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 Mochizuki 3-cocycle invariant. As another result, we determine the third quandle homology group of the dihedral quandle of odd order.

As for classical links, the quandle cocycle invariants were much studied (see, e.g., [15,23,25,31]), and are known to be derived from the quandle homotopy invariant above (see [6,31]). The study of the homotopy group π 2 (B X) was useful to understood a topological meaning of some quandle cocycle invariants [15].
In this paper, we introduce and study a quandle homotopy invariant of oriented linked surfaces valued in a group ring Z[π Table 1 Some homotopy groups π 2 (B X Q ) obtained from H Here ω = ±1 In particular, π 2 (B X) is a direct summand of Z ⊕ H Q 3 (X ; Z). In conclusion, regarding H Q 3 (X ; Z) as a second homology with local coefficients (see Remark 4.1), Theorem 3.9 implies that the quandle homotopy invariant is completely determined as a linear sum of quandle cocycle invariants through H 2 Q (X ; A)⊕ H 3 Q (X ; A) of the local system (Remark 5.5). Furthermore, the above sequence enables us to compute π 2 (B X). For several quandles X of order ≤ 9, we determine π 2 (B X) exactly (see Table 1), following the values of H Q 2 (X ; Z) and H Q 3 (X ; Z) presented in [17]. Furthermore, since Mochizuki [19,20] has computed H 2 Q (X ; F q ) ⊕ H 3 Q (X ; F q ) for X = F q [T ]/(T − ω), we determine π 2 (B X) ⊗ Z F p and have made the quandle homotopy invariants computable (see Appendix A). One of the noteworthy results is that if X is the product h-copies of the dihedral quandle, i.e., X = Z[T ]/( p, T + 1) h , then the dimension of π 2 (B X) ⊗ Z F p is a quadratic function of h: to be precise, dim F p (π 2 (B X) ⊗ 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 [15] computed π 2 (B X) ∼ = Z ⊕ Z/m, it follows from Theorem 3.9 above that H Q 3 (X ; Z) ∼ = Z/m (Corollary 3.11). When m is an odd prime, the isomorphism H Q 3 (X ; Z) ∼ = Z/m was conjectured by Fenn, Rourke and Sanderson (see [27,Conjceture 5.12]) and solved by Niebrzydowski and Przytycki using homological algebra [25]. On the other hand, our proof is another approach from π 2 (B X), and further gives a generalization of the conjecture. This paper is organized as follows. In Sect. 2, we review quandles and the rack spaces B X, and define the quandle homotopy invariant. In Sect. 3, we state our theorems and corollaries. In Sect. 4, we review the quandle homology groups, and prove Theorem 3.1 on the rational homotopy group π Q 3 (B X) ⊗ Q. In Sect. 5, we show Theorems 3.5 and 3.9. In Appendix A, we compute π 2 (B X) ⊗ Z/ pZ for some Alexander quandles X on finite fields. Notational convention Throughout this paper, F q is a finite field of characteristic p > 0. 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 finitely generated Z-module N , N ( p) means the localization at a prime p. Further, * (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 oriented, closed and of C ∞ -class.

Quandle homotopy invariants of linked surfaces
We review quandles and the rack spaces in Sect. 2.1, and introduce quandle homotopy invariants of linked surfaces in Sect. 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 a unique element w ∈ X such that w * y = x. For example, a Z[T ±1 ]-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) (x, y ∈ X ) . 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 acted on by As(X ). For example, when Y = X , the quandle X is itself an X -set by means of the quandle operation. We next review X -colorings [4,Definition 5.1.]. Let D be a broken diagram of a linked surface L. In this paper, any broken diagram is regarded being 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 Fig. 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 Fig. 1). We denote the set consisting of such X -colorings of D by Col X (D). It is widely known (see, e.g., [4,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 ±1 ]/(h(T )) and L is a knotted surface, the set Col X (D) is obtained from the Alexander polynomials of L (see [16] for details).
Finally, we review the rack space introduced by Fenn-Rourke-Sanderson [13]. Fix an X -set 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 Here our description divides the element of Y from the elements of ([0, 1] × X ) n by a semicolon ';'. Then the rack space, B X Y , is defined by the quotient space (we later explain the 3-skeleton for more details in Sect. 2.2). By construction, we have a cell decomposition of Fig. 1 Coloring conditions around double-point curves and triple-points. Here a, b, c ∈ X B X Y , by regarding the projection n≥0 Y × ([0, 1] × X ) n → B X Y as the characteristic maps. We easily see that if the action on Y of As(X ) is transitive, the space B X Y is pathconnected (see [8,Proposition 2.12] for details). When Y is a single point, we denote the path-connected space B X Y by B X for short. Note that π 1 (B X) ∼ = As(X ) by the definition of the 2-skeleton of B X.

