Homotopical interpretation of link invariants from finite quandles
aa r X i v : . [ m a t h . G T ] M a y Homotopical interpretation of link invariants from finite quandles
Takefumi Nosaka
Abstract
This paper demonstrates a topological meaning of quandle cocycle invariants of links with respect to finiteconnected quandles X , from a perspective of homotopy theory: Specifically, for any prime ℓ which doesnot divide the type of X , the ℓ -torsion of this invariants is equal to a sum of the colouring polynomial anda Z -equivariant part of the Dijkgraaf-Witten invariant of a cyclic branched covering space. Moreover, ourhomotopical approach involves application of computing some third homology groups and second homotopygroups of the classifying spaces of quandles, from results of group cohomology. Keywords
Quandle, group homology, homotopy group, link, branched covering,bordism group, orthogonal and symplectic group, mapping class group
A quandle, X , is a set with a binary operation ⊳ : X → X the axioms of which weremotivated by knot theory. The operation is roughly the conjugation of a group; In analogyof group (co)homology, Carter et. [CJKLS, CEGS] defined quandle (co)homology, and usedsome cocycles to define some knot-invariants, which we call quandle cocycle invariants . Asmentioned in [CJKLS], the definition is an analogue of the Dijkgraaf-Witten invariant [DW]of closed oriented 3-manifolds M , which is constructed from a finite group G and a 3-cocycle κ ∈ H ( G ; A ): Precisely, the invariant is topologically expressed as the formal sum of pairingsDW κ ( M ) := X f ∈ Hom gr ( π ( M ) ,G ) h f ∗ ( κ ) , [ M ] i ∈ Z [ A ] , (1)where [ M ] is the orientation 3-class in H ( M ; Z ). While this DW κ ( M ) is far from beingcomputable by definition, the cocycle invariants from quandles can be diagrammatically com-putable if we find an explicit description of a quandle cocycle. However, since the cocycleinvariants are defined from link diagrams, their topological meanings are mysterious.To solve such topological problems, this paper is inspired by Quillen-Friedlander theory[Fri] in group homology. Roughly speaking the theory, if a finite group G is of Lie type andthe abelianization of G is almost zero, the group homology H gr ∗ ( G ; Z ) in a stable range canbe computed from homotopy groups π ∗ ( K ( G, + ). Here, the space K ( G, + is so-called theQuillen plus construction and, in many cases, is homotopic to an infinite loop space, i.e., Ω-spectrum. For example, the Chern-Simons invariant which is a ploto model of DW κ ( M ) canbe universally interpreted as a plus construction or as the algebraic K -theory (see [DW, JW]).In this paper, from such a homotopical viewpoint, we shall focus on the quandle homotopy (knot-)invariant, which is valued in the group ring Z [ π ( BX )]. Here, BX is the rack space BX of a quandle X and was introduced by Fenn, Rourke and Sanderson [FRS1, FRS2], as ananalogy of the classifying spaces of groups. Further, as is known [FRS2, RS], the homotopyinvariant is universal in the sense that any cocycle invariant is derived from the homotopy nvariant via the Hurewicz map π ( BX ) → H ( BX ; A ) with local system (see [N1, §
2] forthe detailed formula). Therefore, it is natural to address π ( BX ) for the study the requiredtopological meaning of the cocycle invariants.The main theorem demonstrates a topological meaning of the homotopy invariants as ageneral statement, together with computing quantitatively the homotopy groups π ( BX ). Tostate the theorem, we set two simple notation: A quandle X is said to be connected , if any x, y ∈ X admit some a , . . . , a n ∈ X such that ( · · · ( x ⊳ a ) ⊳ · · · ) ⊳ a n = y ; The type t X of X is the minimal N such that x = ( · · · ( x ⊳ y ) ⊳ · · · ) ⊳ y [ N -times on the right with y ] for any x, y ∈ X . Theorem 1.1 (Theorem 3.5 ) . Let X be a connected quandle of type t X and of finite order.Let H X : π ( BX ) → H ( BX ; Z ) be the Hurewicz map as a common sense. Then there is ahomomorphism Θ X from π ( BX ) to the third group homology H gr3 ( π ( BX ); Z ) for which thesum Θ X ⊕ H X : π ( BX ) −→ H gr3 ( π ( BX ); Z ) ⊕ H ( BX ; Z ) is an isomorphism after localization at ℓ ∈ Z , where ℓ is relatively prime to t X . Moreover, the sum Θ X ⊕ H X which we call the TH-map plays an important role as follows:We will show (Corollary 3.6) that, for any link L ⊂ S , the homotopy invariant Ξ X ( L ) ∈ Z [ π ( BX )] is sent to a sum of two invariants via the TH-map H X ⊕ Θ X : precisely, theoriginal quandle 2-cycle invariant ∈ Z [ H ( BX )] in [CJKLS] and “a Z -equivariant part” ofDijkgraaf-Witten invariant of b C t X L (see (9) for the definition), where b C t X L is the t X -fold cycliccovering space of S branched over the link L . To summarize, this corollary 3.6 is figurativelysummarized to the following equality in the group ring Z [ π ( BX )] up to t X -torsion: (cid:18) Quandle homotopyinvariant of L (cid:19) = (cid:18) Z -equivariant partof DW κ ( b C t X L ) (cid:19) + (cid:18) Quandle 2cycleinvariant of L (cid:19) Here remark that, as is shown [E2], the second term is topologically characterized by longitudesof X -colored links (see § X ( L ) without t X -torsion is reduced to the two topological invariants as desired.Further, we should compare with the previous works and explain the type t X in somedetails. Two papers [Kab, HN] tried to extract a topological meaning of the cocycle invariants;however, as in [FRS2, Cla1, N2] as some topological approaches to BX , the second homology H ( BX ; Z ) is an obstacle to study the space BX . Accordingly successful works were only forthe simplest quandle of the form X = Z / (2 m −
1) with x ⊳ y := 2 y − x . Furthermore, asseen in §
7, the type t X is often smaller than the order of H ( BX ) in many cases. Hence, theabove theorem is quite general, and implies that a minimal obstacle in studying π ( BX ) andtopological meanings is the type t X of X , rather than the second homology H ( BX ).In addition, this paper further determine π ( BX ) of several quandles X : “regular Alexanderquandles”, most “symplectic quandles over F q ”, “Dehn quandle D g ”, extended quandles and After §
2, we employ reduced notation of BX for simplicity. For example, we use two groups, As( X ) and Π ( X ), such that π ( BX ) ∼ = As( X ) and π ( BX ) ∼ = Z ⊕ Π ( X ) (see (18)). Furthermore, instead of the homology H ∗ ( BX ), we mainly deal withthe quandle homology H Q ∗ ( X ; Z ) introduced in [CJKLS], and remark two known isomorphisms H ( BX ) ∼ = Z ⊕ H Q ( X ) and H ( BX ) ∼ = Z ⊕ H Q ( X ) ⊕ H Q ( X ) (see (14) and (28)). he connected quandles of order ≤ π ( BX ) follows from computing H ( BX ) and H gr3 ( π ( BX )), as in Theorem1.1. In fact, for this computation, another paper [N4] gave an algorithm to determine thegroup π ( BX ) and the third homology H gr3 ( π ( BX )) up to t X -torsion.Finally, we emphasize two applications of our study on π ( BX ). First, our approach to π ( BX ) involves a new method for computing the third homology H ( BX ), and establishesa relation between third quandle homology and group homology. As a general result, withrespect to a finite connected quandle X , we solve some torsion subgroups of H ( BX ) in termsof group homology (Theorem 3.9). In application, as examples, we compute most torsionsof the third homology groups H ( BX ) of the regular Alexander quandles, the symplecticquandles and spherical quandles over F q in a stable range (see Sections 8.2 and 8.4). Here, weshould note that most of known methods for computing H ( BX ) was a result of Mochizuki[Moc], where X is an Alexander quandles on F q . Although his presentation of H ( BX ; F q )was a little complicated, our result implies that the complexity of H ( BX ; F q ) stems fromthat of H gr3 ( π ( BX )).On the other hand, our theorem suggests an approach for computing the Dijkgraaf-Witteninvariant. In contract to the simple formula (1), it is not so easy to compute this invariantexactly. Actually, most known computations of the invariants are in the cases where G areabelian (see, e.g., [DW, Kab, HN]). However, as mentioned above, the TH-map Θ X ⊕ H X implies that we can deal with some Z -equivariant parts of this invariants. In fact, in thesubsequent paper [N3] via the quandle cocycle invariants, we will compute the invariants ofsome knots using Alexander quandles X over F q , whose π ( BX ) are nilpotent. As a result,triple Massey products of some Brieskorn manifolds Σ( n, m, l ) will be calculated; see [N3, § X , and Section 5reviews the second quandle homology. Section 6 proves the Theorem 1.1. Section 7 computes π ( BX ) of some quandles, as examples. Section 8 is devoted to computing the third homology H ( BX ). Conventional notation.
