An explicit description of the second cohomology group of a quandle
aa r X i v : . [ m a t h . QA ] S e p AN EXPLICIT DESCRIPTION OF THE SECONDCOHOMOLOGY GROUP OF A QUANDLE
AGUSTÍN GARCÍA IGLESIAS, LEANDRO VENDRAMIN
Abstract.
We use the inflation–restriction sequence and a result ofEtingof and Graña on the rack cohomology to give a explicit descriptionof -cocycles of finite indecomposable quandles with values in an abeliangroup. Several applications are given. Introduction and main results
Quandles are non-associative algebraic structures introduced indepen-dently by Joyce [17] and Matveev [20] in connection with knot theory. Theyproduce powerful invariants similar to those obtained by coloring [6], [22].Quandles turned out to be useful in different branches of algebra, topologyand geometry since they have connections to several different topics suchas permutation groups [16], quasigroups [24], symmetric spaces [25], Hopfalgebras [2], etc.Quandles have a very interesting cohomology theory that first appearedin [4] and independently in [12]. This theory is somewhat based on the rackcohomology introduced in [11] by Fenn, Rourke and Sanderson. As in thecase of groups, 2nd quandle cohomology groups can be used to produce newquandles by means of extensions.The explicit computation of quandle cohomology groups is an importantproblem relevant to different areas of current research. The 2nd quandlecohomology group is particularly important since it has many applicationsgoing from knot theory to Hopf algebras.In [4], Carter, Jelsovsky, Kamada, Langford and Saito used quandle co-homology classes to produce powerful invariants of classical links and theirhigher dimensional analogs. The invariants based on quandle -cocycles im-prove the effectiveness of the quandle-coloring invariants since, for example,they distinguish knots from their mirror images. These invariants require anexplicit description of -cocycles.In the Hopf algebra context, quandles and their cohomology parametrizeYetter-Drinfeld modules. In turn these modules are crucial ingredients in theclassification problem of finite-dimensional Hopf algebras with non-abeliancoradical. Indeed, an important step of the Lifting Method proposed byAndruskiewitsch and Schneider to solve this classification problem is theexplicit computation of the 2nd cohomology of finite quandles, see [1]. In this work we give an explicit description of the second cohomologygroup of a finite indecomposable quandle. Our presentation is made by
This work is partially supported by CONICET, FONCyT PICT-2013-1414 and PICT-2014-1376, Secyt (UNC), ICTP and MATH-AmSud. means of the characters of a certain finite group. This reduces the problemof computing -cocycles of a quandle to an easy manipulation involving cosetsin a finite group. Our method is based on a result of Etingof and Graña [9]which relates the 2nd cohomology of a quandle and the first cohomology ofan infinite group. We now review the basics of our construction. Let X be a finite quandle.Recall that the enveloping group of X is the group(1.1) G X = h x ∈ X : xy = ( x ⊲ y ) x i . Assume that X is indecomposable and fix x ∈ X . Under the identification h x i ≃ Z we show in Lemma 2.3 that G X ≃ N X ⋊ Z , where N X is thecommutator group [ G X , G X ] of G X . The group G X acts transitively on X in a natural way, hence so does N X , see Corollary 2.4. We denote by N thestabilizer of N X on x : this is a finite group cf. Lemma 2.1.Fix an abelian group A and let M = Fun( X, A ) be the right G X -moduleof functions X → A , i.e. ( f · x )( y ) = f ( x ⊲ y ) for x, y ∈ X and f ∈ M .We prove that there is a commutative diagram with exact columns (cid:15) (cid:15) (cid:15) (cid:15) H ( Z , M N ) ∼ / / inf (cid:15) (cid:15) A ι (cid:15) (cid:15) H ( X, A ) ∼ H ( G X , M ) res (cid:15) (cid:15) A × H ( N X , M ) π (cid:15) (cid:15) H ( N X , M ) Z (cid:15) (cid:15) H ( N X , M ) ∼ / / (cid:15) (cid:15) Hom( N , A )0 0 where the isomorphism(1.2) H ( X, A ) ≃ H ( G X , M ) , q f q , is [9, Corollary 5.4], see also (2.3); inf and res denote the inflation-restrictionmaps and ι and π denote the canonical inclusion and projection.See Lemmas 3.1, 3.4 and Proposition 3.7 for a proof of the isomorphismsand the equality in the rows of the diagram. The exactness of the first columnis a well-known fact cf. Lemma 2.9. We show that it splits in Lemma 2.10.By diagram chasing, we derive an isomorphism H ( X, A ) ≃ A × Hom( N , A ) . From this isomorphism we obtain an explicit description of rack and quandle -cocycles with values in any abelian group A , see Theorem 1.1.We denote by f f the map H ( G X , M ) → Hom( N , A ) deduced fromthe diagram above.Our first main result reads as follows, see §3 for a proof. HE 2ND RACK COHOMOLOGY GROUP 3
Theorem 1.1.
Let X be a finite indecomposable quandle, x ∈ X and A anabelian group with trivial G X -action. Then (1.3) H ( X, A ) ≃ A × Hom( N , A ) , q ( q x ,x , ( f q ) ) . In particular this shows that the non-constant 2-cocycles on X are con-trolled by a finite group. Our second main result is a precise recipe to reconstruct a cocycle q ∈ H ( X, A ) from a datum ( a, g ) ∈ A × Hom( N , A ) . That is, we give aconverse to the map in (1.3) to build all explicit 2-cocycles of a given quandle.To do this, we need to introduce some extra notation.First, we fix a good coset decomposition N X = k G i =0 σ i N , into N -cosets, i.e. the representatives σ , . . . , σ k are chosen so that:(1) σ = 1 ;(2) for each i ∈ { , . . . , k } there is j ∈ { , . . . , k } such that x ⊲ σ i = σ j ;(3) for each x ∈ X there is j ∈ { , . . . , k } such that σ j ⊲ x = x .The existence of such a decomposition is given in Proposition 4.1, togetherwith a recursive method for constructing it.We define σ : N → { σ , . . . , σ k } , σ ( n ) = σ i if n ∈ σ i N . We set, cf. (3.4), c ( n ) = σ ( n ) − n ∈ N . Given y ∈ X and j ∈ { , . . . , k } such that σ j ⊲ x = y we write σ y := σ j . Our second main result is the following, see §4 for the proof.
Theorem 1.2.
Let X be a finite indecomposable quandle, x ∈ X and A an abelian group with trivial G X -action. Let N X = F ki =0 σ i N be a gooddecomposition of N X into N -cosets. For each a ∈ A and g ∈ Hom( N , A ) ,the map q : X × X → A given by q x,y = a + g ( c ( xσ y x − )) (1.4) is a -cocycle of X with values in A . Combining Theorems 1.1, 1.2 and the isomorphism (1.2), namely q x,y = f q ( x )( y ) , q ∈ H ( X, A ) , we immediately obtain the following corollary. Corollary 1.3.
Let X be a finite indecomposable quandle, x ∈ X and A an abelian group with trivial G X -action. Let N X = F ki =0 σ i N be a gooddecomposition of N X into N -cosets and let q ∈ H ( X, A ) . Then there exists a ∈ A and g ∈ Hom( N , A ) such that (1.4) holds for all x, y ∈ X . Corollary 1.3 has many applications and can be used for explicit calcula-tions of rack cohomology groups of quandles. In particular, if the commuta-tor subgroup N X acts regularly on X , then N = 1 and hence we obtain thefollowing corollary. A. GARCÍA IGLESIAS, L. VENDRAMIN
Corollary 1.4.