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 [14, §3] (see also [23, §2]). For the purpose, given an X -set Y , we first define a quandle space as follows. Define a subspace by We denote this by X Y D , and consider a composite ι Y D : Here the second map is the projection mentioned above.
We refer to the space as quandle space. When Y is a single point, we denote B X Y Q by B X Q for short. The homology H * (B X Q ; Z) coincides with the homology of the quandle complex introduced in [4] (see also Sect. 4.1). Roughly speaking, the space B X Q is a geometric realization of the quandle complex.
We now describe the 3-skeleton of the rack space B X and the 4-skeleton of the quandle space B X Q in more details. The 1-skeleton of B X is a bouquet of |X |-circles labeled by elements of X . The 2-skeleton of B X 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 Fig. 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 Fig. 2, to the corresponding 2-cells. The resulting space is the 3-skeleton of B X.
We next describe the 4-skeleton of B X Q . To begin, the 3-skeleton of B X Q is obtained from the 3-skeleton of B X by attaching 3-cells which bound 2-cells labeled by (a, a) for a ∈ X . The 4-skeleton of B X Q is obtained from the 3-skeleton of B X Q 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 4cell, and the 4-dimensional cube labeled by (a, b, c, d) ∈ X 4 bounds the eight corresponding   Fig. 3. Note that the preceding quandle space considered in [23, §2] 2 is the 3-skeleton of the space B X Q , and the higher skeleton was not defined in [23].

3-cells as shown in
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, branch-points and triple-points (see, e.g., [9] [4, §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 on the left of Fig. 4. Put a square D xyz which is bounded by the three intervals, and further attach a 3-cell which is bounded by the square D xyz and contains the branch-point P B (see the right of Fig. 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 Fig. 5, we here attach 2-cells and 3-cells to the dual decomposition shown as the right of Fig. 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 X Q , as follows. We take the 0-cells of the modified decomposition to the single 0-cell of B X. We send 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 Fig. 6. Furthermore, 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 Fig. 4). By collecting them, we obtain a cellular map ξ D,C : S 3 → B X Q . We denote the homotopy class of ξ D,C by X (D; C) ∈ π 3 (B X Q ). By the construction of 4-cells of B X Q , we can verify that the homotopy class X (D; C) is invariant under the Roseman moves (cf. [4,Theorem 5.6]). For example, the invariance of the move in the bottom right of [4, Figure 7] corresponds to the 4-cell in Fig. 3.

Remark 2.2
It is known that any oriented surface-link is represented by a broken diagram D without branch-points (see [9]). Recall that we defined the group π Q 3 (B X) to be the quotient of π 3 (B X) 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 X Q 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 X Q factors through the rack space B X, which implies that the homotopy class X (D; C) is derived from π 3 (B X). Therefore, for the study of the invariant, it sufficies to analyze the image of (i X ) * : In conclusion, we then define Definition 2.3 Let X be a finite quandle. We define the group π Q 3 (B X) by the image of the induced map π 3 (B X) → π 3 (B X Q ) by the inclusion B X → B X Q . We furthermore 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.
We later study the container π Q 3 (B X) concretely in Sect. 3.1.

Reconstrucion of generalized quandle cocycle invariants
We now reformulate the (generalized) quandle cocycle invariants in [3,4,6] from our quandle homotopy invariant. We first remark that (1) has no 1-cell by definition. Given an action of π 1 (B X Q ) = As(X ) on an abelian group A, we set a 3cocycle ψ ∈ H 3 (B X Q ; A) with local coefficients. The homology H 3 (B X Q ; A) coincides with the quandle homology H To summarize, 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 [3, §7] by the definition of the Hurewicz homomorphism and the construction of the map ξ D,C . Therefore the equality (2) means that by knowing a concrete presentation of ψ we can calculate some parts of the quandle homotopy invariant. However, since there are many choices of actions π 1 (B X) A, it is a problem to determine those of of local systems for which the associated quandle cocycle invariants pick out the quandle homotopy invariant. Section 3.1 gives an answer with respect to 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 (B X)-module. We then consider a 4-cocycle ψ ∈ H 4 (B X Q ; A) to be a cocycle of H 3 (B X Q ; A X ) (see Remark 4.1 for detail); The quandle cocycle invariant of ψ coincides with the shadow cocycle invariant described in [6, §5].

