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

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.

arXiv comment29 pages, 4 figure, 2 tables. Replaced Sec 4. Removed former Sec 10, 11. Added Sec 8