### RIMS-1707

### Some topological aspects of

### 4-fold symmetric quandle invariants of 3-manifolds

### By

### Eri HATAKENAKA and Takefumi NOSAKA

### October 2010

### R _{ESEARCH} I NSTITUTE FOR M ATHEMATICAL S _{CIENCES}

### KYOTO UNIVERSITY, Kyoto, Japan

**Some topological aspects of**

**4-fold symmetric quandle invariants of 3-manifolds**

### Eri Hatakenaka and Takefumi Nosaka

**Abstract**

The paper relates the 4-fold symmetric quandle homotopy (cocycle) invariants with topological
objects. We show that the 4-fold symmetric quandle homotopy invariants are at least as
powerful as the Dijkgraaf-Witten invariants. As an application, for an odd prime*p, we show*
that the quandle cocycle invariant of a link in *S*^{3} using the Mochizuki 3-cocycle is equivalent
to the Dijkgraaf-Witten invariant with respect toZ*/p*Zof the double covering of*S*^{3} branched
along the link. We also reconstruct the Chern-Simons invariant of closed 3-manifolds as a
quandle cocycle invariant via the extended Bloch group, in analogy to [IK].

**Keywords** Quandle, symmetric quandle, quandle cocycle invariant, link,
3-manifold, branched covering, Dijkgraaf-Witten invariant,
bordism group, Chern-Simons invariant, the extended Bloch group.

**Contents**

**1** **Introduction** **2**

**2** **Review: 4-fold symmetric quandle homotopy invariant** **4**
**3** **4-fold symmetric quandle homotopy invariants as natural transforma-**

**tions** **6**

3.1 Preliminaries: the symmetric link quandle and the associated group . . . . 7
3.2 *G*e* _{c}*-colorings of labeled diagrams . . . 8
3.3 A fundamental symmetric quandle of a 3-manifold . . . 11
3.4 4-fold symmetric quandle homotopy invariants as natural transformations . 12

**4**

**Formulas for the connected sum and the opposite orientation**

**13**

**5** **Dijkgraaf-Witten invariant** **16**

5.1 Preliminary: bordism Dijkgraaf-Witten invariant . . . 16
5.2 From Π^{4f}_{2,ρ}(*G*e*c*) to the oriented bordism group Ω3(G, c) . . . 18
5.3 Applications . . . 21
**6** **Chern-Simons invariant as a quandle cocycle invariant** **23**
6.1 Review: 4-fold symmetric quandle cocycle from normalized group cocycle 23
6.2 Review: Cheeger-Chern-Simons class from the extended Bloch group. . . . 25
6.3 Chern-Simons invariant as a quandle cocycle invariant . . . 26
6.4 Recovery of Chern-Simons invariant from the multiplication by 6. . . 27

**1** **Introduction**

Our motivation stems from a topic of Dijkgraaf-Witten invariant described in (1) be-
low. Let *M* be an oriented compact 3-manifold. Dijkgraaf and Witten [DW] discussed
the relation between Chern-Simons action and Wess-Zumino-Witten term on *M* through
the cohomology of the Eilenberg-MacLane space *H*^{3}(K(G,1);Z), where *G* is a compact
Lie group. If *M* has no boundary, then the quantum ﬁeld theory of the Chern-Simons
functional in *H*^{3}(K(G,1);C*/*Z) can be interpreted as an obstruction class in the oriented
bordism group Ω_{3}(K(G,1)). Furthermore, one of their noteworthy results is that, when *G*
is ﬁnite, the path integral of the distribution function of a 3-cocycle *ψ* *∈H*^{3}(K(G,1);*A)*
reduces to a ﬁnite sum

DW* _{ψ}*(M) = ∑

*f**∈*Homgrp(π1(M),G)

*⟨f** ^{∗}*(ψ),[M]

*⟩ ∈*Z[A], (1)

where [M] *∈* *H*_{3}(M;*A) is the fundamental class of* *M*. From a mathematical viewpoint,
Wakui [W] rigorously formulated DW* _{ψ}*(M) of certain 3-cocycles

*ψ*

*∈*

*H*

^{3}(G;

*A) in terms*of a triangulation of

*M*. Recently, the ﬁrst author [H1] reformulated the Wakui formula as a “quandle cocycle invariant” of links, using the fact that any closed 3-manifold is a 4-fold branched covering of

*S*

^{3}.

Inspired by her work, the second author [N2] introduced a 4-fold symmetric quandle
homotopy invariant of 3-manifolds using a quandle*G*e* _{c}*. Here,

*G*is a ﬁnite group and

*c∈G*is its central element satisfying

*c*

^{2}=

*e, and*

*G*e

*c*is a certain quandle deﬁned with respect to the pair of (G, c). The invariant of

*M*is deﬁned to be a set of “

*G*e

*-colorings” with a grading by an abelian group Π*

_{c}^{4f}

_{2,ρ}(

*G*e

*). Here, if*

_{c}*M*is a 4-fold branched covering of

*S*

^{3}along a link

*L, aG*e

*-coloring is roughly a homomorphism*

_{c}*π*

_{1}(S

^{3}

*\L)→G*

^{4}oS

_{4}compatible with the monodromy

*π*

_{1}(S

^{3}

*\L)→*S

_{4}. On the other hand, the group Π

^{4f}

_{2,ρ}(

*G*e

*) is deﬁned to be a certain link bordism group of*

_{c}*G*e

*c*-colorings. The previous paper [N2] studied the quandle structure of

*G*e

*and estimated Π*

_{c}^{4f}

_{2,ρ}(

*G*e

*) in an algebraic viewpoint. It is shown [N2,*

_{c}*§*7]

that the invariant produces the above quandle cocycle invariants considered in [H1].

In this paper, we topologically study the 4-fold symmetric quandle homotopy invariant.

For this, we give a topological interpretation of the *G*e* _{c}*-colorings, and relate the group
Π

^{4f}

_{2,ρ}(

*G*e

*c*) with some topological objects. We ﬁrst show a natural bijection between

*G*

^{3}

*×*Hom

_{grp}(π

_{1}(M), G) and the set of

*G*e

*-colorings (Theorem 3.3). Note that the set of*

_{c}*G*e

*- colorings is a classical invariant and is independent of the central element*

_{c}*c∈G.*

We next give our invariants some functoriality (*§*3.4). We introduce a fundamental
symmetric quandle *SQ(M*) of *M* (Deﬁnition 3.5). *SQ(M*) is roughly deﬁned to be a
universal quandle representing the all*G*e*c*-colorings. Using the bijection of Theorem 3.3 and
the universality, we show that the quandle *SQ(M*) is quandle isomorphic to *G(M*^)* _{c(M)}*,
where

*G(M*) :=

*π*

_{1}(M)

*×*Z

*/2*Z and

*c(M*) := (e,1)

*∈*

*G(M*) (Corollary 3.6). Further, we deﬁne the fundamental class to be a canonical class of a link bordism using

*SQ(M*)

(Deﬁnition 3.7). Consequently, the study of the 4-fold homotopy invariant of *M* is a
research of the fundamental class using relativity to other 4-fold symmetric quandles *G*e*c*

(see *§*3.4 for detail).

