数理解析研究所講究録
第 1777 巻 2012 年 21-30

21A note on quantum fundamental groups andquantum representation varieties for 3-manifolds
Kazuo HabiroRIMS, Kyoto University
September 25, 2011
This informal note is based on the author talk “Quantum fundamentalgroups and quantum representation varicties for 3-manifolds“ givcn in the workshop “Geometric and analytic approaches to representations of a group and representation spaces‘’, held at RIMS during June 20–June 24, 2011. Details of
this note will appear in papers in preparation.
$s$1

Cobordism categories and embedding categories

1.1

Cobordism categories and TQFTs

In Quantum Topology, one considers the cobordism category $Cob_{d}$ , whose objects are compact, oriented $(d-1)$ -manifolds and whose morphisms are homeomorphism classes of d-dimensional cobordisms.

A d-dimensional Topological Quantum Field Theory is a functor$F:Cob_{d}arrow$

Vectfrom $Cob_{d}$ to the category Vect of vector spaces.

1.2

Embedding category $Emb_{d}$In this note, we consider embedding categories, which are another type of categories closely related to manifold topology. Let $d\geq 1$ be an integer. The
d-diniensional embedding category $Emb_{d}$ is the category whose objects are compact, oriented d-manifolds, and whose morphisms are isotopy classes of embeddings. Composition of morphisms is induced by composition of embeddings,
and the identity morphisms is represented by the identity homeomorphisms.
In what follows, we often
an embedding $f:Marrow N$ and its isotopy
class $[f];Marrow N$ .
$(^{\backslash },ont_{l1}se$221.3

Relation between embedding categories and cobordism categories

The embedding category is related to the cobordism category as follows. Thereis a functor$\partial:Emb_{d}arrow Cob_{d}$ ,$M\mapsto\partial M$$[f:MLarrow N]\mapsto[N\backslash (intf(M))]$.

,
More precisely, the fumctor maps each $(d-1)$ -manifold $Af$ to its boumdaryand each morphism $[f]:Marrow N$ (represented by an embedding $f:Marrow N$ ) toits “complement“ $[N\backslash (int f(M))]$ , where is chosen so that $f(M)$ is containedin the interior of .
$\partial M$$\partial$$f$$\Lambda f$

1.4

Functors from $Emb_{d}$Note that a homeomorphism $f:Marrow M’$ between two d-manifolds $M,$ $M’\in$
Ob $(Emb_{d})$ represents an isomorphism in $Emb_{d}$ . Therefore, for each functor, thc isomorphism class of $F(M)\in$$F:Emb_{d}arrow C$ from $Emb_{d}$ to a catcgory$(Emb_{d})$$M\in$$(C)$Ob
is an invariant of $M$ .
Obfor$C$

1.4.1

The functor $U:Emb_{d}arrow$ TophLet$U:Emb_{d}arrow$Toph:$=$Top/homot $opy$

denote the functor which maps $M\in$ Ob $(Emb_{d})$ to thc underlying topologicalspace of $M$ and which maps each morphism $[f]:Marrow M$ ‘, which is an isotopy
class. to the homotopy class of . Composing with various fUnctors fromToph defined in Algebraic Topology, one obtains many functors $fi\cdot omEmb_{d}$ .
For example,
$U$

$f$

$Emb_{d}arrow^{U}$

Toph$Emb_{d}^{*}arrow^{U}$Toph“$H_{k}\underline{(-}’ Z$$\rangle$$arrow^{\pi_{1}}$)

AbGrp$Hom(-\prime G)arrow$Setop

Here Emb7 and Toph“ are tlie basepointed versions of $Emb_{d}$ and Toph, respectively, and is a fixed group.
$G$

1.4.2

Skein modules

Another important class of functors defined on $Emb_{d}$ is defined by skein modules. Roughly speaking, a skein module associated with a manifold $M$ is$A(M)=k$ { links” in M}/(ambient isotopy and local relations)23
Here “links“ are a certain kind of subcomplexes in $M$ , possibly with framings,coloring, etc. It is clear that $A(M)$ is functorial in embeddings, and hence wehave a functor$A:Emb_{d}arrow k$2The category-Mod.