Let X be a finite indecomposable quandle. If the action of N X on X is regular, then H ( X, C × ) ≃ C × . The paper is organized as follows. Preliminaries on racks and quandles,cohomology theory of groups, and cohomology theories of racks and quan-dles appear in Section 2. Our first main result, Theorem 1.1, is proved inSection 3. Theorem 1.2 is proved in Section 4. Applications of our theoryare given in Section 5. These applications include the calculations of the 2ndrack cohomology group of: (a) the quandle associated with the conjugacyclass of transpositions, see Theorem 5.5; (b) affine racks of size p and p ,where p is a prime number, see Propositions 5.8, 5.10, 5.11 and 5.12; and(c) another proof of Eisermann’s formula for computing the 2nd quandlehomology group of a quandle, see Theorem 5.6. Preliminaries
For a set X we denote by S X the group of permutations X → X . If X is finite of cardinal | X | ∈ N , then we identify S | X | = S X .For any group G we denote by [ G, G ] its commutator subgroup and G ab itsabelianization, i.e. G ab = G/ [ G, G ] . In addition, Z ( G ) is the center of G and G G ( g ) = { h ∈ G : hg = gh } for g ∈ G . We denote by Aut( G ) the groupof automorphisms G → G ; if γ ∈ Aut( G ) , then ord( γ ) is the order of γ .Let M be an abelian group equipped with a G -action. We denote by H n ( G, M ) , n ≥ , the n th cohomology group of G with coefficients on M . We denote by Z n ( G, M ) , resp. B n ( G, M ) , the groups of cocycles, resp.cobordisms, of G with values on M . We refer the reader to [3] for unexplainednotation and terminology. A rack is a non-empty set X together with a binary operation ⊲ : X × X → X such that the maps ϕ x = x ⊲ − : X → X , y x ⊲ y ,are bijective for each x ∈ X , and x ⊲ ( y ⊲ z ) = ( x ⊲ y ) ⊲ ( x ⊲ z ) for all x, y, z ∈ X . A quandle is a rack that further satisfies x ⊲ x = x for all x ∈ X .A prototypical example of a rack is a group G with ⊲ given by conjugation.A rack is indecomposable if the inner group Inn( X ) = h ϕ x : x ∈ X i ≤ S X acts transitively on X .The enveloping group G X cf. (1.1) also acts on X , and this action is readilyseen to be transitive when X is indecomposable. The group G X is infinite.There is a finite analogue of this group, which is constructed as follows: Foreach x , let n x = ord ϕ x . Then the subgroup Z X = h x n x , x ∈ X i ≤ G X is normal and the quotient F X = G X /Z X is finite, see [14, §2]. We write N X = [ G X , G X ] to denote the commutator subgroup of G X . Lemma 2.1. [15, Lemma 1.10]
Let X be an indecomposable quandle. Then N X ≃ [ F X , F X ] . In particular, N X is finite. The last claim of Lemma 2.1 also follows from the following result and atheorem of Schur, see for example [23, Theorem 5.32].
Lemma 2.2.
Let X be a finite indecomposable quandle. Then all conjugacyclasses of G X are finite. HE 2ND RACK COHOMOLOGY GROUP 5
Proof.
Since G X acts transitively on X and the center Z ( G X ) is the kernelof this action, it follows that the index [ G X : Z ( G X )] is finite. This impliesthat all conjugacy classes of G X are finite as [ G X : C G X ( g )] ≤ [ G X : Z ( G X )] , where C G X ( g ) denotes the centralizer of g in G X . (cid:3) We consider the unique surjective group homomorphism d : G X → Z (2.1)satisfying d( x ) = 1 for all x ∈ X . In particular, this homomorphism showsthat G X is infinite and induces a notion of degree on G X . Lemma 2.3.
Let X be an indecomposable finite quandle and x ∈ X . Thenthe following hold: (1) G X = ker d ⋊ h x i . (2) ker d = N X if X is indecomposable.Proof. Since ker d is a normal subgroup of G X , ker d h x i is a subgroup of G X .It is clear that ker d ∩ h x i = 1 comparing degrees. Finally G X = ker d h x i since x = ( xx − ) x ∈ ker d h x i for all x ∈ X .It is clear that N X ⊆ ker d . Next we prove the equality when X is in-decomposable. Let ℓ : G X → Z be defined as ℓ ( g ) = n , if g = x ǫ i . . . x ǫ n i n , ǫ i ∈ {± } , i ∈ { , . . . , n } , is a a reduced expression of g in terms of thegenerators of G X . We show that ker d ⊆ N X by induction on ℓ ( g ) , g ∈ ker d .If ℓ ( g ) = 2 , then g = x ± i x ∓ j . So we may assume that g = x i x − j (if not,take inverses). Now, as X is indecomposable, there is h ∈ G X such that h · x j = x i . Hence g = hx j h − x − j ∈ N X . Now, if ℓ ( g ) > , then there isa reduced expression of g (or g − ) in which g = g x i x − j g , x i , x j ∈ X and g , g ∈ G X . Now, on the one hand, g ) = d( g ) + d( g ) and thus g g ∈ N X as ℓ ( g g ) < ℓ ( g ) . On the other, g = ( g x i x − j g − )( g g ) andtherefore g ∈ N X . (cid:3) Corollary 2.4.
The restriction of the action of G X on X to N X is transitive.Proof. Let x, y ∈ X and let g ∈ G X such that g · x = y and let ℓ = d( g ) .Then g ′ = y − ℓ g ∈ N X by Lemma 2.3 and g ′ · x = y . (cid:3) A cohomology theory for racks was introducedin [10] and independently in [12]. A cohomology theory for quandles wasdeveloped in [4]. These theories were further developed and generalized forexample in [2] and [18].We briefly recall these cohomology theories next. Let X be a rack and let M be a right G X -module. Set C n = C n ( X, M ) = Fun( X n , M ) , n ≥ , theset of functions from X n to M . Consider the differential d : C n → C n +1 df ( x , . . . , x n +1 ) = n X i =1 ( − i − (cid:16) f ( x , . . . , x i − , x i +1 , . . . , x n +1 ) − f ( x , . . . , x i − , x i ⊲ x i +1 , . . . , x i ⊲ x n +1 ) · x i (cid:17) . A. GARCÍA IGLESIAS, L. VENDRAMIN
The rack cohomology H • ( X, M ) of X with coefficients in M is the cohomol-ogy of the complex ( C • , d ) [9, Definition 2.3]. The groups of cocycles resp.cobordisms, are denoted by Z n ( X, M ) , resp. B n ( G, M ) . When A is anabelian group and no reference to a G X -action on A is specified, H • ( X, A ) stands for the cohomology of X with values in the trivial module M = A . If q is a class in H ( X, A ) , we set q x,y := q ( x, y ) . Hence q ∈ H ( X, A ) if andonly if q x⊲y,x⊲z q x,z = q x,y⊲z q y,z , ∀ x, y, z ∈ X (2.2)and two classes q, q ′ ∈ H ( X, A ) are equivalent if and only if there exists γ : X → A such that q ′ x,y = q x,y γ ( x ⊲ y ) γ ( y ) − for all x, y ∈ X .The rack homology H • ( X, A ) with values in an abelian group A is definedanalogously, by considering the free abelian group F n ( X ) on X n , n ≥ ,and setting C n ( X, A ) := F n ( X ) ⊗ A . If X is a quandle, then the subgroup F Dn ( X ) ≤ F n ( X ) generated by n -tuples ( x , . . . x n ) with x i = x i +1 for some i , defines a subcomplex C D • = C D • ( X, A ) of C • . The quandle homology H Q • ( X, A ) of X is the homology of the quotient complex C Q • = ( C n /C Dn ) n ≥ .In this work we give a description of the group H ( X, A ) of 2-cocycles on X with values in an abelian group A , which allows us to compute cocyclesexplicitly. We recall next some identifications between the (co)homologytheories described above that will be useful for our goal. Lemma 2.5. [5, Proposition 3.4] H ( X, A ) ≃ Hom( H ( X, Z ) , A ) , via H ( X, A ) ∋ q ([ x, y ] q x,y ) ∈ Hom( H ( X, Z ) , A ) . The following is a particular case of [19, Theorem 7].
Lemma 2.6.
Assume X is an indecomposable quandle. Then H ( X, Z ) ≃ H Q ( X, Z ) × Z . Explicitly, if ( x, y ) ∈ X , then the isomorphism is induced by the map ( x, y ) ( ( x, y ) × , if x = y × , if x = y. Etingof and Graña found a deep relation between group cohomology andrack cohomology.