Next, in order to study the group Π^{4f}_{2,ρ}(*G*e* _{c}*) mentioned above, we compare the4-fold
symmetric quandle homotopy invariants with the Dijkgraaf-Witten invariants given in
(1). For this, we take a perspective of a bordism group rather than the state sum formula.

We then work with the bordism Dijkgraaf-Witten invariant deﬁned by using an oriented
bordism group Ω_{3}(G, c) of the pair of (G, c) (see *§*5.1). From the viewpoint regarding
Π^{4f}_{2,ρ}(*G*e* _{c}*) as a certain link bordism, we canonically obtain an epimorphism Π

^{4f}

_{2,ρ}(

*G*e

*)*

_{c}*→*Ω

_{3}(G, c), and show that the bordism invariant is derived from the 4-fold symmetric quandle homotopy invariant (Theorem 5.4). We remark that, when

*c*=

*e, the bordism Dijkgraaf-*Witten invariant produces DW

*(M) in (1) for*

_{ψ}*any*3-cocycle

*ψ*

*∈*

*H*

^{3}(K(G,1);

*A) (see*Remark 5.1); hence, so does the 4-fold symmetric quandle homotopy invariant. For the future, it is a problem whether the epimorphism is isomorphic or not (Problem 5.7). If this is isomorphic, the two invariants in Theorem 5.4 are equivalent.

As an application of bordism groups, we succeed in giving a topological interpretation
of two combinatorial invariants of links. For an odd *m* *∈* Z, the dihedral quandle *R** _{m}* is
deﬁned to beZ

*/m*Z with a binary operation

*x∗y*= 2y

*−x. Then the quandle homotopy*invariant of oriented links

*L⊂S*

^{3}with respect to

*R*

*m*is deﬁned by a combinatorial method (see, e.g., [FRS, N1]). Let

*M*

*be the double branched cover of the link*

_{L}*L. Then we show*that the quandle homotopy invariant of

*L*is equivalent to the Dijkgraaf-Witten invariant of

*M*

*with respect to*

_{L}*G*= Z

*/m*Z (Corollary 5.9). As a special case, if

*m*is a prime

*p,*then it is known [N1] that “the quandle cocycle invariant Φ

_{θ}*(L)*

_{p}*∈*Z[t]/(t

^{p}*−*1) using Mochizuki 3-cocycle” is equivalent to the quandle homotopy invariant of links, leading to an equality Φ

_{θ}*(L) =*

_{p}*at*

*DW*

^{n}*(M*

_{ψ}*) for some*

_{L}*a, n∈*Z(Corollary 5.11). As a corollary, we compute the Dijkgraaf-Witten invariants of some 3-manifolds (Example 5.12, 5.13, 5.14), using known values of the quandle cocycle invariants Φ

_{θ}*(L) in [AS, Iwa1, Iwa2].*

_{p}In another direction, when*G*=*SL(2;*C) or*P SL(2;*C), we discuss the Cheeger-Chern-
Simons class *C*b2 *∈* *H*^{3}(G;C*/4π*^{2}Z). Given a homomorphism *f* : *π*1(M)*→* *G, the Chern-*
Simons invariant of *M* is deﬁned to be the pairing *⟨f** ^{∗}*(

*C*b

_{2}),[M]

*⟩ ∈*C

*/4π*

^{2}Z. It had been a long-standing problem to provide a computation of the Cheeger-Chern-Simons class and the Chern-Simons invariant. Dupont [Dup] gave an answer modulo

*π*

^{2}Q. Lifting his formula, Neumann [Neu] has obtained an explicit formula for

*C*b

_{2}with

*G*=

*P SL(2;*C) via the extended Bloch group

*B*b(C), and a computation of the Chern-Simons invariant in term of a triangulation of

*M*. Further, Dupont, Goette and Zickert succeeded in an extension of the formula suitable for

*G*=

*SL(2;*C) [DG, DZ]. From quandle theory, Inoue-Kabaya [IK] in ’09 reconstructed the Chern-Simons invariant of knot complements

*S*

^{3}

*\K*as a quandle cocycle invariant, using

*B*b(C). In this paper, as an analogy, for

*G*=

*SL(2;*C), we reconstruct the Chern-Simons invariant of closed 3-manifolds as a quandle cocycle

invariant through*B*b(C), using the Dupont formula [Dup] and a result in [H1] (Theorem 6.5
and *§*6.4). Similar to Inoue-Kabaya’s result [IK], a beneﬁt of our reformulation is that we
can combinatorially compute the Chern-Simons invariant only from the homomorphism
*f* : *π*_{1}(M) *→* *SL(2;*C) and the monodromy *ϕ* : *π*_{1}(S^{3} *\L)* *→* S_{4} of a 4-fold branched
covering *M* *→S*^{3}, without using triangulation of *M*.

Lastly, we outline the reconstruction. The point follows from the Dupont formula
[Dup] rather than *B*b(C). He reformulated 6*C*b_{2} as a function on a conﬁguration space
Conf4(SL(2;R)). The reformulation is adequate for a result of the ﬁrst author [H1];

hence, we succeed in reconstructing 6 multiple of the Chern-Simons invariant (Theorem
6.5). Finally, the Chern-Simons invariant makes a recovery from the multiplication by 6,
using the Dijkgraaf-Witten invariants with respect to cyclic groups (*§*6.4).

This paper is organized as follows. In *§*2, we review some notation of 4-fold sym-
metric quandle homotopy invariants. In *§*3, we give a topological interpretation of the
*G*e* _{c}*-colorings. In

*§*4, we show the connected sum formula. In

*§*5, we compare the 4-fold symmetric quandle homotopy invariant with the bordism Dijkgraaf-Witten invariant. In

*§*6, we reformulate the Chern-Simons invariant as a quandle cocycle invariant.

**2** **Review: 4-fold symmetric quandle homotopy invariant**

We brieﬂy review some notation of 4-fold symmetric quandle homotopy invariants in [N2,

*§*2 and 3]. Throughout this paper, manifolds are assumed to be *C** ^{∞}*-smooth, oriented,
connected and compact. Unless

*§*5, we assume that manifolds have no boundary.

