Quandle homotopy invariants of knotted surfaces

Given a finite quandle, we introduce a quandle homotopy invariant of knotted surfaces in the 4-sphere, modifying that of classical links. This invariant is valued in the third homotopy group of the quandle space, and is universal among the (generalized) quandle cocycle invariants. We compute the second and third homotopy groups, with respect to"regular Alexander quandles". As a corollary, any quandle cocycle invariant using the dihedral quandle of prime order is a scalar multiple of the Mochizuki 3-cocycle invariant. As another result, we determine the third quandle homology group of the dihedral quandle of odd order.


Introduction
A quandle X is an algebraic system satisfying axioms that correspond to the Reidemeister moves. Given a quandle X, Fenn, Rourke and Sanderson [FRS] defined the rack space BX, in analogy to the classifying spaces of groups; Further, an invariant of framed links in S 3 was proposed in [FRS2,§4], which is called a quandle homotopy invariant, and is valued in the second homotopy group π 2 (BX) [see [N1] for some computations of π 2 (BX)]. In addition, as a modification of the homology H * (BX; Z), Carter et. al [CJKLS] introduced its quandle homology denoted by H Q n (X; A), and further quandle cocycle invariants of classical links (resp. linked surfaces) using cocycles of the cohomology H 2 Q (X; A) (resp. H 3 Q (X; A)). For its application, the homology groups H * (BX; A) and H Q * (X; A) of some quandles X have been computed [Cla,CJKS,M1,M2,N2,NP2]. Furthermore the quandle cocycle invariants were generalized to allow the cohomology H * Q (X; A) with local coefficients [CEGS]. As for classical links, the quandle cocycle invariants were much studied (see, e.g., [NP1,RS,N1,HN]), and are known to be derived from the quandle homotopy invariant above (see [CKS, RS]). A topological meaning of some quandle cocycle invariants was understood from the study of the homotopy group π 2 (BX) [HN].
In this paper, we introduce and study a quandle homotopy invariant of oriented linked surfaces valued in a group ring Z[π Q 3 (BX)] (Definition 2.3), modifying the above quandle homotopy invariant of links in S 3 . Here the group π Q 3 (BX) is defined by a quotient group of the third homotopy group π 3 (BX) (see Remark 2.2 for details). Similar to the quandle homotopy invariant of classical links, that of linked surfaces is shown to be universal among the quandle cocycle invariants with local coefficients of linked surfaces (see §2.3). It is therefore significant to estimate and determine π Q 3 (BX); Moreover, from the study of π Q 3 (BX), we address a problem of determining for what kinds of local coefficients the associated quandle cocycle invariants pick out completely the quandle homotopy invariant. dimension of π 2 (BX) ⊗ Z F p is a quadruple function of h: to be precise, dim Fp (π 2 (BX) ⊗ Z F p ) = 1 + h 2 (h 2 + 11)/12 (Corollary A.4).
As a corollary of our work, for an odd m, we determine H Q 3 (X; Z) of the dihedral quandle X of order m, i.e., X = Z[T ]/(m, T + 1). Since Hatakenaka and the author [HN] computed π 2 (BX) ∼ = Z⊕Z/mZ, it follows from Theorem 3.9 above that H Q 3 (X; Z) ∼ = Z/mZ (Corollary 3.11). When m is an odd prime, H Q 3 (X; Z) ∼ = Z/mZ was conjectured by Fenn, Rourke and Sanderson (see [Oht,Conjceture 5.12]), and solved by Niebrzydowski and Przytycki using homological algebra [NP1]. On the other hand, our proof is another approach from π 2 (BX), and further gives a generalization of the conjecture (see also Remark 3.12).
This paper is organized as follows. In §2, we review quandles and the rack spaces BX, and define the quandle homotopy invariant. In §3, we state our theorems and corollaries. In §4, we review the quandle homology groups, and prove Theorem 3.1 on the rational homotopy group π Q 3 (BX) ⊗ Q. In §5, we show Theorems 3.5 and 3.9. In §6, we compute π 2 (BX) ⊗ Z/pZ of some Alexander quandles X on finite fields.
Notational convention Throughout this paper, p is a prime and F q is a finite field such that | F q | = q = p h . We denote by Z m the cyclic group of order m ∈ Z. A symbol "⊗" is always the tensor product over Z. For a group G, we denote by Z[G] the group ring of G over Z. For a Z-module N, N (p) means the localization at a prime p. Further, Λ * (N) denotes the * -part of the exterior algebra over N. A symbol "pt." stands for a single point.
All embeddings of surfaces into S 4 are assumed to be C ∞ -class. Furthermore, we assume that a surface is connected, oriented and closed.

Quandle homotopy invariants of linked surfaces
We review quandles and the rack spaces in §2.1, and introduce quandle homotopy invariants of linked surfaces in §2.2.

