Bordism from quasi-isomorphism

Let $X$ be a graded Liouville domain. Fix a pair of infinite loop spaces $\Psi = (\Theta \to \Phi)$ living over $(BO \to BU)$. This determines a spectral Fukaya category $\mathcal{F}(X;\Psi)$ whenever $TX$ lifts to $\Phi$, containing closed exact Lagrangians $L$ for which $TL$ lifts compatibly to $\Theta$; and by Bott periodicity and index theory, a Thom spectrum $R$ with bordism theory $R_*$. Suppose that $L$ and $K$ are quasi-isomorphic in the Fukaya category over $\mathbb{Z}$. We prove that: (a) if both lift to $\mathcal{F}(X;\Psi)$, then there is a rank one $R$-local system $\xi: L \to BGL_1(R)$ over $L$ so that $(L,\xi)$ and $K$ are quasi-isomorphic in the spectral Fukaya category; (b) when $X$ is polarised and $\Psi = (BO \times F \to BO)$, if only $K$ lifts to $\mathcal{F}(X;\Psi)$, then the composition $L \to B^2GL_1(R)$ of the stable Gauss map of $L$ and the delooped $J$-homomorphism is nullhomotopic. Combined with the computation of the open-closed fundamental class associated to $(L,\xi)$ in \cite{PS3}, these results have applications to bordism and stable homotopy types of quasi-isomorphic Lagrangians, to Hamiltonian monodromy groups, and to smooth structures on nearby Lagrangians. A key ingredient in the proofs is a new form of obstruction theory for flow categories `lying over' a manifold $L$, closely related to a `spectral Viterbo restriction functor' also introduced here.