We ﬁrst review symmetric quandles introduced by Kamada [Kam]. A*symmetric quan-*
*dle* is a triple of a set *X, a binary operation* *∗* on *X* and an involution *ρ* : *X* *→* *X*
satisfying that, for any *x, y, z* *∈* *X, x∗x* =*x,* (x*∗y)∗z* = (x*∗z)∗*(y*∗z), ρ(x∗y) =*
*ρ(x)∗y,* (x*∗y)∗ρ(y) =x.* For example,*S* :=*{*(ij)*∈*S4*}* with*x∗y*:=*yxy* and*ρ(x) =* *x*
is a symmetric quandle. We give another example introduced in [N2, Example 4.1] as
follows. We consider a pair of a group *G* and its central element *c∈G* such that *c*^{2} = *e,*
where*e∈G*is the identity element. Such a pair of (G, c) is called a *cored group. Putting*
*T*_{12} :=*{*(i, j)*∈*Z^{2}*|*1*≤i, j* *≤*4, i *̸*=*j}*, we deﬁne*G*e* _{c}*to be a quotient set

*G×T*

_{12}

*/∼*, where the equivalence relation

*∼*on

*G×T*12 is deﬁned by (g, i, j)

*∼*(g

^{−}^{1}

*c, j, i), for (i, j)*

*∈*

*T*12

and *g* *∈G. We equip* *G*e* _{c}* with an operation

*∗*:

*G*e

_{c}*×G*e

_{c}*→G*e

*deﬁned by (g, i, j)*

_{c}*∗*(g

^{′}*, i, j) = (g*

^{′}*g*

^{−}^{1}

*g*

^{′}*, i, j),*(g, i, j)

*∗*(g

^{′}*, j, k) = (gg*

^{′}*, i, k),*(g, i, j)

*∗*(g

^{′}*, k, l) = (g, i, j),*

where *i, j, k, l* are distinct indices. Further, deﬁne *ρ* : *G*e_{c}*→* *G*e* _{c}* by

*ρ(g, i, j) = (gc, i, j).*

Then (*G*e*c**, ρ) is a symmetric quandle. Moreover, putting a projection* *p*_{G}_{e}

*c* : *G*e*c* *→ S*
which sends (g, i, j) to (ij)*∈ S*, the triple of (*G*e_{c}*, ρ, p*_{G}_{e}

*c*) satisﬁes the axioms of the *4-fold*
*symmetric quandle* (see [N2, Deﬁnition 3.1] for detail). Remark that if *G* = *{e}*, then
*G*e_{e}*∼*=*S*.

For simplicity, in this paper we denote three elements (e,1,2), (e,2,3), (e,3,4) *∈* *G*e* _{c}*
by

*e*

_{12},

*e*

_{23},

*e*

_{34}, respectively.

We review *X**ρ*-colorings. Let *D*be an unoriented link diagram on R^{2}. For a symmetric
quandle (X, ρ), an*X** _{ρ}*-coloringof

*D*is a map

*C*:

*{*the two normal orientations on arcs of

*D}*

*→X* satisfying the following two conditions:

(X1) For the two orientations *α*_{1}, *α*_{2} of the same arc as shown in Figure 1, the colors
satisfy *C(α*_{1}) =*ρ*(

*C(α*_{2}))

. (Hence, we will later draw the only one color of the two).

(X2) At each crossing shown in Figure 1, the three orientations satisfy*C(γ) =* *C(α)∗C(β).*

*α*

_{1}

*α*

_{2}

*C(α*_{1}) =*ρ*(
*C(α*_{2}))

*α* *β*

*γ*

*C(γ) =**C(α)**∗**C(β)*

Figure 1: The conditions of symmetric colorings on orientations.

Note that the conditions (X1)(X2) are well-deﬁned by the axioms of symmetric quandles.

Denote by Col*X,ρ*(D) the set of all *X**ρ*-colorings of *D. It is known [KO, Proposition 6.2]*

that, if two diagrams*D*_{1}and*D*_{2}are related by Reidemeister moves, there exists a bijection
between Col* _{X,ρ}*(D

_{1}) and Col

*(D*

_{X,ρ}_{2}).

For a symmetric quandle (X, ρ), an (X, ρ)-set is a set Λ equipped with a map *∗* :
Λ*×X* *−→* Λ satisfying (λ*∗x)∗x** ^{′}* = (λ

*∗x*

*)*

^{′}*∗*(x

*∗x*

*) and (λ*

^{′}*∗x)∗ρ(x) =*

*λ*for any

*λ∈*Λ and

*x, x*

^{′}*∈X. For example,X*is an (X, ρ)-setitself by the quandle operation. An

*X*

_{Λ}-coloring of D is deﬁned to be an

*X*

*-coloring of*

_{ρ}*D*with an assignment of elements of Λ to each complementary regions of

*D*such that, for each regions separated by the arc with a color

*x∈X, the colors and assignments satisfy the following picture.*

*x* [λ]

[λ* ^{′}*]

*λ∗x*=*λ*^{′}*.* (λ, λ^{′}*∈*Λ)

We will interpret 3-manifolds as *S*id-colorings, where *S* = *{*(ij) *∈* S_{4}*}* as above. It is
well-known that any 3-manifold *M* is a 4-fold simple covering of *S*^{3} branched over a link
*L* with its monodromy *ϕ* :*π*_{1}(S^{3}*\L)* *→*S_{4}. Remark that *ϕ* is surjective and sends each
meridian of *L* to a transposition inS_{4} (see, e.g., [N2,*§*2.2]). A link diagram *D* of *L*with
such a monodromy*ϕ* is called a*labeled diagram*and denoted by*D** _{ϕ}*. Notice that a labeled
diagram can be regarded as an

*S*id-coloring of

*D*by Wirtinger presentation (see [PS,

*§*24]

for detail). A labeled diagram *D**ϕ* is said to be *3-fold, if its subdiagram of* *D**ϕ* labeled by
(34) is a single unknot as shown in Figure 2. It is known that *M* can be regarded as a
3-fold labeled diagram *D** _{ϕ}* (see, e.g., [R,

*§*10.D]). We ﬁx three orientations

*α*

_{12}

*, α*

_{23}

*, α*

_{34}of three distinct arcs in

*D*

*labeled by (12),(23),(34)*

_{ϕ}*∈*S

_{4}as shown in Figure 2, respectively.

We denote by Col^{e}_{e}^{12}^{,e}^{23}^{,e}^{34}

*G**c**,ρ* (D* _{ϕ}*) the subset of

*G*e

*-colorings*

_{c}*C*of

*D*such that

*p*

_{G}_{e}

*c*(C) =*D** _{ϕ}*
and these orientations

*α*

*are colored by*

_{ij}*e*

*= (e, i, j)*

_{ij}*∈G*e

_{c}*,*for (ij)

*∈ {*(12),(23),(34)

*}*.

*D* _{ϕ}

_{ϕ}

^{(34)}

(23) (12)

*α*_{34}
*α*_{23}

*α*_{12}

Figure 2: A 3-fold labeled diagram with three orientations*α*12*, α*23*, α*34*.*

For a cored group (G, c), we review an abelian group Π^{4f}_{2,ρ}(*G*e* _{c}*) deﬁned in [N2,

*§*3.1].

