ON RACK COHOMOLOGY

P. ETINGOF & M. GRAÑA

Abstract. We prove that the lower bounds for Betti numbers of the rack, quandle and degeneracy cohomology given in [CJKS] are in fact equalities. We compute as well the Betti numbers of the twisted cohomology introduced in [CES]. We also give a group-theoretical interpretation of the second cohomology group for racks.

1. Introduction

A rack is a pair (X, ▷) where X is a set and ▷ : X × X → X is a binary operation such that:

(1) The map ϕ_{x} : X → X, ϕ_{x}(y) = x ▷ y, is a bijection for all x ∈ X, and

(2) x ▷ (y ▷ z) = (x ▷ y) ▷ (x ▷ z) ∀x, y, z ∈ X.

It is easy to show that (X, ▷) is a rack if and only if the map R : X^{2} → X^{2} given by R(x, y) = (x, x ▷ y) is an invertible solution of the quantum Yang-Baxter equation R^{12}R^{13}R^{23} = R^{23}R^{13}R^{12}.

Racks have been studied by knot theorists in order to construct invariants of knots and links and their higher dimensional analogs (see [CS] and references therein). A basic example of a rack is a group with the operation x ▷ y = xyx^{−1} (or, more generally, a conjugation invariant subset of a group).

Several years ago, Fenn, Rourke and Sanderson [FRS] proposed a cohomology theory of racks. Namely, for each rack X and an abelian group A, they defined cohomology groups H^{n}(X, A). This cohomology is useful for knot theory and also, as was recently found, for the theory of pointed Hopf algebras [G]. There have been a number of results about this cohomology [LN, M, CJKS], in particular it was shown in [CJKS] that for a finite rack X and a field k of characteristic zero, the Betti numbers dim H^{n}(X, k) are bounded below by |X/ ∼ |^{n}, where ∼ is the equivalence relation on X generated by the relation z ▷ y ∼ y ∀y, z ∈ X. The equality was anticipated in [CJKS], and proved in a number of cases [LN, M], but not in general.

The main result of this paper implies that the Betti numbers of a finite rack are always equal to |X/ ∼ |^{n}. The proof is based on a group-theoretical approach to racks, originating from the works [LYZ], [S] on settheoretical solutions of the quantum Yang-Baxter equation. Namely, we use the structure group G_{X} and the reduced structure group G_{X}^{0} of a rack X considered in [LYZ, S].

We also give a group-theoretic interpretation of the second cohomology group H^{2}(X, A), which is used in the theory of Hopf algebras. Namely, we show that this group is isomorphic to the group cohomology H^{1}(G_{X}, Fun(X, A)), where Fun(X, A) is the group of functions from X to A. This is a relatively explicit description, since it is shown by Soloviev [S] that for a finite rack X, the group G_{X} is a central extension of the finite group G_{X}^{0} by a finitely generated abelian group. Thus the cohomology of G_{X} can be studied using the Hochschild-Serre sequence.

Acknowledgments. The work of P.E. was supported by the NSF grant DMS-9988796. The work of M.G. was supported by Conicet.

2. Definitions and notation

Definition 2.1. The structure group of a rack X is the group G_{X} with generators being the elements of X and relations x · y = (x ▷ y) · x ∀x, y ∈ X. ^{1}

The group G_{X} acts on X from the left by ▷. Consider the quotient G_{X}^{0} of G_{X} by the kernel of this action, i.e. the group of trasformations of X generated by x▷. This group is called the reduced structure group of X.

Remark 2.2. The groups G_{X}, G_{X}^{0} were studied by Soloviev [S] (we note that in his work, racks are called “derived solutions”). In particular, he showed that the category of racks is equivalent to the category of quadruples (G, X, ρ, π), where G is a group, X a set, ρ : G × X → X a left action, and π : X → G an equivariant mapping (where G acts on itself by conjugation), such that π(X) generates G and the G-action on X is faithful. Namely, the quadruple corresponding to X is simply (G_{X}^{0}, X, ρ, π), where ρ and π are obvious.