Properties of quandle homotopy invariants
We state some properties of quandle homotopy invariants of linked surfaces, similar to those of classical links in [23, §5]. For this, we review connected components of a quandle X . By definition of quandles, we note that (• * y) : X → X is bijective for any y ∈ X . Let Inn(X ) denote the subgroup of S |X | generated by all the right actions (• * y), 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, let T denote the trivial quandle of order given by the operation x * y = x for x, y ∈ T ; Each point in T is its own orbit. 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 [17, Proposition 1] that an Alexander quandle X is connected if and We give a slight reduction of quandle homotopy invariants: the subset of Col X (D) such that the sheet is colored by x ∈ X. Then, Proof For any y ∈ X , We see that the bijection (• * y) : [14,Proposition 5.2]). Therefore, since X is connected, for any z ∈ X , we have 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 [23, §5] on the quandle homotopy invariant of classical links.

Proposition 2.5 (cf. [23, 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 .
. The similar formulas to Propositions 2.4, 2.5 hold for the quandle cocycle invariants via the equality (2).

(BX) and π 2 (BX Q )
In Sect. 2.2, we defined an invariant valued in Z[π Q 3 (B X)]. We will state our results on the free and torsion parts of the group π Q 3 (B X) in Sect. 3.1. We also compute π 2 (B X Q ) in Sect. 3.2. Here recall from Definition 2.1 that π 2 (B X Q ) denotes the second homotopy group of the quandle space. Meanwhile the container π To begin with, we determine the free subgroup of π Q 3 (B X) as follows: Theorem 3.1 Let X be a finite quandle with -connected components (see Sect. 2.4 for the definition). Then π Q 3 (B X) is finitely generated. Furthermore, the rational homotopy groups are determined by the following equalities: In particular, if X is connected, then π Q 3 (B X) is a finite abelian group.
The proof will appear in Sect. 4.2. Furthermore we later discuss a topological meaning of the quandle homotopy invariant after tensoring with Q valued in π Furthermore, we give a corollary: Tables 3,4 and 5], for a connected quandle of order 3, a non-trivial quandle cocycle invariant 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 (B X). However, in general, it is difficult to compute explicitly homotopy groups of spaces. In this paper, we then confine ourselves to π Q 3 (B X) of regular Alexander quandles X . First, we give an estimate of π 3 (B X) by homologies of the rack space B X in homological contexts. The homology of B X is usually called rack homology of X (see Sect. 4.1 for details).

Proposition 3.4 Let X be a regular Alexander quandle of odd order. Let H
. Then there exists an exact sequence Here H gr 3 (H R 2 (X ); Z) is the third group homology of the abelian group H R 2 (X ). Proposition 3.4 will be proven by a routine calculation of Postnikov tower in Sect. 5. We see later that π 2 (B X) is determined by H R 2 (X ) and H R 3 (X ) [Theorem 3.9 and (5)]; in conclusion, π 3 (B X) and its quotient π Q 3 (B X) 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 (B X) and π Q 3 (B X) exactly. However, with an additional assumption, we determine π Q 3 (B X) of regular Alexander quandles X : There are many Alexander quandles satisfying the assumption. For example, we We defer the proof until Sect. 4.2. As a corollary, we now determine π Q 3 (B X) 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 now consider the dihedral quandle, i.e., the case ω = −1. The third coho- [19,20], 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.

Computation of the second homotopy groups of quandle spaces π 2 (B X Q )
Changing the subject, we discuss the quandle homotopy invariant of 1-dimensional links considered in [23]. This invariant is valued in the group ring Z[π 2 (B X Q )] and is universal among the generalized cocycle invariants in [3, §6], similar to the equality (2) (see [23, §2] for details). For a finite connected quandle X , it is shown [23, Theorem 3.6 and Proposition 3.12] that π 2 (B X Q ) is finite and π 2 (B X) = π 2 (B X Q ) ⊕ Z; The author further computed π 2 (B X Q ) for connected quandles of order ≤ 6 [23, §5].
In this paper, we explicitly compute π 2 (B X Q ) of regular Alexander quandles X of odd order in the following theorem, which we will prove in Sect. 5: 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 X Q ) than [23, Theorem 3.6].

