The LMO Spectrum: Factorization Homology and the E_3-Structure of the Jacobi Diagram Algebra
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arXiv
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| Natura: | Preprint |
| Pubblicazione: |
2025
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| _version_ | 1866915534558199808 |
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| author | Kuriya, Takahito |
| author_facet | Kuriya, Takahito |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2508_18985 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | The LMO Spectrum: Factorization Homology and the E_3-Structure of the Jacobi Diagram Algebra Kuriya, Takahito Geometric Topology Mathematical Physics Algebraic Topology Quantum Algebra Primary 57K18, 57R56, Secondary 18F30, 55P48, 17B65, 81T45 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. |
| title | The LMO Spectrum: Factorization Homology and the E_3-Structure of the Jacobi Diagram Algebra |
| topic | Geometric Topology Mathematical Physics Algebraic Topology Quantum Algebra Primary 57K18, 57R56, Secondary 18F30, 55P48, 17B65, 81T45 |
| url | https://arxiv.org/abs/2508.18985 |