Normal invariant of nearby Lagrangians via twisted derivative

TL;DR

This work analyzes obstructions to realizing nearby Lagrangians in cotangent bundles by studying the normal invariant in $[M,G/O]$. It introduces a twisted, tube-based framework, factoring the normal invariant through $M o B(oldsymbol{ au},oldsymbol{ heta}) o B(G/O)$ and then through a twisted $S$-duality map to $G/O$, which is shown to be 2‑torsion. By passing to topological tubes and almost quadratic twisted generating functions, the authors define a twisted derivative $B(oldsymbol{ au},oldsymbol{ heta}) o B(G,O)$ and prove that the corresponding twisted duality map $B(G/O) o G/O$ is 2‑torsion. These results yield new hard obstructions in the simple structure sets $ ext{S}^s(M)$, with concrete implications for homotopy spheres, products of spheres, and fake complex projective spaces, and provide a general criterion to identify when obstructions are hard. The framework combines generating function techniques, Waldhausen derivatives, and surgery-theoretic invariants to extract deep constraints on the smooth topology of nearby Lagrangians.”

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.

Figures (1)

• Figure 1: Three auxiliary functions

Theorems & Definitions (131)

• Definition 1.1
• Theorem 1
• Theorem 2
• Theorem 3
• Corollary 1.2
• proof
• Example 1.3
• Example 1.4
• Theorem 4
• Corollary 1.5