Corollary 3.10 Let X be a regular Alexander quandle of odd order. Any element of
Proof It is known [24, Corollary 6.2] that H Q 3 (X ; Z) is annihilated by |X |. We next give two applications. First, we explicitly compute homotopy groups π 2 (B X Q ) 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 X Q ). For example, the computations of all regular Alexander quandles X with 5 ≤ |X | ≤ 9 are listed in Table 1 Table 1]. For more example, in Appendix A we present a computation of π 2 (B X Q ) ⊗ Z p for all Alexander quandles on F q with p > 2.

Proof of Theorem 3.1
Our purpose is to prove Theorem 3.1 in Sect. 4.2 about the rational homotopy groups π Q * (B X) ⊗ Q. A computation of π 3 (B X) will involve a preliminary analysis of the homology H i (B X G ; Z). For this, in Sect. 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 [13] (see also [10]) and the quandle homologies defined in [3]. 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, by  We next review basic properties of the complexes. Let X be a finite quandle withconnected components and M = Z. According to [17,Theorem 2

.2], we have
In addition, the following long exact sequence splits (see [17, 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 X Y as a covering as follows. Note that the collapsing map Y → {pt.} induces a continuous map B X Y → B X. It is known (see [13,Theorem 3.7]) that this is a covering; hence, so is the induced map B X Y Q → B X Q . As a special case, if Y = G is a quotient group of As(X ) (hence B X G is pathconnected), then the two coverings B X G → B X and B X G Q → B X Q are principal G-bundles over B X and B X Q , respectively (see [8,Proposition 2.11]). Recalling π 1 (B X) ∼ = As(X ) from Sect. 2.1, the group π 1 (B X G ) is exactly the kernel of p G : As(X ) → G The action of π 1 (B X G ) on the higher homotopy groups π * (B X G ) is known to be trivial (see [14,Proposition 5.2] and [8,Proposition 2.16]). Furthermore, the action of π 1 (B X) on the higher homology group H * (B X G ) is also trivial (see [10,Lemma 3.1] or [8,Remark 7]); hence, the action on H * (B X G Q ) is also trivial, since π 1 (B X) = π 1 (B X Q ) and H * (B X G Q ) is a quotient of H * (B X G ) by (7).
Finally, we discuss topological monoids on some rack spaces introduced by Clauwens [8]. Let G be either As(X ) or Inn(X ), and let G act on X canonically. Then, he introduced an operation given by where n, m ∈ Z >0 ; this further gives rise to a topological monoid structure on B X G (see [8, §2.5]). In particular, π 1 (B X G ) is an abelian group. Since B X G is a so-called nilpotent space, we often deal with the localization of B X G at a prime p.

Proof of Theorem 3.1
Proof Consider G = Inn(X ), which is a finite group. Since the rack space B X G is a topological monoid and is a path-connected CW-complex of finite type, B X G is homotopic to a (based) loop space of some simply connected CW-complex W of finite type, that is, B X G W as an H -space (see [22,Theorem 1.5 and §2] for details). In particular, it is without saying that π * (B X G ) is finitely generated; hence, so is the quotient π Q 3 (B X). Let us focus on rational homologies of B X G and B X G Q . Recall that the projections p G Q : B X G Q → B X Q and p G : B X G → B X are coverings of degree |G|, and that the actions of π 1 (B X) on H * (B X G Q ) and on H * (B X G ) are trivial. It then follows from the transfer maps that p G Q and p G induce two isomorphisms: One discusses π * (B X G ) ⊗ Q. Since B X G W mentioned above, we recall the known following formula (see [12, §33(c) where we put r i = dim(π i (B X G ) ⊗ Q). By (6) and (8), we then easily have r 2 = ( 2 + )/2 and For this, we equip H * (B X G ; Q) with a Hopf algebra arising from the monoid structure, and denote by P * (B X G ) the set of primitive elements of H * (B X G ; Q). By the Milnor-Moore theorem (see, e.g., [12,Theorem 21.5]), π * (B X G ) ⊗ Q ∼ = P * (B X G ) and the dual Hurewicz map (H) * ⊗Q : We now explain the isomorphism (10) below. By the isomorphisms (8) and the splitting in (7), we notice that the induced map (i X ) * : Furthermore, we consider the long exact sequences of cohomology and homotopy groups of the pair (B X G Q , B X G ), and have the following commutative diagram: (9) Here the top and bottom sequences are exact, and the vertical arrows are the Hurewicz homomorphisms. By the Milnor-Moore theorem again, we obtain We will calculate the right hand side P 3 (B X G )∩Ker(δ * ) using the complexes in Sect. 4.1. Inspired by the monoid structure on B X G , 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 G D : ). Let us identify the above map δ * with the induced map (i G D ) * by definitions. By (7) we then notice that In conclusion, the dimension of P 3 (B X G ) ∩ Ker(δ * ) 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 on each connected components, where T is the trivial quandle of order . We see that this map induces π

