Homotopy equivalence and Bott periodicity comparison for the Floer-theoretic operator loop map

Establish that the map Ω_{A_{−},A_{+}}𝔄(ℋ) → Fred(ℋ′, ℋ″) defined by A_t ↦ (∂_s + A_s), where A_t is a loop of bounded self-adjoint operators, is a homotopy equivalence. Further, determine and prove the precise sense in which this map agrees with the Bott periodicity isomorphism (U/O ≃ Ω Fred) used in KO-theory, providing a rigorous comparison between the Floer-theoretic construction and classical Bott maps.

Background

The paper uses a Floer-theoretic construction relating loops of self-adjoint operators to Fredholm operators via the assignment A_t ↦ (∂_s + A_s). This construction is central for producing framings and KO-theory classes in the Floer setting.

Classically, Bott periodicity identifies U/O with loop spaces of Fredholm operators, and the authors note that a proof establishing that the Floer-theoretic map is a homotopy equivalence and comparing it to Bott periodicity is not known to them. A rigorous proof would bridge Floer constructions with established KO-theoretic machinery and justify implicit comparisons used in the literature.