  We generalize the Cohen-Jones-Segal construction to the Morse-Bott setting.
In other words, we define framings for Morse-Bott analogues of flow categories
and associate a stable homotopy type to this data. We use this to recover the
stable homotopy type of a closed manifold from Morse-Bott theory, and the
stable equivariant homotopy type of a closed manifold with the action of a
compact Lie group from Morse theory.
  We use this machinery in Floer theory to construct a genuine circle
equivariant model for symplectic cohomology with coefficients in the sphere
spectrum. Using the formalism of relative modules, we define equivariant maps
to (Thom spectra over) the free loop space of exact, compact Lagrangians. We
prove that this map is an equivalence of Borel equivariant spectra when the
Lagrangian is the zero section of a cotangent bundle -- an equivariant Viterbo
isomorphism theorem over the sphere spectrum.
