Symplectic Floer-Donaldson theory and quantum cohomology. (English) Zbl 0874.53031

Piunikhin, S.; Salamon, D.; Schwarz, M.

Thomas, C. B. (ed.), Contact and symplectic geometry. Cambridge: Cambridge University Press. Publ. Newton Inst. 8, 171-200 (1996).

For semi-positive symplectic manifolds the Floer cohomology groups are naturally isomorphic to the ordinary cohomology groups. The paper outlines a new proof for this fact and shows that the isomorphism intertwines the quantum cup product in ordinary cohomology with the “pair-of-pants” product in Floer cohomology. The new proof uses a glueing theorem for J-holomorphic curves. The article provides motivations and sketches the proofs. Technical details are promised to appear elsewhere.
For the entire collection see [Zbl 0852.00028].

Reviewer: C.Günther (Libby)

MSC:

53C15	General geometric structures on manifolds (almost complex, almost product structures, etc.)
57R19	Algebraic topology on manifolds and differential topology
58A14	Hodge theory in global analysis
14F40	de Rham cohomology and algebraic geometry

Keywords:

symplectic cohomology; Floer homology; quantum cup product; quantum cohomology