Theorem 2.7. [9, Corollary 5.4]
Let X be a finite indecomposable rack and A an abelian group with a trivial G X -action. Then H ( G X , Fun(
X, A )) ≃ H ( X, A ) . This equivalence is given as follows: (1) If f ∈ H ( G, Fun(
X, A )) then a 2-cocycle q f ∈ H ( X, A ) arises as q fx,y = f ( x )( y ) , x, y ∈ X. (2) Conversely, q ∈ H ( X, A ) determines f q ∈ H ( G, Fun(
X, A )) byextending q recursively via f q ( xy )( z ) = q x,y⊲z + q y,z , x, y, z ∈ X. (2.3) HE 2ND RACK COHOMOLOGY GROUP 7
Remark . Let G be a (non-abelian) group and fix Z ( X, G ) ⊂ Fun( X , G ) as the subset of all q : X → G satisfying (2.2). We say that q is equivalentto q ′ , and we write q ∼ q ′ , in Z ( X, G ) if and only if there is γ ∈ Fun(
X, G ) such that q ′ x,y = γ ( x ⊲ y ) q x,y γ ( y ) − . If H ( X, G ) := Z ( X, G ) / ∼ , thenTheorem 2.7 holds, see [9, Remark 5.6]. Let G be a group, N ⊳ G a normal subgroupand M a right G -module. Recall cf. [3, 3.8] that there is a right G/N -actionon H ( N, M ) , induced by ( f · g )( n ) = f ( gng − ) · g, g ∈ G, n ∈ N, f ∈ H ( N, M ) . (2.4)Indeed, let f ∈ Z ( N, M ) . If g ∈ N , then ( f · g )( n ) = f ( gn ) − f ( g ) = f ( g ) · n + f ( n ) − f ( g ) by the cocycle condition. Hence ( f · g )( n ) − f ( n ) = f ( g ) · n − f ( g ) and thus f · g = f ∈ H ( N, M ) . The inflation-restriction sequence is(2.5) → H ( G/N, M N ) ι → H ( G, M ) r → H ( N, M ) G/N → H ( G/N, M N ) → H ( G, M ) where the inflation map ι ( h ) , h ∈ H ( G/N, M N ) , is the composition G ։ G/N h → M N ֒ → M and the restriction map r ( g ) , g ∈ H ( G, M ) , is the composition N ֒ → G g → M. In the case where
G/N ≃ Z one obtains the following result, see loc.cit. Lemma 2.9.
Assume that
G/N ≃ Z . Then (1) H ( G/N, M N ) = 0 . (2) H ( G/N, M N ) = M N / h m · g − m i , (class of ) f (class of ) f (1) .In particular, the exact sequence (2.5) reduces to → H ( G/N, M N ) inf → H ( G, M ) res → H ( N, M ) G/N → . (2.6) Lemma 2.10.
Assume that N is finite and G/N ≃ Z . Then (2.6) splits. Aretraction for inf : H ( G/N, M N ) → H ( G, M ) is given by j : H ( G, M ) → H ( G/N, M N ) , j ( f )( ℓ ) = 1 | N | X n ∈ N (cid:16) f ( nx ℓ ) − f ( n ) (cid:17) . Proof.
We need to check that j is well-defined, that is:(1) If f ∈ Z ( G, M ) , then j ( f )( G/N ) ⊆ M N .(2) If f ∈ Z ( G, M ) , then j ( f ) ∈ Z ( G/N, M N ) .(3) If f ∈ B ( G, M ) , then j ( f ) ∈ B ( G/N, M N ) . A. GARCÍA IGLESIAS, L. VENDRAMIN
Let f ∈ Z ( G, M ) and set ϕ := j ( f ) . For (1), using the cocycle condition, ϕ ( ℓ ) · n = 1 | N | X m ∈ N (cid:16) f ( mx ℓ ) · n − f ( m ) · n (cid:17) = 1 | N | X m ∈ N (cid:16) f ( mx ℓ n ) − f ( n ) − f ( mn ) + f ( n ) (cid:17) = 1 | N | X m ∈ N (cid:16) f ( mx ℓ nx − ℓ x ℓ ) − f ( mn ) (cid:17) . By reordering the sum, ϕ ( ℓ ) · n = ϕ ( ℓ ) for all n ∈ N , ℓ ∈ Z . Hence (1) holds.In (2), we get ϕ ( ℓ + r ) = 1 | N | X n ∈ N (cid:16) f ( nx ℓ + r ) − f ( n ) (cid:17) = 1 | N | X n ∈ N (cid:16) f ( nx ℓ ) · x r + f ( x r ) − f ( n ) (cid:17) = 1 | N | X n ∈ N (cid:16) f ( nx ℓ ) · x r − f ( n ) · x r + f ( n ) · x r + f ( x r ) − f ( n ) (cid:17) = ϕ ( ℓ ) · r + 1 | N | X n ∈ N ( f ( nx r ) − f ( x r ) + f ( x r ) − f ( n ))= ϕ ( ℓ ) · r + ϕ ( r ) . Thus (2) holds. If f ∈ B ( G, M ) , then there exists ψ ∈ M such that f ( g ) = ψ · g − ψ . Hence, j ( f )( ℓ ) = 1 | N | X n ∈ N (cid:16) ψ · nx ℓ − ψ − ψ · n + ψ (cid:17) = 1 | N | X n ∈ N (cid:16) ψ · nx ℓ − ψ · n (cid:17) and thus j ( f )( ℓ ) = γ · ℓ − γ for γ = 1 | N | X n ∈ N ψ · n ∈ M N . This shows (3). Finally we prove that j ◦ inf = id . For this, recall that if ϕ ∈ H ( G/N, M N ) and g ∈ G , then inf( ϕ )( g ) = ϕ (¯ g ) , where ¯ g is the classof g in G/N ≃ Z . Then ( j ◦ inf)( ϕ )( ℓ ) = 1 | N | X n ∈ N (cid:16) inf( ϕ )( nx ℓ ) − inf( ϕ )( n ) (cid:17) = 1 | N | X n ∈ N ϕ ( ℓ ) = ϕ ( ℓ ) for all ℓ ∈ Z . This completes the proof. (cid:3) Proof of Theorem 1.1
Assume that X is a finite indecomposable rack. We write G = G X , N = [ G X , G X ] . Let A be an abelian group with trivial G -action and set HE 2ND RACK COHOMOLOGY GROUP 9 M = Fun( X, A ) . Fix x ∈ X and G ≃ N ⋊ Z as in Lemma 2.3. It followsfrom Lemma 2.10 that → H ( G/N, M N ) inf −→ H ( G, M ) res −−→ H ( N, M ) G/N → splits. We first identify the first term of this sequence. Lemma 3.1. H ( Z , M N ) ≃ A , via f f (1)( x ) .Proof. Recall from Lemma 2.9(2) that H ( Z , M N ) ≃ M N /F , where F is thesubmodule generated by { ϕ · x p − ϕ : p ∈ Z , ϕ ∈ M N } . Since X = N ⊲ { x } by Corollary 2.4 and n ⊲ x = x ∈ X for some n ∈ N , ϕ ( x ) = ϕ ( n ⊲ x ) = ( ϕ · n )( x ) for all ϕ ∈ M . Hence, if ϕ ∈ M N , then ϕ ( x ) = ϕ ( x ) , x ∈ X . Conse-quently, F = { } and H ( Z , M N ) ≃ M N . But M N ≃ A as any ϕ ∈ M N isdetermined by its value ϕ ( x ) ∈ A . Hence the lemma follows. (cid:3) As for the third term, we will show in Proposition 3.7 that H ( N, M ) Z ≃ Hom( N , A ) . (3.1)To do so, we first need several lemmas. Lemma 3.2.
