Second homology of some quandles

Introduction and main results

A quandle $Q$ is a set equipped with a binary operation $Q \times Q \to Q$ satisfying some axioms. They originally appeared in knot theory [1]. To a quandle one can associate an enveloping group $G(Q)$ which is infinite for $Q$ non-empty. For quandles one has quandle homology and rack homology. Here, we consider quandle homology, more specifically the second quandle homology group. Study of second quandle homology is present in the literature. We cite [6], [7],[8] and [9] where the authors consider quandles of transpositions in $S_n$, Alexander quandles, generalized alexander quandles, the conjugacy quandle of the symmetric group and other examples. More generally, Eisermann ([5]), gives a formula to compute the second quandle homology from the enveloping group. In this text we derive formulas from this formula.

A representation group $R(Q)$ for $Q$ is a type of finite quotients of $G(Q)$ ([10]). We derive from Eisermann’s formula that for $Q$ a connected injective quandle, the second quandle homology is given by:

where $q$ is an element in $R(Q)$ coming from $Q$ and $C_{R(Q)}(q)$ is the centralizer of $q$ in $R(Q)$. This formula is derived in section 1, where we also give some basic definitions (quandles, enveloping groups, representation groups etc.) and derive some facts used in sections 2, 3 and 4.

The set of $k$-cycles of $S_n$ forms a quandle with the operation $x \triangleright y = xyx^{-1}$. It is a subquandle of the conjugacy quandle of $S_n$. Such quandles are connected except when $n = k$, $k + 1$ for $k$ odd. We determine representation groups for these injective quandles when they are connected (with the exception of $n = 6, 7$ for $k$ odd) and compute their second quandle homology using the above formula. The enveloping group for $k$ odd is determined. This is done in § 3, where we treat the cases $k$ even and $k$ odd separately.

In § 4, we show that for a finite injective quandle $Q$, the second quandle homology is given by:

where $n_Q$ is the number of orbits of $Q$ and $x_i$ represent in $R(Q)$ the orbits of $Q$. We use this formula to compute the second quandle homology for the non connected quandles of $k$-cycles still with the exception of $n=6,7$. We also determine the enveloping groups of these quandles. We note that the last formula for connected differ in form from the one given before. We give a direct proof that both formulas agree.

In § 5, we consider $G$ a Schur cover of a centerless group. A conjugacy class of $G$ is a quandle with respect to the binary operation $g\triangleright h=ghg^{-1}$. We show that if the conjugacy class of $x\in G$ ($x$ not in the center) generates $G$, then the conjugacy class form a connected injective quandle, $G$ is a representation group for this quandle and the first formula given here for quandle homology apply. As an application, we determine the second homology of the quandle defined by the conjugacy class of $x\in SL(n,q)$ for $x$ diagonalizable, with some exceptions on $n,q$.

In § 6, we show that if a finite group $G$ has trivial Bogomolov multiplier, then the second quandle homology of the conjugacy quandle $Conj(G)$ of $G$ is given by:

where $Cl(G)$ is the set of conjugacy classes of $G$, $g_C$ is a representative of the class $C$ and $C_G(g_C)$ is the centralizer of $g_C$ in $G$. When $G$ is moreover perfect the sum over $Cl(G)$ correspond to the first Hochschild homology group of the group ring $\mathbb{Z}[G]$. We apply the above formula to compute $H_2(Conj(GL(2,q)))$ for $q\ne2,3$.

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Quandles, second quandle homology and representation groups

We recall some facts about quandles, second quandle homology and representation groups. We also prove that for an injective connected quandle, the second quandle homology can be computed in a representation group by giving a formula for that. A criterion on wether a Schur cover of the inner automorphism group of a quandle is a representation group for the quandle is given.

Quandles

1.1. Quandles. A quandle is a set $Q$ equipped with a binary operation $\triangleright$ such that:

• $x\triangleright x=x$ for $x\in Q$.

• For all $x,y\in Q$ there exist a unique $z\in Q$, such that $x\triangleright z=y$.

• $x\triangleright(y\triangleright z)=(x\triangleright y)\triangleright(x\triangleright z)$, for $x,y,z\in Q$.

A subquandle of $Q$ is a subset stable under $\triangleright$. One can construct a quandle from a group $G$ using the binary operation defined by $x \triangleright y = xyx^{-1}$ for $x,y \in G$. The defined quandle is called the conjugacy quandle and is usually denoted by $Conj(G)$. A conjugacy class of $G$ is a subquandle of $Conj(G)$. A map $f:Q \to Q'$ is a quandle morphism if $f(x \triangleright y)=f(x)\triangleright f(y)$. It is called isomorphism if $f$ is a bijection.

Inner automorphisms

For $Q$ a quandle and $x \in Q$ denote by $L_x$ the bijection given by $L_x(y)=x \triangleright y$ for $y \in Q$. $L_x$ is called the left translation by $x$. The inner automorphism group $Inn(Q)$ of $Q$ is the subgroup of bijections of $Q$ generated by the bijection $L_x$ for $x \in Q$. The orbits of $Q$ are the orbits of the action of $Inn(Q)$. The morphism $x \mapsto L_x$ is a quandle morphism. A quandle is called faithful if the last map is injective. A quandle is called connected if $Q$ consist of one orbit under the natural action of $Inn(Q)$. For $Q=Conj(G)$ the group $Inn(Q)$ correspond to $Inn(G)$ and the orbits are the conjugacy classes of $G$.

Proof. $Q$ is stable by $Inn(G)$ and hence we have a morphism $\phi$ from $Inn(G)$ to the bijections of $Q$. Since $Q$ generates $G$, any element of $G$ is a product of elements of $Q$. It follows that any element of $Inn(G)$ restricted to $Q$ is a product of left translations of $Q$. Hence, $\phi$ is in fact a morphism $Inn(G) \to Inn(Q)$. $\phi$ is clearly onto. Now, let $c_g$ be the conjugacy by $g \in G$. Since $Q$ generates $G$, $c_g$ restricts trivially to $Q$ if and only if $g$ is central in $G$. This proves that $\phi$ is injective. We have proved that $\phi:Inn(G) \to Inn(Q)$ is an isomorphism. $Q$ is a single orbit under $Inn(G)$. This proves that $Q$ is connected. □

Enveloping group

The enveloping group $G(Q)$ of a quandle $Q$ is the group:

The enveloping group is also known as: the associated group, adjoint group or structure group. The group $G(Q)$ is infinite.

A quandle $Q$ is called injective if $\varphi_Q:Q \to G(Q)$ is injective.

Proof. This follows from the universal property of $\varphi_Q$ applied to the inclusion $Q \to Conj(G)$. □

As we have mentioned $Q \to Inn(Q)$, $x \mapsto L_x$ is a quandle morphism. Hence, we have a unique group morphism $\theta_Q:G(Q) \to Inn(Q)$ given by $e_x \mapsto L_x$ for $x \in Q$.