polyfold setting. In the case of S 1-action with fixed locus, we give a sufficient condition for the existence of equivariant transverse perturbations. We outline the application to Hamiltonian-Floer cohomology and a proof of the weak Arnold conjecture for general symplectic manifolds, assuming the existence of Hamiltonian-Floer cohomology polyfolds. Contents

A polyfold proof of the Arnold conjecture

“…The second chain homotopy is as claimed in Theorem 1.3 (iii) but with ι = id, which requires arguing that the only isolated holomorphic spheres with two marked points evaluating to an unstable and stable manifold are constant. This again requires S 1equivariant transversality (which we expect to be able to achieve with the techniques in [39]). (iii) Theorem 1.3 is proven by following the [30]-approach as above but avoiding the use of new polyfold technology such as equivariant or strata-avoiding perturbations.…”

“…This requires S 1 -equivariant transversality to argue that isolated Floer trajectories must be S 1 -invariant, hence Morse trajectories. A conceptually transparent construction of equivariant and transverse perturbationsunder transversality assumptions at the fixed point set which are met in this settingcan be found in [39], assuming a polyfold description of Floer trajectories. (ii) The second approach to Theorem 1.1 by [30] is to construct a direct isomorphism between the Floer homology of the given Hamiltonian and the Morse homology for some unrelated Morse function.…”

Polyfold regularization of constrained moduli spaces

“…We note some additional properties of the objects constructed in the proof of Lemma 4.2. These properties are required for the construction of quotients of polyfolds by group actions in [24]. The sc-Banach space K is given by…”

“…The tame strong bundle charts on a tame strong bundle ρ : E → B induce a double filtration E m,k for m ≥ 0 and 0 ≤ k ≤ m + 1 from the double filtration (24) on the bundle retracts in the charts. From this we distinguish the M -polyfolds E[i] for i = 0, 1, by the filtrations (54)…”

A landscape of contact manifolds via rational SFT

“…We use it to obtain automatic closedness of a chain in the computation of planarity for any augmentation. In Remark 7.20, we explain how one can drop this condition using polyfold techniques in [82]. On the other hand, the role of k < 3n−1 2 is more mysterious.…”