Now let us define rack cohomology. Let X be a rack. Let G_{X} be its structure group. Let M be a right G_{X}-module. We define a cochain complex (C^{•}(X, M), d), where C^{n}(X, M) = Fun(X^{n}, M), n ≥ 0, with differential

df(x_{1}, …, x_{n+1}) = ∑_{i=1}^{n}(−1)^{i−1}(f(x_{1}, …, x_{i−1}, x_{i+1}, …, x_{n+1}) − f(x_{1}, …, x_{i−1}, x_{i} ▷ x_{i+1}, …, x_{i} ▷ x_{n+1}) · x_{i})

(Here X^{0} is a set of one element, and Fun(Y, Z) is the set of functions from Y to Z for any sets Y, Z).

Definition 2.3. The cohomology of C^{•}(X, M) is called the rack cohomology of X with coefficients in M.

This includes the ordinary rack cohomology with coefficients in an abelain group A, introduced in [FRS] (this corresponds to taking M = A with the trivial action of G_{X}), as well as the twisted rack cohomology introduced in [CES] (in this case one needs to take a ℤ[T, T^{−1}] module M, and define a right action of G_{X} on it by vx = T v, x ∈ X).

Remark 2.4. One can also define the dual notion of rack homology. As usual, it is completely analogous to cohomology, so we will not consider it.

Remark 2.5. In [AG] there is a more general definition of cohomology, with coefficients in objects of a wider category than that of G_{X}-modules. When restricted to G_{X}-modules, the definition there takes as differential the map d′, defined by

d′f(x_{1}, …, x_{n+1}) = ∑_{i=1}^{n}(−1)^{i−1}(f(x_{1}, …, x_{i−1}, x_{i+1}, …, x_{n+1})(x_{1} ▷ (x_{2} ▷ (⋯ x_{i})))^{−1}
− f(x_{1}, …, x_{i−1}, x_{i} ▷ x_{i+1}, …, x_{i} ▷ x_{n+1}))

1This group appears already in the work of Joyce [J], who pointed out that the functor X → G_{X} is adjoint to the functor assigning to a group the underlying rack (with the conjugation operation). Thus the group G_{X} can be viewed as the “enveloping group” of X.

This complex is isomorphic to the one we consider here, by means of the map

T : (C^{•}(X, M), d) → (C^{•}(X, M), d′),

defined by (T f)(x_{1}, …, x_{n}) = f(x_{1}, …, x_{n})(x_{1} ⋯ x_{n})^{−1}.

3. The structure of rack cohomology

Let M be a right G_{X}-module. Then C^{n}(X, M) = Fun(X^{n}, M) is also a right G_{X}-module, with the action defined on the generators by

(f · y)(x_{1}, …, x_{n}) = f(y ▷ x_{1}, …, y ▷ x_{n}) · y.

Lemma 3.1. (1) The coboundary operator d : C^{n}(X, M) → C^{n+1}(X, M) is a map of G_{X}-modules. In particular, there is a natural right action of G_{X} on the groups of cocycles Z^{n}(X, M), coboundaries B^{n}(X, M), and cohomology H^{n}(X, M).

(2) H^{n}(X, M) is a trivial G_{X}-module.

Proof.

(1) Straightforward.

(2) Let f ∈ Z^{n}(X, M) and consider f_{y} ∈ C^{n−1}(X, M), defined by the formula

f_{y}(x_{2}, …, x_{n}) = f(y, x_{2}, …, x_{n}).

Notice that

(3.2)  d(f_{y})(x_{1}, …, x_{n}) = (f − f · y)(x_{1}, …, x_{n}) − (df)(y, x_{1}, …, x_{n}) = (f − f · y)(x_{1}, …, x_{n}).

Then f · y = f in H^{n}(X, M).

□

Remark 3.3. The action f · y and the assignments f ↦ f_{y}, as well as (3.2), appear in [LN].

By Lemma 3.1 we can consider the subcomplex C_{inv}^{•}(X, M) = C^{•}(X, M)^{G_{X}}. We define the invariant rack cohomology H_{inv}^{•}(X, M) = H^{•}(C_{inv}^{•}(X, M)). Clearly, we have a natural map

ξ : H_{inv}^{•}(X, M) → H^{•}(X, M),

induced by the inclusion of complexes.

