Normal invariant of nearby Lagrangians via twisted derivative

Let $L$ and $M$ be closed, connected, smooth manifolds and let $L\hookrightarrow T^{\ast }M$ be an exact Lagrangian embedding. The induced map $L\to M$ is known by earlier work to be a homotopy equivalence. We show that the associated normal invariant $M\to G\mathrm{/}O$ factors through a map $B(\mathcal{T},\mathcal{Q})\to G\mathrm{/}O$ which is a twisted version of the Waldhausen derivative $\mathcal{T}\to G$ on the space $\mathcal{T}$ of tubes. Further, we show that this twisted derivative map itself factors though a map $B(G\mathrm{/}O)\to G\mathrm{/}O$ which is a twisted version of the $S$-duality map $BG\to G$. In particular we deduce that the normal invariant of the homotopy equivalence $L\to M$ is 2-torsion.