Review of quandles and of rack spaces
A quandle is a set X with a binary operation (x, y) → x * y such that, for any x, y, z ∈ X, x * x = x, (x * y) * z = (x * z) * (y * z) and there exists uniquely w ∈ X satisfying w * y = x. For example, a Z[T ± ]-module M has a quandle structure given by x * y := T x + (1 − T )y, called an Alexander quandle. For a quandle X, the associated group of X is defined by the group presentation As(X) := x ∈ X | x · y = y · (x * y) . Remark that, for any quandle X, As(X) is of infinite order, since As(X) has an epimorphism onto Z obtained by the length of words. An X-set is defined to be a set Y with an action of As(X). For example, when Y = X, the quandle X is an X-set itself by the quandle operation.
We next review X-colorings [CJKLS,Definition 5.1]. Let D be a broken diagram of a linked surface L. In this paper, a broken diagram is in S 3 . An X-coloring is a map C from the set of sheets of D to X such that, for every double-point curve as shown in the left of Figure 1, the three sheets α, β, γ satisfy C(γ) = C(α) * C(β). This definition is compatible with triple-points (see the right of Figure 1). We denote the set consisting of such X-colorings of D by Col X (D). It is widely known (see, e.g., [CJKLS,Theorem 5.6]) that if two diagrams D and D ′ are related by the Roseman moves, then there naturally exists a 1-1 correspondence between Col X (D) and Col X (D ′ ). Furthermore, when X is an Alexander quandle of the form X = Z p [T ± ]/(h(T )) and L is a knotted surface, the set Col X (D) is obtained from the Alexander polynomials of L (see [Ino] for details). Figure 1: Coloring conditions around double-point curves and triple-points. Here a, b, c ∈ X Finally, we review the rack space introduced by Fenn-Rourke-Sanderson [FRS]. Fix an Xset Y . Equipping X and Y with their discrete topology, we start with n≥0 Y ×([0, 1]×X) n , and consider the equivalence relations given by (y; t 1 , x 1 , . . . , x j−1 , 1, x j , t j+1 , . . . , t n , x n ) ∼ (y·x j ; t 1 , x 1 * x j , . . . , t j−1 , x j−1 * x j , t j+1 , x j+1 , . . . , t n , x n ), (y; t 1 , x 1 , . . . , x j−1 , 0, x j , t j+1 , . . . , t n , x n ) ∼ (y; t 1 , x 1 , . . . , t j−1 , x j−1 , t j−1 , x j+1 , . . . , t n , x n ).
Here our description divides the element of Y from the elements of ([0, 1] × X) n by a semicolon ';'. Then the rack space B Y X is defined by the quotient space (we later explain the 3-skeleton for more details in §2.2). By construction, we have a cell decomposition of B Y X, by regarding the projection n≥0 Y × ([0, 1] × X) n → B Y X as the characteristic maps. We easily see that if the action on Y of As(X) is transitive, the space B Y X is pathconnected (see [Cla,Proposition 7] for details). When Y is a single point, we denote the path-connected space B Y X by BX for short. We remark that π 1 (BX) ∼ = As(X) by the definition of the 2-skeleton of BX.

Definition of a quandle homotopy invariant of linked surfaces
Our purpose is to define a quandle homotopy invariant of linked surfaces. This is an analogue of the quandle homotopy invariant of (framed) links in S 3 [FRS2, §3] (see also [N1, §2]). For the purpose, given an X-set Y , we first define a quandle space as follows. Define a subspace by n≥2 (y; t 1 , x 1 , . . . , t n , We denote this by X D Y , and consider a composite ι D Y : Here the second map is the projection mentioned above. Definition 2.1. Let X be a quandle. Let Y be an X-set. We define a space B Q Y X by the cone of the composite map ι D Y : X D Y → B Y X. We refer to the space as quandle space. When Y is a single point, we denote B Q Y X by B Q X for short. The homology H * (B Q X; Z) coincides with the homology of the quandle complex introduced in [CJKLS] (see also §4.1). Roughly speaking, the space B Q X is a geometric realization of the quandle complex.
We now explain the 3-skeleton of the rack space BX and the 4-skeleton of the quandle space B Q X in more details. The 1-skeleton of BX is a bouquet of |X|-circles labeled by elements of X. The 2-skeleton of BX is obtained from the 1-skeleton by attaching 2-cells of squares labeled by (a, b) for any a, b ∈ X, where the 4 edges with X-labels as shown in Figure 2 are attached to the corresponding 1-cells. In addition, we attach |X 3 |-cubes labeled by (a, b, c) ∈ X 3 , whose six faces are labeled as shown in Figure 2, to the corresponding 2-cells. The resulting space is the 3-skeleton of BX. We next explain the 4-skeleton of B Q X. To begin, the 3-skeleton of B Q X is obtained from the 3-skeleton of BX by attaching 3-cells which bound 2-cells labeled by (a, a) for a ∈ X. The 4-skeleton of B Q X is obtained from the 3-skeleton of B Q X by attaching 4-cells as follows: for a, b ∈ X, a 3-cell labeled by either (a, a, b) or (a, b, b) is bounded by a 4-cell, and the 4-dimensional cube labeled by (a, b, c, d) ∈ X 4 bounds the eight corresponding 3cells as shown in Figure 3. Remark that the precceding quandle space considered in [N1, §2] 2 is the 3-skeleton of the space B Q X, and the higher skeleton was not defined in [N1].