Remark 3.4. If f ∈ Z_{inv}^{n}(X, M), by the proof of Lemma 3.1 part 2, it is clear that f_{y} ∈ Z^{n−1}(X, M) ∀y ∈ X.

For M, N right G_{X}-modules, consider the natural multiplication map

C^{a}(X, M) × C^{b}(X, N) → C^{a+b}(X, M ⊗ N).

This map will be denoted by f, g → f ⊗ g.

Lemma 3.5. Suppose that A is a trivial G_{X}-module. Then for any f ∈ C^{i}(X, A), g ∈ C_{inv}^{j}(X, N), one has

d(f ⊗ g) = df ⊗ g + (−1)^{i}f ⊗ dg.

Proof. The proof is straightforward. We note that the statement becomes false if A is nontrivial as a G_{X}-module or g is not invariant. □

Lemma 3.5 shows that if f ∈ Z^{i}(X, A) and g ∈ Z_{inv}^{j}(X, N) then f ⊗g ∈ Z^{i+j}(X, A⊗N). Furthermore, by the same Lemma, the cohomology class of f ⊗ g depends only of the cohomology classes of f and g. Thus, we have a product

H^{•}(X, A) × H_{inv}^{•}(X, N) → H^{•}(X, A ⊗ N).

In particular, if R is a (unital) ring with the trivial G_{X}-action, then H_{inv}^{•}(X, R) is a graded algebra, and for any left R-module M with a compatible G_{X}-action, H_{inv}^{•}(X, M) is a graded left H_{inv}^{•}(X, R)-module.

4. Cohomology of finite racks

In this section we will assume that X is a finite rack.

Let M be a right G_{X}-module, such that the kernel K of the action of G_{X} on M has finite index. Let L be the intersection of K with the kernel Γ of the action of G_{X} on X, and let G = G_{X}/L (notice that G is finite). Assume that the multiplication by |G| is an isomorphism M → M.

Lemma 4.1. Under these conditions the map ξ : H_{inv}^{•}(X, M) → H^{•}(X, M) is an isomorphism.

Proof. The complex C^{•}(X, M) is a complex of G-modules. On each term of this complex we have a projector given by P = \frac{1}{|G|} ∑_{g∈G} g, which projects to G_{X}-invariants. This projector commutes with the differential, so the complex C^{•}(X, M) is representable as a direct sum of complexes:

C^{•}(X, M) = C_{inv}^{•}(X, M) ⊕ C^{•}(X, M)(1 − P).

By Lemma 3.1, the second summand is acyclic: indeed, any cohomology class in it satisfies cP = 0, while the lemma says that cP = c, hence c = 0. This implies the desired statement. □

In particular, for any ring R with trivial G_{X}-action, such that N = |G_{X}^{0}| is invertible in R (for example, R = ℤ[1/N] or R = ℚ), the cohomology H^{•}(X, R) is an algebra, and if M is an R-module with a compatible G_{X} action then H^{•}(X, M) is a left module over this algebra.

Let Orb(X) = X/G_{X} be the set of G_{X}-orbits on X, and m = | Orb(X)|. The main result in this section is

Theorem 4.2. Under the conditions of Lemma 4.1, we have

H^{•}(X, R) ≃ T_{R}^{•}(H^{1}(X, R)) ≃ T_{R}^{•}(Fun(Orb(X), R)) ≃ Fun(Orb(X)^{•}, R)

as an algebra (where T_{R}^{•}(B) denotes the tensor algebra of an R-bimodule B), and if M is an R-module with a compatible G_{X} action then

H^{•}(X, M) ≃ T_{R}^{•}(H^{1}(X, R)) ⊗_{R} M^{G_{X}} ≃ T_{R}^{•}(Fun(Orb(X), R)) ⊗_{R} M^{G_{X}} ≃ Fun(Orb(X)^{•}, M^{G_{X}})

as a left module over the algebra H^{•}(X, R).

Before proving the theorem, we will derive a corollary.