The map (3.2) Z ( N, M ) → Hom( N , A ) , f f , where f ( n ) = f ( n )( x ) for n ∈ N , is well-defined and factors to a map H ( N, M ) → Hom( N , A ) .Proof. We first prove that f is indeed a group homomorphism: f ( n n ′ ) = f ( n n ′ )( x ) = ( f ( n ) · n ′ )( x ) + f ( n ′ )( x )= f ( n )( n ′ ⊲ x ) + f ( n ′ )( x )= f ( n )( x ) + f ( n ′ )( x ) = f ( n ) + f ( n ′ ) , n , n ′ ∈ N . We now show that the map factors to a map H ( N, M ) → Hom( N , A ) .Let f ∈ B ( N, M ) , that is f ( n ) = ϕ · n − ϕ for some ϕ ∈ M . Then f ( n ) = f ( n )( x ) = ( ϕ · n )( x ) − ϕ ( x )= ϕ ( n ⊲ x ) − ϕ ( x ) = ϕ ( x ) − ϕ ( x ) = 0 . This completes the proof. (cid:3)
Lemma 3.3.
The map H ( N, M ) → Hom( N , A ) , f f , is an injectivegroup homomorphism.Proof. It is clear that f f is a group homomorphism.Let f ∈ H ( N, M ) be such that f = 0 . That is, f ( n )( x ) = 0 for every n ∈ N . We claim that there is ϕ ∈ M such that f ( m ) = ( ϕ · m ) − ϕ andthus f = 0 in H ( N, M ) . Set ϕ ( x ) := f ( n )( x ) if x = n ⊲ x . Let us check that this is well-defined: if x = n ⊲ x = n ′ ⊲ x , then n − n ′ ∈ N . Since f (1) = 0 , one obtains that f ( n − ) = − f ( n ) · n − . Then f ( n − n ′ ) = f ( n − n ′ )( x )= − f ( n )( n − n ′ ⊲ x ) + f ( n ′ )( x ) = − f ( n )( x ) + f ( n ′ )( x ) , and thus ϕ ( x ) does not depend on n ∈ N such that x = n ⊲ x . Finally foreach m ∈ N and every x = n ⊲ x ∈ X with n ∈ N , ( ϕ · m − ϕ )( x ) = ϕ ( m ⊲ x ) − ϕ ( x ) = ϕ ( m ⊲ n ⊲ x ) − ϕ ( n ⊲ x )= f ( mn )( x ) − f ( n )( x ) = ( f ( m ) · n )( x ) + f ( n )( x ) − f ( n )( x )= f ( m )( n ⊲ x ) = f ( m )( x ) , and therefore f = 0 . (cid:3) Recall the definition of the Z -action on H ( N, M ) from (2.4). Lemma 3.4.
Assume X is a quandle. Then H ( N, M ) = H ( N, M ) Z .Proof. Let f ∈ H ( N, M ) and set g = f − f · x . If n ∈ N , then g ( n ) = f ( n )( x ) − f ( x n x − )( x ⊲ x ) = 0 . Thus g = 0 and hence f = f · x for all f ∈ H ( N, M ) by Lemma 3.3, sincethe group homomorphism g g is injective. (cid:3) In order to show the surjectivity of the map f f from Lemma 3.3, weneed to fix a decomposition of N into N -cosets N = k G i =0 σ i N , where σ i ∈ N is a representative, σ N = N . We define σ : N → { σ , . . . , σ k } , σ ( n ) = σ i if n ∈ σ i N . (3.3)For n ∈ N we consider c ( n ) ∈ N defined by n = σ ( n ) c ( n ) . (3.4) Remark . For all n ∈ N and n ∈ N it follows that c ( nn ) = c ( n ) n .Indeed, nn = σ ( n ) c ( n ) n = σ ( nn ) c ( nn ) and thus the claim holds sinceeach m ∈ N decomposes uniquely as m = σ ( m ) c ( m ) . Lemma 3.6.
The map H ( N, M ) → Hom( N , A ) , f f , is surjective.Proof. Let g : N → A be a group homomorphism; we shall construct an f ∈ Z ( N, M ) such that f = g . We claim that the map f : N → M , n f ( n ) , given by f ( n )( x ) = g ( c ( nm )) − g ( c ( m )) = g ( c ( nm ) c ( m ) − ) , (3.5)where m ∈ N is such that x = m ⊲ x , is well-defined. Indeed, if m ′ ∈ N also satisfies x = m ′ ⊲ x , then m − m ′ ∈ N and thus σ ( m ) − σ ( m ′ ) ∈ N .That is σ ( m ) = σ ( m ′ ) and thus σ ( nm ) = σ ( nm ′ ) for every n ∈ N since ( nm ′ ) − nm ∈ N . As g is a group homomorphism, g ( c ( nm ) c ( m ) − ) − g ( c ( nm ′ ) c ( m ′ ) − ) = g ( c ( nm ) c ( m ) − c ( m ′ ) c ( nm ′ ) − ) . HE 2ND RACK COHOMOLOGY GROUP 11
Now, c ( nm ) c ( m ) − c ( m ′ ) c ( nm ′ ) − is, by definition, ( σ ( nm ) − nm )( m − σ ( m ))( σ ( m ′ ) − m ′ )( m ′ − n − σ ( nm ′ )) = 1 . Hence g ( c ( nm ) c ( m ) − ) − g ( c ( nm ′ ) c ( m ′ ) − ) = g (1) = 0 and thus f does notdepend on the choice of m .Now we show that f ∈ Z ( N, M ) . Let x ∈ X , n, n ′ ∈ N and m ∈ N besuch that x = m ⊲ x . On the one hand, we have f ( nn ′ )( x ) = g ( c ( nn ′ m )) − g ( c ( m )) . On the other, ( f ( n ) · n ′ )( x ) + f ( n ′ )( x ) = f ( n )( n ′ ⊲ x ) + f ( n ′ )( x )= g ( c ( nn ′ m )) − g ( c ( n ′ m )) + g ( c ( n ′ m )) − g ( c ( m ))= f ( nn ′ )( x ) . Finally we see that g = f , that is f ( n ) = g ( n ) for n ∈ N . Now, if n ∈ N ,then c ( n ) = c (1 · n ) = c (1) n cf. Remark 3.5. Also, as as x = 1 ⊲ x , f ( n ) = f ( n )( x ) = g ( c ( n · c (1) − ) = g ( c (1 · n )) − g ( c (1))= g ( c (1) n ) − g ( c (1)) = g ( c (1)) + g ( n ) − g ( c (1)) = g ( n ) and the lemma follows. (cid:3) Now we proceed to show (3.1).
Proposition 3.7.
The map Z ( N, M ) → Hom( N , A ) given by f f ,where f ( n ) = f ( n )( x ) for n ∈ N , induces a group isomorphism H ( N, M ) Z → Hom( N , A ) . Proof.
Lemma 3.4 implies that H ( N, M ) Z ≃ H ( N, M ) and Lemmas 3.3and 3.6 yield H ( N, M ) ≃ Hom( N , A ) , as desired. (cid:3) This allows us to complete the proof of Theorem 1.1.
Proof of Theorem 1.1.
Using the cocycle condition, we get j ( f )( ℓ ) = 1 | N | X n ∈ N (cid:16) f ( n ) · x ℓ + f ( x ℓ ) − f ( n ) (cid:17) = f ( x ℓ ) + 1 | N | X n ∈ N (cid:16) f ( n ) · x ℓ − f ( n ) (cid:17) . Hence, as X is a quandle, for each ℓ ∈ Z ,(3.6) j ( f )( ℓ )( x ) = f ( x )( x ) . Since H ( G/N, M N ) ≃ A by Lemma 3.1 and by Proposition 3.7 thereexists an isomorphism ζ : H ( N, M ) Z ≃ Hom( N , A ) , we write the inflation-restriction sequence (2.6) as → A inf −−→ H ( G, M ) res −−→ Hom( N , A ) → , (3.7)where res ( f ) = res( f ) for all f ∈ H ( G, M ) and inf is the composition A ≃ M N ≃ H ( G/N, M N ) . We set f = res ( f ) by abuse of notation, i.e. f ( n ) = f ( n )( x ) , n ∈ N . (3.8) A retraction for inf is given by the composition j : H ( G, M ) j → H ( G/N, M N ) ≃ A, using Lemmas 2.10 and 3.1, that is j ( f ) = j ( f )(1)( x ) = f ( x )( x ) , (3.9) cf. (3.6). Hence H ( G, M ) ≃ A × Hom( N , A ) via f ( f ( x )( x ) , f ) . (3.10)This completes the proof. (cid:3) Proof of Theorem 1.2
In this section we show the Reconstruction Theorem 1.2. We fix x ∈ X and write N ≤ N X for the stabilizer of x in N X . By Lemma 2.1, N is afinite group.The key for the proof of Theorem 1.2 lays in the existence of a particularclass of decompositions N X = k G i =0 σ i N of N X into N -cosets, which are good in our context. Proposition 4.1.
