Approximability for Lagrangian submanifolds

TL;DR

This work defines categorical metric approximability for metric spaces of Lagrangian submanifolds equipped with the spectral metric $d_{\gamma}$, viewing approximability as a categorification of Turing-type finite-type approximability. It develops a robust framework of triangulated persistence categories (TPCs) and filtered $A_{\infty}$-Fukaya categories, employing perturbation data, Lefschetz fibrations, and Abouzaid-style split-generation in a persistence setting. The authors prove three main geometric instances are $\text{TPC}$-approximable: (i) closed exact Lagrangians in unit cotangent disk bundles $D^{*}N$ (via cotangent/Lefschetz technology); (ii) equators on $S^2$ (via retract-approximability in a sphere setup); and (iii) non-contractible Lagrangians on $\mathbb{T}^2$ (via retract-approximation on the torus). They also show the associated Lagrangian spaces are not totally bounded and derive corollaries linking approximability to complexity and persistence-barcode data, including weighted cone-length and persistence Hochschild invariants. The results illustrate how categorical persistence tools yield concrete approximability phenomena in symplectic topology and open pathways to quantifying complexity and entropy in Lagrangian dynamics. Overall, the work provides a flexible, persistence-driven approach to approximating and measuring complexity of spaces of Lagrangians, with rigorous ties to open-closed maps and ambient quantum cohomology.

Abstract

This paper introduces a notion of categorical approximability for metric spaces that can be viewed as a categorification of approximability for metric groups, as defined by Turing in 1938. Approximability as introduced here is a property of metric spaces that is more general than precompactness. It is shown that several classes of Lagrangian submanifolds - closed Lagrangian submanifolds in a cotangent disk bundle; equators on the sphere; weakly exact Lagrangians on the torus-endowed with the spectral metric are approximable in this sense. Among other geometric applications, we show that there are such examples of spaces of Lagrangians that are approximable but are not precompact.

Figures (13)

• Figure 1: The diffeomorphism $\psi_{z}$.
• Figure 2: The graphs of the functions $\phi_{1}$ and $\phi_{2}$.
• Figure 3: The graphs of the functions $\hat{\phi}_{1}$ and $\hat{\phi}_{2}$.
• Figure 4: The fibration $\bar{h}:E\to \mathbb{C}$.
• Figure 5: Examples of configurations contributing to the coproduct $\Delta^{\mathcal{B},K}_q\colon \Delta_{\mathcal{B}_p}\to \mathcal{Y}_{p}^l(K)\otimes \mathcal{Y}_{p}^r(K)$. The colored strips indicate the support of the non-trivial parts of Hamiltonian perturbation data.

Theorems & Definitions (180)

• Definition 1.1.1
• Remark 1.1.2
• Theorem 1
• Corollary 1.1.3
• Corollary 1.1.4
• Corollary 1.1.5
• Remark 1.1.6
• Remark 2.1.1
• Lemma 2.1.2
• Definition 2.2.1