Corollary 4.3. The Betti numbers of X are dim H^{i}(X, ℚ) = m^{i}. Furthermore, the only primes which can appear in the torsion of H^{•}(X, ℤ) are those dividing N.

Proof. The first assertion is clear taking R = ℚ. For the second one, take R = ℤ[\frac{1}{N}] (or R = ℤ/p, p ∤ N) and apply the universal coefficent theorem. □

Remark 4.4. This, together with the lower bounds for the Betti numbers of the quandle and degeneracy cohomology in [CJKS] and the splitting result of [LN], implies that those lower bounds are in fact equalities.

Proof. (of Theorem 4.2). Since M^{G_{X}} = H^{0}(X, M), for any M we have an obvious multiplication mapping μ : T^{•}(H^{1}(X, R)) ⊗_{R} M^{G_{X}} → H^{•}(X, M), which is compatible with the algebra and module structures. Thus, all we have to show is that μ is an isomorphism.

Let us first show that μ is injective. This is in fact the lower bound of [CJKS], but we will give a different proof. The proof is by induction in degree. The base of induction is clear. Assume the statement is known in degrees < n, and c ∈ Fun(Orb(X)^{n}, M^{G_{X}}) is such that μ(c) = 0. This means that the pullback f : X^{n} → M of the function c is a coboundary: f = dg. Because f is invariant (under the diagonal action of G_{X}), and C^{•} = C_{inv}^{•} ⊕ C^{•}(1 − P), we can assume that g is invariant. This means that for any y ∈ X, we have (dg)_{y} = d(g_{y}) (we recall that g_{y}(x_{1}, …, x_{l}) := g(y, x_{1}, …, x_{l})). Thus, f_{y} = dg_{y}. But f_{y} is a pullback of a function c_{y} ∈ Fun(Orb(X)^{n−1}, M^{G_{X}}), so by the induction assumption c_{y} = 0. Hence c = 0.

Now let us prove that μ is surjective. For this it suffices to show that H^{n}(X, M) ⊂ H^{1}(X, R)H^{n−1}(X, M). Let c ∈ H^{n}(X, M). By Lemma 4.1, the element c can be represented by an invariant cycle, f ∈ Z_{inv}^{n}(X, M). By remark 3.4, f_{y} ∈ Z^{n−1}(X, M) for all y ∈ X. For each y ∈ X, decompose f_{y} as f_{y} = (f_{y})^{+} + (f_{y})^{−}, where

(f_{y})^{+} = f_{y} · P ∈ Z_{inv}^{n−1}(X, M) and (f_{y})^{−} = f_{y} · (1 − P) ∈ Z^{n−1}(X, M).

These functions give rise to unique functions f^{+}, f^{−} ∈ C^{n}(X, M) such that (f^{±})_{y} = (f_{y})^{±} ∀y ∈ X. Moreover, it is clear that f = f^{+} +f^{−}. Since (f^{+})_{y} ∈ Z_{inv}^{n−1}(X, M) ∀y, it is easy to see that f^{+} ∈ Z^{n}(X, M). Thus, also f^{−} ∈ Z^{n}(X, M). Let us see now that f^{±} are invariant: for any h ∈ C^{n}(X, M), g ∈ G_{X}, we have the equality h_{y} · g = (h · g)_{g^{−1}y}, which implies that

f_{gy}^{+} = f_{gy} · P = f_{gy} · g^{−1}P = (f · g^{−1})_{y} · P = f_{y}^{+},

and thus (f^{+} · g)_{y} = (f_{gy}^{+}) · g = (f^{+})_{y}. Since this equality holds ∀y ∈ X, we have f^{+} ∈ Z_{inv}^{n}(X, M) as claimed. Since f ∈ Z_{inv}^{n}(X, M), we also have f^{−} ∈ Z_{inv}^{n}(X, M). Now, as G_{X} acts trivially on cohomology, there exists h ∈ C^{n−1}(X, M) such that d(h_{y}) = f_{y}^{−} for each y ∈ X. Take h̃ = hP. We have

d((h · g)_{y}) = d(h_{gy} · g) = d(h_{gy}) · g = f_{gy}^{−} · g = (f^{−} · g)_{y} = f_{y}^{−},

