Spectral Floer theory and tangential structures

In \cite{PS}, for a stably framed Liouville manifold $X$ we defined a Donaldson-Fukaya category $\mathcal{F}(X;\mathbb{S})$ over the sphere spectrum, and developed an obstruction theory for lifting quasi-isomorphisms from $\mathcal{F}(X;\mathbb{Z})$ to $\mathcal{F}(X;\mathbb{S})$. Here, we define a spectral Donaldson-Fukaya category for any graded tangential pair' $Θ\to Φ$ of spaces living over $BO \to BU$, whose objects are Lagrangians $L\to X$ for which the classifying maps of their tangent bundles lift to $Θ\to Φ$. The previous case corresponded to $Θ= Φ= \{\mathrm{pt}\}$. We extend our obstruction theory to this setting. The flexibility to tune’ the choice of $\mathrm{\Theta }$ and $\mathrm{\Phi }$ increases the range of cases in which one can kill the obstructions, with applications to bordism classes of Lagrangian embeddings in the corresponding bordism theory ${\mathrm{\Omega }}_{\ast }^{(\mathrm{\Theta },\mathrm{\Phi }),\circ }$. We include a self-contained discussion of when (exact) spectral Floer theory over a ring spectrum $R$ should exist, which may be of independent interest.