$\mathcal{E}$In the rest of this note, we restrict to the case $d=3$ .
In this section, we define the “category of disc-based 3-manifolds and discbased embeddings”, denoted by . which is the main object of study in thisnote.
$\mathcal{E}$In what follows, all manifolds are oriented and all codimension$0$

are orientation-preserving.

2.1

embeddings

Disk-based 3-manifolds and disk-based embeddings

A disk-based 3-manifold $(M, i)$ consists of1$\bullet$a connected 3-manifold $M$ , andan embedding$i:D^{2_{a}}arrow\partial M$.

The embedding is called the disc-basing.
A (disk-based) embeddin9 $f:(M, i)arrow(N,j)$ is an embedding $f:Mrightarrow N$
which is compatible with the disc-basing, i.e., $j=(f|_{\partial M})oi$ .
$i$

2.2

The category$\mathcal{E}$Define to be the category as follows, The objects are disc-based 3-manifolds,
the morphisms are the equivalence classes of disk-based embeddings, where
two disc-based $3arrow manifolds$ are equivalent if tliere is an isotopy between themthrough disk-based embeddings. The composition in is induced by composition of embeddings. The identity morphisms are defined by $1_{(M,i)}=[id_{M}]$ .
For simplicity, we often write $M$ for $(M, i)$ by dropping the disc-basing ,and we often confuse embeddings and their isotopy classes.
$\mathcal{E}$$\mathcal{E}$

$i$

2.3

Based-homeomorphisms as isomorphisms in

Clearly, a based-homeomorphism is an isomorphism in$\mathcal{E}$$\mathcal{E}$.

Thus, given a functor $F:\mathcal{E}arrow C$ from to a category $C$ , the isomorphismclass of $F(M)$ in $C$ is a topological invariant of $M$ .
$\mathcal{E}$24Proposition 1. A morphism $f:Marrow M’$ in is an isomorph,
if is represented by a disk-based homeomorphism.

$\dot{u}m$$\mathcal{E}$

if and only$f$Corollary 2. For $M\in$ Ob , the group Aut $e(M)$ is isornorphic to the “discbased mapping class group of $M$ ”, i.e., the group of the disc-based ambient
isotopy classes of the disc-based self-homeomorphisms of $M$ .
$(\mathcal{E})$Braided monoidal structure of

2.4

The category$\bullet$$\mathcal{E}$$\mathcal{E}$ha,s a braided monoidal category structure.

The tensor functor$\otimes:\mathcal{E}\cross \mathcal{E}arrow \mathcal{E}$is given by a kind of boundary connected sum.
$\bullet$$\bullet$

The monoidal unit given by the 3-ball$B^{3}$.

The braidings$\psi_{M,M’}:M\otimes M’arrow\Lambda;I’\otimes M\underline{\simeq}$

is represented by a homeomorphism which switches the M-part and theM’-part in $M\otimes M’$ and $M’\otimes M$ .

of handlebody embeddings

3The category

3.1

The full subcategory

Let$?t$$’\kappa$$\mathcal{H}$denote the full subcategory ofof$\mathcal{E}$$\mathcal{E}$such that$Ob(\mathcal{H})=\{V_{0}, V_{1}, V_{2}, \ldots\}$,

where is a fixed genus 9 handlebody obtained from a cylinder$g1$ -handles on the top.
with $\{0,1,2, \ldots\}$ .
We identify Ob
In other words, $H$ is the category with Ob $(\mathcal{H})=\{0,1,2, \ldots\}$ and$D^{2}\cross[0_{!}1]$$V_{g}$$(\mathcal{H})$} $/isotopy${ -component bottom tangles in } $/isotopy$ .

$?t(m,n)=$ {d.b. embeddings
$=$

$m$

$V_{m}rightarrow V_{;}$$V_{n}$

with

25
3.2
LetRelations of$C$$\bullet$$\bullet$denote the$\prime kt$

and other categories$Cranarrow Kei\backslash ler$-Yettcr (CKY) cobordism category [3, 5]:objects –surface with boundary parametrized by$S^{1}$.