and thus, by (3.2), (dh̃)_{y} = d(h̃_{y}) = f_{y}^{−}, whence dh̃ = f^{−}. Thus, f^{−} is a coboundary, and we can assume that f = f^{+}. In other words, f ∈ Fun(Orb(X), Z^{n−1}(X, M)^{G_{X}}). This means that f = ∑_{s∈Orb(X)} 1_{s} ⊗ f(s), where 1_{s} is the characteristic function of s with values in R. Since 1_{s} is a cocycle, we have proved that c ∈ H^{1}(X, R)H^{n−1}(X, M), as desired. □

Now let M be a semisimple finite dimensional G_{X}-module over a field k of characteristic zero (but we do not require the image of G_{X} to be finite). In this case, we have

Theorem 4.5. Lemma 4.1 and Theorem 4.2 are true for such M.

Proof. By a Chevalley’s theorem [C], the representations C^{n}(X, M) = Fun(X, k)^{⊗n} ⊗ M are semisimple (as tensor products of semisimple representations). Therefore, there exists an invariant projector P : C^{•} → (C^{•})^{G_{X}}. The rest of the proof is the same as in the previous case. □

Recall [S] that G_{X} is a central extension of the finite group G_{X}^{0} with kernel being the finitely generated abelian group Γ.

Corollary 4.6. If M is a finite dimensional ℚ[G_{X}]-module and M(1) the generalized eigenspace for the trivial character of Γ, then H^{•}(X, M) = H^{•}(X, M(1)).

Proof. Write M = ⊕_{χ}M(χ), where χ runs over the characters of Γ. We have H^{•}(X, M) = ⊕_{χ}H^{•}(X, M(χ)). Now, we prove by induction on the dimension of M(χ) that if χ is non-trivial then H^{•}(X, M(χ)) = 0. If dim M(χ) = 0, the cohomology clearly vanishes. Suppose now that dim M(χ) = n > 0 and for smaller dimensions the statement is known. Let M_{0} be a simple submodule of M(χ). We have then the short exact sequence of complexes

0 → C^{•}(X, M_{0}) → C^{•}(X, M(χ)) → C^{•}(X, M(χ)/M_{0}) → 0.

The first complex is acyclic by Theorem 4.5, the third one is acyclic by the induction assumption, so by the long exact sequence in cohomology, the complex in the middle is also acyclic. The induction step and the corollary are proved. □

Corollary 4.7. Let M be a finite dimensional ℚ[T^{±1}]-module. Then the twisted rack cohomology H_{T}^{i} (X, M) equals the twisted rack cohomology H_{T}^{i} (X, M(1)), where M(1) is the generalized eigenspace of T in M with eigenvalue 1. □

To compute the Betti numbers of twisted cohomology, the only lacking case is that in which the elements of the rack X act on M by a Jordan block with 1 on the diagonal.

Proposition 4.8. Let M be an ℚG_{X}-module with basis {v_{1}, …, v_{k}} on which the elements of X act by v_{i} ↦ v_{i−1} + v_{i} (v_{0} := 0). Then dim H^{n}(X, M) = m^{n}, where m = | Orb(X)|.

Before proving the Proposition we state two easy lemmas:

Lemma 4.9. Let (C^{•}, d) be a complex and suppose that C^{•} = C_{1}^{•} ⊕ C_{2}^{•} and that the differential d has the

form ( \begin{matrix}d_{1} & α \\ 0 & d_{2} \end{matrix} ) for this decomposition. Then α induces a map α_{∗}^{n} : H^{n−1}(C_{2}^{•}) → H^{n}(C_{1}^{•}). Consider then

the short exact sequence of complexes

0 ⟶ C_{1}^{•} ⟶^{i} C^{•} ⟶^{p} C_{2}^{•} ⟶ 0

and let β^{n} : H^{n−1}(C_{2}^{•}) → H^{n}(C_{1}^{•}) be the connecting homomorphism. Then β^{n} = α_{∗}^{n}.