Let X be a finite indecomposable quandle. Then thereexists a decomposition N X = F ki =0 σ i N of N X into N -cosets such that thefollowing hold: (1) σ = 1 . (2) For each i ∈ { , . . . , k } there is j ∈ { , . . . , k } such that x ⊲ σ i = σ j . (3) For each x ∈ X there is j ∈ { , . . . , k } such that σ j ⊲ x = x .Proof. Fix a decomposition into cosets N X = F ki =0 σ i N . Recall from (3.3)and (3.4) the definition of the corresponding assignments σ : N → { σ , . . . , σ k } and c : N → N . Since σ = 1 , (1) holds. Condition (3) also holds trivially: If x ∈ X , thereis n ∈ N X is such n ⊲ x = x by Corollary 2.4. Now, there is j ∈ { , . . . , k } such that n ∈ σ j N , that is n = σ j n for some n ∈ N . Then x = n ⊲ x = σ j ⊲ ( n ⊲ x ) = σ j ⊲ x .For Condition (2), set S = { σ , . . . , σ k } . We define t j = t j ( S ) := min { t ≥ x t ⊲ σ j = σ j } for all j ∈ { , . . . , k } . Observe that ≤ t j ( S ) ≤ ord ϕ x , cf. §2.2. For i ∈ { , . . . , k } and t ∈ { , . . . , t j ( S ) − } we define τ j,t = x t ⊲ σ j and let T = { τ j,t : 1 ≤ j ≤ k, ≤ t < t j ( S ) } . It is clear that S ⊆ T , as σ j = τ j, by definition, and that if S = T , then weare done. Notice that this is not a multi-set: we may have τ j,t = τ j ′ ,t ′ , fordifferent ( j, t ) , ( j ′ , t ′ ) . On the other hand, if t = t ′ , then τ j,t = τ j,t ′ for every j , since t < t j ( S ) . In other words, there are r ≤ k , i < i < · · · < i r and s j ≤ t i j , ≤ j ≤ r such that T = { τ i j ,t : 1 ≤ j ≤ r, ≤ t < s j } HE 2ND RACK COHOMOLOGY GROUP 13 and τ i j ,t = τ i j ′ ,t ′ if j = j ′ or t = t ′ . We reorder the set S so i j = j , j = 1 , . . . , r . If S = T , then we proceed inductively: we order T by: τ i,s ≺ τ j,t ⇐⇒ i < j or i = j and s < t. Let τ = min { τ j,t : τ j,t S } and let ℓ be such that σ ( τ ) = σ ℓ , i.e. τ j,t = x t ⊲ σ j ∈ σ ℓ N and τ j,t = σ ℓ . Observe that if τ = τ j,t , then ℓ = j . Set S = S and T = T . We make a new choice of representatives replacing the originalset S by S = ( S \ { σ ℓ } ) ∪ { τ } = { σ , . . . , σ ℓ − , τ, σ ℓ +1 , . . . , σ k } . Define t j ( S ) and ( T , ≺ ) accordingly. We claim that t j ( S ) ≤ t j ( S ) for all j . Indeed, equality holds if j = ℓ and it readily follows that t ℓ ( S ) = t ℓ ( S ) − t < t ℓ ( S ) . In particular, it follows that | S | = | S | ≤ | T | < | T | . (This also follows aswhen constructing T we are removing all the τ ℓ,t .) If T = S , then we aredone. Otherwise, we repeat this procedure until we end up with S p = T p forsome p > . Then S p becomes the set of representatives we searched for. (cid:3) We say that a decomposition of N X into N -cosets satisfying the condi-tions in Proposition 4.1 is good .If N X = F ki =0 σ i N is a good decomposition, then for each y ∈ X we set(4.1) σ y := σ j . for j ∈ { , . . . , k } such that σ j ⊲ x = y . Lemma 4.2. If N X = F ki =0 σ i N is good, then c ( x ⊲ n ) = c ( n ) . Proof.
Indeed, x ⊲ n = x σ ( n ) x − c ( n ) , as c ( n ) ∈ N and x σ ( n ) x − = σ i ,for some i ∈ { , . . . , k } . (cid:3) Recall the definition of the group homomorphism d : G X → Z from (2.1). Lemma 4.3.
For each u ∈ G X and y ∈ X , σ u⊲y = uσ y x − d( u )0 c (cid:16) uσ y x − d( u )0 (cid:17) − . In particular if n ∈ N , then σ n⊲y = σ ( nσ y ) .Proof. Since σ u⊲y ⊲ x = u ⊲ y = u ⊲ ( σ y ⊲ x ) = ( uσ y ) ⊲ x = ( uσ y x − d( u )0 ) ⊲ x and uσ y x − d( u )0 ∈ N , it follows that σ u⊲y = σ ( uσ y x − d( u )0 ) . Then uσ y x − d( u )0 = σ u⊲y c (cid:16) uσ y x − d( u )0 (cid:17) , and the first claim follows. If n ∈ N , then d( n ) = 0 and therefore it followsthat σ n⊲y = nσ y c ( nσ y ) − = σ ( nσ y ) cf. (3.4). (cid:3) We can now proceed to prove Theorem 1.2.
Proof of Theorem 1.2.
We need to define an inverse to the map (3.10). Fix a ∈ A , g ∈ Hom( N , A ) and set f : G → M as f ( u )( y ) := d( u ) a + g (cid:16) c ( uσ y x − d( u )0 ) (cid:17) , for each u ∈ G . We show that f ∈ Z ( G, M ) and f ( a, g ) via (3.10).On the one hand, as σ x = σ = 1 , f ( x )( x ) = a + g ( c ( x x − )) = a. On the other, if n ∈ N , then d( n ) = 0 and thus f ( n ) = f ( n )( x ) = g ( c ( n )) = g ( n ) . Now we check the cocycle condition. First, f ( uu ′ )( y ) = d( uu ′ ) a + g (cid:16) c ( uu ′ σ y x − d( uu ′ )0 ) (cid:17) . Second, ( f ( u ) · u ′ )( y ) + f ( u ′ )( y ) = f ( u )( u ′ ⊲ y ) + f ( u ′ )( y )= d( u ) a + g ( c ( uσ u ′ ⊲y x − d( u )0 )) + d( u ′ ) a + g ( c ( u ′ σ y x − d( u ′ )0 ))= f ( uu ′ )( y ) , since A is abelian, d and g are a group homomorphisms and c (cid:16) uσ u ′ ⊲y x − d( u )0 (cid:17) = c (cid:16) uu ′ σ y x − d( uu ′ )0 (cid:17) c (cid:16) u ′ σ y x − d( u ′ )0 (cid:17) − by Lemma 4.3. Hence f ∈ Z ( G, M ) . (cid:3) Applications
Our method for computing the 2nd cohomology group of an indecompos-able quandle X involves the group N , see §1.3. In several important cases,this group can be obtained applying the following lemma. Lemma 5.1.
