1999/05/14 by Dusa McDuff, McDuff, Dusa
Mathematics · #53 C 15 #Algebraic structures and combinatorial models #Differential Geometry (math.DG) #FOS: Mathematics #Geometric and Algebraic Topology #Homotopy and Cohomology in Algebraic Topology #Symplectic Geometry (math.SG) #math.DG #math.SG #msc:15 #msc:53
paper · pdf · doi:10.48550/arxiv.math/9905092
53 pages, Latex document
arxiv created 1999/05/14 · openalex publication_date 1999/05/14 · arxiv updated 2009/11/30 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
This paper studies the (small) quantum homology and cohomology of fibrations p: P→ S2 whose structural group is the group of Hamiltonian symplectomorphisms of the fiber (M,\om). It gives a proof that the rational cohomology splits additively as the vector space tensor product H^*(M)⊗ H^*(S2), and investigates conditions under which the ring structure also splits, thus generalizing work of Lalonde-McDuff-Polterovich and Seidel. The main tool is a study of certain operations in the quantum homology of the total space P and of the fiber M, whose properties reflect the relations between the Gromov-Witten invariants of P and M. In order to establish these properties we further develop the language introduced in [Mc3] to describe the virtual moduli cycle (defined by Liu-Tian, Fukaya-Ono, Li-Tian, Ruan and Siebert).