Proof. Since d^{2} = 0, we have d_{1}α = −αd_{2}, whence it induces a map in cohomology. The second assertion follows in a straightforward way from the definition of the connecting homomorphism. □

Lemma 4.10. Let C^{•} = C_{1}^{•} ⊕ C_{2}^{•} be as in Lemma 4.9. Suppose that (C_{2}^{•′}, d′_{2}) is a complex and that f : C_{2}^{•′} → C_{2}^{•} is a quasi-isomorphism. Then (id ⊕f) : C_{1}^{•} ⊕ C_{2}^{•′} → C^{•} is a quasi-isomorphism, where the

first complex has differential given by ( \begin{matrix}d_{1} & αf \\ 0 & d′_{2} \end{matrix} ).

Proof. This follows easily from the 5-lemma. □

Proof of Proposition 4.8. The proof is by induction on k. If k = 1 the assertion is Corollary 4.3. Assume that the result is true for dimensions < k. Let us decompose C^{•} = C^{•}(X, M_{1}) ⊕ C^{•}(X, M_{2}), where M_{1} is generated by v_{1}, …, v_{k−1} and M_{2} is generated by v_{k}. Notice that the differential d in C^{•} can be written as

( \begin{matrix}d_{1} & α \\ 0 & d_{2} \end{matrix} ), where d_{i} : C^{•}(X, M_{i}) → C^{•}(X, M_{i}) are the differentials of the same complex we are considering

for M of dimension k − 1 and 1 respectively.

Let us take C_{2}^{•′} = T^{•}(Fun(Orb(X), ℚ)). By Theorem 4.2, the inclusion i : C_{2}^{•′} → C_{2}^{•} is a quasiisomorphism, and thus by Lemma 4.10 we can work with C^{•}(X, M_{1}) ⊕ T^{•}(Fun(Orb(X), ℚ)). We consider the long exact sequence

(4.11)  → H^{n−1}(C_{2}^{•′}) ⟶^{β^{n}} H^{n}(C_{1}^{•}) ⟶^{i^{n}} H^{n}(C_{1}^{•} ⊕ C_{2}^{•′}) ⟶^{p^{n}} H^{n}(C_{2}^{•′}) ⟶^{β^{n+1}} H^{n+1}(C_{2}^{•′}) →

Let ᾱ = α|_{C_{2}^{•′}} and consider the induced map in cohomology ᾱ_{∗}, i.e.,

ᾱ_{∗}^{n} : H^{n−1}(C_{2}^{•′}) = T^{n−1}(Fun(Orb(X), ℚ)) → H^{n}(C_{1}^{•}) = H^{n}(X, M_{1}).

By Lemma 4.9, β^{n} = ᾱ_{∗}^{n}. We claim that rk ᾱ_{∗} = rk ᾱ. To see this, it suffices to prove that Im ᾱ^{n}∩B^{n}(C_{1}^{•}) = 0. Suppose that ᾱ^{n}(f) ∈ B^{n}(C_{1}^{•}), then it has the form ᾱ^{n}(f) = ∑_{i=1}^{k−1} b_{i}v_{i}, where b_{i} ∈ C^{n}(X, ℚ). Furthermore, it is clear that b_{k−1} ∈ B^{n}(X, ℚ). On the other hand, if π : X → Orb(X) is the canonical projection, we have

α^{n}(f)(x_{1}, …, x_{n}) = ∑_{i=1}^{n}(−1)^{i}f(π(x_{1}), …, π(x_{i−1}), π(x_{i+1}), …, π(x_{n}))v_{k−1},

which shows that b_{k−1} ∈ T^{n}(Fun(Orb(X), ℚ)). But it is shown in the injectivity part of the proof of Theorem 4.2 that T^{n}(Fun(Orb(X), ℚ)) ∩ B^{n}(X, ℚ) = 0, and the claim is proved.

