# Kuperberg invariants for sutured 3-manifolds

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## Involutivity and semisimplicity

We now discuss some important consequences of the involutivity condition. Theorem 1.1.16. ([LR88a,LR88b]) Let H be a finite dimensional (ungraded) Hopf algebra over a field of characteristic zero. Then H is involutive if and only if H is semisimple. This extends to positive characteristic as follows: H is involutive and dim(H) 6= 0 if and only if H is semisimple and cosemisimple.
As far as the author knows, all examples of semisimple Hopf algebras (in characteristic zero) are somehow built from group algebras. Therefore, no interesting Hopf algebras (for our purposes) will be found in such case. Fortunately, the aforementioned theorem of Larson-Radford is no longer true for Hopf superalgebras. Indeed, the opposite holds [AEG01, Corollary 3.1.2]:
Proposition 1.1.17. If H = H0 ⊕H1 is a finite dimensional involutive Hopf superalgebra over a field of characteristic zero, then H is non-semisimple if and only if H1 6= 0.
Proof. We sketch the proof given in [AEG01]. If H1 = 0, then H is semisimple by the theorem above. For the converse, let H′ be the bosonization of H. If H is involutive, then (S′)2(x) = (−1)|x|x so H′ is non-involutive if H1 6= 0. Therefore, Larson-Radford’s theorem implies that H′ is non-semisimple and hence so is H, since their representation categories are equivalent by Proposition 1.1.15.

### Integrals and cointegrals

We now turn to introduce one of the fundamental notions of Hopf algebra theory, that of cointegrals and integrals. This notion has its origins in the theory of Lie groups, where the property of right invariance of the Haar integral can be appropriately abstracted leading to the notion of Hopf algebra integral. We begin by giving the basic definitions, and then we discuss the unimodularity condition and some properties.

#### Balanced sutured 3-manifolds

Sutured manifolds were introduced by Gabai in order to study foliations of 3-manifolds [Gab83]. We use a slightly less general definition, as in [Juh06, JTZ12].
Definition 2.1.1. A sutured manifold is a pair (M, γ) where M is a 3-manifoldwith- boundary and γ is a collection of pairwise disjoint annuli contained in ∂M. Each annuli in γ is supposed to be the tubular neighborhood of an oriented simple closed curve, called a suture, the set of which is denoted by s(γ). We further suppose that each component of R(γ) := ∂M \ int (γ) is oriented and we require that each (oriented) component of ∂R(γ) is oriented-parallel to a suture. We denote by R+(γ) (resp. R−(γ)) the union of the components of R(γ) whose orientation coincides (resp. is opposite) with the induced orientation of ∂M.
Note that the last condition in the above definition implies that for each annuli A in γ, one component of ∂A is contained in R−(γ) while the other is contained in R+(γ).
Definition 2.1.2. A balanced sutured manifold is a sutured manifold (M, γ) in which M has no closed components, χ(R−(γ)) = χ(R+(γ)) and every component of ∂M has at least one suture.
If M has no closed components and every component of ∂M contains a suture (a proper sutured manifold as in [JTZ12]), then s(γ) determines R±(γ). Thus, we only need to specify s(γ) in order to define a balanced sutured manifold. We now give some examples of balanced sutured manifolds.

The Reidemeister-Singer theorem

The Reidemeister-Singer theorem describes how two Heegaard diagrams of a same sutured manifold are related. Since we work with embedded diagrams, we will use a slightly stronger form than the usual theorem. For this, we make a few definitions. If H1 = (1,1, 1),H2 = (2,2, 2) are two Heegaard diagrams, a diffeomorphism d : H1 → H2 consists of an orientation-preserving diffeomorphism d : 1 → 2 such that d(1) = 2 and d(1) = 2.
Definition 2.2.7 ([JTZ12, Definition 2.34]). Let H1 = (1,1, 1),H2 = (2,2, 2) be two Heegaard diagrams of (M, γ) and denote by ji : i → M the inclusion map, for i = 1, 2. A diffeomorphism d : H1 → H2 is isotopic to the identity in M if j2 ◦ d : 1 → M is isotopic to j1 : 1 → M relative to s(γ).
Definition 2.2.8. Let H = (,, ) be a Heegaard diagram. Let δ be an arc embedded in int () connecting a point of a curve αj to a point of a curve αi and such that int (δ) ∩ = ∅. There is a neighborhood of αj ∪ δ ∪ αi which is a pair of pants embedded in and whose boundary consists of the curves αj , αi and a curve α′j. We say that α′j is obtained by handlesliding the curve αj over αi along the arc δ. Similarly, we can handleslide a β curve over another along an arc δ ⊂ int () such that int (δ) ∩ = ∅. See Figure 2.2 below.