Π^{4f}_{2,ρ}(*G*e* _{c}*) is deﬁned by a quotient set of all

*G*e

*-colorings of all link diagrams*

_{c}*D*modulo Reidemeister moves, symmetric concordance relations and MI, II moves. Here,

*symmetric*

*concordance relations*and

*MI, II moves*are local moves as shown in Figure 3 and 4, respec- tively. Then we impose an abelian structure on Π

^{4f}

_{2,ρ}(

*G*e

*c*) by letting disjoint union deﬁne our multiplication. For a labeled diagram

*D*

*, we put a map Ξ*

_{ϕ}^{4f}

_{e}

*G**c*(D* _{ϕ}*;

*•*) : Col

^{e}_{e}

^{12}

^{,e}^{23}

^{,e}^{34}

*G**c**,ρ* (D* _{ϕ}*)

*→*Π

^{4f}

_{2,ρ}(

*G*e

*c*) which sends a

*G*e

*c*-coloring of

*D*

*ϕ*to the canonical class (see [N2, (3)]).

Let *M* be a 3-manifold presented by a 3-fold labeled diagram *D** _{ϕ}*. When

*G*is ﬁnite, the second author deﬁned a

*4-fold symmetric quandle homotopy invariant*of

*M*by the following formula (see [N2, Deﬁnition 3.3 and Lemma 4.6]).

Ξ^{4f}_{e}

*G**c*(M) = *|G|*^{3} ∑

*C**∈*Col^{e}_{e}^{12}^{,e}^{23}^{,e}^{34}

*Gc,ρ* (D*ϕ*)

Ξ^{4f}_{e}

*G**c*(D* _{ϕ}*;

*C)∈*Z[Π

^{4f}

_{2,ρ}(

*G*e

*)]. (2) The deﬁnition does not depend on the choice of labeled diagrams and the three arcs (see [N2, Theorem 3.4 and Lemma 4.6]).*

_{c}*a* *ρ(a)*

*a* *ρ(a)*

*a* *ρ(a)* *ρ(a)*

*a*

*a ρ(a)*

*∅*

Figure 3: symmetric concordance relations. Here*a**∈**G*e* _{c}*.

**3** **4-fold symmetric quandle homotopy invariants as natural trans-** **formations**

In*§*3.2, we give a topological meaning of*G*e* _{c}*-colorings. In

*§*3.3, we introduce a fundamental symmetric quandle

*SQ(M) of a 3-manifold*

*M*, and give an interpretation of

*G*e

*-colorings*

_{c}(g, i, j) (h, j, k) (g, i, j) (h, j, k)

(g, i, j) (h, j, k) (gh, i, k)

(f, k, l) (g, i, j)

(f, k, l) (g, i, j)

(g, i, j) (f, k, l)

Figure 4: MI, II moves of *G*e* _{c}*-colorings. Here, (g, i, j), (h, j, k), (f, k, l)

*∈*

*G*e

*, and*

_{c}*i, j, k, l*are distinct.

as a representable functor using*SQ(M*). Further, we interpret a 4-fold symmetric quandle
homotopy invariant as a natural transformation.

**3.1** **Preliminaries: the symmetric link quandle and the associated group**
Let *L* be an unoriented link in *S*^{3}. We brieﬂy review the symmetric link quandle of *L*
introduced by Kamada [Kam]. Let *SQ(L) be the set of homotopy classes of all pairs of*
(*D, γ), where* *D* means an oriented meridian disk of *L* and *γ* means a path in *S*^{3} *\* *L*
starting from a point of the boundary *∂D* and ending at a ﬁxed base point in *S*^{3}*\L. We*
equip *SQ(L) with a binary operation given by*

[(*D*1*, γ*_{1})]*∗*[(*D*2*, γ*_{2})] := [(*D*1*, γ*_{1}*·γ*_{2}^{−}^{1}*·∂D*2*·γ*_{2})],

and an involution *ρ* of *SQ(L) given by* *ρ([(D, γ)]) = [(−D, γ)], where* *−D* stands for the
disk *D* with the opposite orientation. Then *SQ(L) turns out to be a symmetric quandle*
(see also [KO, Example 2.4]).

We will explain the correspondence (3) below. Let (X, ρ) be a symmetric quandle and
*D*an unoriented link diagram of *L. Let us denote by Hom*_{sQnd}(SQ(L), X) the set of maps
*SQ(L)* *→* *X* preserving the operations *∗* and *ρ, which are called (symmetric quandle)*
*homomorphisms. Kamada [Kam] gave a canonical bijection*

*Q*(*•*) : Col* _{X,ρ}*(D)

*−→*Hom

_{sQnd}(SQ(L), X), (3)

where, for an*X** _{ρ}*-coloring

*C,Q*(C) is deﬁned to be a homomorphism sending the meridian disk of an arc

*α*to the color of

*C*on

*α. This bijection is analogous to [Joy,*

*§*16].

We study labeled diagrams from a view of*SQ(L). Recall that any labeled diagramD**ϕ*

can be regarded as an*S*id-coloring. By substituting the bijection (3) to*X* =*S*, we see that
the map *ϕ* *∈* Hom_{sQnd}(SQ(L),*S*) associated to *D** _{ϕ}* through (3) is surjective. We denote
three meridian disks obtained from the previous arcs

*α*

_{12}

*, α*

_{23}

*, α*

_{34}, by

*D*12

*,*

*D*23

*,*

*D*34

*∈*

*SQ(L), respectively. Then, as a restriction of the bijection (3), we obtain a bijection*

Col^{e}_{e}^{12}^{,e}^{23}^{,e}^{34}

*G**c**,ρ* (D* _{ϕ}*)

*≃*Hom

^{(}

_{4sQnd}

^{D}^{12}

^{,}

^{D}^{23}

^{,}

^{D}^{34}

^{)(e}

^{12}

^{,e}^{23}

^{,e}^{34}

^{)}(SQ(L),

*G*e

*), (4) where Hom*

_{c}^{(}

_{4sQnd}

^{D}^{12}

^{,}

^{D}^{23}

^{,}

^{D}^{34}

^{)(e}

^{12}

^{,e}^{23}

^{,e}^{34}

^{)}(SQ(L),

*G*e

*) is deﬁned to be the set of symmetric homo- morphisms*

_{c}*f*:

*SQ(L)*

*→*

*G*e

*satisfying*

_{c}*f(D*

*ij*) =

*e*

*= (e, i, j)*

_{ij}*∈*

*G*e

*and*

_{c}*p*

_{G}_{e}

*c* *◦f* = *ϕ* *∈*
Hom_{sQnd}(SQ(L),*S*).