Then, rk β^{n} = rk ᾱ^{n}. But the latter is not difficult to compute: if we consider the complex (D^{•}, d̂), where D^{n} = Fun((Orb(X))^{n}, ℚ) and d̂ is given by

d̂(f)(a_{1}, …, a_{n}) = ∑_{i=1}^{n}(−1)^{i}f(a_{1}, …, a_{i−1}, a_{i+1}, …, a_{n}),

then it is clear that ᾱ^{n} and d̂^{n} have the same rank. Furthermore, it is well known that D^{•} is acyclic (it gives the reduced cohomology of a simplex of dimension m − 1). It is easy then to compute the rank of d̂; we have rk d̂^{n} = m^{n−1} − m^{n−2} + m^{n−3} − ⋯ ± 1. We add this computation to the long exact sequence (4.11) and we are done: we have rk β^{n} = m^{n−1} − m^{n−2} + ⋯ ± 1, and since by the inductive assumption dim H^{n}(C_{1}^{•}) = m^{n}, then rk i^{n} = m^{n} − m^{n−1} + ⋯ ± 1. Also, we have rk β^{n+1} = m^{n} − m^{n−1} + ⋯ ± 1 and since dim H^{n}(C_{2}^{•′}) = m^{n}, we get rk p^{n} = m^{n−1} − m^{n−2} + ⋯ ± 1. Thus, dim H^{n}(C^{•}) = rk i^{n} + rk p^{n} = m^{n}, proving the inductive step. □

Since for M as above we have dim M^{G_{X}} = 1, we have proved:

Corollary 4.12. Let M be a right ℚG_{X}-module on which all the elements of X act by the same operator. Then dim H^{n}(X, M) = m^{n} × dim M^{G_{X}}.

Remark 4.13. It is interesting to study the graded algebra H_{inv}^{•}(X, k), where k is a field of characteristic p dividing |G_{X}^{0}|, to which Theorem 4.2 does not apply. One may ask the following questions about this ring:

• Is it finitely generated?

• What is its Poincaré series? Is it a rational function?

5. A relation with group cohomology

In this section, for any rack X, we want to give a group theoretical interpretation of the group H^{2}(X, A) (where A is a trivial G_{X}-module). This group is useful in the theory of pointed Hopf algebras [G].

We start with the following obvious, but useful proposition.

Proposition 5.1. Let A be a trivial G_{X}-module. Then one has a natural isomorphism of complexes J : C^{n}(X, A) → C^{n−1}(X, Fun(X, A)), n ≥ 1, where we consider the action of G_{X} on Fun(X, A) given by (hy)(x) = h(y ▷ x). It is given by (Jf)(x_{1}, …, x_{n−1})(x_{n}) = f(x_{1}, …, x_{n}). In particular, it induces an isomorphism H^{n}(X, A) → H^{n−1}(X, Fun(X, A)). □

Remark 5.2. We note that this proposition becomes false if the action of G_{X} on A is not trivial.

Now we give the main result of this section. Let M be a right G_{X}-module.

Proposition 5.3. H^{1}(X, M) ≃ H^{1}(G_{X}, M).

Propositions 5.1 and 5.3 imply

Corollary 5.4. If A is a trivial G_{X}-module, then H^{2}(X, A) ≃ H^{1}(G_{X}, Fun(X, A)). □

Proof. (of Proposition 5.3) Let C^{•}(G, M) be the standard complex of a group G with coefficient in a right Gmodule M. Let η : C^{1}(G_{X}, M) → C^{1}(X, M) be the homomorphism induced by the natural map X → G_{X}. It is easy to show that this homomorphism maps cocycles to cocycles and coboundaries to coboundaries. Thus, it induces a homomorphism η : H^{1}(G_{X}, M) → H^{1}(X, M). Thus, our job is to show that any f ∈ Z^{1}(X, M) lifts uniquely to a 1-cocycle on G_{X}.

To do this, recall that a map π : G_{X} → M is a 1-cocycle iff the map π̂ : G_{X} → G_{X} ⋉ M given by g → (g, π(g)) is a homomorphism. On the other hand, we have a map ξ_{f} : X → G_{X} ⋉ M given by ξ_{f}(x) = (x, f(x)). So we need to show that ξ_{f} extends to a homomorphism G_{X} → G_{X} ⋉ M. But the group G_{X} is generated by X with relations xy = (x ▷ y)x. Thus, we only need to check that ξ_{f}(x) satisfy the same relations. But it is easy to check that this is exactly the condition that df = 0. We are done. □

Another, more conceptual, proof runs as follows: let N be a right X-module (i.e, a right G_{X}-module) and consider on X × N the following structure:

(x, n) ▷ (y, m) = (x ▷ y, n(1 − (x ▷ y)^{−1}) + mx^{−1}).