Proofs of Theorems 3.and 3.9
Our purpose in this section is to prove Theorem 3.5 in Sect. 5.1 and Theorem 3.9 in Sect. 5.2.
We now outline the proof of Theorem 3.5. To see this, we first aim to study π 3 (B X) since π Q 3 (B X) is a quotient of π 3 (B X). Let B X be the universal covering of B X. By the monoid structure of B X explained in Sect. 4.1, B X is a loop space of a 2-connected CWcomplex. Thanks to the fact [2] that the second k-invariant of B X is annihilated by 2, we will show π 3 (B X) = π 3 ( B X) ∼ = H 3 ( B X; Z) ⊕ π 3 ( S 2 ) by a routine argument of the Postnikov tower. In Sect. 5.1, we see H 3 ( B X; Z) ∼ = H Q 4 (X ; Z) using techniques of quandle homologies (see (11)). Finally, we show that H as stated in Theorem 3.5. Theorem 3.9 is also proven in a similar way.
To carry out the outline, we here prepare two propositions, which are proven in Sect. 5.3. Proof From now on, X is assumed to be a regular Alexander quandle with H Q 2 (X ; Z) ∼ = 0. By Proposition 5.2, the kernel of the epimorphism As(X ) → Inn(X ) is entirely Z. Hence, setting G = Inn(X ), we have B X G S 1 × B X by Lemma 5.3; thus it follows from Proposition 5.1 and the Kunneth formula that

Proposition 5.1 Let X be a regular Alexander quandle of finite order. Let G = Inn(X ). Then
Let us calculate the localizations of π 3 ( B X) at every prime p. Since B X is also a topological monoid, the B X is a loop space. Let B X i denote the i-th stage of Postnikov tower. It immediately follows from [2, Theorem 3.2] that the third k-invariant k 4 ∈ H 4 (K (π 2 ( B X), 2); π 3 ( B X)) satisfies 2k 4 = 0 (see also [30,Proposition 3]). Hence, since the order |X | is odd, the localization of B X 3 at a prime p > 2 is reduced to be Hence, the Hurewicz map H ( p) passes to an isomorphism π 3 (B X) ( p) ∼ = H 3 ( B X; Z) ( p) , since H 3 (K (π 2 (B X), 2); Z) vanishes (see [18, §8 bis ]). As a consequence, combining with (11) immediately gives π 3 (B X) We consider the last case p = 2. It is known [14,Theorem 5.12] that BT 1 is homotopic to S 2 , where T 1 is the trivial quandle of order 1. Let X → T 1 be the collapsing map. This induces a topological monoid homomorphism f : B X G → S 2 . By Lemma 5.7 below, the torsion subgroup of H * (B X G ; Z) is annihilated by |X |; therefore, ( f (2) ) * : H * (B X G ; Z) (2) → H * ( S 2 ; Z) (2) Hence, by the Whitehead theorem, the cellular map f gives rise to a homotopy equivalence f (2) : B X G (2) → S 2 (2) . In particular, Furthermore, by Theorem 3.1 and (6), we note Next, we discuss π Considering the covering g : B X → B X G and the inclusion i X : B X G → B X G Q , we have a commutative diagram For p > 2, we remark that the localization of the Hurewicz map H 3 is an isomorphism by (12). By Lemma 5.3, the map g * is injective; hence, so is H R 3 . We note that, by Proposition 5.1, the above map (i X ) * : is the isomorphism. To complete the proof, it suffices to show that the direct summand Z 2 of π 3 (B X) is sent to the zero via (i X ) * : π 3 (B X) → π 3 (B X Q ). By the above discussion of the direct summand Z 2 , we may assume X = T 1 . It is known [7,28] that π 3 (BT 1 ) ∼ = Z 2 is isomorphic to the framed link bordism group ( ∼ = Z 2 ), and that the generator of the bordism group is represented by an embedding of a pair of unknotted tori (see [28,Example 1.12] for details). Hence, from the definition of B X Q , for any T 1 -coloring C of the tori, the cellular map ξ D,C in Sect. 2.2 is bounded by a 3-cell of the subspace X D Y in Sect. 2.1. Namely, the homotopy class [ξ D,C ] vanishes in π 3 (B X Q ), which completes the proof.