We review the *associated group* [KO] of a symmetric quandle (X, ρ) deﬁned by the
following presentation:

As(X)* _{ρ}* =

*⟨x∈X*

*|*

*y·*(x

*∗y) =x·y, ρ(x) =*

*x*

^{−}^{1}(x, y

*∈X)⟩.*

Notice that a symmetric quandle homomorphism *f* : (X, ρ) *→* (X^{′}*, ρ** ^{′}*) induces a group
homomorphism As(f) : As(X)

*ρ*

*→*As(X

*)*

^{′}*ρ*

*.*

^{′}Lastly, we discuss a canonical map *i** _{X}* :

*X*

*→*As(X)

*deﬁned by*

_{ρ}*i*

*(x) =*

_{X}*x. Note that*if

*X*=

*G*e

*, then*

_{c}*i*

_{G}_{e}

*c* is injective (cf. [N2, Lemma 3.8]). Further, we consider the case where
*X* is the link quandle *SQ(L). Put a map* *i** _{L}* :

*SQ(L)*

*→*

*π*

_{1}(S

^{3}

*\L) given by that, for a*meridian disk

*D*,

*i*

*(*

_{L}*D*)

*∈*

*π*

_{1}(S

^{3}

*\L) corresponds with a loop of the boundary*

*∂D*. Then

*i*

*L*passes to a group homomorphism As(SQ(L))

*ρ*

*→*

*π*1(S

^{3}

*\L). It can be veriﬁed that*this is an isomorphism by Wirtinger presentation (cf. [Joy,

*§*14 and 15]). Moreover, we can check that

*i*

*is entirely the above map*

_{L}*i*

*:*

_{X}*X*

*→*As(X)

*with*

_{ρ}*X*=

*SQ(L), and that*the image Im(i

*) consists of the conjugacy classes of the meridians of*

_{L}*L.*

**3.2** *G*e_{c}**-colorings of labeled diagrams**
Our objective is to show Theorem 3.3.

Given a labeled diagram *D**ϕ* which presents a 3-manifold *M*, we ﬁrst introduce the
associated cored group denoted by (*G**ϕ**, c** _{ϕ}*) as follows. Recall the associated monodromy
representation

*ϕ*:

*π*

_{1}(S

^{3}

*\L)−→*S

_{4}in

*§*3.1. Let (S

_{4})

_{1}denote a subgroup

*{σ*

*∈*S

_{4}

*|σ(1) =*1

*}*(

*∼*= S3). Let

*S*^

^{3}

*\L*be the 4-fold (unbranched) covering associated to

*ϕ. Then an*isomorphism

*π*

_{1}(

*S*^

^{3}

*\L)∼*=

*ϕ*

*((S*

^{−1}_{4})

_{1}) is known (see, e.g., [H1,

*§*3.1]) and the boundary of

*S*^

^{3}

*\L*consists of 3♯L-tori. Let

*D*be a link diagram of

*L, and*

*l*the number of the arcs of

*D. For 1*

*≤*

*t*

*≤*

*l, we let*

*m*e

_{t,j}*∈*

*π*

_{1}(

*S*^

^{3}

*\L) be the meridian associated obtained from*each lifted arcs, where

*j*

*∈ {*1,2,3

*}*. We may assume that, for 1

*≤t*

*≤l,*

*m*e

_{t,1}*∈π*

_{1}(

*S*^

^{3}

*\L)*is obtained from the branched locus of index 2. Then, we deﬁne a normal subgroup of

*π*

_{1}(

*S*^

^{3}

*\L) by*

*N** _{ϕ}* =

*⟨*[

*m*e

_{t,1}*, π*

_{1}(

*S*^

^{3}

*\L)],*

*m*e

*(*

_{t,1}*m*e

_{t}*′*

*,1*)

^{−}^{1}

*,*(

*m*e

*)*

_{t,1}^{2}

*,*

*m*e

_{t,2}*,*

*m*e

*(1*

_{t,3}*≤t, t*

^{′}*≤l)⟩,*(5) where [

*m*e

_{t,1}*, π*

_{1}(

*S*^

^{3}

*\L)] are their commutator subgroups and the symbol ‘⟨ ⟩*’ temporarily stands for the normal closure in

*π*

_{1}(

*S*^

^{3}

*\L). Deﬁne a group*

*G*

*ϕ*=

*π*

_{1}(

*S*^

^{3}

*\L)/N*

*. Notice*

_{ϕ}*π*

_{1}(M)

*∼*=

*π*

_{1}(

*S*^

^{3}

*\L)/⟨m*e

*(1*

_{t,j}*≤*

*t*

*≤*

*l,*1

*≤*

*j*

*≤*3)

*⟩*by Van Kampen theorem, leading a canonical epimorphism

*G*

*ϕ*

*−→*

*π*

_{1}(M). We can verify the epimorphism is a central extension, and the kernel is either Z

*/2*Z or 0. Let us denote by

*c*

*ϕ*a generator of the kernel. We deﬁne a cored group (

*G*

*ϕ*

*, c*

*) as required.*

_{ϕ}**Proposition 3.1.** *Let* (G, c) *be a cored group, andD*_{ϕ}*a labeled diagram which presents a*
*3-manifold* *M. Then, there is a canonical bijection*

Col^{e}_{e}^{12}^{,e}^{23}^{,e}^{34}

*G**c**,ρ* (D*ϕ*)*≃*Hom**Grp*** _{c}*(

*G*

*ϕ*

*, G),*(6)

*where* Hom_{Grp}* _{c}*(

*G*

*ϕ*

*, G)*

*is defined to be the set of group homomorphisms*

*f*:

*G*

*ϕ*

*→*

*G*

*satisfying*

*f*(c

*ϕ*) =

*c∈G.*

*Proof.* In the case of*c*=*e, the ﬁrst author proved this theorem (see [H1, Proposition 3.5]).*

In general of*c, our proof is analogous to her proof. Then, we sketch a proof of Proposition*
3.1. We often regard*C* *∈*Col^{e}_{e}^{12}^{,e}^{23}^{,e}^{34}

*G**c**,ρ* (D* _{ϕ}*) as a homomorphism

*ψ*:

*SQ(L)→G*e

*by (4).*

_{c}First, given a symmetric quandle homomorphism *ψ* : *SQ(L)* *→* *G*e* _{c}*, we construct
Ψ

*∈*Hom

_{Grp}*(*

_{c}*G*

*ϕ*

*, G) as follows. We consider an injection*

*χ*:

*G*e

_{c}*→*

*G*

^{4}oS

_{4}deﬁned by

*χ(g, i, j) = (a*

_{1}

*, a*

_{2}

*, a*

_{3}

*, a*

_{4}

*,*(ij)), where

*g*=

*a*

*=*

_{i}*ca*

^{−}

_{j}^{1}and

*a*

*=*

_{k}*e*(k

*̸*=

*i, j).*Here S

_{4}acts on

*G*

^{4}by the transformations of components of

*G*

^{4}. The map

*χ*induces a group homomorphism ¯

*χ*: As(

*G*e

*)*

_{c}*→*

*G*

^{4}oS

_{4}. Further, by the previous subsection, we have a commutative diagram:

*SQ(L)*

*i**L*

*ψ* // *G*e_{ _}_{c}

*i*_{Gc}_{e}

v

*χ*

((R

RR RR RR RR RR RR RR R

*π*_{1}(S^{3}*\L)*

As(ψ) //As((*G*e* _{c}*)

*)*

_{ρ}

_{χ}_{¯}

^{//}

*G*

^{4}oS

_{4}

Denote by ¯Ψ a composite group homomorphism ¯*χ◦* As(ψ). Hence, for each meridian
m*∈π*_{1}(S^{3}*\L), if the associated arc of* *D** _{ϕ}* is colored by (g, i, j)