Throughout this paper, most homology groups are with (trivial)integral coefficients; so we often omit writing coefficients, e.g., H n ( X ). We write H gr n ( G ) forthe group homology of a group G . Furthermore, we denote a Z -module M localized at a prime ℓ by M ( ℓ ) . Moreover, a homomorphism f : A → B between abelian groups is said to be a [1 /N ] -isomorphism , denoted by f : A ∼ = [1 /N ] B , if the localization of f at ℓ is an isomorphismfor any prime ℓ that does not divide N . In addition, we assume that every manifolds are in C ∞ -class and oriented, and that any fields is not of characteristic 2. To establish our results in §
3, we will review quandles in § § .1 Review of quandles A quandle is a set, X , with a binary operation ⊳ : X × X → X such that(i) The identity a ⊳ a = a holds for any a ∈ X .(ii) The map ( • ⊳ a ) : X → X defined by x x ⊳ a is bijective for any a ∈ X .(iii) The identity ( a ⊳ b ) ⊳ c = ( a ⊳ c ) ⊳ ( b ⊳ c ) holds for any a, b, c ∈ X. We refer the reader to § X is said to be of type t X , if t X > N such that x = x ⊳ N y for any x, y ∈ X , where we denote by • ⊳ N y the N -times on the right operation with y . Note that, if X is of finite order, it is oftype t X for some t X ∈ Z . A map f : X → Y between quandles is a (quandle) homomorphism ,if f ( a ⊳ b ) = f ( a ) ⊳ f ( b ) for any a, b ∈ X .Next, we review the associated group [FRS1], denoted by As( X ). This group is is definedby the group presentationAs( X ) = h e x ( x ∈ X ) | e − x ⊳ y · e − y · e x · e y ( x, y ∈ X ) i . We fix an action As( X ) on X defined by x · e y := x ⊳ y for x, y ∈ X . Note the equality e x · g = g − e x g ∈ As( X ) ( x ∈ X, g ∈ As( X )) , (2)by definitions. The orbits of the above action of As( X ) on X are called connected componentsof X , denoted by O( X ) . If the action of As( X ) on X is transitive, X is said to be connected .Note that the group As( X ) is of infinite order. Actually, there is a splitting epimorphism ε X : As( X ) −→ Z (3)which sends each generators e x to 1 ∈ Z . Furthermore, if X is connected, this ε X gives theabelianization As( X ) ab ∼ = Z because of (2). The reader should keep in mind this epimorphism ε X . We begin reviewing X -colorings. Let X be a quandle, and D an oriented link diagram of a link L ⊂ S . An X - coloring of D is a map C : { arcs of D } → X such that C ( γ k ) = C ( γ i ) ⊳ C ( γ j ) atany crossing of D such as Figure 1. Let Col X ( D ) denote the set of all X -colorings of D . As iswell known, if two diagrams D and D are related by Reidemeister moves, we easily obtaina canonical bijection Col X ( D ) ≃ Col X ( D ); see, e.g., [Joy, CJKLS]. γ i γ j γ k γ j γ k γ i Figure 1: Positive and negative crossings.4 e now study a topological meaning of an X -coloring C of D . Let us correspondence eacharc γ to the generator Γ C ( γ i ) := e C ( γ i ) ∈ As( X ), which defines a group homomorphismΓ C : π ( S \ L ) −→ As( X ) (4)by Wirtinger presentation. The reader should keep this map Γ C in mind, for the sake of laterdiscussion. Then, we have a map Col X ( D ) → Hom( π ( S \ L ) , As( X )) which carries C to Γ C ,leading to a topological meaning of the set Col X ( D ) as follows: Proposition 2.1 (cf. [E2, Lemma 3.14] in the knot case) . Let X be a quandle. Let D be adiagram of an oriented link L . We fix a meridian-longitude pair ( m i , l i ) ∈ π ( S \ L ) of eachlink-component which is compatible with the orientation. Then, the previous map which sends C to Γ C gives rise to a bijection between the Col X ( D ) and a set { ( x , . . . , x L , f ) ∈ X L × Hom gr ( π ( S \ L ) , As( X )) | f ( m i ) = e x i , x i · f ( l i ) = x i } . (5)This proof will appear in Appendix B, although it is not so difficult.Next, we briefly recall the quandle homotopy invariant of links (our formula is a modificationthe formula in [FRS1]). Let us consider the set, Π ( X ), of all X -colorings of all diagramssubject to Reidemeister moves and the concordance relations illustrated in Figure 2. Then,disjoint unions of X -colorings make Π ( X ) into an abelian group, which is closely related to ahomotopy group π ( BX ); see (18). For any link diagram D , we have a map Ξ X,D : Col X ( D ) → Π ( X ) taking C to the class [ C ] in Π ( X ). If X is of finite order and D is a diagram of a link L , then the quandle homotopy invariant of L is defined as the expressionΞ X ( L ) := X C∈ Col X ( D ) Ξ X,D ( C ) ∈ Z [Π ( X )] . (6)Moreover, as is known [RS] (see also [N1, § ( X ) in details. aa aa a φ Figure 2: The concordance relations
Finally, we briefly review the original quandle cocycle invariant [CJKLS]. Given a finitequandle X , we set its quandle homology H Q ( X ) with trivial coefficients, which is a quotientof the free module Z h X × X i ; see § C ∈
Col X ( D ), weconsider a sum P τ ǫ τ ( C ( γ i ) , C ( γ j )) ∈ Z h X × X i , where τ runs over all crossing of D as shownin Figure 1 and the symbol ǫ τ ∈ {± } denotes the sign of the crossing τ . As is known (see RS, N1]), this sum is a 2-cycle, and homology classes of these sums in H Q ( X ) are independentof Reidemeister moves and the concordance relations; Hence we obtain a homomorphism H X : Π ( X ) −→ H Q ( X ) . (7)Using the formula (6), the quandle cycle invariant of a link L , denoted by Φ X ( L ), is thendefined to be the image H X (Ξ X ( L )) valued in the group ring Z [ H Q ( X )]. Namely,Φ X ( L ) := H X (Ξ X ( L )) ∈ Z [ H Q ( X )] . (8)As is known [RS, N1, CKS], given a quandle 2-cocycle φ : X → A , the pairing betweenthis φ and the cycle invariant Φ X ( L ) coincides with the original cocycle invariant in [CJKLS,Theorem 4.4].Although this Φ X ( L ) is constructed from link diagrams, in § H Q ( X ) following from Eisermann [E1, E2]. The purpose in this section is to state our results on the group Π ( X ). In § X . In § ( X ). As an application,we will see the computations of third quandle homology groups in § Θ X Before describing our main results, we will set up a homomorphism Θ X in Theorem 3.1, whichplays a key role in this paper. Furthermore we discuss Corollary 3.4 which proposes a necessarycondition to obtain topological interpretations of the quandle homotopy invariants.To describe the theorem, for any link L ⊂ S , we denote by p : ^ S \ L → S \ L the t X -foldcyclic covering associated to the homomorphism π ( S \ L ) → Z /t X sending each meridianof L to 1. Furthermore, given an X -coloring C and using the map Γ C in (4), we consider thecomposite p ∗ ◦ Γ C : π ( ^ S \ L ) → As( X ). Then Theorem 3.1.
Let X be a connected quandle of type t X . Then, for any diagram D of anylink L ⊂ S , the composite p ∗ ◦ Γ C : π ( ^ S \ L ) → As( X ) induces a group homomorphism θ X,D ( C ) : π ( b C t X L ) −→ As( X ) , where we denote by b C t X L the t X -fold cyclic covering space of S branched over the link L .Moreover, consider the pushforward of the fundamental class [ b C t X L ] by this θ X,D ( C ) , that is, (cid:0) θ X,D ( C ) (cid:1) ∗ ([ b C t X L ]) ∈ H gr3 (As( X )) . Then, these pushforwards with running over all X -coloringsof all diagrams D yield an additive homomorphism Θ X : Π ( X ) → H gr3 (As( X )) . As is seen in the proof in §
4, we later reconstruct these maps θ X,D and Θ X concretely. By onstruction, they provide a commutative diagram, functorial in X , described asCol X ( D ) θ X,D / / Ξ X,D (cid:15) (cid:15)
Hom gr (cid:0) π ( b C t X L ) , As( X ) (cid:1) ( • ) ∗ ([ b C tXL ]) (cid:15) (cid:15) Π ( X ) Θ X / / H gr3 (As( X )) . Remark 3.2.
As is seen in §
4, for any X -coloring C ∈
Col X ( D ), the homomorphism θ X,D ( C ) : π ( b C t X L ) → As( X ) factors through the kernel Ker( ε X ) ⊂ As( X ) in (3), and the coveringtransformation Z y b C t X L is Z -equivariant to the action Z y
Ker( ε X ) from the splitting (3).The map plays an important role in this paper. Thus we fix terminologies: Definition 3.3.
Let X be a connected quandle of type t X . We call the map Θ X T-map , andthe sum Θ X ⊕ H X TH-map . Here H X is the map H X : Π ( X ) → H Q ( X ) defined in (7).To state Corollary 3.4 below, let us consider the image of the θ X,D in Theorem 3.1, comparedwith the Dijkgraaf-Witten invariant (1). Strictly speaking, when X is of finite order, we definethe formal sumDW Z As( X ) ( b C t X L ) := X C∈ Col X ( D ) [Θ X (Ξ X,D ( C ))] = X C∈ Col X ( D ) θ X,D ( C ) ∗ ([ b C t X L ]) ∈ Z [ H gr3 (As( X ))] . (9)We call it a Z -equivariant part of the Dijkgraaf-Witten invariant of branched covering spaces b C t X L . Then, we obtain the conclude that the quandle homotopy invariant under an assumptionis topologically characterized as follows: Corollary 3.4.
Let X , b C t X L , Θ X be as above and let ℓ ∈ Z be a prime. Let | X | < ∞ . If theTH-map Θ X ⊕ H X : Π ( X ) → H gr3 (As( X )) ⊕ H Q ( X ) is an isomorphism after ℓ -localization,then the ℓ -torsion of the quandle homotopy invariant of any link L is decomposed as (Θ X ⊕ H X ) ( ℓ ) (Ξ X ( L )) = DW Z As( X ) ( b C t X L ) ( ℓ ) + Φ X ( L ) ( ℓ ) ∈ Z [Π ( X ) ( ℓ ) ] . (10)In conclusion, under the assumption on the TH-map, we succeed in providing a topologicalinterpretation of the quandle homotopy invariant Ξ X ( L ) as mentioned in the introduction.Actually, the two invariants in the right hand side of (10) are topologically defined. Θ X ⊕ H X . Following Corollary 3.4, it is thus significant to find quandles such that the localized TH-map(Θ X ⊕ H X ) ( ℓ ) are isomorphisms. The following main theorem is a general statement. To bespecific, Theorem 3.5.
Let X be a connected quandle of type t X < ∞ . If the homology H gr3 (As( X )) is finitely generated (e.g., if X is of finite order), then the TH-map is a [1 /t X ] -isomorphism. As a result, combining this theorem with Corollary 3.4, we have obtained the interpretationof some torsion of the quandle homotopy (cocycle) invariant. Precisely, orollary 3.6. Let X be a finite connected quandle of type t X . Then the equality (10) inCorollary 3.4 holds for any prime ℓ which does not divide t X . Note that there are many quandles whose types t X are powers of some prime, e.g., the quandlesin Examples 7.2 and 7.3. In conclusion, for such quandles X , we determine most subgroupsof π ( BX ) by Theorem 3.5. Remark 3.7.
Corollary 3.6 is a strong generalization of some results in [Kab, HN]. Indeed,the results dealt with only the Alexander quandle of the from X = Z [ T ± ] / (2 n − , T + 1),and were based on peculiar properties of the quandle.We moreover address some t X -torsion subgroups of Π ( X ). First we discuss an easy con-dition of vanishing of these t X -torsions: Proposition 3.8.
Let X be a connected quandle of type t X . If the t X -torsion part of theimage H gr3 (As( X )) ⊕ H Q ( X ) is zero, then that of the TH-map is also zero. As seen in §
7, there are several such quandles. In conclusion, we obtain a topological meaningof the quandle homotopy (cocycle) invariants, from the viewpoint of Corollary 3.4.
As an application of computing the (homotopy) group Π ( X ), we develop a new method forcomputing the third quandle homology H Q ( X ); see § the inner automorphism group , Inn( X ), of aquandle X . Recalling the action of As( X ) on X , we thus have a group homomorphism ψ X from As( X ) to the symmetric group S X . The group Inn( X ) is defined to be the image ( ⊂ S X ).Hence we have a group extension0 −→ Ker( ψ X ) −→ As( X ) ψ X −−−→ Inn( X ) −→ . (11)We should notice that, by the equality (2), this kernel Ker( ψ X ) is contained in the center.Further, as was shown [N4] (see Theorem 6.1), if X is of type t X and connected, then there isa [1 /t X ]-isomorphism Ker( ψ X ) ∼ = Z ⊕ H gr2 (Inn( X )).The following theorem is an estimate on the third homology H Q ( X ); the proof will be donein § Theorem 3.9.
Let X be a connected quandle of finite order. Then, there is the followingisomorphism up to | Inn( X ) | / | X | -torsion: H Q ( X ) ∼ = H gr3 (As( X )) ⊕ (cid:0) Ker( ψ X ) ∧ Ker( ψ X ) (cid:1) . Here note from [N4, Lemma 3.7] that the type t X divides the order | Inn( X ) | / | X | . In summary,many torsion subgroups of the third quandle homology are determined after computing thegroup homologies of As( X ) and Inn( X ). Remark 3.10.
The isomorphism does not hold in the 2-torsion subgroup. See Remark 7.11 fora counterexample. Furthermore, as mentioned in the introduction, the most known results on he third H Q ( X ) are with respect to the Alexander quandles over F q and are due to Mochizuki[Moc]. So this theorem is expected to be applicable to other quandles.In a subsequent paper [N3], this theorem on the quandle e X will be used to understand theT-map Θ X from a viewpoint of complexes of groups and quandles. Θ X From now on, we will prove the results mentioned in the previous section.Our purpose in this section is to prove Theorem 3.1 and, is to construct a homomorphismfrom Π ( X ) to a bordism group (Lemma 4.1), which plays a key role in this paper. Theconstruction is a modification of a certain map in [HN, § π ( b C tL ), where b C tL denotes the t -fold cyclic covering of S branched along a link L . Put a link diagram D of L . Let γ , . . . , γ n be the arcs of this D . Let ^ S \ L be the t -fold cyclic covering space of S \ L associated to the homomorphism π ( S \ L ) → Z /t which sends each γ i to 1. For any index s ∈ Z /t , we take a copy γ i,s of the arc γ i . Then, by Reidemeister-Schreier method (see, e.g.,[Rol, Appendix A] and [Kab, § π ( ^ S \ L ) can be presented bygenerators : γ i,s (0 ≤ i ≤ n, s ∈ Z ) , relations : γ k,s = γ − j,s − γ i,s − γ j,s for each crossings such as Figure 1 and ,γ , = γ , = · · · = γ ,t − = 1 . Further we can define the inclusion p ∗ : π ( ^ S \ L ) ֒ → π ( S \ L ) by ι ( γ i,s ) = γ s − γ i γ − s , witha choice of an appropriate base point. Moreover, the fundamental group π ( b C tL ) is obtainedfrom this presentation by adding the relation γ ,t − = 1.Next, given a quandle X of type t , we now construct a map (12) below. By the abovepresentation of π ( b C tL ), the map Γ C induces a homomorphism b Γ C : π ( b C tL ) → Ker( ε X ), where ε X : As( X ) → Z is the homomorphism which sends the generators e x to 1 ∈ Z [Recall (3)].Precisely, this b Γ C is defined by the formula b Γ C ( γ i,s ) = e s − C ( γ ) e C ( γ i ) e − s C ( γ ) (we use [N4, Lemma 3.5]for the well-definedness). In summary, we obtain the map stated in Theorem 3.1: θ X,D : Col X ( D ) −→ Hom gr ( π ( b C tL ) , Ker( ε X )) , ( C 7−→ b Γ C ) . (12)We here remark that this map depends on the choice of the arc γ ; however it does not up toconjugacy of Ker( ε X ) by construction, if X is connected.Finally, in order to state Lemma 4.1 below, we briefly recall the oriented bordism group ,Ω n ( G ), of a group G . Let us consider a pair consisting of a closed connected oriented n -manifold M without boundary and a homomorphism π ( M ) → G . Then the set, Ω n ( G ), isdefined to be the quotient set of such pairs ( M , π ( M ) → G ) subject to G -bordant equivalence.Here, such two pairs ( M i , f i : π ( M i ) → G ) are G - bordant , if there exist an oriented ( n + 1)-manifold W which bounds the connected sum M − M ) and a homomorphism ¯ f : π ( W ) → so that f ∗ f = ¯ f ◦ ( i W ) ∗ , where i W : M − M ) → W is the natural inclusion. An abeliangroup structure is imposed on Ω n ( G ) by connected sum. Note that this Ω n ( G ) agrees withthe usual oriented ( SO -)bordism group of the Eilenberg-MacLane space K ( G, Lemma 4.1.
