The LMO Spectrum: Factorization Homology and the E_3-Structure of the Jacobi Diagram Algebra

TL;DR

This paper introduces the LMO spectrum, a categorification of the 3-manifold invariant, using factorization homology and establishing an $E_3$-structure on Jacobi diagrams, with applications to distinguishing lens spaces.

Contribution

It establishes an $E_3$-algebra structure on Jacobi diagrams and develops a universal surgery formula for the LMO spectrum, advancing the understanding of 3-manifold invariants.

Findings

The algebra of Jacobi diagrams has a homotopy $E_3$-algebra structure.

A universal surgery formula for the LMO spectrum is derived.

An $H_1$-decorated LMO invariant distinguishes certain lens spaces.

Abstract

We define the LMO spectrum, a categorification of the Le-Murakami-Ohtsuki (LMO) invariant for 3-manifolds, using factorization homology. The theoretical foundation is our main algebraic result (Theorem A): the algebra of Jacobi diagrams, $\AJac$, possesses a homotopy $E_3$-algebra structure. This is a necessary condition for consistency within factorization homology, and the proof relies on the formality of the little 3-disks operad. A universal surgery formula is derived from the excision axiom (Theorem B), providing a computational basis independent of conjectural models. As an application (Theorem C), we construct an ``$H_1$-decorated LMO invariant'' that distinguishes the lens spaces $L(156, 5)$ and $L(156, 29)$, a pair that the classical LMO invariant fails to separate.