Let X be a finite indecomposable quandle and x ∈ X . Assumethat the canonical map X → G X is injective. Then N ≃ [ F X , F X ] ∩ C F X ( ψ ( x )) , where ψ : X → G X → F X is the composition of the canonical maps and C F X ( ψ ( x )) is the centralizer of ψ ( x ) in F X .Proof. Since X → G X is injective and X is indecomposable, X can beidentified with the conjugacy class of x in G X . By [15, Lemma 1.8], X canalso be identified with the conjugacy class of ψ ( x ) in F X . From Lemma 2.1one obtains that N X = [ G X , G X ] ≃ [ F X , F X ] and thus the claim follows. (cid:3) Remark . If X is a conjugation quandle, then the canonical map X → G X is injective. Thus Lemma 5.1 gives a nice description of N in the case offinite indecomposable conjugation quandles. Example . The claim of Lemma 5.1 does not hold for arbitrary quandles.Let X be the quandle { x , x , x , x } with the structure given by ϕ x = ( x x x ) , ϕ x = ( x x x ) , ϕ x = ( x x x ) , ϕ x = ( x x x ) . HE 2ND RACK COHOMOLOGY GROUP 15
This quandle is isomorphic to the conjugacy class of -cycles in A . Let f : X × X → C × be the map given by f ( x, y ) = ( if x = x or y = x or x = y , − otherwise . Then f is a -cocycle of X with values in {− , } ≃ Z , see [2, Example 2.2].Let Y = X × {− , } be the quandle given by ( x, i ) ⊲ ( y, j ) = ( x ⊲ y, jf ( x, y )) , x, y ∈ X, i, j ∈ {− , } . Then the canonical map Y → G Y is not injective. Indeed, ( x , x , −
1) = ( x , x ,
1) = ( x , x ,
1) = ( x , x , implies that ( x , −
1) = ( x , in G Y .Fix y ∈ Y . A straighforward calculation shows that F Y ≃ SL (2 , and [ F Y , F Y ] ∩ C F Y ( ψ ( y )) ≃ Z . However, since [ F Y , F Y ] and Y both have eightelements, N is the trivial group. S n . Let X = (12) S n be the quandle of transpo-sitions in the symmetric group S n . For n ≥ a non-constant 2-cocycle χ ∈ H ( X, C × ) was constructed in [21]. This cocycle is given by χ ( σ, τ ) = ( if σ ( i ) < σ ( j ) , − otherwise , (5.1)where τ = ( ij ) , ≤ i < j ≤ n . Lemma 5.4.
Let X = (12) S n , n ≥ , and fix x = (12) ∈ X . (1) F X ≃ S n . Hence N X ≃ A n . (2) N ≃ Z ⋉ A n − . In particular, N / [ N , N ] ≃ Z .Proof. Recall that S n = h σ , . . . , σ n − i with relations σ i σ i +1 σ i = σ i +1 σ i σ i +1 , ≤ i < n − ,σ k σ j = σ j σ k , ≤ j, k < n, | j − k | > ,σ i = 1 , ≤ i < n. Set ι : X ֒ → S n the canonical inclusion let ϕ : h X i → S n the unique grouphomomorphism with ϕ | X = ι . This is in fact an epimorphism. Observe that ϕ ( x ) = ι ( x ) = 1 and ϕ ( xy ) = ι ( x ) ι ( y ) = ι ( x ) ι ( y ) ι ( x ) − ι ( x ) = ι ( x ⊲ y ) ι ( x ) = ϕ (( x ⊲ y ) x ) . Thus, ϕ factors through φ : F X ։ S n . Now, set S be the free group on s , . . . , s n − and let ψ ′ : S → F X be the group epihomomorphism given by s i ( i i + 1) . Now ψ ′ factors through ψ : S n ։ F X and it is clear that φ and ψ are inverses to each other.Let us prove the second claim. By the first part, we identify N with A n . Consider A n − ≤ A n as those permutations fixing 1 and 2 and set t = (12)(34) . Then tσt − ∈ A n − for all σ ∈ A n − . Clearly h t i ⋉ A n − ≤ N .Since A n is generated by { (34 ℓ ) | ≤ ℓ ≤ n, ℓ = 3 , } , the group N is generated by the subgroups A n − and A ≃ h (134) , (234) i . Notice that h (134) , (234) i ∩ N ≃ h t i . We have |h t i ⋉ A n − | = ( n − and { σ (12) σ − : σ ∈ A n } = (12) S n . Thus | N | = | N | / | (12) S n | = ( n − and hence N = h t i ⋉ A n − .Finally, since the commutator subgroup of some group A ⋉ B is the groupgenerated by [ A, A ] ∪ [ A, B ] ∪ [ B, B ] and N = h t i ⋉ A n − , it follows that [ N , N ] ≃ A n − and hence N / [ N , N ] ≃ Z . (cid:3) Theorem 5.5.
Let n ≥ and X = (12) S n be the conjugacy class of trans-positions. Then H ( X, C × ) ≃ C × × h χ i .Proof. Set x = (12) ∈ S n . Since N ≃ Z ⋉ A n − and N / [ N , N ] ≃ Z by Lemma 5.4, it follows that Hom( N , C × ) ≃ Z . Applying the isomor-phism (1.3) of Theorem 1.1 to the -cocycle χ given in (5.1), χ ( − , ( f χ ) ) , where ( f χ ) : N → C × , n f χ ( n )( x ) , n ∈ N . Now the claim followssince ( f χ ) generates Hom( N , C × ) . Indeed, f χ = 1 since ( f χ ) ((12)(34)) = f χ ((12)(34))(12) (2.3) = χ ((12) , (34) ⊲ (12)) χ ((34) , (12))= χ ((12) , (12)) χ ((34) , (12)) = − . This completes the proof. (cid:3)
We give a new proof of a formula of Eisermannas a consequence of our results.
Theorem 5.6. [8, Theorem 1.12]
Let X be a finite indecomposable quandleand x ∈ X . Then H Q ( X, Z ) ≃ ([ G X , G X ] ∩ C G X ( x )) ab ≃ ( N ) ab , where N is the stabilizer of a given x ∈ X of the action of [ G X , G X ] on X .Proof. The claim follows by “chasing” the chain of equivalences A × Hom( N , A ) ≃ H ( X, A ) ≃ Hom( H ( X, Z ) , A ) given by the application of Theorem 1.2 and Lemma 2.5. More explicitly, if ( a, g ) ∈ A × Hom( N , A ) , then it defines q ∈ H ( X, A ) via (1.4), which inturn defines a morphism H ( X, Z ) → A by Lemma 2.5: [ x, y ] q x,y = a + g ( c ( xσ y x − )) ∈ A, cf. Theorem 1.2. Now, H ( X, Z ) ≃ H Q ( X, Z ) × Z by Lemma 2.6 and so thisassignment becomes a map in Hom( H Q ( X, Z ) × Z , A ) : ([ x, y ] , ℓ ) ℓ a + g ( c ( xσ y x − )) . Thus we see that the restriction of this map to H Q ( X, Z ) × { } gives anequivalence Hom( H Q ( X, Z ) , A ) ≃ Hom( N , A ) ≃ ( N ) ab for any abeliangroup A . Hence we derive Eisermann’s formula H Q ( X, Z ) ≃ ( N ) ab . (cid:3) If we combine this fact with Lemma 2.6, we obtain the following.
Corollary 5.7.