Let X be a connected quandle of type t . Then, by considering all link diagrams D , the maps θ X,D in (12) give rise to an additive homomorphism Θ ΠΩ : Π ( X ) −→ Ω (Ker( ε X )) . (13) A sketch of the proof.
Since the proof is analogous to [HN, Lemma 5.3 and Proposition 4.3]essentially, we will sketch it. To obtain the homomorphism Θ ΠΩ , it suffices to show that themaps take the concordance relations to the bordance ones.First, to deal with the local move in the right of Figure 2, we recall that the t -fold cycliccovering of S branched over the 2-component trivial link T is S × S → S (see [Rol, § f : π ( S × S ) → Ker( ε X ) is G -bordant. Indeed, f : π ( B × S ) → Ker( ε X ) provides its bordance, where B is a ball.Next, for two X -colorings C and C related by the left in Figure 3, we will show that theconnected sum θ X,D ( C −C ) ∗ ) : π ( b C tL b C tL ) → Ker( ε X ) is null-bordance. Let N C i ⊂ S bea neighborhood around the local move. Then we put a canonical saddle F in N C × [0 ,
1] whichbounds the four arcs illustrated in Figure 3. Define an embedded surface W ⊂ S × [0 ,
1] tobe (cid:0) ( L \ N C ) × [0 , (cid:1) ∪ F . Then the t -fold cyclic covering W → S × [0 ,
1] branched over W bounds b C tL ⊔ b C tL . Moreover, we can verify that the sum θ X,D ( C −C ) ∗ ) extends to a grouphomomorphism π ( W ) → Ker( ε X ), which gives the desired null-bordance. F aa a a Figure 3: F is a saddle in the neighborhood N C i × [0 , Finally, in order to prove Theorem 3.1, let us review
Thom homomorphism T G : Ω n ( G ) → H n ( K ( G, H gr n ( G ) obtained by assigning to every pair ( M, f : π ( M ) → G ) the image ofthe orientation class under f ∗ : H n ( M ) → H n ( K ( G, n = 3, themap T G is an isomorphism Ω ( G ) ∼ = H gr3 ( G ). Proof of Theorem 3.1.
Let G be As( X ). Put the inclusion ι : Ker( ε X ) ֒ → As( X ). Conse-quently, defining the T-map Θ X to be the composite T As( X ) ◦ ι ∗ ◦ Θ ΠΩ : Π ( X ) → H gr3 (As( X )),we can see that this Θ X satisfies the desired properties. As a preliminary, we now review some properties of the (co)homology of the rack space, anda topological interpretation of the cocycle invariants. There is nothing new in this section. e start by reviewing the (action) rack space introduced by Fenn-Rourke-Sanderson [FRS1,Example 3.1.1]. Let X be a quandle. We further fix a set Y acted on by As( X ), which iscalled X -set . For example, the quandle X is itself an X -set, referred as to the primitive X -set ,from the canonical action X x As( X ) mentioned in §
2. We further equip a quandle X andthe X -set Y with their discrete topology. Let us put a union S n ≥ (cid:0) Y × ([0 , × X ) n (cid:1) , andconsider the relations given by ( y, t , x , . . . , x j − , , x j , t j +1 , . . . , t n , x n ) ∼ ( y · e x j , t , x ⊳ x j , . . . , t j − , x j − ⊳ x j , t j +1 , x j +1 , . . . , t n , x n ) , ( y, t , x , . . . , x j − , , x j , t j +1 , . . . , t n , x n ) ∼ ( y, t , x , . . . t j − , x j − , t j − , x j +1 , . . . , t n , x n ) . Then the rack space B ( X, Y ) is defined to be the quotient space. When Y is a single point, wedenote the space by BX for short. By construction, we have a cell decomposition of B ( X, Y )by regarding the projection S n ≥ (cid:0) Y × ([0 , × X ) n (cid:1) → B ( X, Y ) as characteristic maps.Furthermore, we briefly review the rack and quandle (co)homologies (our formula relies on[CEGS]). Let X be a quandle, and Y an X -set. Let C Rn ( X, Y ) be the free right Z -modulegenerated by Y × X n . Define a boundary ∂ Rn : C Rn ( X, Y ) → C Rn − ( X, Y ) by ∂ Rn ( y, x , . . . , x n ) = X ≤ i ≤ n ( − i (cid:0) ( y ⊳ x i , x ⊳ x i , . . . , x i − ⊳ x i , x i +1 , . . . , x n ) − ( y, x , . . . , x i − , x i +1 , . . . , x n ) (cid:1) . The composite ∂ Rn − ◦ ∂ Rn is known to be zero. The homology is denoted by H Rn ( X, Y ) and iscalled rack homology . As is known, the cellular complex of the rack space B ( X, Y ) above isisomorphic to the complex ( C R ∗ ( X, Y ) , ∂ R ∗ ). Remark 5.1. If Y is the primitive X -set Y = X , we have the chain isomorphism C Rn ( X, X ) → C Rn +1 ( X, pt) induced from the identification X × X n ≃ X n +1 ; see, e.g., [Cla1, Proposition 2.1].In particular, we obtain an isomorphism H Rn ( X, X ) ∼ = H Rn +1 ( X, pt).Furthermore, let C Dn ( X, Y ) be a submodule of C Rn ( X, Y ) generated by ( n +1)-tuples ( y, x , . . . , x n )with x i = x i +1 for some i ∈ { , . . . , n − } . It can be easily seen that the submodule C Dn ( X, Y )is a subcomplex of C Rn ( X, Y ) . Then the quandle homology , H Qn ( X, Y ), is defined to be thehomology of the quotient complex C Rn ( X, Y ) /C Dn ( X, Y ).We will review some properties of these homologies in the case where Y is a single point (Insuch a case, we omit the symbol Y ). Let us decompose X as X = ⊔ i ∈ O( X ) X i by the connectedcomponents. The following isomorphisms were shown [LN]: H R ( X ) ∼ = Z O( X ) , H R ( X ) ∼ = H Q ( X ) ⊕ Z O( X ) . (14)Furthermore, Eisermann [E2] gave a computation of the second quandle homologies H Q ( X )with trivial Z -coefficients. To see this, for i ∈ O( X ), define a homomorphism ε i : As( X ) → Z by (cid:26) ε i ( e x ) = 1 ∈ Z , if x ∈ X i ,ε i ( e x ) = 0 ∈ Z , if x ∈ X \ X i . (15)Note that the sum ⊕ i ∈ O( X ) ε i yields the abelianization As( X ) ab ∼ = Z O( X ) by (2). Furthermore heorem 5.2 ([E2, Theorem 9.9]) . Let X be a quandle. Decompose X = ⊔ i ∈ O( X ) X i as theorbits by the action of As( X ) . Fix an element x i ∈ X i for each i ∈ O( X ) . Let Stab( x i ) ⊂ As( X ) be the stabilizer of x i . Then the quandle homology H Q ( X ) is isomorphic to the directsum of the abelianizations of Stab( x i ) ∩ Ker( ε i ) : Precisely, L i ∈ O( X ) (cid:0) Stab( x i ) ∩ Ker( ε i ) (cid:1) ab . Eisermann showed topologically this result by using a certain CW-complex. However, in § Awe later give another proof as a slight application of Proposition 8.2.Furthermore, we now turn into the study of the cycle invariant Φ X ( L ) mentioned in (8),and briefly explain a topological interpretation of this invariant shown by Eisermann [E1, E2].Decompose X = ⊔ i ∈ O( X ) X i as above. Given an X -coloring C ∈
Col X ( D ), with respect to alink component of L , we fix an arc γ j on D for 1 ≤ j ≤ L . Let x j := C ( γ j ) ∈ X j , andfix a longitude l j of the component. Recall from (4) the associated group homomorphismΓ C : π ( S \ L ) → As( X ). Remark that each longitude l j commutes with the meridian inthe same link component. Accordingly, Γ C ( l j ) commutes with e x j in As( X ): in other wards,Γ C ( l j ) ∈ Stab( x j ). Furthermore, since the class of the longitude l j in H ( S \ L ) is zero,Γ C ( l j ) is contained in the kernel Ker( ε j ) [see (15)]. Therefore, Γ C ( l j ) lies in Stab( x j ) ∩ Ker( ε j ).Further, consider the class Γ C ( l j ) in the abelianization of this Stab( x j ) ∩ Ker( ε j ). In summary,we obtain (cid:0) [Γ C ( l )] , . . . , [Γ C ( l L )] (cid:1) ∈ M ≤ j ≤ L (cid:0) Stab( x j ) ∩ Ker( ε j ) (cid:1) ab . (16)Here note from Theorem 5.2 that each direct summand in the right side is contained in H Q ( X ).We then put the product [Γ C ( l ) · · · Γ C ( l L )] ∈ H Q ( X ). By the discussion in [E1, Theorems3.24 and 3.25], it can be seen that the product coincides with the value H X ( C ) in (7) exactly.Hence, when | X | < ∞ , the cycle invariant Φ X ( L ) written in (8) is reformulated asΦ X ( L ) = X C∈ Col X ( D ) [Γ C ( l ) · · · Γ C ( l L )] ∈ Z [ H Q ( X )] . (17)This Φ X ( L ) was called “colouring polynomials” in [E1, § BX . Recall the map H X : Π ( X ) → H Q ( X ) in (7). Using the isomorphisms (14) and (18),we consider a composite π ( BX ) ∼ = Π ( X ) ⊕ Z O( X ) proj . −−→ Π ( X ) H X −−→ H Q ( X ) ֒ → H Q ( X ) ⊕ Z O( X ) ∼ = H ( BX ) . From the definitions of the maps H X and the 2-skeleton of the rack space BX , we can easilyverify that this composite coincides with the Hurewicz map of BX modulo the direct summand Z O( X ) (see [RS] and [N1, Proposition 3.12] for details). In conclusion, the formula (17) enablesus to compute the Hurewicz map of BX . This section proves Theorem 3.5. Since the proof is ad hoc, the hasty reader may read onlythe outline in § tates a t X -vanishing theorem and some properties of quandle coverings. In Section 6.3, wewill investigate the homomorphism Θ X in terms of relative bordism groups, and complete theproof. We will fix notation throughout this section. Notation.
