A Morse-Bott approach to the equivariant homotopy type

Morse theory provides an effective way to calculate the homology of smooth manifolds, in terms of critical points of a function and its gradient flow. Floer applied this idea in the infinite dimensional setting to produce new invariants in symplectic and low dimensional topology, and motivated by this, Cohen, Jones and Segal has shown how to obtain finer information about the topology of a smooth manifold from the Morse theory, thus providing a framework for refining Floer's invariant too. However, neither Morse theory nor the framework of Cohen-Jones-Segal are compatible with the compact group actions on the underlying manifold. In this talk, I will explain how to define a new framework for Morse-Bott functions in order to extract information about the equivariant stable homotopy type. In the remaining time, I will discuss applications to equivariant Floer theory. Joint work with Laurent Coté.