*∈G*e

*, then*

_{c}Ψ(m) = (a¯ _{1}*, a*_{2}*, a*_{3}*, a*_{4}*,*(ij)), where *g* =*a** _{i}* =

*ca*

^{−}

_{j}^{1}and

*a*

*=*

_{k}*e*(k

*̸*=

*i, j).*(7) Let us recall that

*π*

_{1}(

*S*^

^{3}

*\L)*

*∼*=

*ϕ*

^{−}^{1}((S

_{4})

_{1}) is a subgroup of

*π*

_{1}(S

^{3}

*\L). Denote by*Ψe the restriction of ¯Ψ on

*π*

_{1}(

*S*^

^{3}

*\L). Then, the image of*Ψ is contained in the subgroupe

*G*

^{4}o(S4)1

*⊂G*

^{4}oS4. Let

*π*

^{1}

*:*

_{G}*G*

^{4}o(S4)1

*−→G*be the projection on the ﬁrst component in

*G*

^{4}. Then by (7) we can check that the composite homomorphism

*π*

_{G}^{1}

*◦*Ψ sends eache meridians

*m*e

_{t,1}*∈*

*π*

_{1}(

*S*^

^{3}

*\L) to*

*c*

*∈*

*G. Therefore*

*π*

_{G}^{1}

*◦*Ψ induces a homomorphism Ψ :e (

*G*

*ϕ*

*, c*

*)*

_{ϕ}*−→*(G, c) as desired.

Conversely, given Ψ*∈*Hom_{Grp}* _{c}*(

*G*

*ϕ*

*, G), we will construct aG*e

*-coloring. Put the canon- ical section*

_{c}*s*

^{1}

*:*

_{G}*G−→*(S4)1n

*G*

^{4}of

*π*

_{G}^{1}. We deﬁneΨ to be a compositee

*s*

^{1}

_{G}*◦*Ψ

*◦π*

_{G}*, where*

_{ϕ}*π*

_{G}*is the canonical projection*

_{ϕ}*π*

_{1}(

*S*^

^{3}

*\L)−→ G*

*ϕ*. According to [H1], we have known the group presentations of

*π*

_{1}(S

^{3}

*\L) andπ*

_{1}(

*S*^

^{3}

*\L), although we do not write them (see [H1,*

*§*3.1] for more details). Then we can verify that *π*1(S^{3} *\L) is generated by elements of*
*π*_{1}(*S*^^{3}*\L) and three elements* *i** _{L}*(

*D*12), i

*(*

_{L}*D*23), i

*(*

_{L}*D*34)

*∈π*

_{1}(S

^{3}

*\L), that is,*

*π*_{1}(S^{3}*\L) =* *⟨π*_{1}(*S*^^{3}*\L), i** _{L}*(

*D*12), i

*(*

_{L}*D*23), i

*(*

_{L}*D*34)

*⟩.*

Moreover, we can verify that Ψ uniquely extends to a homomorphism ¯e Ψ :*π*_{1}(S^{3}*\L)−→*

*G*^{4}oS_{4} given by ¯Ψ(i* _{L}*(

*D*

*ij*)) =

*χ(e*

*)*

_{ij}*∈G*

^{4}oS

_{4}(see [H1, Page 278]). Then ¯Ψ satisﬁes the condition (7). Hence, Im( ¯Ψ) is contained in Im(χ). We thus have a map

*ψ*:

*SQ(L)−→G*e

*given by (χ)*

_{c}

^{−}^{1}

*◦*Ψ

*◦i*

*L*(see the diagram below). Further, since

*χ(G*e

*c*)

*⊂*

*G*

^{4}oS4 is the

conjugacy class of (c, e, e, e,(12)), we can check that the map *ψ* is a symmetric quandle
homomorphism *SQ(L)→G*e*c*. By the bijection (4), we obtain the required *G*e*c*-coloring of
*D** _{ϕ}* whose arcs

*α*

*are colored by (e, i, j)*

_{ij}*∈G*e

*.*

_{c}The two constructions give the required 1 : 1 correspondence. For an understanding of the proof, we put the following commutative diagram:

*G**ϕ*
Ψ

*π*_{1}(*S*^^{3}*\L)*

*π*_{Gϕ}

oo

Ψe

//*π*_{1}(S^{3}*\L)*

Ψ¯

*SQ(L)*

*ψ*

*i**L*

oo

*G* ^{π}*G*^{4}o(S_{4})_{1}

1

oo *G* //*G*^{4}oS_{4}oo * ^{χ}* ? _

*G*e

*c*

We next discuss the projection *G**ϕ* *→π*_{1}(M) mentioned above.

**Lemma 3.2.** *Let* *M* *be a 3-manifold. There exists a 3-fold labeled diagram* *D*_{ϕ}*which*
*presents* *M* *such that the projection has a splitting:* *G**ϕ**∼*=*π*1(M)*×*Z*/2*Z*.*

*Proof.* It is shown [HMT] that there exists a 3-fold irregular branched covering*p*:*M* *→S*^{3}
such that the set of points at which *p*fails to be a local homeomorphism bounds a disk in
*M*. We put the associated 3-fold labeled diagram*D** _{ϕ}*.

Let us construct a homomorphism *f* : *G**ϕ* *→* Z*/2*Z as follows. Let *ϕ* : *π*_{1}(S^{3} *\K)* *→*
S_{3} be the monodromy, where *K* is a knot in *S*^{3}. Let ˜*l* (resp. *m)*˜ *∈* *π*_{1}(*S*^^{3}*\K) be*
the longitude (resp. meridian) of a torus in *S*^^{3}*\K* of local index 2. Put a map *ι** _{M}* :

*H*

_{1}(

*S*^

^{3}

*\K*;Z)

*→*

*H*

_{1}(M;Z) induced by the inclusion

*S*^

^{3}

*\K ,→*

*M*. Since ˜

*l*bounds a disk in

*M, we have*

*ι*

*(˜*

_{M}*l) = 0∈*

*H*

_{1}(M;Z). Therefore, by a Mayer-Vietoris argument, we conclude that

*H*

_{1}(

*S*^

^{3}

*\K;*Z)

*∼*=

*H*

_{1}(M;Z)

*⊕*Z, where the direct summand Z is generated by ˜

*m*

*∈H*

_{1}(

*S*^

^{3}

*\K;*Z). Putting the projection

*H*

_{1}(

*S*^

^{3}

*\K*;Z)

*→*Z, we deﬁne a composite homomorphism by

*π*_{1}(*S*^^{3}*\K)−−−→H*_{1}(*S*^^{3}*\K;*Z)*−−−−→*^{proj.} Z*−−−−→*^{proj.} Z*/2*Z*,*

where the ﬁrst map is the abelinization. Therefore, from the deﬁnition of*G**ϕ*, the composite
induces a required homomorphism *f* :*G**ϕ**→*Z*/2*Z satisfying *f(c** _{ϕ}*) = 1.