Let X be a connected quandle of type t X < ∞ (possibly, X is of infinite order).Furthermore, we write a symbol ℓ for a prime which is relatively prime to the integer t X . We roughly outline the proof of Theorem 3.5 to compute Π ( X ).As an approach to the homotopy group π ( BX ), the reader should keep in mind the follow-ing isomorphism shown by [FRS2] (see also [N1, Theorem 6.2] for the detailed description): π ( BX ) ∼ = Π ( X ) ⊕ Z ⊕ O( X ) , (18)where the symbol O( X ) is the set of the connected components of X . According to theisomorphism (18), to compute Π ( X ), we will change a focus on computing the homotopygroup π ( BX ) from a standard discussion of “Postnikov tower on BX ”. To illustrate, let c : BX ֒ → K ( π ( BX ) ,
1) be an inclusion obtained by killing the higher homotopy groups of BX . Notice that the homotopy fiber of c is the universal covering of BX . Thanks to the fact[FRS2] that the action of π ( BX ) on π ( BX ) is trivial (see also [Cla1, Proposition 2.16]), theLeray-Serre spectral sequence of the map c provides an exact sequence H ( BX ) c ∗ −→ H gr3 ( π ( BX )) τ −→ π ( BX ) H −→ H ( BX ) c ∗ −→ H gr2 ( π ( BX )) → , (19)where H is the Hurewicz map of BX and the τ is the transgression (see, e.g., [McC, § . bis ],[Bro, § II.5] for details).We now reduce this (19) to (20) below. Recalling the isomorphism H ( BX ) ∼ = Z O( X ) ⊕ H Q ( X ) (see (14)), the restriction of the Hurewicz map H on the summand Z O( X ) ⊂ π ( BX )is shown to be an isomorphism [N1, Proposition 3.12]. Therefore, recalling the isomorphismAs( X ) ∼ = π ( BX ), the sequence (19) is reformulated as H ( BX ) c ∗ −→ H gr3 (As( X )) τ −→ Π ( X ) H −→ H Q ( X ) c ∗ −→ H gr2 (As( X )) → . (20)Since this paper often uses this sequence, we call it the P -sequence (of X ).Using the P -sequence, we outline the proof of Theorem 3.5. Let X be connected and of type t X < ∞ . We later see Theorem 6.1 which says that the maps c ∗ : H n ( BX ) → H gr n ( π ( BX ))in (19) are annihilated by t X for n ≤
3. Thus, the P -sequence (20) becomes a short exactsequence up to t X -torsion (Corollary 6.2). Hence, in order to show that the TH-map Π ( X ) → H gr3 (As( X )) ⊕ H Q ( X ) is a [1 /t X ]-isomorphism as stated in Theorem 3.5, we shall show thatthe T-map Θ X : Π ( X ) → H gr3 (As( X )) constructed in Theorem 3.1 turns out to be a splittingof the exact sequence (20).To this end, we first show the splitting with respect to the extended quandles e X (Proposition6.6). So we will review properties of the e X in § τ in (20) can be regarded as an inverse mapping of the T-map Θ e X in a relative) bordism theory. After that, returning connected quandles X , the functoriality ofthe projection e X → X completes the proof of Theorem 3.5.Before going to the next subsection, we now immediately prove Proposition 3.8: Proof of Proposition 3.8.
Since the t X -torsion of H gr3 (As( X )) ⊕ H Q ( X ) is zero by assumption,that of Π ( X ) vanishes because of (20). Hence, that of the TH-map Θ X ⊕ H X is zero asdesired. t X -vanishing of the map c ∗ , and the quandle covering. Following the preceding outline, we will review the results in [N4]: First,
Theorem 6.1 ([N4]) . Let X be a connected quandle of type t X , and let t X < ∞ . For n = 2 and , the induced map c ∗ : H n ( BX ) → H gr n (As( X )) in (19) is annihilated by t X .Furthermore, the second group homology H gr2 (As( X )) is annihilated by t X . This theorem yields two corollaries which are useful for the P -sequences as follows. Corollary 6.2.
Let X be as above, and ℓ be a prime which is relatively prime to the t X . Thenthe P -sequence localized at ℓ is reduced to a short exact sequence −→ H gr3 (As( X )) ( ℓ ) −→ π ( BX ) ( ℓ ) H X −−−→ H ( BX ) ( ℓ ) −→ . Corollary 6.3.
Let X and ℓ ∈ Z be as above. Let X be of finite order. Then the quandle cycleinvariant Φ X in (8) is non-trivial in the ℓ -torsion part. That is, for any class [ O ] ∈ H ( BX ) ( ℓ ) ,there exists some X -coloring C of some link such that H X ([ C ]) = [ O ] .Proof. By Corollary 6.2, the map H X localized at ℓ is surjective. Since the Π ( X ) is generatedby X -colorings of links by definition, we have H X ([ C ]) = [ O ] for some X -coloring C . Remark 6.4.
In general, we can similarly see that, for a quandle X with H gr2 (As( X )) = 0,any class [ O ] ∈ H ( BX ) ensures some X -coloring C such that H X ([ C ]) = [ O ].Changing the subject, we will mention extended quandles considered in [Joy, § ε X : As( X ) → Z in (3). Given a connected quandle X with a ∈ X , we equipthe kernel Ker( ε X ) with a quandle operation by setting g ⊳ h := e − a gh − e a h for g, h ∈ Ker( ε X ) . Let us denote the quandle (Ker( ε X ) , ⊳ ) by e X , and call the extended quandle (of X ) . We easilysee the independence of the choice of a ∈ X up to quandle isomorphisms. Using the restrictedaction X x Ker( ε X ) ⊂ As( X ), the canonical map p : e X → X sending g to a · g is known tobe a quandle homomorphism (see [Joy, Theorem 7.1]), and is called the universal (quandle)covering of X .The previous paper [N4] showed their interesting properties, which will be used later. Theorem 6.5 ([N4, § . Let X be a connected quandle of type t X , and let p ∗ : As( e X ) → As( X ) be the epimorphism induced from the covering p : e X → X . Fix the identity element e X ∈ Ker( ε X ) = e X . Then, i) The quandle e X is connected and is of type t X . If X is of finite order, then so is e X .(ii) Under the canonical action of As( e X ) on e X , the stabilizer Stab(1 e X ) of e X is equal to Z × Ker( p ∗ ) in As( e X ) . Furthermore, the summand Z is generated by e X .(iii) The second quandle homology of e X is isomorphic to the abelian kernel of the projection p ∗ : As( e X ) → As( X ) . Namely H Q ( e X ) ∼ = Ker( p ∗ ) .(iv) The homology H Q ( e X ) ∼ = Ker( p ∗ ) is annihilated by the type t X .(v) The above map p ∗ induces a [1 /t X ] -isomorphism H gr3 (As( e X )) ∼ = H gr3 (As( X )) . Θ X as a splitting We first prove Theorem 3.5 by using the following key proposition.
Proposition 6.6.
Let p : e X → X be the universal covering. If the homology H gr3 (As( X )) is finitely generated, then the T-map Θ e X : Π ( e X ) → Ω (As( e X )) in Theorem 3.1 is a [1 /t X ] -splitting in the short exact sequence in Corollary 6.2.Proof of Theorem 3.5. Take the P -sequences associated to the covering p : e X → X : H gr3 (As( e X )) ( ℓ ) ˜ τ ∗ / / p ∗ (cid:15) (cid:15) Π ( e X ) ( ℓ ) / / p ∗ (cid:15) (cid:15) H Q ( e X ) ( ℓ ) p ∗ (cid:15) (cid:15) / / H gr2 (As( e X )) ( ℓ ) = 0 p ∗ (cid:15) (cid:15) H gr3 (As( X )) ( ℓ ) τ ∗ / / Π ( X ) ( ℓ ) / / H Q ( X ) ( ℓ ) / / H gr2 (As( X )) ( ℓ ) = 0 . Since the left p ∗ between group homologies is a [1 /t X ]-isomorphism (see Theorem 6.5 (iv) and(v)), the TH-map Θ X ⊕ H X : Π ( X ) ( ℓ ) → H gr3 (As( X )) ( ℓ ) ⊕ H Q ( X ) ( ℓ ) is an isomorphism by thefunctoriality of Θ X and Proposition 6.6.Thus, we shall aim to prove Proposition 6.6 with respect to the extended quandles e X .For the proof, we will review a classical bordism theory (see [CF, § Y, A ) with A ⊂ Y , consider a continuous map f : ( M, ∂M ) −→ ( Y, A ) , where M is an oriented compact n -manifold. Such two maps f , f are G -bordant , if thereexist an oriented compact manifold W of dimension n + 1 and a map F : W → Y for which(I) There is an n -dimensional submanifold M ′ ⊂ ∂W satisfying F ( ∂W \ M ′ ) ⊂ A. (II) There is a diffeomorphism g : ( − M ) ∪ M → M ′ preserving orientation such that( − f ) ∪ f = ( F | M ′ ) ◦ g .Then the bordism group of ( Y, A ), denoted by Ω n ( Y, A ), is defined to be the set of all suchmap f subject to the G -bordant relations. We make this Ω n ( Y, A ) into an abelian group bydisjoint union. Furthermore, { Ω n ( Y, A ) } n ≥ gives a homology theory (see [CF, § n ( Y, A ) ∼ = H n ( Y, A ), for n ≤
3, is obtained by the Atiyah-Hirzebruch spectral equence. Furthermore, if Y is the Eilenberg-MacLane space K ( G,
1) and A is the empty set,then Ω n ( Y, A ) is isomorphic to the group Ω n ( G ) introduced in § e X -coloring C with L link components, we take t X -copies of C , and denote them by C j for1 ≤ j ≤ m . Let us fix an arc of each link component of C , and consider a connected sum ofthese C , . . . , C m (see Figure 4). Denote the resulting link by L and the associated e X -coloringof L by C . We then set a homomorphism Γ C : π ( S \ L ) → As( e X ) discussed in (4). Note thateach meridian of L is sent to be e g for some g ∈ e X by definition. Furthermore each longitudes l j ∈ π ( S \ L ) are sent to be zero. Actually the formula (16) says that the Γ C ( l j ) lies inKer( ε e X ) ∩ Stab(1 e X ), which is equal to the abelian kernel Ker( p ∗ ) and is annihilate by t X (seeTheorem 6.5). Consequently, the map Γ C sends every boundaries of S \ L to a 1-cell of B e X .Here note that the 1-skeleton B e X is, by definition, a bouquet of circles labeled by elementsof e X . In the sequel, considering all e X -coloring C and such homomorphisms Γ C modulo thebordance relations, the map C 7→ Γ C defines the desired homomorphismΥ e X : Π ( e X ) −→ Ω ( K (As( e X ) , , B e X ) . (21)Hereafter, we denote by Ω rel3 ( e X ) this relative bordism Ω ( K (As( e X ) , , B e X ), for simplicity. γ γ γ γ γ γ γ γ C C C C m C · · · · · · · · ··· · · ·· · · Figure 4: Construction of C from C , when the link components of C are two. We now prove Proposition 6.6 by using the following lemma:
Lemma 6.7.
The homomorphism Υ e X : Π ( e X ) → Ω rel3 ( e X ) is surjective up to t X -torsion.Proof of Proposition 6.6. We first explain the following diagram: / / Ω (As( e X )) ( ℓ ) (cid:31) (cid:127) δ ∗ / Ω rel3 ( e X ) ( ℓ ) / / Ω ( B e X ) ( ℓ ) = 0 / / H (As( e X )) ( ℓ ) e τ ∗ / / Π ( e X ) ( ℓ )Υ e X O O / / H Q ( e X ) ( ℓ ) = 0Here the upper sequence is derived by the homology theory Ω n with considering the pair B e X ֒ → K (As( e X ) , P -sequence of e X with Theo-rems 6.1 and 6.5. Since Π ( e X ) is [1 /t X ]-isomorphic to the finitely generated module Ω (As( e X ))by assumption, the localized map of Υ e X is an isomorphism by Lemma 6.7.Therefore, to accomplish the proof, it is sufficient to show the equality t X · (cid:0) δ ∗ ◦ Θ e X ([ C ]) (cid:1) = t X · Υ e X ([ C ]) ∈ Ω rel3 ( e X ) , (22) or any e X -coloring C . For this, put the resulting link L and coloring C explained above.Furthermore, take the t X -fold cyclic covering p : C t X L → S \ L , and consider the naturalinclusion i C : C t X L ⊂ b C t X L by gluing the 2-handles along the boundary tori. Here notice thatthe composite θ e X,D ( C ) ◦ ( i C ) ∗ : π ( C t X L ) → As( X ) coincides with Γ C ◦ p ∗ by the definition (12).Furthermore notice that the inclusion i C gives a bordance relation between the θ e X,D ( C ) andthis composite θ e X,D ( C ) ◦ ( i C ) ∗ . Since the above map δ ∗ comes from the correspondences with( M, f ) to (
M, f ) itself by definition, we thus have t X · δ ∗ ◦ Θ e X ([ C ]) = δ ∗ ◦ Θ e X ([ C ]) = [ θ e X,D ( C )] = [ θ e X,D ( C ) ◦ ( i C ) ∗ ] = [Γ C ◦ p ∗ ] ∈ Ω rel3 ( e X ) , where the first equality is derived from t X [ C ] = [ C ] ∈ Π ( e X ) from the definition of L . Wenotice [Γ C ◦ p ∗ ] = t X [Γ C ] ∈ Ω rel3 ( e X ) since the projection p takes the (relative) fundamental classof C t X L to the t X -multiple of that of S \ L . Hence, since Υ e X ([ C ]) = Γ C by definition, we havethe desired equality (22).To conclude this section, we will work out the proof of Lemma 6.7. Proof of Lemma 6.7.
