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State-Sum Invariants of 4-Manifolds I

Louis Crane, Louis H. Kauffman, David N. Yetter

TL;DR

The paper develops a rigorous, category-theoretic framework for 4D state-sum invariants constructed from artinian semisimple tortile categories, generalizing Crane and Yetter's earlier work. It proves invariance under triangulation via blob techniques and connects the resulting invariants to Broda-style surgery constructions, as well as to 3D Reshetikhin–Turaev theory through TL translations and Roberts' chainmail method. It also analyzes the role of the center of the braided category, showing reductions to topological quantities like Euler characteristic and signature under suitable non-degeneracy, and outlines extensions to manifolds carrying (co)homology data. The work establishes a robust algebraic toolkit for 4-manifold invariants and sets up the framework for the sequel on insertions related to (co)homology classes.

Abstract

We provide, with proofs, a complete description of the authors' construction of state-sum invariants announced in [CY], and its generalization to an arbitrary (artinian) semisimple tortile category. We also discuss the relationship of these invariants to generalizations of Broda's surgery invariants [Br1,Br2] using techniques developed in the case of the semi-simple sub-quotient of $Rep(U_q(sl_2))$ ($q$ a principal $4r^{th}$ root of unity) by Roberts [Ro1]. We briefly discuss the generalizations to invariants of 4-manifolds equipped with 2-dimensional (co)homology classes introduced by Yetter [Y6] and Roberts [Ro2], which are the subject of the sequel. (citations refer to bibliography in the paper)

State-Sum Invariants of 4-Manifolds I

TL;DR

The paper develops a rigorous, category-theoretic framework for 4D state-sum invariants constructed from artinian semisimple tortile categories, generalizing Crane and Yetter's earlier work. It proves invariance under triangulation via blob techniques and connects the resulting invariants to Broda-style surgery constructions, as well as to 3D Reshetikhin–Turaev theory through TL translations and Roberts' chainmail method. It also analyzes the role of the center of the braided category, showing reductions to topological quantities like Euler characteristic and signature under suitable non-degeneracy, and outlines extensions to manifolds carrying (co)homology data. The work establishes a robust algebraic toolkit for 4-manifold invariants and sets up the framework for the sequel on insertions related to (co)homology classes.

Abstract

We provide, with proofs, a complete description of the authors' construction of state-sum invariants announced in [CY], and its generalization to an arbitrary (artinian) semisimple tortile category. We also discuss the relationship of these invariants to generalizations of Broda's surgery invariants [Br1,Br2] using techniques developed in the case of the semi-simple sub-quotient of ( a principal root of unity) by Roberts [Ro1]. We briefly discuss the generalizations to invariants of 4-manifolds equipped with 2-dimensional (co)homology classes introduced by Yetter [Y6] and Roberts [Ro2], which are the subject of the sequel. (citations refer to bibliography in the paper)

Paper Structure

This paper contains 13 sections, 18 theorems, 39 equations, 37 figures.

Table of Contents

  1. Introduction
  2. Tortile Categories
  3. $U_q(sl_2)$ Recoupling Theory
  4. General semi-simple tortile categories
  5. Coloring Triangulations and State-Sum Invariants
  6. Surgical Versions and an Aside About 3-Manifold Invariants
  7. TL translation of $CY(W)$ and Roberts' Chainmail Method
  8. Translation
  9. Roberts' Chain Mail
  10. The Center and Roberts' Chainmail Method
  11. Extensions to Manifolds with (Co)homology Classes
  12. Appendix on Diagrammatic Notation
  13. References

Key Result

Proposition 2.1

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Figures (37)

  • Figure 1: Axioms for the Bracket Polynomial
  • Figure 2: Regular Isotopy---Reidemeister Moves II and III
  • Figure 3: The Bracket under Reidemeister I
  • Figure 4: The Framed First Reidemeister Move
  • Figure 5: Examples of cabling
  • ...and 32 more figures

Theorems & Definitions (31)

  • Proposition 2.1
  • Definition 2.2
  • Definition 2.3
  • Definition 2.4
  • Proposition 2.5
  • Theorem 2.6
  • Definition 2.7
  • Definition 2.8
  • Lemma 2.9
  • Lemma 2.10
  • ...and 21 more