Remark 5.4
In a similar manner, we can calculate the forth homotopy groups π 4 (B X) of regular Alexander quandles X of odd order with H Q 2 (X ; Z) = 0 as follows. In fact, it is shown [2,Theorem 4.6] that the forth k-invariant of B X is annihilated by 16; hence the forth stage ( B X) 4 localized at p > 2 is presented by About the case p = 2, it follows from the proceeding map f (2) that π 4 (B X) (2) concludes that the group π 4 (B X) can be calculated from π 2 (B X), π 3 (B X) and H 4 (B X G ; Z). For instance, let us compute π 4 (B X) in the dihedral case X = Z[T ]/( p, T +1) with p > 2. Recall that π 3 (B X) = Z 2 p from Corollary 3.7, and that π 2 (B X) = Z⊕Z p from Theorem 3.9. In addition, by Lemma 5.3. Therefore, by (13) we conclude π 4 (B X) = Z 2 (cf. Corollary 3.8).

Proofs of Theorem 3.9 and Proposition 3.4
We let X be a regular Alexander quandle without the assumption H Q 2 (X ; Z) = 0, and |X | be odd. We fix a notation G = Inn(X ). To prove Theorem 3.9, we use the Postnikov tower of the rack space B X (see, e.g., [18, Lemma 8 bis .27] with n = 2), which is an exact sequence Here H gr * (π 1 (B X G ); Z) is the group homology of π 1 (B X G ).
Proof of Theorem 3.9 (i) By assumption of H Q 2 (X ; Z) = 0, we have π 1 (B X G ) ∼ = Z by Proposition 5.2. Noting H gr r (Z; Z) ∼ = 0 for r ≥ 2, the Hurewicz map H in (14) is isomorphic. Recall H 2 (ii) To obtain the required sequence (4), we will show that the map H in (14) is a split injection. Recall that B X G is a path-connected loop space. It immediately follows from [2, Theorem 3.2] that the second k-invariant k 3 ∈ H 3 gr (π 1 (B X G ); π 2 (B X G )) satisfies 2k 3 = 0. By Proposition 5.2, π 1 (B X G ) ∼ = H R 2 (X ; Z), which implies that the group cohomology H 3 gr (π 1 (B X G ); π 2 (B X G )) is annihilated by |X |. Hence, the second k-invariant is zero. That is, the second stage of the Postnikov tower is homotopic to K (π 1 (B X G ); 1) × K (π 2 (B X G ); 2), and the map H is a split injection (cf. [2, pp 3]). We now calculate each terms in the sequence (14) above. Using Proposition 5.2, the second group homology in (14) is expressed as (14) is then rewritten in Since π 2 (B X) ∼ = Z ⊕ π 2 (B X Q ) and the restriction of H to the Z-part is known to be isomorphic (see [23,Proposition 3.12]), we immediately obtain the required sequence (4).

Remark 5.5
As is seen in the proof, the Hurewicz homomorphism H in the sequence (14) is injective. Hence, via H, any element of π 2 (B X Q ) can be evaluated by some cocycles such that any quandle cocycle invariant constructed from X is a linear sum of the quandle cocycle invariants associated with ψ 1 , . . . , ψ m . Furthermore, by a similar discussion, we now show Proposition 3.4 as follows: Proof of Proposition 3.4 Let Y i denote the i-th stage of the Postnikov tower of B X G . We recall the following exact sequence in [18,Lemma 8 bis .27].
Let us calculate the each terms in (15). By Lemma 5.9, the third homology H 3 (B X G ; Z) is isomorphic to the rack homology H R 4 (X ; Z). Moreover, it follow from the previous proof of Theorem 3.9 that Y 2 is homotopic to K (π 1 (B X G ), 1) × K (π 2 (B X G ), 2). Furthermore, recall the isomorphism H R 2 (X ; Z) ∼ = π 1 (B X G ) from Proposition 5.2. Therefore, we easily obtain the required sequence stated in Proposition 3.4 from the sequence (15) and the Kunneth formula.

