Künneth formulae and cross products for the symplectic Floer cohomology

For a symplectic monotone manifold (P,ω) and φ∈Symp0(P,ω), we define a Z-graded symplectic Floer cohomology (a local invariant) over integral coefficients. There is a spectral sequence which arises from a filtration on the Z-graded symplectic Floer cochain complex. The spectral sequence converges to the Z2N-graded symplectic Floer cohomology (a global invariant). We show that there are cross products on the Z-graded symplectic Floer cohomology and on the spectral sequence, hence on the usual Z2N-graded symplectic Floer cohomology. The Künneth formula for the Z-graded symplectic Floer cohomology is proved and similar results for the spectral sequence are obtained.