Equivariant Floer homotopy via Morse-Bott theory

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.