To begin, we claim that the Z ( ℓ ) -module Ω rel3 ( e X ) ( ℓ ) is generated by bor-dism classes represented by (normal) 3-submanifolds in S with torus boundary components.For this purpose, we first set up the isomorphism (23) below. Here refer to the fact [Cla1, § B e X is a topological monoid; hence, it is a based loop spaceof some space L X . We therefore have two homotopy fibrationsΩ L X −→ B e X c ∗ −→ K (As( e X ) , , B e X c ∗ −→ K (As( e X ) , P L −→ L X . From the right map P L , we obtain an isomorphism after localization at ℓ :( P L ) ∗ : Ω ( B e X, K (As( e X ) , ( ℓ ) ∼ = Ω ( L X ) ( ℓ ) . (23)However, since the L X is 2-connected by definition, the Hurewicz theorem π ( L X ) ∼ = [1 /t X ] Ω ( L X ) implies that this Ω ( L X ) is generated by maps S → L X . Noticing that the map( P L ) ∗ can be regarded as a map coming from collapse of each boundaries of manifolds, thisisomorphism (23) implies that generators of the Ω ( K (As( e X ) , , B e X ) ( ℓ ) are derived from3-submanifolds in S .Next, so as to verify the claim above, consider the inclusions S ⊂ B e X ⊂ B e X c ֒ −→ K (As( e X ) , , where the first is obtained by taking the circle labeled by a ∈ e X . Notice from Theorem 6.1that these inclusions S ⊂ B e X ⊂ K (As( e X ) ,
1) induce isomorphisms H ( S ) ( ℓ ) ∼ = H ( B e X ) ( ℓ ) ∼ = H gr2 (As( e X )) ( ℓ ) ∼ = 0 . Therefore, they yield isomorphismsΩ ( K (As( e X ) , , S ) ( ℓ ) ∼ = Ω rel3 ( e X ) ( ℓ ) ∼ = Ω ( K (As( e X ) , , B e X ) ( ℓ ) . ere note that, since the last term is generated by classes from 3-manifolds in S as observedabove and π ( S ) ∼ = Z is abelian, the first term is generated by classes from 3-manifolds in S with torus boundaries . Hence, so is the Ω rel3 ( e X ) ( ℓ ) as claimed.Finally, to show the required surjectivity of Υ e X , we will prove that any generator O ofΩ rel3 ( e X ) ( ℓ ) comes from some e X -coloring via the bijection (5). By the previous claim, t − X · O isrepresented by a homomorphism f : π ( S \ L ) → As( e X ) for some link L ⊂ S . Furthermoreput the resulting link L in constructing Υ e X in (21). By repeating the process, we haveanother L . Then the f extends to two maps f : π ( S \ L ) → As( e X ) and f : π ( S \ L ) → As( e X ) canonically, where because the class of the latter in Ω rel3 ( e X ) ( ℓ ) equals the generator O . Notice that, for each link component 1 ≤ i ≤ L , with a choice of meridian element m i , the f ( m i ) ∈ As( e X ) is conjugate to e n i a for some n i ∈ N , from the definition of Ω rel3 ( e X ) ( ℓ ) :Namely, in Wirtinger presentation of π ( S \ L ), each arc α is labeled by e n i y α for some y α ∈ e X ;see (2). Accordingly, replacing the i -th component of the link L by n i -parallel copies of thecomponent, we have another link L P and, then, can construct a canonical homomorphism f P : π ( S \ L P ) → As( e X ) by which each meridian of L P is sent to e y for some y ∈ e X (see Figure 5). We remark that, this f P sends the associated longitude l i of L P to e t X n y y forsome n y ∈ Z , by the reason similar to the construction of Υ e X in (21). In particular, we have y · f P ( l i ) = y · e t X n y y = y ∈ e X . Hence, the bijection (5) admits an e X -coloring C f such that Γ C f = f P : π ( S \ L P ) → As( e X ). Consequently, we have the equality Υ e X ([ C f ]) = O ∈ Ω rel3 ( e X ) ( ℓ ) byconstruction, which implies the surjectivity. e n i a e n j b e n i a ⊳ nj b e a e b e a ⊳ nj b n j -strands z }| { n i -strands z}|{ Figure 5: Construction from the f : π ( S \ L ) → As( e X ) to f P : π ( S \ L P ) → As( e X ). Π ( X ) . We will compute the group Π ( X ) with respect to same quandles. and do Π ( X ) of connectedquandles of order ≤ § P -sequence. Actually, the proofs result fromcomputations of the terms H gr3 (As( X )) and H Q ( X ) with the TH maps concretely includingthese t X -torsions. To be precise, since the first homology of any closed surface is generated by homology classes from some tori, given asubmanifold M ⊂ S with f : π ( M ) → As( X ) such that f ( π ( ∂M )) ⊂ Z , we can obtain another M ′ ⊂ S with torus boundariesby attaching some 2-handles to M , and the maps f extend to ¯ f : π ( M ′ ) → As( X ) such that ¯ f ( π ( ∂M ′ )) ⊂ Z . .1 On Alexander quandles We start reviewing Alexander quandles. Any Z [ T, T − ]-module M is a quandle with theoperation x ⊳ y = y + T ( x − y ) for x, y ∈ M , which is called an Alexander quandle . The typeis the minimal N such that T N = id M since x ⊳ n y = y + T n ( x − y ). As a typical example, witha choice of an element ω ∈ F q \{ , } , the quandle of the form X = F q [ T ] / ( T − ω ) is calledan Alexander quandle on a finite field F q . Furthermore, an Alexander quandle X is said to be regular , if X is connected and its type is relatively prime to the order | X | , e.g., the Alexanderquandles on F q owing to ω q − = 1.We now discuss regular Alexander quandles X of finite order. Corollary 7.1.
Let X be a regular Alexander quandle of finite order. Then, the TH-map isan isomorphism.Proof. As is known [N4, § ℓ -torsion subgroups of the homologies H Q ( X ) and H gr3 (As( X ))are zero. Hence, the proof is immediately obtained from Theorem 3.5.We however remark that the torsion Π ( X ) ⊗ Z /p of the Alexander quandles on F q with p = 2were already computed from another direction (see [N2, Appendix]). Thus, we omit examplingconcrete computations. F q We will show Theorem 7.4 that computes Π ( X ) of symplectic and orthogonal quandles X over finite fields. Let us begin introduce the two quandles in details: Example 7.2 (Symplectic quandle) . Let K be a field, and Σ g be the closed surface of genus g .Define X to be the first homology with K -coefficients away from 0, that is, H (Σ g ; K ) \ { } = K g \{ } . Denote the symplectic form by h , i : H (Σ g ; K ) → K . Then, this set X is made intoa quandle by the operation x ⊳ y := h x, y i y + x ∈ X for any x, y ∈ X , and is called a symplecticquandle (over K ) . The operation • ⊳ y : X → X is called the transvection of y , in the commonsense. Note that the quandle X is of type p =Char( K ) since x ⊳ N y = N h x, y i y + x . When K is a finite field F q , we denote the quandle by Sp gq . Example 7.3 (Spherical quandle) . Let K be a field of characteristic = 2. Let h , i : K n +1 ⊗ K n +1 → K be the standard symmetric bilinear form. Consider a set of the form S nK := { x ∈ K n +1 | h x, x i = 1 } . We define the operation x ⊳ y to be 2 h x, y i y − x ∈ S nK . This pair ( S nK , ⊳ ) is a quandle of type2, and is referred to as a spherical quandle (over K ). In the finite case K = F q , we denote thequandle by S nq .As observed above, quandle consists of, figuratively speaking, ‘operations itself centered at y ∈ X ’, which can be described as homogenous spaces (see [Joy, §
7] for detail).To describe the theorem, we call q = p d ∈ N exceptional , if the q is one of { , , , , } ,that is, d ( p − ≤ heorem 7.4. Let q = p d be odd, and be not exceptional.(I) Let X be the symplectic quandle Sp nq over F q . Then, the TH-map is an isomorphism.Furthermore, Π ( X ) ∼ = (cid:26) Z / ( q − , i f n > , Z / ( q − ⊕ ( Z /p ) d , i f n = 1 . (II) Let X be the spherical quandle S nq over F q . The TH-map is a [1 / -isomorphism. More-over, Π ( X ) ∼ = [1 / (cid:26) Z / ( q − , i f n > , Z / ( q − ⊕ Z / ( q − δ q ) , i f n = 2 . Here δ q = ± is according to q ≡ ± .Proof. The proof essentially relies on some works of Quillen and Friedlander, who had cal-culated group homologies of some groups of Lie type over F q . We list their works after theproof.For (I), we will mention some homologies associated with the symplectic quandle X = Sp nq .As is shown [N4, § X ) ∼ = Z × Sp (2 n ; F q ); hence Theorems 7.6 and 7.7 below tellus the group homology H gr3 (As( X )) ∼ = Z / ( q − n ≥ H Q ( X ) vanishes; see [N4]; Therefore the P -sequence (19) is reduced to bean epimorphism Z / ( q − → Π ( X ) →
0. Since X is of type p , Theorem 3.5 with n ≥ Z / ( q − ∼ = Π ( X ) as required.Next, we work out the case n = 1. Note from [N4] that the second quandle homology H Q ( X ) is ( Z /p ) d . Thereby the P -sequence (19) is rewritten as Z / ( q − −→ Π ( X ) −→ ( Z /p ) d −→ . (24)Using Theorem 3.5 again, we have Π ( X ) ∼ = Z / ( q − ⊕ ( Z /p ) d as required.For (II), we similarly deal with the spherical quandle X = S nq with n ≥
2. As is shown[N4, § H gr3 (As( X )) ∼ = H gr3 ( O ( n + 1)). Then, it follows fromTheorem 7.6 below that H gr3 (As( X )) ∼ = Z / ( q −
1) without 2-torsion. Moreover, it is shown[N4, § n ≥ H Q ( X ) is an elementary abelian 2-group, and that if n = 2 thequandle homology H Q ( X ) is Z / ( q − δ q ). Hence, the desired isomorphism Π ( X ) ∼ = Z / ( q − P -sequence (19) and Theorem 3.5.As mentioned above, we review the group homologies of the symplectic groups Sp (2 g, F q )and the orthogonal groups O ( n ; F q ). There is nothing new until the end of this subsection.We start by recalling the homologies of Sp (2 , F q ) and O (3; F q ). Proposition 7.5. If p = 2 and q = 3 , , then the first and second homologies of Sp (2 g ; F q ) vanish, i.e., H gr1 ( Sp (2 g, F q )) ∼ = H gr2 ( Sp (2 g, F q )) ∼ = 0 . Furthermore, the ℓ -torsions of the third homology of Sp (2 , F q ) are expressed as H gr3 ( Sp (2 , F q )) ( ℓ ) ∼ = (cid:0) Z / ( q − (cid:1) ( ℓ ) , for ℓ = p. n the other hand, the homologies H gr1 ( O (3 , F q )) and H gr2 ( O (3 , F q )) are annihilated by 2.Furthermore, the ℓ -torsions of the third homology of O (3 , F q ) are expressed by H gr3 ( O (3 , F q )) ( ℓ ) ∼ = (cid:0) Z / ( q − (cid:1) ( ℓ ) , for ℓ = p, . Proof.
See [FP, VIII. §
4] or [Fri], noting the order | O (3 , F q ) | = 2 q ( q − Sp (2 g ; F q ) and O ( n, F q ) as follows: Theorem 7.6 ([FP, Fri]) . Let q = p d be odd. The inclusion Sp (2 , F q ) ֒ → Sp (2 n, F q ) induceisomorphisms H gr3 ( Sp (2 , F q )) ∼ = ( ℓ ) H gr3 ( Sp (2 n, F q )) ∼ = ( ℓ ) Z / ( q − localized at ℓ = p . Further-more, for n ≥ , the inclusion O (3 , F q ) ֒ → O ( n, F q ) induces isomorphisms H gr3 ( O (3 , F q )) ∼ = ( ℓ ) H gr3 ( O ( n, F q )) ∼ = ( ℓ ) Z / ( q − localized at ℓ = p, .Proof. According to [FP], the inclusions induces isomorphisms their cohomology with Z /ℓ -coefficients. Taking limits as n → ∞ , their homologies are known to be H gr3 ( Sp ( ∞ , F q )) ∼ = ( ℓ ) Z / ( q −
1) and H gr3 ( O ( ∞ , F q )) ∼ = ( ℓ ) Z / ( q − ℓ are isomorphisms.In addition, we focus on these p -torsion parts, and state the vanishing theorem. Theorem 7.7 (Quillen and [Fri, § . Let q = p d be odd. If d ( p − > , then the p -torsionparts H gr3 ( Sp (2 n, F q )) ( p ) and H gr3 ( O ( n + 2 , F q )) ( p ) vanish for any n ≥ . Furthermore, if n isenough large, the p -vanishing holds even for d ( p − ≤ . Remark 7.8.
