Spectral Floer theory and tangential structures

For a stably framed Liouville manifold X X we [ Bordism of flow modules and exact Lagrangians , Preprint] defined a Donaldson-Fukaya category F ( X ; S ) \mathscr {F}(X;\mathbb {S}) over the sphere spectrum, and developed an obstruction theory for lifting quasi-isomorphisms from F ( X ; Z ) \mathscr {F}(X;\mathbb {Z}) to F ( X ; S ) \mathscr {F}(X;\mathbb {S}) . Here, we define a spectral Donaldson-Fukaya category for any ‘graded tangential pair’ Θ → Φ \Theta \to \Phi of spaces living over B O → B U BO \to BU , whose objects are Lagrangians L → X L\to X for which the classifying maps of their tangent bundles lift to Θ → Φ \Theta \to \Phi . The previous case corresponded to Θ = Φ = { p t } \Theta = \Phi = \{\mathrm {pt}\} . We extend our obstruction theory to this setting. The flexibility to ‘tune’ the choice of