Let X be a finite indecomposable quandle, x ∈ X . Then H ( X, Z ) ≃ ( N ) ab × Z . HE 2ND RACK COHOMOLOGY GROUP 17
Let L be an abelian group and γ ∈ Aut( L ) . The affine (or Alexander ) quandle
Aff(
L, γ ) is the set L together with the action x ⊲ y = γ ( y ) + x − γ ( x ) , x, y ∈ L. In [7] Clauwens described the enveloping group of an affine quandle; wereview his construction next. Set τ γ : L ⊗ Z L → L ⊗ Z L, ( x, y ) ( x, y ) − ( y, γ ( x )) ,S ( L, γ ) := coker τ γ = L ⊗ Z L/ h ( x, y ) − ( y, γ ( x )) i . (5.2)We write [ x, y ] ∈ S ( L, γ ) for the class of an element x ⊗ y ∈ L ⊗ Z L . Set X = Aff( L, γ ) ; then G X is the set L ⋊ Z × S ( L, γ ) with multiplication ( x, m, [ p, q ]) ( y, n, [ r, s ]) = ( x + γ m ( y ) , m + n, [ p + r + x, q + s + γ m ( y )]) , for m, n ∈ Z , x, y ∈ L , [ p, q ] , [ r, s ] ∈ S ( L, γ ) .The rack X identifies with the subset L ⋊ { } × with the rack actiongiven by conjugation: ( x, , y, ,
0) = ( x + γ ( y ) , , [ x, γ ( y )]) = ( x + γ ( y ) , , [ x ⊲ y, γ ( x )])= ( x + γ ( y ) − γ ( x ) + γ ( x ) , , [ x ⊲ y, γ ( x )])= ( x ⊲ y, , x, , since [ x ⊲ y, γ ( x )] = [ γ ( y ) , γ ( x )] + [ x, γ ( x )] − [ γ ( x ) , γ ( x )] = [ x, γ ( y )] , as [ x, γ ( x )] = [ γ ( x ) , γ ( x )] . We fix x = (0 , , ; then N X = L × { } × S ( L, γ ) , N = { } × { } × S ( L, γ ) . (5.3)Let { x , x , . . . , x n } be an enumeration of the elements of L . In particular, N X = G i ∈{ ,...,n } σ i N ≃ L × coker τ γ , σ i = ( x i , , , (5.4)is a good decomposition of N into N -cosets, cf. Proposition 4.1. Indeed,(1) σ = (0 , , coincides with the unit element in G X ;(2) fix j ∈ { , . . . , n } and let k ∈ { , . . . , n } be such that x k = γ ( x j ) .Then x ⊲ σ j = (0 , , x j , , , − ,
0) = ( γ ( x j ) , ,
0) = σ k ; and(3) if i ∈ { , . . . , n } and x j = (1 − γ ) − ( x i ) , then σ j ⊲ x = x i .Recall from (4.1) the definition of the elements σ y , y ∈ X , and from (3.4)the map c : N X → N . We see from Item (3) above that in this case σ y = (cid:0) (1 − γ ) − ( y ) , , (cid:1) , y ∈ X. As a direct consequence of Theorem 1.2, we obtain the following.
Proposition 5.8.
Let L be an abelian group, γ ∈ Aut( L ) and X = Aff( L, γ ) be the corresponding affine quandle and set Γ = S ( L, γ ) as in (5.2) . Fix x = 0 ∈ X and let A be an abelian group with trivial G X -action. Considera decomposition of N X into N -cosets as in (5.4) . For each a ∈ A and g ∈ Hom(Γ , A ) , the map q : X × X → A given by q x,y = a + X By Theorem 1.2 and Corollary 1.3, any 2-cocycle is of the form q x,y = a + g ( c ( xσ y x − )) . for some a ∈ A and g ∈ Hom(Γ , A ) . Using the identifications above, we have xσ y x − = ( x, , − γ ) − ( y ) , , , − , x + γ (1 − γ ) − ( y ) , , [ x, γ (1 − γ ) − ( y )])= σ k (0 , , [ x, γ (1 − γ ) − ( y )]) ∈ σ k N for k ∈ { , . . . , n } such that x + γ (1 − γ ) − ( y ) = x k . Hence γ (1 − γ ) − ( y ) = (1 − γ ) − ( y ) − y = X L, γ ) . Lemma 5.9. Let p be a prime number and = ω ∈ F × p , set X = Aff( p, ω ) .Then G X ≃ L ⋊ Z , N X ≃ L and N is trivial.Proof. Indeed, S ( L, γ ) is a quotient of Z p ≃ Z p ⊗ Z Z p and we have that = (1 − ω ) ⊗ ∈ Im( τ γ ) , hence S ( L, γ ) = 0 and the lemma follows. (cid:3) We recover the following result from [13, Lemma 5.1]. Proposition 5.10. H (Aff( p, ω ) , C × ) ≃ C × . Proof. It follows from Theorem 1.1, using Lemma 5.9. (cid:3) p . Let p be a prime numberand let X be an indecomposable quandle of size p . By [13], X is one of thefollowing affine quandles ( L, γ ) in the following list: L = Z p ⊕ Z p , γ α,β ( x, y ) = ( α x, β y ) , α, β ∈ Z ∗ p \ { } ; (5.6) L = Z p ⊕ Z p , γ α ( x, y ) = ( α x, α y + x ) , α ∈ Z ∗ p \ { } ; (5.7) L = F p , γ α ( x ) = α x, α ∈ F p \ F p ; (5.8) L = Z p , γ α ( x ) = α x, α , p ) . (5.9)We identify F p ≃ F p ⊕ F p as abelian groups for notational reasons. For α = ( α , α ) ∈ F p we set d α := (1 − α + α )(1 − α − α )(1 − α + α ) . (5.10)Assume α ∈ F p \ F p , so α = 0 . If d α = 0 , then α = 1 and we set t α := ( α − α + α )(1 − α ) − , s α := (1 − α ) α − . (5.11) HE 2ND RACK COHOMOLOGY GROUP 19 Proposition 5.11. The 2nd homology groups of the indecomposable quan-dles of order p are as follows: H (( Z p ⊕ Z p , γ α,β ) , Z ) ≃ ( Z × Z p , if αβ = 1 , Z , if αβ = 1 .H (( Z p ⊕ Z p , γ α ) , Z ) ≃ ( Z × Z p , if α = 1 , Z , if α = 1 .H (cid:0) ( F p , γ α ) , Z (cid:1) ≃ ( Z × Z p , if d α = 0 , Z , if d α = 0 .H (cid:0) ( Z p , γ α ) , Z (cid:1) ≃ Z . Proof. By Corollary (5.7), if X = ( L, γ ) and τ γ : L ⊗ Z L → L ⊗ Z L as in(5.2), then H ( X, Z ) = ( N ) ab × Z = coker τ γ × Z We compute coker τ γ case by case. We will use the identifications(5.12) (cid:0) Z p ⊕ Z p (cid:1) ⊗ Z (cid:0) Z p ⊕ Z p (cid:1) ≃ Z p , ( a, b ) ⊗ ( c, d ) ( ac, ad, bc, bd ) F p ⊗ Z F p ≃ F p ⊗ F p F p ≃ F p , ( a, b ) ⊗ ( c, d ) ( ac, ad, bc, bd ) Z p ⊗ Z Z p ≃ Z p , a ⊗ b ab. Case (5.6): We have that τ α,β := τ γ α,β is τ α,β (( a, b ) ⊗ ( c, d )) = ( a, b ) ⊗ ( c, d ) − ( c, d ) ⊗ ( α a, β b ) . With the identifications above this yields τ α,β : Z p → Z p , ( x, y, z, w ) ((1 − α ) x, y − β z, z − α y, (1 − β ) w ) . Next, we compute the image I α,β of this map: For ( a, b, c, d ) ∈ Z p to be inthis subgroup, we need x = a (1 − α ) − , w = d (1 − β ) − (recall α, β = 1 ) and y, z to be a solution of y − β z = b , − α y + z = c . This system has always asolution if αβ = 1 . If αβ = 1 , then I α,β = { ( a, b, − α b, d ) | a, b, d ∈ Z p } ≃ Z p , hence coker τ α = ( , if αβ = 1 , Z p , if αβ = 1 . In case (5.7), we have τ α := τ γ α : Z p → Z p is given by ( x, y, z, w ) (cid:0) (1 − α ) x, y − α z + x, z − α y, (1 − α ) w − y (cid:1) . For ( a, b, c, d ) to be in the image I α of τ α , we need x = a (1 − α ) − (recall α = 1 ) and ( y, z, w ) to be a solution of y − α z = b − a (1 − α ) − , − α y + z = c, − y + (1 − α ) w = d. This system has always a solution if α = 1 . If α = 1 , then I α = { ( a, b, α b − α (1 − α ) a, d ) | a, b, d ∈ Z p } ≃ Z p , hence coker τ α = ( , if α = 1 , Z p , if α = 1 . In case (5.8), if α = ( α , α ) ∈ F p \ F p (hence α = 0 ), then the map τ α ∈ End( F p ) is represented by the matrix [ τ α ] = − α − α − α − α − α − α − α − α ! , with det[ τ α ] = d α , see (5.10). Let I α denote the image of this map. Now,the rank of this matrix is ≥ , as det (cid:18) − α − α − α − α (cid:19) = − α = 0 . Hence, coker τ α = ( , if det[ τ α ] = 0 , Z p , if det[ τ α ] = 0 . If det[ τ α ] = 0 , i.e. d α = 0 , then we set t α , s α ∈ F p as in (5.11) and thus I α = { ( a, b, c, − t α ( a + b ) − s α c ) | a, b, c ∈ Z p } ≃ Z p . In case (5.9), τ α : Z p → Z p is x (1 − α ) x ; hence coker τ α = 0 . (cid:3) Next we apply Proposition 5.8 to compute all non-constant 2-cocycles for the affine quandles X described in (5.6)–(5.9). Moreprecisely, we focus on those affine quandles in that list admitting a non-constant 2-cocyle, as stated in Proposition 5.11: L = Z p ⊕ Z p , γ α ( x, y ) = ( α x, α − y ) , α ∈ Z ∗ p \ { } ; (5.13) L = Z p ⊕ Z p , γ ( x, y ) = ( − x, x − y ); (5.14) L = F p , γ α ( x ) = α x, α ∈ F p \ F p , d α = 0 . (5.15)Recall our identification F p ≃ F p , x ( x , x ) , and t α , s α ∈ Z p from (5.11).For x, y ∈ L and j ∈ N we set, for X is as in (5.13), ζ j ( x, y ) = α j x y + α − j x y ; for X is as in (5.14), ζ j ( x, y ) = ( j + 2( − j ) x y + ( − j ( x y − x y ); and, for X is as in (5.15), ζ j ( x, y ) = x ( α j y ) + t α (cid:0) x ( α j y ) + x ( α j y ) (cid:1) + s α x ( α j y ) . Next, we define the map h , i : L × L → Z as h x, y i = X Let X = ( L, γ ) be an indecomposable affine rack of order p . If q ∈ H ( X, k ∗ ) is non-constant, then X belongs to the list (5.13) – (5.15) and there are < ℓ < p and λ ∈ k ∗ such that q x,y = λ exp (cid:18) π i ℓp h x, y i (cid:19) , x, y ∈ X. (5.16) HE 2ND RACK COHOMOLOGY GROUP 21 Proof. Fix x = 0 ∈ L and a good decomposition N ≃ L × coker τ γ of N X into N -cosets, see (5.4). In this case, N = x × coker τ γ ≃ Z p , by Proposition5.11. More precisely, if we denote by ϕ : N → Z p this isomorphism, then itfollows from the proof of Proposition 5.11 that, for t α , s α as in (5.11): ϕ ([( a, b ) , ( c, d )]) = bc + αad ∈ Z p , X as (5.13) ; bc + ad − ac ∈ Z p , X as (5.14) ; bd + t α ( ac + ad ) + s α bc ∈ Z p , X as (5.15) . (5.17)On the other hand, if g ∈ Hom( N , k ∗ ) , then there is ≤ ℓ < p such that g is the morphism g ℓ given by exp (cid:16) π i ℓp (cid:17) . By Proposition 5.8, any q ∈ H ( X, k ∗ ) is thus of the form q x,y = λ Y We thank G. García and M. Kotchetov for interesting discussions. We alsothank N. Andruskiewitsch for his constant guidance and support. This workwas initiated while the authors were visiting María Ofelia Ronco, at Univer-sidad de Talca, Chile. We are grateful for her warm hospitality. The authorsare grateful to the reviewer for useful remarks, interesting suggestions andcorrections. References [1] N. Andruskiewitsch, F. Fantino, G. A. García, and L. Vendramin. On Nichols alge-bras associated to simple racks. In Groups, algebras and applications , volume 537 of Contemp. Math. , pages 31–56. Amer. Math. Soc., Providence, RI, 2011.[2] N. Andruskiewitsch and M. Graña. From racks to pointed Hopf algebras. Adv. Math. ,178(2):177–243, 2003.[3] K. S. Brown. Cohomology of groups , volume 87 of Graduate Texts in Mathematics .Springer-Verlag, New York, 1994. Corrected reprint of the 1982 original.[4] J. S. Carter, D. Jelsovsky, S. Kamada, L. Langford, and M. Saito. Quandle coho-mology and state-sum invariants of knotted curves and surfaces. Trans. Amer. Math.Soc. , 355(10):3947–3989, 2003.[5] J. S. Carter, D. Jelsovsky, S. Kamada, and M. Saito. Quandle homology groups, theirBetti numbers, and virtual knots. J. Pure Appl. Algebra , 157(2-3):135–155, 2001.[6] W. E. Clark, M. Elhamdadi, M. Saito, and T. Yeatman. Quandle colorings of knotsand applications. J. Knot Theory Ramifications , 23(6):1450035, 29, 2014.[7] F. J. B. J. Clauwens. The adjoint group of an alexander quandle. Preprint:arXiv:1011.1587 .[8] M. Eisermann. Quandle coverings and their Galois correspondence. Fund. Math. ,225:103–168, 2014.[9] P. Etingof and M. Graña. On rack cohomology. J. Pure Appl. Algebra , 177(1):49–59,2003.[10] R. Fenn, C. Rourke, and B. Sanderson. Trunks and classifying spaces. Appl. Categ.Structures , 3(4):321–356, 1995.[11] R. Fenn, C. Rourke, and B. Sanderson. James bundles. Proc. London Math. Soc. (3) ,89(1):217–240, 2004. [12] M. Graña. On Nichols algebras of low dimension. In New trends in Hopf algebra theory(La Falda, 1999) , volume 267 of Contemp. Math. , pages 111–134. Amer. Math. Soc.,Providence, RI, 2000.[13] M. Graña. Indecomposable racks of order p . Beiträge Algebra Geom. , 45(2):665–676,2004.[14] M. Graña, I. Heckenberger, and L. Vendramin. Nichols algebras of group type withmany quadratic relations. Adv. Math. , 227(5):1956–1989, 2011.[15] I. Heckenberger and L. Vendramin. Nichols algebras over groups with finite rootsystem of rank two II. J. Group Theory , 17(6):1009–1034, 2014.[16] A. Hulpke, D. Stanovský, and P. Vojtěchovský. Connected quandles and transitivegroups. J. Pure Appl. Algebra , 220(2):735–758, 2016.[17] D. Joyce. A classifying invariant of knots, the knot quandle. J. Pure Appl. Algebra ,23(1):37–65, 1982.[18] V. Lebed. Homologies of algebraic structures via braidings and quantum shuffles. J.Algebra , 391:152–192, 2013.[19] R. A. Litherland and S. Nelson. The Betti numbers of some finite racks. J. Pure Appl.Algebra , 178(2):187–202, 2003.[20] S. V. Matveev. Distributive groupoids in knot theory. Mat. Sb. (N.S.) ,119(161)(1):78–88, 160, 1982.[21] A. Milinski and H.-J. Schneider. Pointed indecomposable Hopf algebras over Cox-eter groups. In New trends in Hopf algebra theory (La Falda, 1999) , volume 267 of Contemp. Math. , pages 215–236. Amer. Math. Soc., Providence, RI, 2000.[22] T. Nosaka. Quandle homotopy invariants of knotted surfaces. Math. Z. , 274(1-2):341–365, 2013.[23] J. J. Rotman. An introduction to the theory of groups , volume 148 of Graduate Textsin Mathematics . Springer-Verlag, New York, fourth edition, 1995.[24] D. Stanovský. A guide to self-distributive quasigroups, or Latin quandles. QuasigroupsRelated Systems , 23(1):91–128, 2015.[25] H. Tamaru. Two-point homogeneous quandles with prime cardinality. J. Math. Soc.Japan , 65(4):1117–1134, 2013. Departamento de Matemática – FCEN, Universidad de Buenos Aires, Pa-bellón I, Ciudad Universitaria (1428) Buenos Aires, República Argentina E-mail address : [email protected] FaMAF–CIEM (CONICET) – Universidad Nacional de Córdoba, MedinaAllende s/n, Ciudad Universitaria (5000) Córdoba, República Argentina. E-mail address ::