As a result in [Fri, Corollary 1.8], the inclusions Sp (2 n ; F q ) ֒ → GL (2 n ; F q ) ֒ → GL ( ∞ ; F q ) induce the isomorphism between the third homology H gr3 ( Sp (2 n, F q )) and theQuillen K -group K ( F q ) ∼ = H gr3 ( GL ( ∞ ; F q )) ∼ = Z / ( q − d ( p − > n is enoughlarge. For instance, following [Fri, § q = p = 5 and n ≥
7, the thirdhomology H gr3 ( Sp (2 n, F q )) is Z / (5 −
1) = Z / . We later use this result in § ≤ . Furthermore, we focus on connected quandles X of order ≤ Theorem 7.9.
For any connected quandle X of order ≤ , the TH-map is an isomorphism Π ( X ) ∼ = H gr3 (As( X )) ⊕ Im( H X ) . This theorem is a generalization of the previous paper [N1, § ( X ) ofonly quandles X with | X | ≤ ( X )from the list of connected quandles of order ≤
8. In the proof, we will often use the T-mapΘ ΠΩ : Π ( X ) → Ω (Ker( ε X )) in Lemma 4.1. We first determine the Π ( X ) of the connected quandle of order 4 as follows: roposition 7.10. If X is the Alexander quandle of the form Z [ T ] / (2 , T + T + 1) , then Π ( X ) ∼ = Z / ⊕ Z / . Proof.
We first recall the fact [N1, Proposition 4.5] which says that Π ( X ) is either Z / ⊕ Z / Z / ⊕ Z / H Q ( X ) ∼ = Z /
2. By the main theorem 3.5, hence it suffices to constructa [1 / H gr3 (As( X )) ∼ = [1 / Z /
8. For this, note As( X ) ∼ = Q ⋊ Z , where Q is thequaternion group of order 8 (see [N1, Lemma 4.8] for details). Noting Type( X ) = 3, considerthe quotient Q ⋊ Z /
3, which is isomorphic to Sp (2; F ). By using Proposition 7.5 and thetransfer, we thus have H gr3 (As( X )) ∼ = H gr3 ( Sp (2; F )) ∼ = Z / Remark 7.11.
We here note a relation between Π ( X ) and quandle homology groups. Asis known, H Q ( X ) ∼ = Z / H Q ( X ) ∼ = Z / Z / ( X ) is evaluated not by the quandle cohomology, but by the group cohomology H ( Q ; Z / ( X ) in general.Next, let us consider two Alexander quandles of order 8 of the forms X = Z [ T ] / (2 , T + T + 1) and X = Z [ T ] / (2 , T + T + 1). Then the both Π ( X ) were shown to be Z / X of order 4 or 8 is one of the three quandles above; hence Π ( X ) has been determined. In this subsection, we will calculate Π ( X ) of two quandles S , S ′ of order 6. Here the quandle S (resp. S ′ ) is defined to be the set of elements of a conjugacy class in the symmetric group S including (12) ∈ S (resp. (1234) ∈ S ) with the binary operation x ⊳ y = y − xy . (1432) (1342)(1423) (12)(23)(24)(12)Figure 6: An S -coloring C of the trefoil knot 3 , and an S ′ -coloring C of T , . Proposition 7.12.
For the quandle S , Π ( S ) ∼ = Z / ⊕ Z / . The first summand Z / isgenerated by Ξ S , ( C ) , where C is a coloring of the trefoil knot shown as Figure 6.On the other hand, for another quandle S ′ , we have Π ( S ′ ) ∼ = Z / . The generator isrepresented by Ξ S ′ ,T , ( C ) , where C is a coloring of the torus knot T , shown as Figure 6.Proof. We will show the sequence (25) below. It follows from the proof of [N1, Proposition4.9] that H Q ( S ) ∼ = Z / H gr2 (As( S )) ∼ = 0, and that H gr3 (As( S )) is a quotient of Z / P -sequence (20) becomes Z / −→ Π ( S ) H −→ H Q ( X ) (cid:0) ∼ = Z / (cid:1) −→ . (25) ext we will show that Π ( S ) surjects onto Z /
24. It is shown [N1, Lemma 4.10] thatthe kernel Ker( ε X ) is the binary tetrahedral group D = Sp (2; F ) whose third homologyis Z /
24 (see Proposition 7.5). Let D act canonically the 3-sphere S . Since the 4-foldcovering branched over the trefoil is S /D (see [Rol, § θ X,D in (12) sends the X -coloring C to an isomorphism π ( S /D ) ∼ → Sp (2; F ). SinceΩ ( D ) ∼ = Z /
24 is known to be generated by the pair ( S /D , id D ) [see [Bro, VI. Examples9.2]], the T-map Θ ΠΩ : Π ( S ) → Ω ( D ) in Lemma 4.1 turns out to be surjective.Finally, for proving the decomposition Π ( S ) ∼ = Z / ⊕ Z /
24, it is enough to show that theclass [ C ] ∈ Π ( S ) is sent to 2 ∈ H Q ( X ) ∼ = Z / H X . By the formula (17), H X ([ C ]) = Γ C ( l ) = e − e (1432) e (1423) ∈ Ker( ε X ) ∩ Stab( x ) ∼ = Z / . An elementary calculation can show the square H ([ C ]) = 1 and H ([ C ]) = 1 in As( X ),although we will not go into the details. In the sequel, H ([ C ]) = 2 = 0 as desired.Changing the subject, we will compute another Π ( S ′ ). For this, we now explain thesequence (26) below. It is shown [N1, Lemma 4.12 and Appendix A.2] that H Q ( S ′ ) ∼ = H gr2 (As( S ′ )) ∼ = Z /
2, and to estimate the order | H gr3 (As( S ′ )) | ≤
12. Hence, as a routinework, the P -sequence (20) becomes H gr3 (As( S ′ )) −→ Π ( S ′ ) H −→ Z / −→ Z / −→ . (26)For the sake of proving Π ( S ′ ) ∼ = Z /
12, it is sufficient to show the surjectivity of the T-mapΘ ΠΩ : Π ( S ′ ) → Z /
12. As is shown [N1, Lemma 4.12], the kernel Ker( ε X ) is the alternatinggroup A of order 12 whose third homology is Z /
12. Since the quandle S ′ is of type 4 and thedouble cover branched over the knot T , is S /D (see [Rol, §
10. D and E]), we can show thatthe map θ X,D in (12) sends the X -coloring C to the epimorphism π ( S /D ) = D → A ,which is a central extension. Hence, the class [ θ X,D ( C )] is a generator of Ω ( A ) ∼ = Z / ΠΩ : Π ( S ′ ) → Z /
12; Recalling | H gr3 (As( S ′ )) | ≤
12 and thesequence (26) above, the T-map Θ ΠΩ is an isomorphism Π ( S ′ ) ∼ = Z /
12. Furthermore, by thisprocess, the generator is represented by the coloring C . Finally, the rest of connected quandle of order 8 is the (extended) quandle e X explained in § X is the Alexander quandle of the form X = Z [ T ] / (2 , T + T +1) . Since As( X ) ∼ = Q ⋊Z (see the proof of Proposition 7.10), we have | e X | = 8 by definition. Proposition 7.13.
Let e X be the above quandle of order . Then Π ( e X ) ∼ = Z / .Proof. We first show that the induced map p ∗ : As( e X ) → As( X ) is an isomorphism. As isseen in the proof of Proposition 7.10, we note H gr2 (As( X )) ∼ = 0 and H gr1 (As( X )) ∼ = Z . Sincethe map p ∗ is a central extension (see Theorem 6.5(iii)), p ∗ is an isomorphism. Next we willshow Π ( e X ) ∼ = Z / H Q ( e X ) ∼ = 0 from Theorem 6.5 (iii), and H gr3 (As( e X )) ∼ = H gr3 ( Q ⋊ Z ) ∼ = Z / P -sequence is reducedto be an epimorphism Z / → Π ( e X ). By Theorem 3.5 again, we conclude Π ( e X ) ∼ = Z / roof of Theorem 7.9. One first deals with quandles X with | X | = 3 , ,
7. Then X is shownto be an Alexander quandle over the finite field F | X | [EGS]. By Theorem 7.1, the isomorphismΘ X ⊕ H X holds for such X . Next, a connected quandle of even order with | X | ≤ ≥ Notation.
Denote by Σ g,k the closed surface of genus g with k boundaries as usual. Let M g,k denote the mapping class group of Σ g,k which is the identity on the k -boundaries. In the case k = 0, we often suppress the symbol k , e.g., Σ g, = Σ g .We now review Dehn quandles [Y]. Consider the set, D g , defined to be D g := { isotopy classes of (unoriented) non-separating simple closed curves γ in Σ g } . For α, β ∈ D g , we define α ⊳ β ∈ D g by τ β ( α ), where τ β ∈ M g is the positive Dehn twistalong β . The pair ( D g , ⊳ ) is a quandle, and called (non - separating) Dehn quandle . As is well-known, any two non-separating simple closed curves are related by some Dehn twists. Hence,the quandle D g is connected, and is not of any type t X . In addition, since the Dehn twists aretransvections in the view of the cohomology H (Σ g ; F p ), for any prime p , we have a quandleepimorphism P p from D g to the symplectic quandle Sp gp . Concisely, P p : D g → Sp gp . The Dehnquandle D g is applicable to study 4-dimensional Lefschetz fibrations (see, e.g., [Y, N4]).We now aim to compute the second homotopy groups π ( B D g ) in a stable range as follows: Theorem 7.14.
Let g ≥ . The group Π ( D g ) is isomorphic to either Z / or Z / . Fur-thermore, a generator of Π ( D g ) is represented by a D g -coloring in Figure 7.Proof. We first observe homologies of the associated group As( D g ). Note the well-knownfacts H gr1 ( M g ) ∼ = 0 and H gr2 ( M g ) ∼ = Z (see [FM]); Gervais [Ger] showed the isomorphismAs( D g ) ∼ = Z × T g , where T g is the universal central extension of M g . Then, Lemma 7.15 belowand Kunneth theorem immediately imply H gr2 (As( D g )) ∼ = 0 and H gr3 (As( D g )) ∼ = Z / P -sequences in respect to the above epimorphism P : D g → Sp g with p = 5 . Let us use the facts H Q ( D g ) ∼ = Z / H Q ( Sp g ) ∼ = 0; see [N4, § § P -sequences are written in Z / ( P ) ∗ (cid:15) (cid:15) / / Π ( D g ) ( P ) ∗ (cid:15) (cid:15) / / H Q ( D g ) (cid:0) ∼ = Z / (cid:1) / / ( P ) ∗ (cid:15) (cid:15) H gr3 (As( Sp g )) δ ∗ / / Π ( Sp g ) (cid:0) ∼ = Z / (cid:1) / / H Q ( Sp g ) (cid:0) ∼ = 0 (cid:1) / / . Here the proof of Theorem 3.5 says that the bottom δ ∗ is an isomorphism H gr3 (As( Sp g )) → Π ( Sp g ) ∼ = Z / urthermore, we claim that the middle map ( P ) ∗ : Π ( D g ) → Π ( Sp g ) is surjective. Bycombing the T-map Θ ΠΩ in (13) with the epimorphism P : D g → Sp g above, we have acomposite mapΠ ( D g ) ( P ) ∗ −−−−→ Π ( Sp g ) Θ ΠΩ −−−−→ Ω ( Sp (2 g ; F )) . (27)To show the claim, it is enough to prove the surjectivity of this composite. Let us recall theisomorphisms Ω ( Sp (2 g ; F )) ∼ = H gr3 ( Sp (2 g ; F )) ∼ = Z /
24 from Remark 7.8, and consider the D g -coloring C illustrated below. Notice that the quandle Sp g is of type 5, and that the 5-fold cover of S branched along the trefoil is the Poincar´e sphere Σ(2 , ,
5) (see [Rol, § π is Sp (2; F ) exactly. Using the map θ X,D in (12), we can see that the associatedhomomorphism θ X,D ( C ) : π (Σ(2 , , → Sp (2; F ) is an isomorphism. Hence, the class[ θ X,D ( C )] is a generator of Ω ( Sp (2; F )). It follows from Theorem 7.6 that the inclusion Sp (2; F ) ֒ → Sp (2 g ; F ) induces an isomorphism between these homologies without 5-torsion,which means the claimed surjectivity of ( P ) ∗ . αβ γ C C ( α ) C ( γ ) C ( β ) Figure 7: A D g -coloring of the trefoil knot However, the left vertical map ( P ) ∗ : Z / → H gr3 (As( Sp g )) is not surjective (see Lemma7.16 below). Hence, by carefully observing the above commutative diagram, Π ( D g ) turns outto be either Z /
48 or Z / Lemma 7.15.
