Normal invariant of nearby Lagrangians via twisted derivative

TL;DR

This paper investigates the normal invariant of homotopy equivalences induced by exact Lagrangian embeddings, revealing it factors through twisted derivatives and is 2-torsion, advancing understanding in symplectic topology.

Contribution

It introduces a twisted derivative framework for the normal invariant of Lagrangian embeddings, showing it factors through specific maps and is 2-torsion, which is a novel insight.

Findings

Normal invariant factors through twisted derivatives.

The normal invariant is 2-torsion.

Provides a new perspective on Lagrangian embeddings in symplectic topology.

Abstract

Let $L$ and $M$ be closed, connected, smooth manifolds and let $L \hookrightarrow T^*M$ 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 $M \to G/O$ factors through a map $B(\mathcal{T},\mathcal{Q}) \to G/O$ 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 $B(G/O) \to G/O$ 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.