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

Takahito Kuriya

TL;DR

This work defines the LMO spectrum, a categorification of the LMO invariant via factorization homology, grounded by proving that the Jacobi-diagram algebra $\mathcal{A}_{\text{Jac}}$ carries a homotopy $E_3$-algebra structure. It derives a universal surgery formula from the excision axiom, enabling computation of the spectrum for manifolds presented by Dehn surgery, and demonstrates increased discriminatory power by constructing an $H_1$-decorated invariant that distinguishes $L(156,5)$ and $L(156,29)$, which the classical LMO invariant cannot separate. The construction links graph-complex formality, the Drinfeld associator, and the Kashiwara–Vergne problem to provide a coherent algebraic backbone for factorization-homology-based 3-manifold invariants, including an operator-state formalism that yields a universal $H_1$-decorated weight system. The results open paths to higher-order invariants via Massey products and Johnson homomorphisms and suggest parameterized (character-variety–dependent) refinements that could address LMO completeness. Overall, the paper provides a rigorous, computable, and potentially more powerful framework for 3-manifold invariants rooted in higher-algebraic structures.

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.

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

TL;DR

This work defines the LMO spectrum, a categorification of the LMO invariant via factorization homology, grounded by proving that the Jacobi-diagram algebra carries a homotopy -algebra structure. It derives a universal surgery formula from the excision axiom, enabling computation of the spectrum for manifolds presented by Dehn surgery, and demonstrates increased discriminatory power by constructing an -decorated invariant that distinguishes and , which the classical LMO invariant cannot separate. The construction links graph-complex formality, the Drinfeld associator, and the Kashiwara–Vergne problem to provide a coherent algebraic backbone for factorization-homology-based 3-manifold invariants, including an operator-state formalism that yields a universal -decorated weight system. The results open paths to higher-order invariants via Massey products and Johnson homomorphisms and suggest parameterized (character-variety–dependent) refinements that could address LMO completeness. Overall, the paper provides a rigorous, computable, and potentially more powerful framework for 3-manifold invariants rooted in higher-algebraic structures.

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, , possesses a homotopy -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 ``-decorated LMO invariant'' that distinguishes the lens spaces and , a pair that the classical LMO invariant fails to separate.

Paper Structure

This paper contains 56 sections, 18 theorems, 29 equations, 2 tables.

Table of Contents

  1. Introduction and main results
  2. The LMO invariant and its limitations
  3. Theoretical framework and guiding principles
  4. Main results
  5. The axiomatic framework of factorization homology
  6. Factorization homology as a local-to-global functor
  7. The excision axiom
  8. A universal surgery formula from the excision axiom
  9. Dehn surgery as a homotopy pushout
  10. The LMO spectrum surgery formula
  11. Analogy with the Witten-Reshetikhin-Turaev formula
  12. The algebraic origin of the E3-structure
  13. The axiomatic requirement for an E3-algebra
  14. The little disks operad and the formality theorem
  15. Graph complexes and the IHX relation
  16. ...and 41 more sections

Key Result

Theorem 3.1

Let $M_L$ be a 3-manifold obtained by Dehn surgery on a framed link $L \subset S^3$ with gluing map $\phi_L$. The LMO spectrum of $M_L$ is given by the homotopy pushout: where the term $(\phi_L)_*(\mathcal{A}_{\text{Jac}})$ indicates that the module structure over the boundary is twisted by the action of the gluing map $\phi_L$.

Theorems & Definitions (50)

  • Theorem 3.1
  • Theorem 4.1: Kontsevich99Willwacher15
  • Remark 5.1: Grading convention
  • Definition 5.2: Conant-Vogtmann Operator
  • Proposition 5.3: CV03
  • Definition 5.4
  • Proposition 5.5
  • Definition 5.6: $L_\infty$ Operations on $\mathcal{A}_{\text{Jac}}$
  • Theorem 5.7
  • proof
  • ...and 40 more