In order to construct an invariant of linked surfaces, we prepare a cell decomposition of S 3 . Let L ⊂ S 4 be a linked surface. We fix a broken surface diagram D ⊂ S 3 of L. Regarding  D as a decomposition of S 3 by an immersed surface, we consider the dual decomposition.
Recall that the broken diagram is locally composed of some double-point curves, branchpoints and triple-points (see, e.g., [CS] [CJKLS,§5] for the definitions). We then modify the dual decomposition around each branch-points P B of D as follows. To begin, we add an interval which connects between y and z shown in the middle of Figure 4. Put a square D xyz which bounds the three intervals, and further attach a 3-cell which bounds the square D xyz and contains the branch-point P B (see the right of Figure 4); we here notice that the 3-cell forms a cone on a square. In addition, we attach a cube to the square D xyz from the opposite direction. For example, if the Whitney umbrella crosses another sheet such as the left of Figure 5, we here attach 2-cells and 3-cells to the dual decomposition shown as the right of Figure 5. We orient the modified cell decomposition by using the orientations of D. Next, given an X-coloring C of D, we will construct a map ξ D,C : S 3 → B Q X, as follows. We take the 0-cells of the modified decomposition to the single 0-cell of BX. We send Figure 5: The modification of the dual decomposition II the 1-cells at sheets colored by a to the 1-cell labeled by a. We take the 2-cells around the double-point curves colored by (a, b) to the 2-cell labeled by (a, b). We map the 3-cells around the triple-points colored by (a, b, c) to the 3-cell labeled by (a, b, c) as shown in Figure 6. Further, we take the modified 3-cells around the branch-points colored by a ∈ X to the above cone on the 2-cell labeled by (a, a) (see the right of Figure 4). By collecting them, we obtain a cellular map ξ D,C : S 3 → B Q X. We denote the homotopy class of ξ D,C by Ξ X (D; C) ∈ π 3 (B Q X). By the construction of 4-cells of B Q X, we can verify that the homotopy class Ξ X (D; C) is invariant under the Roseman moves (cf. [CJKLS,Theorem 5.6]). For example, the invariance of the move in the bottom right of [CJKLS,Figure 7] corresponds to the 4-cell in Figure 3. Remark 2.2. It is known that any oriented surface-link is represented by a broken diagram D without branch-point (see [CS]). Recall that we defined the group π Q 3 (BX) to be the quotient of π 3 (BX) by a normal subgroup that takes account of branch-points of knotted surfaces. However, for such a diagram D, we claim that the map ξ D,C : S 3 → B Q X misses the cell of X D Y . Actually, the map ξ D,C can, by definition, be constructed by only the usual dual composition of S 3 by D. Hence, the map ξ D,C : S 3 → B Q X factors through the rack space BX, which implies that the homotopy class Ξ X (D; C) is derived from π 3 (BX). Therefore, for the study of the invariant, it sufficies to analyze the image of (i X ) * : We then define Definition 2.3. Let X be a finite quandle. We define the group π Q 3 (BX) by the image of the induced map π 3 (BX) → π 3 (B Q X) by the inclusion BX → B Q X. Further we define a quandle homotopy invariant of a linked surface L by the following expression: Here D is a broken diagram of L, and Col X (D) denotes the set of X-colorings of D.

Reconstucion of generalized quandle cocycle invariants
We will reformulate the (generalized) quandle cocycle invariants in [CJKLS, CKS, CEGS] from our quandle homotopy invariant. We first remark that (1) has no 1-cell by definition. For an action of π 1 (B Q X) = As(X) on an abelian group A, we put ( This equality implies that the quandle homotopy invariant is universal among the quandle cocycle invariants. We give some remarks on the formula (2). It is not difficult to see that the formula (2) coincides with the combinatorial definition of the (generalized) quandle cocycle invariants in [CEGS,§7], by the definition of the Hurewicz homomorphism and the construction of ξ D,C . Hence, the equality (2) means that, if knowing a concrete presentation of ψ, we can calculate some parts of the quandle homotopy invariant. However there are many choices of actions π 1 (BX) A; then we will approach a problem of determining for what kinds of local systems the associated quandle cocycle invariants pick out the quandle homotopy invariant. §3.1 gives an answer for some Alexander quandles.
Finally, we discuss a local system. Regarding X itself as an X-set, the free A-module A X generated by x ∈ X is a π 1 (BX)-module. We then consider a 4-cocycle ψ ∈ H 4 (B Q X; A) to be a cocycle of H 3 (B Q X; A X ) (see Remark 4.1 for detail); The quandle cocycle invariant of ψ coincides with the shadow cocycle invariant described in [CKS,§5].

