UC Berkeley

In this paper, we propose a general method of defining equivariant theories in symplectic geometry using polyfolds. The construction is twofold, one is for closed theories like equivariant Gromov-Witten theory, the other is for open theories like equivariant Floer cohomology.

When a compact Lie group $G$ acts on a tame strong polyfold bundle $p:W \to Z$, we construct a quotient polyfold bundle $\overline{p}:W/G \to Z/G$ if the $G$-action on $Z$ only has finite isotropy. For a general group action and if $Z$ has no boundary, then every $G$-equivariant sc-Fredholm section $s:Z\to W$ induces a $H^*(BG)$ module map $s_*: H^*_G(Z) \to H^{*-\ind s}(BG)$, which can be viewed as a generalization of the integration over the zero set $s^{-1}(0)$ when equivariant transversality holds. When $Z$ is the Gromov-Witten polyfold, $s_*$ yields a definition equivariant Gromov-Witten invariant for any symplectic manifold. We obtain a localization theorem for $s_*$ if there exist tubular neighborhoods around the fixed locus in the sense of polyfold. For open theories, we first obtain a construction for the Morse-Bott theories under minimal transversality requirement. We discuss the relations between different constructions of cochain complexes for Morse-Bott theory. We explain how homological perturbation theory is used in Morse-Bott cohomology, in particular, both our construction and the cascades construction can be interpreted in that way, In the presence of group actions, we construct cochain complexes for the equivariant theory. Expected properties like the independence of approximations of the classifying spaces and existence of action spectral sequences are proven. We carry out our construction for finite dimensional Morse-Bott cohomology using a generic metric and prove it recovers the regular cohomology. We outline the project of combining our construction with polyfold theory, which is expected to give a general construction for both Morse-Bott and equivariant Floer cohomology.

In the last part, we show that for any asymptotically dynamically convex contact manifold $Y$, the vanishing of symplectic homology $SH(W)=0$ is a property independent of the choice of topologically simple (i.e. $c_1(W)=0$ and $\pi_{1}(Y)\to \pi_1(W)$ is injective) Liouville filling $W$. As a consequence, we answer a question of Lazarev partially: a contact manifold $Y$ admitting flexible fillings determines the integral cohomology of all the topologically simple Liouville fillings of $Y$.