$morphisms-\cdot$ homeomorplxism classes of connected cobordisms.

Remark 3.

$H^{op}$is isomorphic to$\bullet$the “category of bottom tangles in handlebodies“ ([4])$\bullet$the く ‘category of special Lagrangian cobordisms” ([2])

which are (isomorphic to) a subcategory of the CKY category .
$C$

3.3

Some structures of$\mathcal{H}$

Fact. is a braided monoidal subcategory of . In particular,in .
Fact. In , there is a
Hopf algebra structure$tt$$\mathcal{E}$$V_{g}\otimes V_{9’}\cong V_{g+g’}$$\mathcal{E}$$bi\cdot ai(ied$$\prime k\ell$$H=(V_{1},\mu, \eta, \Delta, \epsilon, S)$.

( $Crane\cdot-\cdot Yetter[3]$ and Kerler [5] had introduced the same structiire in $C(\cong C^{op}).$ )4

Quantum fundamental groups

4.1

Definition of quantum fundamental groups

The quantum fundamental group (QFG) of $M\in$ Ob$P(M)=\mathcal{E}(i(-), M):\mathcal{H}^{op}arrow$

Clearly, $P(M)$ is functorial in $M$ . Thus,$\backslash ve$$(\mathcal{E})$is the functor

Set.

have a functor$P:\mathcal{E}arrow\hat{H}:=Set^{H^{\circ p}}$Note t.hat$P(M)(n)=\mathcal{E}(i(n), M)=\mathcal{E}(V_{n}, M)=\{[V_{n^{\sigma}}arrow A\prime I]\}$$=${[n-component bottom tangle in Aq}

maps surjectically onto the direct productof the set.
$\pi_{1}(M)$$\pi_{1}(1|$ノ$I)^{n}$. Thus, $P(M)$ is a refinement

26
4.2

Goal

into QFGs.
I would like to generalize everything about
In the rest of this talk, I will explain attempts to generalizing$\pi_{1}$$\bullet$$\bullet$

representation spaces Rep $G(\pi_{1}M)=Hom_{Grp}(\pi_{1}M, G)$ ,van Kampen Theorem.

Kan extension

5For the definitions and properties of the Kan extensions, see Mac Lane’s book
[7].

Left Kan extension along

5.1
Let
$Q$

$V$

$i:Harrow \mathcal{E}$be a cocomplete category, such as Set, Vect, Grp, Ab, . . . .

If we are given a fumctor
along$Q:\mathcal{H}arrow V$, then there is the left $Kane\prime xtension$ of$i$$Lan_{i}Q:\mathcal{E}arrow \mathcal{V}$

Example 4.

.

1. For the fundamental groups, we have
$Lai\}(\pi_{1} :\mathcal{H}arrow Grp)\cong(\pi_{1}$:$\mathcal{E}arrow$

Grp .
$)$

2. For the QFGs, we have

Lani
Thus, the QFG
.

$is$

へ.

$(Pi=Y:\mathcal{H}arrow\hat{\mathcal{H}})\cong(P:\mathcal{E}arrow \mathcal{H})$the left Kan extension along of the Yoneda embedding$i$$Y:\mathcal{H}arrow’\hat{\kappa}$

5.2

Kan extension as coend

For simplicity, consider the case
Let $k(-)$ : Set$arrow$Vect,$\mathcal{V}=$Vect$S\mapsto k\cdot S$$=$

Vect .
$k$. For $M\in$ Ob$kP(M):\mathcal{H}^{op}arrow$$(\mathcal{E})$, we have a functor

Vect.