Properties of quandle homotopy invariants
We state some properties of quandle homotopy invariants of linked surfaces, similar to those of classical links in [N1, §5]. For this, we review connected components of a quandle X. By definition of X, we note that (• * y) : X → X is bijective for any y ∈ X. Let Inn(X) denote the subgroup of S |X| generated by the all right actions (• * y), and is called the inner automorphism group of X. The connected components of a quandle X are defined by the orbits of the action of Inn(X) on X. For example, we let T ℓ be composed ℓ-points with a binary operation defined by x * y = x for x, y ∈ T ℓ ; then the number of the connected components of T ℓ is ℓ. The quandle T ℓ is called a trivial quandle. Furthermore, a quandle X is said to be connected, if the right action of Inn(X) on X is transitive. For example, it is known [LN, Proposition 1] that a finite Alexander quandle is connected if and only if (1 − T ) is invertible.
We give a slight reduction of quandle homotopy invariants: Proposition 2.4. (cf. [N1,Lemma 5.5]) Let X be a finite connected quandle, x an element of X and D a link diagram of a linked surface L. We fix a sheet of D, and denote Col x X (D) by the subset of Col X (D) satisfying that the sheet is colored by x ∈ X. Then, Proof. For any y ∈ X, We see that the bijection (• * y) : which immediately results the required equality (3).
Furthermore, we can show the formulas for the connected sum and the mirror image of the quandle homotopy invariant. We give the formulas without proofs, since the proofs are similar to the discussions in [N1, §5] on the quandle homotopy invariant of classical links.
Proposition 2.5. (cf. [N1,Proposition 5.1]) Let X be a connected quandle of finite order. Then, for knotted surfaces K 1 and K 2 , where K 1 #K 2 denotes the connected sum of K 1 and K 2 .
Proposition 2.6. (cf. [N1,Proposition 5.7]) Let L be a linked surface. Let −L * denote the mirror image of L with opposite orientation. For any finite quandle X, The similar formulas to Propositions 2.4, 2.5 hold for the quandle cocycle invariants via the equality (2).
3 Results of π Q 3 (BX) and of π 2 (B Q X) In §2.2, we defined an invariant valued in Z[π Q 3 (BX)]. We now state our results on the free and torsion parts of the group π Q 3 (BX) in §3.1. We also compute π 2 (B Q X) in §3.2.
3.1 Results about π Q 3 (BX) To begin with, we determine the free subgroup of π Q 3 (BX) as follows: Theorem 3.1. Let X be a finite quandle with ℓ-connected components (see §2.4 for the definition). Then π Q 3 (BX) is finitely generated. Further, the rational homotopy groups are In particular, if X is connected, π Q 3 (BX) is a finite abelian group.
We later discuss a topological meaning of the quandle homotopy invariant after tensoring (Remark 4.2). Furthermore, we give a corollary: Tables 3,4 and 5], for a connected quandle of order 3, a non-trivial quandle cocycle invariant Φ κ (L) valued in Z[Z 3 ] was proposed. These values are, however, incorrect, e.g., compare Propositions 2.4 and 2.6 with the tables.
Our next step is to study the torsion subgroup of π Q 3 (BX). However, in general, it is difficult to compute explicitly homotopy groups of spaces. In this paper, we then confine ourselves to π Q 3 (BX) of regular Alexander quandles X. Here an Alexander quandle X is said to be regular, if X is connected and the minimal number e satisfying (T e − 1)X = 0 is prime to |X|. For example, given ω ∈ F q with ω = 0, 1, the Alexander quandle on F q obtained by x * y = ωx + (1 − ω)y is regular, since ω q−1 = 1 ∈ F q . Also, for an odd number m ∈ Z, the Alexander quandle of the form D m := Z m [T ]/(T + 1) is regular (D m is called a dihedral quandle). Regular Alexander quandles were partly dealt with in [Cla,§4.1].
First, we give an estimate of π 3 (BX) by homologies of the rack space BX in homological contexts. The homology of BX is usually called rack homology of X (see §4.1 for details).
Proposition 3.4. Let X be a regular Alexander quandle of odd order. Let H R j (X) denote the integral homology H j (BX; Z). Let Y 2 be a product of Eilenberg-MacLane Spaces K(π 2 (BX), 2) × K(H R 2 (X), 1). Then there exists an exact sequence Here H 3 (H R 2 (X); Z) is the third group homology of the abelian group H R 2 (X).
Proposition 3.4 is proven by a routine calculation of Postnikov tower in §5. We see later that π 2 (BX) is determined by H R 2 (X) and H R 3 (X) [Theorem 3.9 and (5)]; in conclusion, π 3 (BX) and its quotient π Q 3 (BX) are estimated from upper bounds using the rack homologies H R j (X) with j ≤ 4, although the exact sequence is not useful to compute π 3 (BX) and π Q 3 (BX). However, under an assumption, we determine π Q 3 (BX) of regular Alexander quandles X: Theorem 3.5. Let X be a regular Alexander quandle of odd order. Let H Q n (X; Z) be the quandle homology with trivial coefficients. Assume that the second homology H Q 2 (X; Z) vanishes. Then π 3 (BX) ∼ = H Q 4 (X; Z) ⊕ Z/2Z and π Q 3 (BX) ∼ = H Q 4 (X; Z).
Remark 3.6. There are many Alexander quandles satisfying the assumption. For example, we later show (Lemma A.6) that, if X is of the form X = F q [T ]/(T − ω) and F q is an extension of odd degree, then H Q 2 (X; Z) ∼ = 0.
