  We associate an invariant called the completed Tate cohomology to a filtered
circle equivariant spectrum and a complex oriented cohomology theory. We show
that when the filtered spectrum is the spectral symplectic cohomology of a
Liouville manifold, this invariant depends only on the stable homotopy type of
the underlying manifold. We make explicit computations for several complex
oriented cohomology theories, including Eilenberg-Maclane spectra, Morava
K-theories, their integral counterparts, and complex K-theory. We show that the
result for Eilenberg-Maclane spectra depends only on the rational homology, and
we use the computations for Morava K-theory to recover the integral homology
(as an ungraded group). In a different direction, we use the completed Tate
cohomology computations for the complex K-theory to recover the complex
K-theory of the underlying manifold from its equivariant filtered Floer
homotopy type. A key Floer theoretic input is the computation of local
equivariant Floer theory near the orbit of an autonomous Hamiltonian, which may
be of independent interest from the perspective of dynamics.