If $Q:\gamma\{arrow$ Vect is a functor, then $(Lai\searrow Q)(M)$ can be computed as theVect
coend, or the “tensor product” of $kP(M):\mathcal{H}^{op}arrow$ Vect and$Q:\mathcal{H}arrow$27
over$\prime \mathcal{H}$$(Lan_{i}Q)(\lambda I)=kP(M)\otimes_{\mathcal{H}}Q$$:= \int^{n\in \mathcal{H}}kP(M)(n)\otimes_{k}Q(n)$$=( \bigoplus_{n\in Ob(\gamma\ell)}kP(M)(n)\otimes_{k}Q(n))$/Relations

where Relations is spanned by$x\otimes Q(f)(y)-kP(M)(f)(x)\otimes y$for $f\in 7\cdot l(n,n’),$ $y\in Q(n),$ $x\in kP(M)(n’)$ , and $n,n’\in$ Ob

5.3

Problem$Lan_{i}Q$

can be denoted$L_{c}\backslash n_{i}Q=kP\otimes_{\mathcal{H}}Q=Ind_{l\ell}^{f}Q:\mathcal{E}arrow$$(\mathcal{H})$

.

Vect.

Problem. Construct interesting functors$Q:\mathcal{H}arrow$

which induce interesting functors on

Vect$\mathcal{E}$Lal$i_{i}Q:\mathcal{E}arrow$

5.4

Vect.

Co-ribbon Hopf algebras

The notion of co-ribbon Hopf algebra is the dual to that of ribbon Hopf algebra:

A co-ribbon Hopf algebm is a Hopf algebrawith$\bullet$$\bullet$

5.5$H=(H, \mu, \eta, \Delta, \epsilon,\cdot S)$

equippeda universal R-form $R:H\otimes Harrow H$ ,a co-ribbon element $r:Harrow k$ .

Examples of co-ribbon Hopf algebras$\bullet$The dual $H^{*}=Hom(H, k)$ of a finite diniensional ribbon Hopf algebra $H$ .

$\bullet$Commutative Hopf algebras.

$(R=\epsilon\otimes\epsilon, r=\epsilon)$-The algebra $Fun_{k}(G)$ of fUnctions on a finite group .
-The algebra $k(G)$ of regular functions on a linear algebraic group$G$$\bullet$

The quantized algebra of regular functions, $k_{q}(G)$ , for $G=SL(N),$$\ldots$

$G$.

.

28
5.6

The category $Comod_{H}$Let $Comod_{H}$ denote tlxe category of left H-comodules.
If $H$ is a Hopf algebra, then $Comod_{H}$ is a rnonoidal category.
Fact.
$\bullet$$\bullet$(Majid) If $H$ is co-quasitriangular, then $Comod_{H}$ is a braided category.
$:=$ ( $H$ , coad)Ob $(Comod_{H})$ has a braided Hopf algebra
The object
structilre. Here$\in$$\underline{H}$coad: $Harrow H\otimes H$,$x \mapsto\sum x_{(1)}S(x_{(_{\backslash }!)})\otimes x_{(2)}$is the left coadjoint coaction.
Theorem 5 (Cf. [6]). If $H$ is a co-ribbon Hopf algebra, then there is a braided
monoidal functor$Q^{H}:\mathcal{H}arrow Comod_{H}$which maps the braided Hopf algebra structure in,$\prime H$

to that in $Comod_{H}$ .

Quantum representation variety

5.7

Since $Comod_{H}$ is cocomplete, we have the following.
Corollary. If $H$ is a co-ribbon Hopf algebra, then we have a functor
Rep

$H$

$:=Lan_{i}Q^{H}:\mathcal{E}arrow Comod_{H}$.

We call Rep $H(M)$ the quantum representation variety of $M$ associated to $H$ .

5.8
1$\bullet$

ExamplesIf $H=F\iota m_{k}\cdot(G)$ with$G$a finite group, tlzen Rep $\mathfrak{N}n(c,)(M)=$ Fun $(Ho\iota n_{Grp}(\pi_{1}M, G))$ .

If $H=k(G)$ with a linear algebraic group, then Rep $k(G)(M)=$ Reg Rep $G(\pi_{1}M))$ ,.
the algebra of regtdar functions on the representation variety Rep$G$$($$c_{(\pi_{1}M)}$

If $H=k_{g}(G)$ , then Rep
is a q-deformation of Rep $k(G)(\pi_{1}M)$ .
It is simply an H-comodule. It is not an algebra.
$k_{q}(G)$$\bullet$)l bracket skein module is a q-deformation
Remark 6. Recall that the(C)-character variety ([1], etc.).
of the$Ka\iota 1ffn$ $aJ$

$SL_{2}$Rep$k_{q}(SL_{2})$is closely related to the Kauffman skein modules.