Let T g be the universal central extension on the group M g . If g ≥ , then H gr2 ( T g ) vanishes. Furthermore, if g ≥ , then H gr3 ( T g ) ∼ = Z / . We now prove Lemma 7.15 by using Quillen plus constructions and Madsen-Tillmann [MT].
Proof.
We first immediately have H gr2 ( T g ) ∼ = 0, since T g is the universal central extension of M g and the group M g is perfect (see, e.g., [Ros, Corollary 4.1.18]).We next focus on H gr3 ( T g ) with g ≥
3. Let B M + g,k denote Quillen plus construction of anEilenberg-MacLane space of M g,k (see, e.g., [Ros, Chapter 5.2] for the definition). Since M g,k is perfect, the space B M + g,k is simply connected. As a basic property of plus constructions(see [Ros, Theorem 5.2.7]), the homotopy group π ( B M + g ) is isomorphic to H gr3 ( T g ).It is therefore sufficient to calculate π ( B M + g ) for g ≥
7. For this, we set up some prelimi-naries. Consider the inclusion M g, → M g +1 , obtained by gluing the surface Σ , along oneof its boundary components. Let M ∞ , := lim g →∞ M g, . Furthermore put an epimorphism δ g : M g, → M g induced by gluing a disc to the boundary component of Σ g, . According to he Harer-Ivanov stability theorem improved by [RW], the inclusion ι ∞ : M g, → M ∞ , in-duces an isomorphism H gr j ( M g, ) ∼ = H gr j ( M ∞ , ), and the map δ g does H gr j ( M g ) ∼ = H gr j ( M g, ),for j ≤ δ + g : B M + g, → B M + g and ι + ∞ : B M + g, → B M + ∞ , inducedby δ g and ι ∞ , respectively. By Whitehead theorem, these maps induce isomorphisms( δ + g ) ∗ : π ( B M + g, ) ∼ = π ( B M + g ) , ( ι + ∞ ) ∗ : π ( B M + g, ) ∼ = π ( B M + ∞ , ) . However the π ( B M + ∞ ) ∼ = Z /
24 was shown by Madsen and Tillmann [MT] (see also [Eb]). Insummary, we have H gr3 ( T g ) ∼ = π ( B M + ∞ , ) ∼ = Z /
24 as required.
Lemma 7.16.
The induced map ( P ) ∗ : H gr3 (As( D g )) → H gr3 (As( Sp g )) with g ≥ is notsurjective.Proof. As mentioned previous, recall the reduction of P : As( D g ) → As( Sp g ) to Z × T g → Z × Sp (2 g ; F ). We easily see that it factors through Z × M g . However H gr3 ( M g ) ∼ = Z /
12 isknown [MT] (see also [Eb, § H gr3 ( T g ) ∼ = H gr3 ( Sp (2 g ; F )) ∼ = Z /
24 as above, the map( P ) ∗ is not a surjection. As an application from the study of the homotopy group π ( BX ), we compute some torsionsubgroups of third quandle homologies H Q ( X ) of finite connected quandles X . First, we proveTheorem 3.9 owing to the facts explained in § § H Q ( X )of some quandles.We briefly explain a basic line to study H Q ( X ) in this section. Let B ( X, X ) be the rackspace associated to the primitive X -set. Note the following isomorphisms: H ( B ( X, X )) ∼ = H R ( X ) ∼ = H Q ( X ) ⊕ H Q ( X ) ⊕ Z , (28)where the first isomorphism is derived from Remark 5.1, and the second was shown [LN,Theorem 2.2]. Composing this (28) with the result on π ( BX ) = π ( B ( X, X )) from Theorem3.5 can compute some torsion of the quandle homology H Q ( X ) [see Lemma (8.4)].Following this line, we now prove Theorem 3.9 as a general statement, which is rewrittenas Theorem 8.1 (Theorem 3.9) . Let X be a connected quandle with | X | < ∞ . Let Ker( ψ X ) be theabelian kernel in (11) . Then an isomorphism H Q ( X ) ∼ = ( ℓ ) H gr3 (As( X )) ⊕ (Ker( ψ X ) ∧ Ker( ψ X )) holds after localization at any prime ℓ which does not divide | Inn( X ) | / | X | .Proof. By Lemma 8.4 and the isomorphism (31) below, we have an isomorphism π ( BX ) ( ℓ ) ⊕ H gr2 (Ker( ψ X )) ( ℓ ) ∼ = H ( B ( X, X )) ( ℓ ) . (29)Recall from Theorem 3.5 the isomorphism π ( BX ) ( ℓ ) ∼ = H gr3 (As( X )) ( ℓ ) ⊕ H Q ( X ) ( ℓ ) ⊕ Z ( ℓ ) . Hence, together with (28) above, the isomorphism (29) is rewritten in H gr3 (As( X )) ( ℓ ) ⊕ H Q ( X ) ( ℓ ) ⊕ Z ( ℓ ) ⊕ H gr2 (Ker( ψ X )) ( ℓ ) ∼ = Z ( ℓ ) ⊕ H Q ( X ) ( ℓ ) ⊕ H Q ( X ) ( ℓ ) . ince the second group homology H gr2 (Ker( ψ X )) is the exterior product V (Ker( ψ X )) [see [Bro, § V.6]], by a reduction of the both hand sides, we reach at the conclusion.
We now recall basic properties of the rack space B ( X, Y ) introduced in § Proposition 8.2 ([FRS1, Theorem 3.7 and Proposition 5.1]) . Let X be a quandle, and Y an X -set. Decompose Y into the orbits as Y = ⊔ i ∈ I Y i . For i ∈ I and an element y i ∈ Y i ,denote by Stab( y i ) ⊂ As( X ) the stabilizer of y i . Then, the subspace B ( X, Y i ) ⊂ B ( X, Y ) ispath-connected, and the natural projection B ( X, Y i ) → BX is a covering. Furthermore, the π ( B ( X, Y i )) is the stabilizer Stab( y i ) , by observing the covering transformation group. We next observe the spaces B ( X, Y ) in some cases of Y , where X is assumed to be con-nected. First, since π ( BX ) ∼ = As( X ) from the 2-skeleton of BX , the projection B ( X, Y ) → BX with Y = As( X ) is the universal covering. Next, we let Y be the inner automor-phism group Inn( X ), and be acted on by As( X ) via (11). Considering the surjectionsInn( X ) → X → { pt } as X -sets, they then yield a sequence of the coverings B ( X, Inn( X )) −→ B ( X, X ) −→ BX. (30)
Remark 8.3.
According to Proposition 8.2, π ( B ( X, X )) is the stabilizer Stab( x ) ⊂ As( X ),and π ( B ( X, Inn( X ))) is the abelian kernel Ker( ψ X ) of ψ X : As( X ) → Inn( X ) in (11).We further observe homologies of the space B ( X, X ). Let ℓ be a prime which does notdivide the order | Inn( X ) | / | X | . As is known, the action of π ( BX ) on the homology group H ∗ ( BX ) is trivial (see [Cla1]). Then the first covering in (30) induces an isomorphism betweentheir homologies localized at ℓ . To be precise H ∗ ( B ( X, Inn( X ))) ( ℓ ) ∼ = H ∗ ( B ( X, X )) ( ℓ ) . (31)Actually the transfer map of the covering is an inverse mapping (see [Cla1, Proposition 4.2]).Finally, we will observe a relation between the homotopy and homology groups of the rackspace B ( X, Inn( X )). Refer to the fact that Clauwens [Cla1, § B ( X, Inn( X )). Lemma 8.4.
Let X be a connected quandle. Let BX G denote the rack space B ( X, Inn( X ))) for short. Then the Hurewicz homomorphism π ( BX ) = π ( BX G ) → H ( BX G ) gives a [1 / -splitting. In particular H ( BX G ) ∼ = [1 / π ( BX ) ⊕ H gr2 (Ker( ψ X )) . Proof.
As is known, the second k -invariants of path-connected topological monoids with CW-structure are annihilated by 2 [Sou, AP]. Namely, the Hurewicz map is a [1 / H .Noting π ( BX G ) ∼ = Ker( ψ X ) by Remark 8.3, we have Coker( H ) ∼ = H gr2 (Ker( ψ X )), whichimplies the required decomposition. .2 Example 1; Alexander quandles. We will compute the third quandle homologies of some quandles in more details than Theorem8.1 shown above. Notice that Theorem 3.9 is of use with respect to quandles X such that theorder | Inn( X ) | / | X | is small. As such examples, we consider the third quandle homologies ofAlexander quandles. Actually, notice that the order | Inn( X ) | / | X | equals Type( X ) exactly. Theorem 8.5.
Let X be a regular Alexander quandle of finite order. Then, there is the[1/2]-isomorphism H Q ( X ) ∼ = H gr3 (As( X )) ⊕ (cid:0)V Ker( ψ X ) (cid:1) .Moreover, if the order of X is odd, then the 2-torsion subgroups of the both sides are zero. Here remark that, the associated group As( X ) and the kernel Ker( ψ X ) have been completelycalculated by Clauwens [Cla2]. In particular, the group As( X ) is known to be a nilpotent groupof degree 2; hence the group (co)homology is not simple, e.g., it contains some Massey products(see [N3, § Proof of Theorem 8.5.
Let K denote the abelian Ker( ψ X ) for short. Consider the rack space B ( X, Inn( X )) whose π is Ker( ψ X ) = Z × K by Remark 8.3. It is shown that the torsionsubgroups of the homology H ( B ( X, Inn( X ))) and H ( B ( X, X )) are annihilated by | X | (see[N2, Lemma 5.7] and [LN, Theorem 1.1]). Hence, H Q ( X ) and H gr3 (As( X )) are annihilatedby | X | , by repeating the proof of Theorem 8.1. Noticing t X = | Inn( X ) | / | X | as above, thepurpose H Q ( X ) ∼ = [1 / H gr3 (As( X )) ⊕ K ⊕ ( ∧ K ) immediately follows from the regularity andTheorem 8.1.Finally, we work out the case where | X | is odd. By repeating the above discussion, thehomologies H ( B ( X, X )) and H ( B ( X, Inn( X ))) have no 2-torsion; so does the H Q ( X ) aswell. We next compute H Q ( X ) for the symplectic and orthogonal quandles over F q , where we excludethe exceptional cases of q , i.e., q = 3 , , , ,
