Floer homotopy type refines the Floer homology by associating a spectrum to an Hamiltonian, whose homology gives the Hamiltonian Floer homology. In particular, one expects the existing structures on the latter to lift as well, such as the circle actions. On the other hand, constructing a genuine circle action even in the Morse theory is problematic: one usually cannot choose Morse-Smale pairs/Floer data that is invariant under the circle action. In this talk, we show how to extend the framework of Floer homotopy theory to the Morse-Bott setting, in order to tackle this problem. In the remaining time, we explain how to relate the Floer homotopy type to the free loop spaces of exact Lagrangian submanifolds equivariantly. Joint work with Laurent Cote.

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