2.4 Homology orientations

Let (,, ) be a balanced Heegaard diagram of (M, γ) and d = || = ||. Denote R− = R−(γ). Let A ⊂ H1(;R) (resp. B) be the subspace spanned by (resp. ). These subspaces have dimension d by [Juh06, Lemma 2.10]. There is a bijection o between orientations of the vector space H∗(M,R−;R) and orientations of the vector space d(A) ⊗ d(B) [FJR11, Sect. 2.4]. This is seen as follows. The Heegaard diagram specifies a handle decomposition of (M, γ) relative to R−×I with no handles of index zero or three. There are d handles of index one and two, so the handlebody complex C∗ = C∗(M,R− × I;R) is just C1 ⊕ C2 where both C1,C2 have dimension d. Now let ω be an orientation of H∗(M,R−;R) and let h11 , . . . , h1 m, h21 , . . . , h2 m be an ordered basis of H∗(M,R−;R) compatible with ω, where hi j ∈ Hi(M,R−;R). Let c11 , . . . , c1 m, c21 , . . . , c2 m ∈ C∗ be chains representing this basis, where ci j ∈ Ci. Then, for any b1, . . . , bd−m ∈ C2 such that c21
, . . . , c2 m, b1, . . . , bd−m is a basis of C2 the collection c11 , . . . , c1 m, ∂b1, . . . , ∂bd−m, c21 , . . . , c2 m, b1, . . . , bd−m is a basis of C∗ whose orientation ω′ depends only on ω. Now, an orientation of C∗ is specified by an ordering and orientation of the handles of index one and two. This is the same as an ordering and orientation of the curves in ∪ or equivalently, an orientation of d(A) ⊗ d(B). This way, the orientation ω induces an orientation of d(A) ⊗ d(B) via ω′. We denote this orientation of d(A) ⊗ d(B) by o(ω). There are a few cases in which there is a canonical orientation of H∗(M,R−(γ);R), and hence there is a canonical sign-ordering of a sutured Heegaard diagram. For example, let Y be a closed oriented 3-manifold. If M = Y \ B where B is an open ball in Y and γ = S1 ⊂ ∂B, then H∗(M,R−(γ);R) = H2(Y ;R) ⊕ H1(Y ;R). Now, any basis of H1(Y ;R) determines a Poincaré dual basis of H2(Y ;R) via the (nondegenerate) intersection pairing H2(Y )⊗H1(Y ) → R. The orientation of the basis of H∗(M,R−(γ)) thus obtained is independent of the basis of H1(Y ) chosen. However, reversing the orientation of Y , multiplies this orientation by (−1)b1(Y ).

Introduction
1 Hopf superalgebras
1.1 Basic notions and examples
1.1.1 Super vector spaces
1.1.2 Graphical notation
1.1.3 Hopf superalgebras
1.1.4 Examples
1.1.5 Bosonization of Hopf superalgebras
1.1.6 Involutivity and semisimplicity
1.2 Integrals and cointegrals
1.2.1 Definitions and examples
1.2.2 Unimodularity
1.3 Hopf G-algebras
1.3.1 Definitions and basic properties
1.3.2 The semidirect product case
2 Sutured 3-manifolds
2.1 Balanced sutured 3-manifolds
2.2 Heegaard diagrams
2.2.1 The Reidemeister-Singer theorem
2.3 Extended Heegaard diagrams
2.3.1 Cut systems of surfaces
2.3.2 Extended diagrams and extended moves
2.3.3 Dual curves
2.4 Homology orientations
3 Kuperberg invariants for sutured 3-manifolds
3.1 Kuperberg invariants of closed 3-manifolds
3.1.1 Tensors associated to Heegaard diagrams
3.1.2 The original construction of Kuperberg
3.1.3 Virelizier’s extension
3.2 Extending to sutured manifolds: the unimodular case
3.2.1 A direct Fox calculus-like formula
3.2.2 Examples of computation
3.2.3 Relation to Virelizier’s extension
3.2.4 Some lemmas
3.2.5 Proof of invariance, special case
3.2.6 The disconnected case
3.3 Extending to sutured manifolds: non-unimodular case
3.3.1 Spinc structures and multipoints
3.3.2 Multipoints and basepoints
3.3.3 Normalizing Z via Spinc
3.3.4 Proof of invariance
3.3.5 The disconnected case with Spinc
3.4 Twisted Kuperberg polynomials
4 Recovering Reidemeister torsion
4.1 Basics of Reidemeister torsion
4.1.1 Algebraic torsion
4.1.2 Twisted Reidemeister torsion
4.1.3 Twisted Alexander polynomials
4.2 Twisted torsion of sutured manifolds
4.2.1 Twisted torsion from a Heegaard diagram
4.2.2 Twisted torsion for link complements
4.3 Reidemeister torsion from Hopf algebra theory
4.3.1 Lemmas on exterior algebras
4.3.2 Proof of Theorem 2
4.3.3 Particular cases of Theorem 2

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