7. We remark that the orders | Inn( X ) | / | X | arenot simple, in contract to Theorem 8.1. Theorem 8.6.
Let q = p d be odd, and be not exceptional.(I) Let X be the symplectic quandle Sp nq over F q . Then, H Q ( X ) ∼ = (cid:26) , if n > Z / ( q − ⊕ ( Z /p ) d ( d +1) / , if n = 1 (II) Let X be the spherical quandle S nq over F q . Then, there are [1 / -isomorphisms H Q ( X ) ∼ = [1 / (cid:26) , if n > Z / ( q − ⊕ Z / ( q − δ q ) , if n = 2 Here δ q = ± is according to q ≡ ± . his theorem mostly settles a problem posed by Kabaya [ILDT] for computing the homology H Q ( Sp nq ) with n = 1. Proof. (I) Recalling As( X ) ∼ = Z × Sp (2 n ; F q ) from the proof of Theorem 7.4, we particularlysee the kernel Ker( ψ X ) ∼ = Z .First, we deal with the case n = 1. Notice Inn( X ) ∼ = Sp (2; F q ) and | Inn( X ) | / | X | = q ( q − /q − q . Therefore the purpose H Q ( X ) ∼ = [1 /p ] Z /q − p -torsion followsfrom Theorem 3.9. So, we now focus on the p -torsion of H Q ( X ) with n = 1. Consider therack space B ( X, X ) associated to the primitive X -set, whose P -sequence is given by H gr3 (Stab( x )) −→ π ( BX ) H −→ H ( B ( X, X )) −→ H gr2 (Stab( x )) −→ . We can easily see the stabilizer Stab( x ) ∼ = Z × ( Z /p ) d . Since π ( BX ) ∼ = Z ⊕ Z / ( q − ⊕ ( Z /p ) d by Theorem 7.4, the exact sequence is rewritten as H gr3 ( Z × ( Z /p ) d ) −→ Z ⊕ Z / ( q − ⊕ ( Z /p ) d H −→ H ( B ( X, X )) −→ H gr2 ( Z × ( Z /p ) d ) −→ . (32)We here claim that the Hurewicz map H ( p ) is a splitting injection. Indeed, it follows from(24) that the composite P ∗ ◦ H : π ( BX ) → H ( BX ) ∼ = Z ⊕ ( Z /p ) d is surjective, where P : B ( X, X ) → BX is the covering. Consequently, this claim made the previous sequenceabove into H ( B ( X, X )) ∼ = ( p ) π ( BX ) ⊕ H gr2 ( Z × ( Z /p ) d ) ∼ = ( p ) Z ( p ) ⊕ ( Z /p ) d ⊕ ( Z /p ) d ( d − / . Hence, compared with the isomorphism (28), we have H Q ( X ) ∼ = ( p ) ( Z /p ) d ( d +1) / as desired.Next, when n ≥
2, we will show H Q ( X ) = 0 as stated. It is shown [N4, § x ) = π ( B ( X, X )) is zero, and that H ( BX ) ∼ = Z .Therefore, the P -sequencesarising from the projection P : B ( X, X ) → BX are written as H gr3 (Stab( x )) P ∗ (cid:15) (cid:15) δ ∗ / / π ( B ( X, X )) ∼ = P ∗ (cid:15) (cid:15) / / H ( B ( X, X )) P ∗ (cid:15) (cid:15) / / H gr3 ( Z × Sp ( n ; F q )) / / π ( BX ) / / H ( BX ) ( ∼ = Z ) / / . By the stability theorem 7.6, the left vertical map P ∗ surjects onto Z / ( q − π ( BX ) ∼ = Z ⊕ Z / ( q −
1) by Theorem 7.4, the delta map δ ∗ is surjective in torsion part. Therefore adiagram chasing can show H ( B ( X, X )) = Z . Using (28) again, we have the goal H Q ( X ) = 0.(II) The calculations of the third homology H Q ( X ) for the spherical quandle X = S nq over F q can be shown in a similar way to the symplectic case. The point is that the homology H gr i (Stab( x )) is [1 / H gr i ( O ( n ; F q )) for i ≤ H Q ( X ) shown in [N4, § § .4 Example 3; the extended quandles. We now discuss the groups H Q ( e X ) and Π ( e X ) of extended quandles e X . Theorem 8.7.
Let X be a connected quandle of type t X . Let p : e X → X be the universal cover-ing mentioned in § H gr3 (As( X )) is finitely generated, then there are [1 /t X ] -isomorphisms H Q ( e X ) ∼ = [1 /t X ] Π ( e X ) ∼ = [1 /t X ] H gr3 (As( X )) . Here the second isomorphism Π ( e X ) ∼ = H gr3 (As( X )) is obtained from the composite Θ X ◦ p ∗ .Proof. We now construct the first isomorphism Π ( e X ) ∼ = [1 /t X ] H Q ( e X ). Consider the rack space B ( e X, e X ) associated with the primitive e X -set. Notice from Remark 8.3 and Theorem 6.5 (ii)that this π is an abelian group Z × Ker( p ∗ ). Then the Postnikov tower is expressed by H gr3 ( Z × Ker( p ∗ )) −→ π ( B e X ) H e X −−−→ H ( B ( e X, e X )) −→ H gr2 ( Z × Ker( p ∗ )) → . Notice from Theorem 6.5 (iii)(iv) that H gr i ( Z × Ker( p ∗ )) is annihilated by t X for i ≥
2. Hence,the Hurewicz map H e X amounts for the claimed [1 /t X ]-isomorphism Π ( e X ) ∼ = H Q ( e X ).To prove the [1 /t X ]-isomorphism Π ( e X ) ∼ = H gr3 (As( X )) and the latter part, we first note H Q ( e X ) ∼ = [1 /t X ] /t X ]-isomorphisms p ∗ : H gr3 (As( e X )) → H gr3 (As( X )) and the T -map Θ X : Π ( e X ) → H gr3 (As( e X )) in Theorem 6.5 (v) and Theorem 3.5respectively, we obtain the composite [1 /t X ]-isomorphism p ∗ ◦ Θ X : Π ( e X ) → H gr3 (As( X )) asdesired.Furthermore, let us show a lemma which will be used in a subsequent paper [N3]. Lemma 8.8.
Let X be a connected Alexander quandle of type t X . Let p : e X → X be theuniversal covering. Then the induced map p ∗ : H Q ( e X ) → H Q ( X ) is injective up to t X -torsion.Proof. By the proof of Theorem 8.5 (see § H X : π ( B ( X, X )) → H ( B ( X, X )) is injective up to 2 t X -torsion. Consider the Postnikov tower with respect to the p : e X → X : H gr3 ( Z × Ker( p ∗ )) ( ℓ ) 0 / / p ∗ (cid:15) (cid:15) π ( B e X ) ( ℓ ) H e X / / p ∗ (cid:15) (cid:15) H ( B ( e X, e X )) ( ℓ ) p ∗ (cid:15) (cid:15) / / H gr2 ( Z × Ker( p ∗ )) ( ℓ ) = 0 p ∗ (cid:15) (cid:15) H gr3 (Stab( x )) ( ℓ ) 0 / / π ( BX ) ( ℓ ) H X / / H ( B ( X, X )) ( ℓ ) / / H gr2 (Stab( x )) ( ℓ ) . Here a prime ℓ is relatively prime to 2 t X . Since the upper H e X is a [1 / t X ]-isomorphism (seethe previous proof of Theorem 8.7), the vertical map p ∗ : H ( B ( e X, e X )) ( ℓ ) → H ( B ( X, X )) ( ℓ ) is injective. Hence, by (28) as usual, the map p ∗ : H Q ( e X ) → H Q ( X ) turns out to be a[1 / t X ]-injection. Proof of Theorem 5.2
We will show Theorem 5.2 as a result of Proposition A.1. This proposition provides analgorithm to compute the first rack homology as follows:
Proposition A.1.
Let X be a quandle, and Y an X -set. Decompose Y into the orbits as Y = ⊔ i ∈ I Y i . For i ∈ I , choose an arbitrary element y i ∈ Y i , and denote by Stab( y i ) ⊂ As( X ) thestabilizer subgroup of y i . Then H R ( X, Y ) is isomorphic to the direct sum of the abelianizationsof Stab( y i ) . Precisely, H R ( X, Y ) ∼ = ⊕ i ∈ I (Stab( y i )) ab . Proof.
Recall from Proposition 8.2, that each connected component of the B ( X, Y ) is B ( X, Y i ),and π ( B ( X, Y )) is the stabilizer Stab( y i ). Thereby H ( B ( X, Y i )) ∼ = π ( B ( X, Y i )) ab ∼ = Stab( y i ) ab .Hence, we conclude H R ( X, Y ) ∼ = H ( B ( X, Y )) ∼ = M i ∈ I H ( B ( X, Y i )) ∼ = M i ∈ I Stab( y i ) ab . Proof of Theorem 5.2.
We first show (33) below. Let Y = X be the primitive X -set. For each x i ∈ X i , we have e x i ∈ Stab( x i ) since x i ⊳ x i = x i . Hence, the restriction of ε i : As( X ) → Z on Stab( x i ) is also surjective, and permits a section s : Z → Stab( x i ) defined by s ( n ) = e nx i .Here we remark that the action of Z on Stab( x i ) ∩ Ker( ε i ) induced by the section is trivial.Indeed, the equality (2) implies g − e x i g = e x i ∈ As( X ) for any g ∈ Stab( x i ). We thereforehave Stab( x i ) ab ∼ = (cid:0) Stab( x i ) ∩ Ker( ε i ) (cid:1) ab ⊕ Z . Hence it follows from Proposition A.1 that H R ( X, X ) ∼ = M i ∈ O( X ) Stab( x i ) ab ∼ = Z ⊕ O( X ) ⊕ M i ∈ O( X ) (cid:0) Stab( x i ) ∩ Ker( ε i ) (cid:1) ab . (33)Finally, it is sufficient to show that H Q ( X ) is isomorphic to the last summand. Recall H R ( X ) ∼ = H R ( X, X ) in Remark 8.3. It is known [LN, Theorem 2.1] that H R ( X ) ∼ = H Q ( X ) ⊕ Z ⊕ O( X ) , and that a basis of the Z ⊕ O( X ) is represented by ( x i , x i ) ∈ C R ( X ) for i ∈ O( X ). Bycomparing the basis with the isomorphisms in (33), we complete the proof. B Proof of Proposition 2.1
Proof of Proposition 2.1.
We first construct an X -coloring from any element ( x , . . . , x L , f )in (5). Let us denote by γ i the oriented arc associated to the meridian m i , and color the γ i by the x i ∈ X . For each i , we consider the path P i along the longitude l i as illustrated in thefigure below. Furthermore, let α (= γ i ) , α , . . . , α N i − , α N i = α be the over-paths on this P i ,and let β k ∈ π ( S \ L ) be the meridian corresponding to the arc that divides the arcs α k − and α k . Then, we define a map C : { over arcs of D } → X by the formula C ( α k ) := x i · (cid:0) f ( β ǫ ) · · · f ( β ǫ k − k − ) (cid:1) ∈ X. Here ǫ j ∈ {± } is the sigh of the crossing of α j and β j . Note C ( α N i ) = C ( α ) = x i since thelongitude l i equals the product β ǫ β ǫ · · · β ǫ Ni − N i − by definition. Here we claim that this C is an X -coloring. For this purpose, using (2), we notice equalities e C ( α k ) = e x i · f ( β ǫ ) ··· f ( β ǫk − k − ) (cid:0) f ( β ǫ ) · · · f ( β ǫ k − k − ) (cid:1) − f ( m i ) (cid:0) f ( β ǫ ) · · · f ( β ǫ k − k − ) (cid:1) = f ( α k ) ∈ As( X ) . (34)Hence, with respect to the crossing between α k and β k with k ≤ N i , we have the followingequality in X : C ( α k ) ⊳ C ( β k ) = C ( α k ) · e ǫ k C ( β k ) = C ( α k ) · f ( β ǫ k k ) = x i · (cid:0) f ( β ǫ ) · · · f ( β ǫ k − k − ) (cid:1) · f ( β ǫ k k ) = C ( α k +1 ) . This equality means that this C turns out to be an X -coloring as desired. Here note theequality f = Γ C for such an X -coloring C coming from the original f (see the previous (34)).To summarize, we obtain a map from the set (5) to the set Col X ( D ) which carries such( x , . . . , x L , f ) to the above C . Moreover, by construction, it is the desired inverse mappingof the Γ • , which proves the desired bijectivity. α = γ i α α P i β β β N i · · · Corollary B.1.
Let X be a quandle such that the map X → As( X ) sending x to e x isinjective. Let D be a diagram of an oriented link L . We fix a meridian m i ∈ π ( S \ L ) ofeach link-component. Then, the set of X -colorings of D is bijective to the following set: { f ∈ Hom gr ( π ( S \ K ) , As( X )) | f ( m i ) = e x i for some x i ∈ X } . (35) Proof.
By Proposition 2.1, it is enough to show that such an f in (35) satisfies the equality x i · f ( l i ) = x i . Actually, noting e x i = f ( m i ) = f ( l i ) − f ( m i ) f ( l i ) = f ( l i ) − e x i f ( l i ) = e x i · f ( l i ) by(2), the injectivity X ֒ → As( X ) concludes the desired x i · f ( l i ) = x i .For example, every quandle in Section 7 satisfies the injectivity of X → As( X ). Acknowledgment
The author expresses his gratitude to Tomotada Ohtsuki and Seiichi Kamada for commentsand advice. He also thanks the referee for kindly reading this paper.
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