In fact, one can define “quantum character variety” $X^{H}(M)\subset$ Rep $H(M)$for a co-ribbon Hopf algebra $H$ andas the H-invariant part of the$H(M)$.
H-comodule Rep$M\in \mathcal{E}$When $H=k_{q}(SL_{2}),$ $X^{k_{q}(SL_{2})}(M)$ seems almost isomorphic to the Kauffianbracket skein module of $Af$ .

29
5.9

Quantum van Kampen (sketch)

There is a gluing formula for the QFG, or “quantum van Kampcn theorem”, ofthe disk-based 3-manifold
obtained fromby gluingalong a connected surface on the boundaries of.
We have$M_{1} \bigcup_{\Sigma A}1I_{2}\in \mathcal{E}$$J\uparrow/I_{1},$$\Sigma$$M_{2}\in \mathcal{E}$$M_{1},$ $M_{2}$.

$P(M_{1} \bigcup_{I^{\backslash },}M_{2})\cong P(M_{1})\otimes_{P(\Sigma)}P(j|/I_{2})$equipped with a monoid structure in the cocompletion, which is a monoidal category.

$HereP(\Sigma)=P(\Sigma x$ $[0,1])$へ$\mathcal{H}=Set^{7\{^{op}}$denotes “tensor product over the monoid
is a cocomplete monoidal category.
$\otimes_{P(\Sigma)}$へ$\mathcal{H}$

5.10$P(\Sigma)$”, which exists sinceGluing formula for QRVs (sketch)

The Quantum van Kampen Theorem for QFGs implies a gluing formula for
QRV associated to a co-ribbon Hopf algebra $H$ .

For a connected surface , there is an (ordinary) algebra structure for$\Sigma$$Rep^{H}(\Sigma\cross[0,1])$.

$A_{\Sigma}$$:=$

If $M$ is a icobordism” fromto , then Rep $H(M)$ is equipped with an-bimodule structure. Thcn the gluing formula for QRVs states that$\Sigma’$$\Sigma$$(A_{\Sigma}, A_{\Sigma’})$Rep$H(ztI_{1^{\cup}\Sigma s}\eta h_{-})\cong$

Rep $H(11$ノ, Rep $H(M_{2})$ .

$I_{1})\otimes_{A_{I}}$

These constructions give a 2-fumctor$Rep^{H}:\{\begin{array}{l}surfacescobordismsembeddings\end{array}\}arrow\{\begin{array}{l}algebrasbimoduleshon1o\mathfrak{n}1\circ rphisms\end{array}\}$References[1] D. Bullock, Rings of $sl_{2}(C)$ -characters and the auffman bracket skeinmodule” , Comment. Math. Helv. 72 (1997) 521-542$I<$[2] D. Cheptea, K. Habiro, G. Massuyeau, A functorial LMO invariant for
Lagrangian cobordismg, Geom. Topol. 12 (2008) 1091-1170[3] L. Crane, D. Yctter, On algebraic structures implicit in topological quantum field theories, J. KnotRamifications 8 (1999) 125-163$\prime I^{\tau}heory$[4] K. Habiro, Bottom tangles and umiversal invariants, Alg. Geom. Topol. 6(2006) 1113-1214

30[5] T. Kerler, Bridged links and tangle presentations of cobordism categories,Adv. Math. 141 (1999) 207-281[6] T. Kerler, Genealogy of non-perturbative quantum invariants of 3manifolds: the surgical family, in “Geonietry and physics (Aarhus, 1995)“,Lecture Notes in Pure and Appl. Math. 184, Dekker, New York (1997),

503-547[7] S. Mac Lane, Categories for the working mathematician, Graduate Texts
in Mathematics 5, Springer, New York (1998)