We defer the proof until §4.2. As a corollary, we now determine π Q 3 (BX) for all Alexander quandles X of prime order: Corollary 3.7. For an odd prime p, we let X be the Alexander quandle of the form Furthermore, we consider the dihedral quandle, i.e., the case of ω = −1. The third cohomology H 3 Q (X; F p ) ∼ = F p is shown [M1, M2], and the generator θ p ∈ H 3 Q (X; F p ) is called Mochizuki 3-cocycle. Any quandle cocycle invariant using the dihedral quandle is summarized to assess the quandle cocycle invariant of the Mochizuki 3-cocycle as follows.
Corollary 3.8. Let X = Z p [T ]/(T + 1) be the dihedral quandle of order p. The group π Q 3 (BX) ∼ = Z p is generated by Ξ X (K; C), where C is an X-coloring of the 2-twist spun (2, p)-torus knot K. Further, any quandle cocycle invariant is equal to a scalar multiple of the quandle cocycle invariant Φ θp (L) ∈ Z[F p ] using the Mochizuki 3-cocycle θ p ∈ H 3 Q (X; F p ).
3.2 Computation of the second homotopy groups of quandle spaces π 2 (B Q X) Changing the subject, we discuss the quandle homotopy invariant of 1-dimensional links considered in [N1]. This invariant is valued in the group ring Z[π 2 (B Q X)], and is universal among the generalized cocycle invariants in [CEGS,§6], similar to the equality (2) (see [N1, §2] for details). For a finite connected quandle X, it is shown [N1, Theorem 3.6 and Proposition 3.12] that π 2 (B Q X) is finite and π 2 (BX) = π 2 (B Q X) ⊕ Z; The author further computed π 2 (B Q X) for connected quandles of order ≤ 6 [N1, §5].
In this paper, we explicitly compute π 2 (B Q X) of regular Alexander quandles X of odd order as the following theorem which we will prove in §5s: Theorem 3.9. Let X be a regular Alexander quandle of finite order.
(ii) If |X| is odd, the following exact sequence splits: Hence we give a stronger estimate of the torsion part of π 2 (B Q X) than [N1, Theorem 3.6].
Corollary 3.10. Let X be a regular Alexander quandle of odd order. Any element of π 2 (B Q X) is annihilated by |X|.
We next give two applications. First, we explicitly compute homotopy groups π 2 (B Q X) for some regular Alexander quandles X. If we concretely know H Q 2 (X; Z) and H Q 3 (X; Z), then Theorem 3.9 enables us to compute π 2 (B Q X). For example, the computations of all regular Alexander quandles X with 5 ≤ |X| ≤ 9 are listed in Table 1 below. In Table 1, the results of H Q 2 (X; Z) and of H Q 3 (X; Z) follow from [LN, Table 1]. For more example, in §A we present a computation of π 2 (B Q X) ⊗ Z p for all Alexander quandles on F q with p > 2. Table 1: Some homotopy groups π 2 (B Q X) obtained from H Q 2 (X; Z) and H Q 3 (X; Z). Here ω = ±1.

Proof of Theorem 3.1
Our purpose is to prove Theorem 3.1 in §4.2. A computation of π 3 (BX) will involve a preliminary analysis of the homology H i (B G X; Z). For this, in §4.1, we review the rack homology and some topological monoids.

Reviews of rack homology, quandle homology and topological monoid
We first review the rack complexes introduced by [FRS] (see also [EG]) and the quandle homologies defined in [CEGS]. For a quandle X, we denote by C R n (X) the free Z[As(X)]module generated by n-elements of X. Namely, C R n (X) = Z[As(X)] X n . Define a boundary homomorphism ∂ n : C R n (X) → C R n−1 (X), for n ≥ 2, to be and ∂ 1 (x 1 ) to be x 1 − 1 ∈ Z[As(X)]. Note that the composite ∂ n−1 • ∂ n is zero. Given a left Z[As(X)]-module M, a complex C R n (X; M) = M ⊗ Z[As(X)] C R n (X) with ∂ n is called the rack complex of X with coefficients M. Next, let C D n (X; M) be a submodule of C R n (X; M) generated by n-tuples (x 1 , . . . , x n ) with x i = x i+1 for some i ∈ {1, . . . , n − 1} if n ≥ 2; otherwise, let C D 1 (X; M) = 0. Since ∂ n (C D n (X; M)) ⊂ C D n−1 (X; M), we denote the homology by H D n (X; Z). Further, a complex C Q * (X; M), ∂ * is defined by the quotient C R n (X; M)/C D n (X; M). The homology H Q n (X; M) is called a quandle homology of X with coefficient M.
For an X-set Y , regarding the free module M = Z Y as a Z[As(X)]-module, the complexes (C R * (X; M), ∂ * ) and (C Q * (X; M), ∂ * ) are chain isomorphic to the cellular complexes of B Y X and of B Q Y X, respectively. As the simplest case, if Y is a single point, then (C R * (X; Z), ∂ * ) and (C Q * (X; Z), ∂ * ) coincide with the rack complex and the quandle complex of X described in [CJKLS], respectively.
