Symplectic cohomology is a powerful invariant associated to (exact) symplectic manifolds, and it is useful in dynamics, mirror symmetry, and so on. Unfortunately, it is not very sensitive to the homotopy type of the underlying manifold: it can vanish for symplectic manifolds with arbitrarily complicated topology. It is known that, if one remembers certain structures on it, one can at least recover the rational homology of the manifold; however, the torsion information is lost. In this talk, we show how to recover further information about the homotopy type of the underlying symplectic manifold, including torsion part of its homology, complex K-theory and Morava K-theory from enhanced versions of symplectic cohomology and the structures on it, via a modified Tate construction. This is joint work with Laurent Cote.
