Heegaard Floer Symplectic homology and Viterbo's isomorphism theorem in the context of multiple particles

Abstract

Given a Liouville manifold $M$, we introduce an invariant of $M$ that we call the Heegaard Floer symplectic cohomology $SH^*_\kappa(M)$ for any $\kappa \ge 1$ that coincides with the symplectic cohomology for $\kappa=1$. Writing $\hat{M}$ for the completion of $M$, the differential counts pseudoholomorphic curves of arbitrary genus in $\mathbb{R} \times S^1 \times \hat{M}$ that are required to be branched $\kappa$-sheeted covers when projected to the $\mathbb{R} \times S^1$-direction; this resembles the cylindrical reformulation of Heegaard Floer homology by Lipshitz. These cohomology groups provide a closed-string analogue of higher-dimensional Heegaard Floer homology introduced by Colin, Honda, and Tian. When $\hat{M}=T^*Q$ with $Q$ an orientable manifold, we introduce a Morse-theoretic analogue of Heegaard Floer symplectic cohomology, which we call the free multiloop complex of $Q$. When $Q$ has vanishing relative second Stiefel-Whitney class, we prove a generalized version of Viterbo's isomorphism theorem by showing that the cohomology groups $SH^*_\kappa(T^*Q)$ are isomorphic to the cohomology groups of the free multiloop complex of $Q$.