Remark 4.1. In the special case Y = X, we can identify C R n+1 (X; Z) with C R n (X; Z X ) from the definitions. Moreover, the identification is a chain map, leading H R n+1 (X; Z) ∼ = H R n (X; Z X ) ∼ = H n (B X X; Z) (see also [FRS2, Theorem 5.12]).
We next review basic properties of the complexes. Let X be a finite quandle with ℓconnected components and M = Z. According to [LN,Theorem 2.2], we have Further, it is known [CJKS,Lemma 3 Furthermore, the following long exact sequence splits (see [LN,Theorem 2.1]): In particular, we have H R n (X; Z) ∼ = H Q n (X; Z) ⊕ H D n (X; Z).
Given a connected quandle X and an X-set Y , we observe the rack space B Y X as a covering as follows. Note that the classing map Y → {pt.} induces a continuous map B Y X → BX. It is known (see [FRS,Theorem 3.7]) that this is a covering; hence, so is the induced map B Q Y X → B Q X. As a special case, if Y = G is a quotient group of As(X) (hence B G X is path-connected), then the two coverings B G X → BX and B Q G X → B Q X are principal G-bundles over BX and B Q X, respectively (see [Cla,Proposition 6]). Recalling π 1 (BX) ∼ = As(X) from §2.1, the group π 1 (B G X) is exactly the kernel of p G : As(X) → G The action of π 1 (B G X) on the higher homotopy groups π * (B G X) is known to be trivial (see [FRS2,Proposition 5.2] and [Cla,Proposition 10]). Furthermore, the action of π 1 (BX) on the higher homology group H * (B G X) is also trivial (see [EG,Lemma 3.1] or [Cla,Remark 7]); hence, the action on H * (B Q G X) is also trivial, since π 1 (BX) = π 1 (B Q X) and H * (B Q G X) is a quotient of H * (B G X) by (7).
Finally, we discuss topological monoids on some rack spaces introduced by Clauwens [Cla]. We consider a natural map X → Inn(X) given by x → (• * x), which yields an epimorphism As(X) → Inn(X). Let G be a quotient group of As(X). We here the epimorphism induces G → Inn(X). Then, Clauwens introduced an operation given by [gh; t 1 , . . . , t n , t ′ 1 , . . . , t ′ m , x 1 · h, . . . , x n · h, x ′ 1 , . . . , x ′ m ], where n, m ∈ Z ≥0 ; this further gives rise to a topological monoid structure on B G X (see [Cla,§2.5]). In particular, π 1 (B G X) is an abelian group. Since B G X is a so-called nilpotent space, we often deal with the localization of B G X at a prime p.

Proof of Theorem 3.1
Proof. Consider G = Inn(X) which is a finite group. Since the rack space B G X is a topological monoid and is a path-connected CW complex of finite type, B G X is homotopic to a (based) loop space of some simply connected CW complex W of finite type, that is, B G X ≃ ΩW as an H-space (see [Mil,Theorem 1.5 and §2] for details). In particular, it is without saying that π * (B G X) is finitely generated; hence, so is the quotient π Q 3 (BX). Next, we discuss rational homologies of B G X and B Q G X. Recall that the projections p Q G : B Q G X → B Q X and p G : B G X → BX are coverings of degree |G|, and that the actions of π 1 (BX) on H * (B Q G X) and on H * (B G X) are trivial. It then follows from the transfer maps that p Q G and p G induce two isomorphisms: Next, we discuss π * (B G X)⊗Q. Since B G X ≃ ΩW , we recall the known following formula (see [FHT,§33(c) where we put r i = dim(π i (B G X) ⊗ Q). By (6) and (8), we then easily have r 2 = (ℓ 2 + ℓ)/2 and r 3 = (ℓ 3 − ℓ)/3. Since π 2 (BX) ∼ = π 2 (B Q X) ⊕ Z ℓ is known (see [N1,Proposition 3.12]), we conclude dim π 2 (B Q X) ⊗ Q = (ℓ 2 − ℓ)/2. We next calculate π Q 3 (BX) ⊗ Q. For this, we equip H * (B G X; Q) with a Hopf algebra arising from the monoid structure, and denote by P * (B G X) the set of primitive elements of H * (B G X; Q). By Milnor-Moore theorem (see, e.g., [FHT,Theorem 21.5]), π * (B G X)⊗Q ∼ = P * (B G X) and the dual Hurewicz homomorphism (H) * ⊗Q : H * (B G X; Q) → Hom(π * (B G X), Q) coincides with the projection H * (B G X; Q) → P * (B G X).