It is easy to verify that this is a rack structure on the product; we shall denote it by (X ⋉ N, ▷) (it is actually the same structure as in [AG] for the left X-module N with x · n = nx^{−1}). We have then, with a straightfoward proof,

Lemma 5.5. Let ω : X → N and define ω̂ : X → X ⋉ N by ω̂(x) = (x, ω(x)x^{−1}). Then ω̂ is a rack homomorphism if and only if ω ∈ Z^{1}(X, N). □

Take α : X ⋉ N → G_{X} ⋉ N, α(x, n) = (x, nx). One can check that in the square

X ⟶^{ω̂} X ⋉ N
↓  ↓^{α}
G_{X} ⟶^{π̂} G_{X} ⋉ N

each of ω, π determines uniquely the other in such a way that the diagram is commutative. □

Remark 5.6. Corollary 5.4 holds also when A is nonabelian. In this case H^{2}(X, A) is the quotient of the set Z^{2}(X, A) = {f : X × X → A | f(x ▷ y, x ▷ z)f(x, z) = f(x, y ▷ z)f(y, z)} by the equivalence relation f ∼ f′ if there is a γ : X → A such that f′(x, y) = γ(x ▷ y)f(x, y)γ(y)^{−1}. The proof is the same as in the abelian case.

References

[AG] N. Andruskiewitsch & M. Graña, From racks to pointed Hopf algebras, Adv. Math, to appear. Also in math.QA/0202084.

[C] C. Chevalley, Théorie des groupes de Lie. Tome III. Théorèmes généraux sur les algèbres de Lie, Hermann & Cie, Paris, 1955.

[CES] J.S. Carter, M. Elhamdadi & M. Saito, Twisted Quandle Cohomology Theory and Cocycle Knot Invariants,

       math.GT/0108051.

[CJKS] J.S. Carter, D. Jelsovsky, S. Kamada & M. Saito, Quandle Homology Groups, Their Betti Numbers, and Virtual Knots, J. Pure Applied Algebra 157 (2001), 135–155.

[CS] J.S. Carter & M. Saito, Quandle Homology Theory and Cocycle Knot Invariants math.GT/0112026.

[FRS] R. Fenn, C. Rourke & B. Sanderson, James bundles and applications, preprint available at http://www.maths.warwick.

       ac.uk/~bjs.

[G] M. Graña, On Nichols algebras of low dimension, in “New Trends in Hopf Algebra Theory” (ed. Andruskiewitsch, Ferrer Santos, Schneider); Contemp. Math. 267 (2000), 111–136.

[J] D. Joyce, A Classifying Invariant of knots, The knot Quandle, J. Pure Appl. Alg. 23 (1982), 37–65.

[LN] R.A. Litherland & S. Nelson, The Betti numbers of some finite racks, math.GT/0106165.

[LYZ] Jiang-Hua Lu, Min Yan & Yong-Chang Zhu, On set-theoretical Yang–Baxter equation, Duke Math. J. 104 (2000), 1–18.

[M] T. Mochizuki, Some calculations of cohomology groups of finite Alexander quandles, preprint available at http://

       math01.sci.osaka-cu.ac.jp/~takuro.

[S] A. Soloviev, Non-unitary set-theoretical solutions to the quantum Yang–Baxter equation, Math. Res. Lett. 7 (2000), no. 5-6, 577–596.

Pavel Etingof:

MIT Math. Dept. Of. 2-176

77 Mass. Ave.

02139, Cambridge, MA, USA.

   E-mail address: etingof@math.mit.edu

Matías Graña:

MIT Math. Dept. Of. 2-155

77 Mass. Ave.

02139, Cambridge, MA, USA.

   E-mail address: matiasg@math.mit.edu