2023/11/28 by Roman Krutowski, Tianyu Yuan, Krutowski, Roman +1 · 1 citation
Mathematics · Medicine · #53D40 (Primary) 57R17 (Secondary) #Botulinum Toxin and Related Neurological Disorders #FOS: Mathematics #Geometric and Algebraic Topology #Homotopy and Cohomology in Algebraic Topology #Symplectic Geometry (math.SG)
paper · pdf · doi:10.48550/arxiv.2311.17031
openalex publication_date 2023/11/28 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
Given a Liouville manifold M, we introduce an invariant of M that we call the Heegaard Floer symplectic cohomology SH^*κ(M) for any κ≥ 1 that coincides with the symplectic cohomology for κ=1. Writing M for the completion of M, the differential counts pseudoholomorphic curves of arbitrary genus in ℝ × S1 × M that are required to be branched κ-sheeted covers when projected to the ℝ × S1-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 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^*κ(T^*Q) are isomorphic to the cohomology groups of the free multiloop complex of Q.