We now explain the isomorphism (10) below. For this, by (8) and the splitting in (7), we notice that the induced map (i X ) * : Further, we consider the long exact sequences of cohomology and homotopy groups of the pair (B Q G X, B G X), and have the following commutative diagram: Here the top and bottom sequences are exact, and the vertical arrows are the Hurewicz homomorphisms. By Milnor-Moore theorem again, we obtain we will calculate P 3 (B G X)∩Ker(∂ * ) using the complexes in §4.1. Inspired by the monoid structure on B G X, we consider a map µ : This provides H * D (X; Q[G]) and H * R (X; Q[G]) with Hopf algebra structures 4 . Further, note that the inclusion i D G : ). Let us identify the above map ∂ * with the induced map (i D G ) * by definitions. By (7) we then notice that In conclusion, the dimension of P 3 (B G X) ∩ Ker(∂ * ) in (10) is equal to Remark 4.2. We roughly explain a topological meaning of the quandle homotopy invariant after tensoring with Q as follows. To illustrate, we put a collapsing map X → T ℓ on each connected components, where T ℓ is the trivial quandle of order ℓ. We see that this map induces π Q 3 (BX)⊗Q ∼ = π Q 3 (BT ℓ )⊗Q by the discussion in the proof; we may assume X = T ℓ . Then we can obtain all cocycles of H 3 R (T ℓ ; Q) ∼ = Q l 3 , since the coboundary maps δ n are zero by definition. Since π Q 3 (BT ℓ ) ⊗ Q is derived from the primitive elements of H R 3 (T ℓ ; Q) by Milnor-Moore theorem, we can evaluate elements of π Q 3 (BT ℓ ) ⊗ Q by the cocycles. On the other hand, we consider the bordism group L 2 ℓ with ℓ-connected components (see [CKSS,§1] for the definition). An isomorphism L 2 is known; further it is shown [CKSS] that L 2 ℓ ⊗ Q is generated by "Hopf 2-links". Recall the map Ξ X (D; •) : Col X (D) → π 3 (B Q X) with X = T ℓ in §2.2. By running over all T ℓ -colorings of all broken diagrams, the maps give rise to a homomorphism L 2 ℓ ⊗ Q → π Q 3 (BX) ⊗ Q. Since we easily compute the rational quandle homotopy invariant of the Hopf 2-links with X = T ℓ by pairing with the previous cocycles, the homomorphism turns to be an isomorphism: 5 Proofs of Theorems 3.5 and 3.9.
Our purpose in this section is to prove Theorem 3.5 in §5.1, and Theorem 3.9 in §5.2. We now outline the proof of Theorem 3.5 as follows. To see this, we first aim to study π 3 (BX), since π Q 3 (BX) is a quotient of π 3 (BX). Let BX be the universal covering of BX. By the monoid structure of BX explained in §4.1, BX is a loop space of a 2-connected CW-complex. Thanks to the fact [AP] that the first k-invariant of BX is annihilated by 2, we thus have π 3 (BX) = π 3 ( BX) ∼ = H 3 ( BX; Z) ⊕ π 3 (ΩS 2 ) by a standard argument of the Postnikov tower. In §5.1, we prove H 3 ( BX; Z) ∼ = H Q 4 (X; Z) using techniques of quandle homologies (see (11)). Finally, we show that H Q 4 (X; Z) does not vanish in π Q 3 (BX), leading π Q 3 (BX) ∼ = H Q 4 (X; Z) as desired. Theorem 3.9 is also proven in a similar way. To carry the outline, we here prepare two propositions, which is proven in §5.3.
Proposition 5.1. Let X be a regular Alexander quandle of finite order. Let G = Inn(X).
(iii) The map H 3 (B G X; Z) → H 3 (B Q G X; Z) induced by the inclusion B G X ֒→ B Q G X coincides with the projection in the sense of the above presentations (i) and (ii).
Proposition 5.2. Let X be a regular Alexander quandle of finite order. Let As(X) → Inn(X) be the epimorphism in §4.1. Then the kernel is isomorphic to π 1 (B G X) = H R 2 (X; Z) = Z ⊕ H Q 2 (X; Z).
Furthermore, we require an elementary lemma:

Proofs of Propositions 5.1 and 5.2
In §5.3 and 5.4, X is assumed to be a finite connected Alexander quandle. We fix G = Inn(X). Recall H n (B G X; Z) ∼ = H R n (X; Z[Inn(X)]) in §4.1. To show Proposition 5.1, we first study Inn(X).
Lemma 5.6. Let e ∈ Z ≥0 be the minimal number satisfying (T e −1)X = 0. Then Inn(X) ∼ = Z/eZ ⋉ X. Here Z/eZ acts on X by the multiplication by T .
As a result, we see that the surjection As(X) → Inn(X) in §4.1 coincides with a homomorphism As(X) → Z/eZ ⋉ X sending x to (1, x). We also observe another lemma: We defer its proof until §5.4. We next study a relation between H Q n (X; M) and H n (B Q G X; Z). To see this, we review a complex C L n (X) introduced in [LN,§2]. This C L n (X) is defined to be a subcomplex of C D n (X) generated by n-tuples (x 1 , . . . , x n ) ∈ X n with x i = x i+1 for some 2 ≤ i < n. It is shown [LN,Lemma 9] that there is an isomorphism of chain complexes: where the direct summand C L * (X; Z) is obtained from the inclusion C L * (X; Z) ֒→ C D * (X; Z).