On the other hand, recall that the projection *G**ϕ* *→π*_{1}(M) is a central extension, and
that the kernel is either 0 or Z*/2*Z. Since *f* gives a crossed section of *G**ϕ* *→* *π*_{1}(M), we
easily see *G**ϕ* *∼*=*π*_{1}(M)*×*Z*/2*Zas desired.

In conclusion, we give a topological interpretation of *G*e* _{c}*-colorings:

**Theorem 3.3.** *For a cored group* (G, c) *and a labeled diagram* *D*_{ϕ}*which presents a 3-*
*manifold* *M, we thus have a bijection*

Col_{G}_{e}

*c**,ρ*(D* _{ϕ}*)

*≃G*

^{3}

*×*Hom

_{grp}(π

_{1}(M), G), (8)

*where* Col_{G}_{e}

*c**,ρ*(D* _{ϕ}*)

*is the set of*

*G*e

_{c}*-colorings*

*C*

*satisfying*

*p*

_{G}_{e}

*c*(C) = *D*_{ϕ}*∈*Col_{S}* _{,id}*(D).

*Proof.* Since the set Col_{G}_{e}

*c**,ρ*(D* _{ϕ}*) depends on only

*M*(see [N2, Proposition 3.2]), we may choose a 3-fold labeled diagram

*D*

*ϕ*in Lemma 3.2. Since

*G*

*ϕ*=

*π*1(M)

*×*Z

*/2*Z, we notice a bijection Hom

_{Grp}*(*

_{C}*G*

*ϕ*

*, G)*

*≃*Hom

_{grp}(π

_{1}(M), G). Further, a bijection Col

_{G}_{e}

*c**,ρ*(D* _{ϕ}*)

*≃*

*G*

^{3}

*×*Col

^{e}_{e}

^{12}

^{,e}^{23}

^{,e}^{34}

*G**c**,ρ* (D* _{ϕ}*) is shown [N2, Lemma 4.6]. Hence, the required bijection is obtained
from Proposition 3.1.

As a result, for a ﬁnite cored group (G, c), the cardinally of *G*e* _{c}*-colorings is a classical
invariant, and does not depend on the central element

*c∈G. Hence, our next step in*

*§*4 is to study the group Π

^{4f}

_{2,ρ}(

*G*e

*).*

_{c}Incidentally, as a corollary, we give a topological interpretation of colorings of core
quandles. Given a group *G, we equip* *G* with a symmetric quandle operation of *g∗h* =
*hg*^{−}^{1}*h* and *ρ*= id* _{G}*, called a

*core quandle.*

**Corollary 3.4.** *LetDbe a link diagram of a linkL, and* *Ga group. Denote by* *Q*_{G}*the core*
*quandle on* *G. LetM**L* *be the double branched covering of* *S*^{3} *branched over the link. Then*
*the set of* *Q*_{G}*-colorings* Col_{Q}_{G}* _{,id}*(D)

*is in 1:1 correspondence with*

*G×*Hom

_{grp}(π

_{1}(M

*), G).*

_{L}*Proof.* By Figure 5, we obtain a labeled diagram *D** _{ϕ}* from

*D, where we equip all arcs*of

*D*with labels (12)

*∈ S*and add two unknots labeled by (23) and (34). Then

*D*

*presents*

_{ϕ}*M*

*. Note that the core quandle*

_{L}*Q*

*is isomorphic to the subquandle composed of*

_{G}*{*(g,1,2)

*∈G*e

*e*

*}*by deﬁnitions. Hence, a

*Q*

*G*-coloring of

*D*is regarded as a

*G*e

*e*-coloring of the labeled diagram

*D*

*, i.e., a homomorphism*

_{ϕ}*π*

_{1}(M

*)*

_{L}*→G*by Theorem 3.3.

*D* *D*

(23) (34)

Figure 5: A labeled diagram*D**ϕ* obtained from a link diagram*D.*

**3.3** **A fundamental symmetric quandle of a 3-manifold**

For a 3-manifold*M*, we will deﬁne a fundamental symmetric quandle of*M* and investigate
its property.

Let *D** _{ϕ}* be a labeled diagram which presents

*M*. Recall the associated symmetric quandle epimorphism

*ϕ*:

*SQ(L)*

*→ S*in

*§*3.1. We consider the following equivalent relations on

*SQ(L):*

*R*_{L}^{3,ϕ} :=*⟨x*_{ij}*∗y** _{jk}* =

*ρ(y*

*)*

_{jk}*∗x*

*(x*

_{ij}

_{ij}*∈ϕ*

^{−}^{1}(ij), y

_{jk}*∈ϕ*

^{−}^{1}(jk))

*⟩*

*R*

^{4,ϕ}

*:=*

_{L}*⟨z*

_{ij}*∗w*

*=*

_{kl}*z*

*(z*

_{ij}

_{ij}*∈ϕ*

^{−}^{1}(ij), w

_{kl}*∈ϕ*

^{−}^{1}(kl))

*⟩*

Then, we deﬁne the quotient symmetric quandle *SQ(L)/⟨R*^{3,ϕ}_{L}*, R*^{4,ϕ}_{L}*⟩*. It goes without
saying that the quotient quandle satisﬁes the axioms of the 4-fold symmetric quandle
by deﬁnition (see [N2, Deﬁnition 4.1]). By a discussion similar to [N2, Proposition 3.2],
if two labeled diagrams *D** _{ϕ}* and

*D*

^{′}

_{ϕ}*are related by some ﬁnite sequences of Reidemeis- ter moves and MI, MII moves with*

_{′}*G*=

*{e}*, then we can obtain a symmetric quandle isomorphism

*SQ(L)/⟨R*

^{3,ϕ}

_{L}*, R*

_{L}^{4,ϕ}

*⟩ ∼*=

*SQ(L*

*)/*

^{′}*⟨R*

^{3,ϕ}

_{L}

_{′}

^{′}*, R*

^{4,ϕ}

_{L}

_{′}

^{′}*⟩*. Thus, by the result in [Apo],

*SQ(L)/⟨R*

_{L}^{3,ϕ}

*, R*

^{4,ϕ}

_{L}*⟩*does depend on only the 3-manifold

*M*(see also [N2, Theorem 2.1]).

