Bott-type symplectic Floer cohomology and its multiplication structures (Q1902215)

The main purpose of this paper is to give a brief description of the construction (\S 2) of the Bott-type symplectic Floer cohomology and its multiplication structures (\S\S 3,4,5) on semipositive symplectic manifolds with details of proofs to appear in a later paper. The ordinary (resp. Bott-type) symplectic Floer cohomology is defined for a Hamiltonian symplectomorphism whose fixed points are nondegenerate (resp. whose fixed point set \(F\) consists of appropriate nondegenerate submanifolds). It should be noted that almost all the analysis in the ordinary case fails in the Bott-type situation. Topologically the Bott-type Floer cohomology is defined as the cohomology of a chain complex consisting of all geometric chains in the fixed point set \(F\). To define the boundary operator for such a complex involves the most difficult and technical work (\S 2) of the paper; namely, to show that the compactification of the moduli space of trajectories is a simplicial complex. Section 3 deals with the generalization of Floer's construction of ``extrinsic multiplication'' to the Bott-type case and section 4 deals with the ``intrinsic multiplication'' defined in terms of pseudo-holomorphic maps of pair-pants. In both cases the multiplication is shown to be independent of the Hamiltonian and, as a consequence, is equivalent to the quantum multiplication. In the last section of the paper the quantum Massey product is defined in terms of the Bott-type Floer cohomology. When the Hamiltonian is nondegenerate it gives the Massey product for the ordinary Floer cohomology.