We later prove this in §5.4. Meanwhile, we consider a complex C R * (X; Z X ) in Remark 4.1. Under the identification C R n (X; Z X ) ∼ = C R n+1 (X; Z), as the restrictions, we have a chain isomorphism C D n (X; Z X ) ∼ = C L n+1 (X; Z). In conclusion, by (7) and (16), we thus have Lemma 5.9. Let X be a regular Alexander quandle of finite order. Then H n (B G X; Proof. Let us regard X as an X-set (see §2.1). We consider the two maps Inn(X) → X and X → {pt.}. They then give rise to two coverings B G X → B X X and B X X → BX (cf. [Cla,§4.1]). Further, note that the covering B G X → B X X is of degree e by Lemma 5.6. Hence, the transfer map yields an isomorphism H n (B G X; Z) (p) ∼ = H n (B X X; Z) (p) where the prime p divides |X| (cf. [Cla,Propositions 22 and 23]). Furthermore, it is evident (6). It is shown [N2, Theorem 6.3] that the torsion subgroup of H n (BX; Z) is annihilated by |X|. Combining with Lemma 5.7, we have H n (B G X; Z) ∼ = H n (B X X; Z) ∼ = H n+1 (BX; Z) ∼ = H R n+1 (X; Z). To prove the latter part, similarly, we can see H * (B Q G X; Z) ∼ = H * (B Q X X; Z). Therefore, we have H n (B Q G X; Z) ∼ = H Q n+1 (X; Z) ⊕ H Q n (X; Z) by (17), which completes the proof.
We now prove Propositions 5.1 and 5.2.
To prove (i), using (7) and (16), Lemma 5.9 results in Therefore, the required isomorphism (i) is obtained from Lemma 5.8. Finally, the proof of (iii) immediately follows from the sequence (7) and the isomorphism (17).

Proofs of Lemmas 5.8 and 5.7
Before proving Lemma 5.8, we remark the condition of H Q 2 (X; Z) ∼ = 0. By (5), we have H R 2 (X; Z) ∼ = Z. It is known [CJKS, LN] that, for any x ∈ X, the 2-cycle of the form (x, x) ∈ C R 2 (X; Z) is a generator of H R 2 (X; Z) ∼ = Z. Namely, any cycle of C R 2 (X; Z) is homologous to α(x, x) for some α ∈ Z.
Proof of Lemma 5.8. Consider a cycle σ ∈ C L 4 (X; Z), i.e., ∂ 4 (σ) = 0. It is enough to show that the cycle σ is homologous to a sum of (y, y, y, y)s for some y ∈ X. Let us expand σ as We show (19) below. For any g i ∈ X, we note that Hence, for any i, we may assume a i = b i . The condition ∂ 3 (σ) = 0 is thus formulated by One deals with the letter term j β j (d j , e j , f j , f j ). We fix φ ∈ X. Therefore C R 1 (X; Z) ∋ 0 = j β j (d j −d j * e j ) = ∂ 2 j β j (d j , e j ) , where j runs satisfying f j = φ. Since H R 2 (X; Z) ∼ = Z, the sum j β j (d j , e j ) is homologous to γ φ (φ, φ) for some γ φ ∈ Z. Notice . We therefore conclude that j β j (d j , e j , f j , f j ) is homologous to φ∈X γ φ (φ, φ, φ, φ).
On the other hand, let us discuss the former term i α i (a i , a i , a i , c i ). By (19), we have Since H R 2 (X; Z) ∼ = Z, the sum i α i (a i , c i ) is homologous to α(y, y) for some α ∈ Z and y ∈ X. By definition, we note In conclusion, i α i (a i , a i , a i , c i ) is homologous to α(y, y, y, y) as desired.
The bijection induces a chain isomorphism from C R n (X; Z[Inn(X)]) to C R U n (X), where we use the identification Inn(X) ∼ = Z/eZ ⋉ X by Lemma 5.6.
Then, we have the induced map |X| · id H R U n (X) = (Z) * : H R U n (X) → H R U n (X). Therefore, for the proof, it suffices to show that any cycle of the form ε a ε (ε, 0; 0, . . . , 0) is contained in the free subgroup of C R U n (X), where a ε ∈ Z. Indeed, the augmentation map C R U n (X) → Z is an n-cocycle and sends (ε, 0; 0, . . . , 0) to 1.
A Appendix: π 2 (B Q X) ⊗ Z p of Alexander quandles X on F q This appendix computes π 2 (B Q X) ⊗ Z p for Alexander quandles of the forms X = F q [T ](T − ω) (see §A.2). For this, in §A.1, we review some results in [M1, M2] and observe b Q i = dim Fp H Q i (X; Z) ⊗ Z p .

A.1 Review and some remarks on Mochizuki's cocycles
Mochizuki explicitly determined the second and third quandle cohomology groups [M1, M2].
To see this, we first recall from [M2, Theorem 2.2] that dim Fq (H 2 Q (X; F q )) = # (i, j) ∈ Z 2 | 1 ≤ p i < p j < q, ω p i +p j = 1 = 0<i<h: where q = p h . In particular, one notices that H 2 Q (X; F q ) vanishes if and only if ω p i +1 = 1 for any i < h, that is, the order of ω is not divisible by p i + 1 for any i < h.