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

Equivariant Floer homotopy via Morse-Bott theory