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Refined Bobtcheva-Messia Invariants of 4-Dimensional 2-Handlebodies

Anna Beliakova, Marco De Renzi

Abstract

In this paper we refine our recently constructed invariants of $4$-dimensional $2$-handlebodies up to $2$-deformations. More precisely, we define invariants of pairs of the form $(W,ω)$, where $W$ is a $4$-dimensional $2$-handlebody, $ω$ is a relative cohomology class in $H^2(W,\partial W;G)$, and $G$ is an abelian group. The algebraic input required for this construction is a unimodular ribbon Hopf $G$-coalgebra. We study these refined invariants for the restricted quantum group $U = U_q \mathfrak{sl}_2$ at a root of unity $q$ of even order, and for its braided extension $\tilde{U} = \tilde{U}_q \mathfrak{sl}_2$, which fits in this framework for $G=\mathbb{Z}/2\mathbb{Z}$, and we relate them to our original invariant. We deduce decomposition formulas for the original invariants in terms of the refined ones, generalizing splittings of the Witten-Reshetikhin-Turaev invariants with respect to spin structures and cohomology classes. Moreover, we identify our non-refined invariant associated with the small quantum group $\bar{U} = \bar{U}_q \mathfrak{sl}_2$ at a root of unity $q$ whose order is divisible by 4 with the refined one associated with the restricted quantum group $U$ for the trivial cohomology class $ω=0$.

Refined Bobtcheva-Messia Invariants of 4-Dimensional 2-Handlebodies

Abstract

In this paper we refine our recently constructed invariants of -dimensional -handlebodies up to -deformations. More precisely, we define invariants of pairs of the form , where is a -dimensional -handlebody, is a relative cohomology class in , and is an abelian group. The algebraic input required for this construction is a unimodular ribbon Hopf -coalgebra. We study these refined invariants for the restricted quantum group at a root of unity of even order, and for its braided extension , which fits in this framework for , and we relate them to our original invariant. We deduce decomposition formulas for the original invariants in terms of the refined ones, generalizing splittings of the Witten-Reshetikhin-Turaev invariants with respect to spin structures and cohomology classes. Moreover, we identify our non-refined invariant associated with the small quantum group at a root of unity whose order is divisible by 4 with the refined one associated with the restricted quantum group for the trivial cohomology class .
Paper Structure (10 sections, 15 theorems, 181 equations)

This paper contains 10 sections, 15 theorems, 181 equations.

Table of Contents

  1. Introduction
  2. Hopf group-coalgebras
  3. Refined invariant
  4. General decomposition formulas
  5. Restricted quantum sl(2)
  6. Refined invariant for restricted quantum sl(2)
  7. Relation between invariants for restricted and small quantum sl(2)
  8. Spin and cohomological decomposition formulas
  9. Rescaling the invariant
  10. Ribbon element and copairing

Key Result

Theorem 3.1

If $W$ is a $4$-di-men-sion-al $2$-han-dle-bod-y equipped with a relative cohomology class $\omega \in H^2(W,\partial W;G)$, if $D$ is a $G$-Kirby diagram representing $(W,\omega)$, and if $x_1 \otimes \ldots \otimes x_k \in H_{\alpha_1} \otimes \ldots \otimes H_{\alpha_k}$ is a bead presentation of is a topological invariant of the pair $(W,\omega)$.

Theorems & Definitions (32)

  • Theorem 3.1
  • proof
  • Remark 3.2
  • Proposition 4.1
  • proof
  • Remark 4.2
  • Lemma 5.1
  • proof
  • Lemma 5.2
  • proof
  • ...and 22 more