Proofs of Propositions 5.1 and 5.2
In Sects. 5.3 and 5.4, X is assumed to be a finite connected Alexander quandle. We fix a notation G = Inn(X ). To show Propositions 5.1 and 5.2, we first study Inn(X ) as follows.
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 Sect. 4.1 coincides with a homomorphism As(X ) → Z/eZ X sending x to (1, x). Recalling the isomorphism We defer its proof until Sect. 5.4. We next study a relation between H Q n (X ; M) and H n (B X G Q ; Z). To see this, we review a complex C L n (X ) introduced in [17, §2]. This C L n (X ) is defined to be the 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. As is shown [17,Lemma 9], 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 Sect. 5.4. Meanwhile, we consider the 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 this restriction, 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 Proof We first show H n (B X G ; Z) ∼ = H R n+1 (X ; Z). Let us regard X as an X -set (see Sect. 2.1). We consider the two maps Inn(X ) → X and X → {pt.}. They then give rise to two coverings B X G → B X X and B X X → B X (cf. [8, §4.1]). Furthermore, note that the covering B X G → B X X is of degree e by Lemma 5.6. Hence the transfer map yields an isomorphism (6). It is shown [24,Theorem 6.3]  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) readily 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 H Q 2 (X ; Z) ∼ = 0. By (5), we have H R 2 (X ; Z) ∼ = Z. It is known [5,17] 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 2-cycle of C R 2 (X ; Z) is homologous to α(x, x) for some α ∈ Z.
Proof of Lemma 5.8 Consider a 4-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 We now show the equality (19) below. For any g i ∈ X , we note that Putting Hence, for any i, we may assume a i = b i . The condition ∂ 4 (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 . We therefore conclude that the sum 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 ∂ 5 (a, a, a, d, f ) = (a, a, a, f ) − (a * d, a * d, a * d, f ) − (a, a, a, f ) In conclusion, the sum i α i (a i , a i , a i , c i ) is homologous to α(y, y, y, y) as desired.
Finally, we will prove Lemma 5.7 as follows. For this, it is convenient to change another "coordinate system" of the complex C R n (X ; Z[Inn(X )]) such as [20, §2.1.3]. We denote elements of Z/eZ by ε. We then define C R U n (X ) to be the free Z-module generated by elements (ε, U 0 ; U 1 , . . . , U n ) of Z/eZ × X × X n , and the boundary map by We can see ∂ n−1 • ∂ n = 0. Furthermore let us consider a bijection given by The bijection yields 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.
Proof of Lemma 5.7 The proof is analogous to [24, Theorem 6.1]. We set a chain homomorphism Z : C R U n (X ) → C R U n (X ) defined by Z( , U 0 ; U 1 , . . . , U n ) := |X | · ( , 0; 0, . . . , 0). Furthermore, we define two homomorphisms D j n,0 : C R U n (X ) → C R U n+1 (X ) and D j n,+ : C In addition, we set D n+1 n,0 = D n+1 n,+ = 0. Then, by direct calculation, we can verify the equality 0≤ j≤n Then, we have the induced map |X | · id 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 any (ε, 0; 0, . . . , 0) to 1.
Acknowledgments The author is grateful to Takuro Mochizuki for helpful communications on quandle cocycles and Remark A.2. He expresses his gratitude to Syunji Moriya for valuable conversations on rational homotopy theory. He is grateful to J. Scott Carter, Inasa Nakamura, Masahico Saito and Kokoro Tanaka for useful discussions on Remark 3.3 and the paper [3]. He sincerely thanks the referee for many suggestions which have improved the exposition Open Access This article is distributed under the terms of the Creative Commons Attribution License which permits any use, distribution, and reproduction in any medium, provided the original author(s) and the source are credited.
Appendix A: π 2 (BX Q ) ⊗ Z p of Alexander quandles X on F q This appendix computes π 2 (B X Q ) ⊗ Z p for Alexander quandles of the forms X = F q [T ]/(T − ω) (see Appendix A.2). For this, in Appendix A.1, we review some results in [19,20] and observe the betti number b A.1 Review and some remarks on Mochizuki's cocycles Mochizuki explicitly determined the second and third quandle cohomology groups [19,20]. To see this, we first recall from [20, Theorem 2.2] that 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.
Remark A. 5 We notice that the dimension is a quadratic function with respect to h. In particular, the second homotopy groups do not preserve the direct products of quandles.