Isomorphisms between spherically framed bordism and flow-category homology/cohomology theories

Establish natural isomorphisms of multiplicative homology theories R^fr_*(·) ≅ Ω^fr_*(·) and R^sfr_*(·) ≅ Ω^sfr_*(·), compatible with the forgetful maps, and additionally establish analogous natural isomorphisms of multiplicative cohomology theories compatible with the same forgetful maps.

Background

The paper uses Abouzaid–Blumberg's framework of flow categories to construct spectra associated to Floer-theoretic data. For framed flow categories, the endomorphism ring spectrum R^fr is known to be equivalent to the sphere spectrum, and its homology theory agrees with classical framed bordism.

For the spherically framed variant, the authors introduce R^sfr and conjecture that it underlies a bordism theory where tangent bundles are trivialized as stable spherical fibrations, expecting natural isomorphisms between R^sfr-based (co)homology theories and the corresponding spherically framed bordism theories. This would place spherically framed Floer homotopy constructions within a well-understood bordism-theoretic context.