**Definition 3.5.** For a labeled diagram *D** _{ϕ}* of a 3-manifold

*M*, we deﬁne a

*fundamental*

*symmetric quandle*of

*M*by the quandle

*SQ(L)/⟨R*

^{3,ϕ}

_{L}*, R*

^{4,ϕ}

_{L}*⟩*. We denote it by

*SQ(M).*

Assume that *D**ϕ* is 3-fold. We use notation *D*12*,* *D*23*,* *D*34 *∈* *SQ(L) in* *§*3.1. Re-
call the category of 4-fold symmetric quandles denoted by **Qnd**** _{4s}** (see [N2, Corollary
4.3]). The objects of

**Qnd****consist of**

_{4s}*G*e

*with respect to cored groups (G, c). Let us denote by Hom*

_{c}

_{Qnd}**(SQ(M),**

_{4s}*G*e

*) the set of morphisms in*

_{c}

**Qnd****from**

_{4s}*SQ(M*) to

*G*e

*(see [N2,*

_{c}*§*4.1] for detail). Remark a natural bijection Hom

_{Qnd}**(SQ(M),**

_{4s}*G*e

*)*

_{c}*≃*Hom

^{(}

_{4sQnd}

^{D}^{12}

^{,}

^{D}^{23}

^{,}

^{D}^{34}

^{)(e}

^{12}

^{,e}^{23}

^{,e}^{34}

^{)}(SQ(M),

*G*e

*) described in [N2, Remark 4.4]. By the correspon- dence (4) and Proposition 3.1, we thus have a bijection*

_{c}Hom_{Qnd}** _{4s}**(SQ(M),

*G*e

*)*

_{c}*≃*Col

^{e}_{e}

^{12}

^{,e}^{23}

^{,e}^{34}

*G**c* (D* _{ϕ}*). (9)

Although the deﬁnition of *SQ(M*) seems ad hoc, we give its concrete presentation as
follows:

**Corollary 3.6.** *For a 3-manifold* *M, there exists a symmetric quandle isomorphism*
*SQ(M)∼*=*G(M*^)_{c(M)}*, where* (

*G(M*), c(M))

*is a cored group* (π_{1}(M)*×*Z*/2*Z*,*(e,1)).

*Proof.* Let *D** _{ϕ}* be a 3-fold labeled diagram which presents

*M*in Lemma 3.2. Recall an equivalence of categories between

**Qnd**_{4s}and the category of cored groups (see [N2, Corol- lary 4.3]). Hence, there exists a bijection

Hom_{Grp}* _{c}*(G(M), G)

*≃*Hom

_{Qnd}**(**

_{4s}*G(M*^)

_{c(M}_{)}

*,G*e

*),*

_{c}for any cored group (G, c). By the canonical bijections (6) and (9), we have a natural
equivalence of the following functors from**Qnd**_{4s} to the category of sets:

Hom_{Qnd}** _{4s}**(SQ(M),e

*•*

*c*)

*≃*Hom

_{Qnd}**(**

_{4s}*G(M*^)

_{c(M}_{)}

*,*e

*•*

*c*).

Hence, by Yoneda embedding, we conclude *SQ(M*)*∼*=*G(M*^)* _{c(M)}*.

**3.4** **4-fold symmetric quandle homotopy invariants as natural transformations**
Furthermore, we deﬁne a fundamental class of *M*, and give an interpretation of the 4-fold
symmetric quandle homotopy invariant as a natural transformation.

We ﬁx a 3-fold labeled diagram *D** _{ϕ}* which presents a 3-manifold

*M*. Let us regard Hom

**Qnd****4s**(SQ(M),e

*•*) as a functor from the category of 4-fold symmetric quandles. Fur- ther, we interpret the group Π

^{4f}

_{2,ρ}(e

*•*) described in

*§*2 as such a functor. We now iden- tify Hom

_{Qnd}**(SQ(M),**

_{4s}*G*e

*) with Col*

_{c}

^{e}_{e}

^{12}

^{,e}^{23}

^{,e}^{34}

*G**c**,ρ* (D* _{ϕ}*) by (9). Thus the map Ξ

^{4f}

_{e}

*G**c*(D_{ϕ}*,†*) :
Col^{e}_{e}^{12}^{,e}^{23}^{,e}^{34}

*G**c**,ρ* (D* _{ϕ}*)

*→*Π

^{4f}

_{2,ρ}(

*G*e

*) can be regarded as a natural transformation:*

_{c}Ξ^{4f}_{e}* _{•}* (D

*;*

_{ϕ}*†*) : Hom

_{Qnd}**(SQ(M),e**

_{4s}*•*)

*−→*Π

^{4f}

_{2,ρ}(e

*•*). (10) Let us consider a set of such natural transformations: by Yoneda lemma, we have a bijection

Nat(

Hom_{Qnd}** _{4s}**(SQ(M),e

*•*),Π

^{4f}

_{2,ρ}(e

*•*))

*≃*Π^{4f}_{2,ρ}(SQ(M)),

which sends Ξ_{e}^{4f}* _{•}* (D

*;*

_{ϕ}*†*) to Ξ

^{4f}

_{SQ(M}_{)}(D

*; id*

_{ϕ}*), where id*

_{SQ(M)}*is the identity map of*

_{SQ(M)}*SQ(M).*

**Definition 3.7.** Let*M* be a 3-manifold, and*SQ(M) the fundamental symmetric quandle*
of *M*. A *fundamental class* of *M* is deﬁned to be Ξ^{4f}* _{SQ(M)}*(D

*ϕ*; id

*)*

_{SQ(M)}*∈*Π

^{4f}

_{2,ρ}(SQ(M)).

By the naturality, we can reformulate the formula (2) of the 4-fold homotopy invariant as
Ξ^{4f}_{e}

*G**c*(M) = *|G|*^{3} *·* ∑

*F**∈*Hom**Qnd**** _{4s}**(SQ(M),

*G*e

*c*)

*F** _{∗}*(

Ξ^{4f}_{SQ(M}_{)}(D* _{ϕ}*; id

*))*

_{SQ(M)}*∈*Z[Π^{4f}_{2,ρ}(*G*e* _{c}*)]. (11)

In conclusion, the study of the 4-fold symmetric quandle homotopy invariant of *M* is
roughly the research of Π^{4f}_{2,ρ}(SQ(M)) and of the fundamental class using relativity to
other 4-fold symmetric quandles *G*e* _{c}*.

**Remark 3.8.** We compare the fundamental classes of knots with those of 3-manifolds. In
the theory of quandle homotopy invariants valued in *π*_{2}(BX), the second author showed
that, for any non-trivial knots*K, the homotopy group of the “knot quandle” is isomorphic*
toZ generated by the “fundamental class” (see [N1, Corollary 4.17]). On the other hand,
on the 4-fold homotopy invariant of 3-manifolds *M*, Π^{4f}_{2,ρ}(SQ(M)) is always neither Znor
generated by the fundamental class, but Π^{4f}_{2,ρ}(SQ(M)) does depend on*M*.

**4** **Formulas for the connected sum and the opposite orientation**

In this section, we show the formulas of the 4-fold quandle homotopy invariant for the connected sum and the opposite orientation.

**Proposition 4.1.** *LetM*_{1} *and* *M*_{2} *be 3-manifolds. LetM*_{1}#M_{2} *denote the connected sum*
*of* *M*_{1} *and* *M*_{2}*. For a finite cored group* (G, c),

Ξ^{4f}_{G}_{e}

*c*(M_{1})*·*Ξ^{4f}_{G}_{e}

*c*(M_{2}) = *|G|*^{3} *·*Ξ^{4f}_{G}_{e}

*c*(M_{1}#M_{2}) *∈*Z[Π^{4f}_{2,ρ}(*G*e* _{c}*)]. (12)