2. Notation and conventions

In this section, we fix notation and conventions. In this article, a group means a multiplicative group if otherwise noted. Let $G$ be a group.

• • The abelianization of $G$ is denoted by $G^{\mathrm{ab}}$ if otherwise noted.

• • The automorphism group $\mathrm{Aut}\,G$ acts on $G$ from the right. For any $\sigma \in \mathrm{Aut}\,G$ and $x \in G$ , the action of $\sigma$ on $x$ is denoted by $x^{\sigma }$ .

• • For a normal subgroup $N$ , we often denote the coset class of an element $g \in G$ by the same $g$ in the quotient group $G/N$ if there is no confusion.

• • For elements $x$ and $y$ of $G$ , the commutator bracket $[x,y]$ of $x$ and $y$ is defined to be $[x,y]\,:\!=xyx^{-1}y^{-1}$ . Then for any $x, y, z \in G$ , we have

  (1) \begin{equation} [xy,z] =[x,[y,z]][y,z][x,z], \hspace {1em} [x,yz] = [x,y][x,z][[z,x],y] \end{equation}
  and
  (2) \begin{equation} [x^{-1},z]=[[x^{-1}, z],x][x,z]^{-1}, \hspace {1em} [x,y^{-1}] = [x,y]^{-1} [y,[y^{-1}, x]]. \end{equation}

• • For subgroups $H$ and $K$ of $G$ , we denote by $[H,K]$ the commutator subgroup of $G$ generated by $[h, k]$ for all $h \in H$ and $k \in K$ .

• • For elements $g_1, \ldots , g_k \in G$ , a left-normed commutator

  \begin{equation*} [[ \cdots [[ g_{1},g_{2}],g_{3}], \cdots ], g_{k}] \in G \end{equation*}
  of weight $k$ is denoted by $[g_{1},g_{2}, \cdots , g_{k}]$ .

• • For any $\mathbf{Z}$ -module $A$ , we denote the $\mathbf{Q}$ -vector space $A \otimes _{\mathbf{Z}} \mathbf{Q}$ by the symbol obtained by attaching a subscript or superscript $\mathbf{Q}$ to $A$ , like $A_{\mathbf{Q}}$ or $A^{\mathbf{Q}}$ . Similarly, for any $\mathbf{Z}$ -linear map $f\,:\, A \rightarrow B$ , the induced $\mathbf{Q}$ -linear map $A_{\mathbf{Q}} \rightarrow B_{\mathbf{Q}}$ is denoted by $f_{\mathbf{Q}}$ or $f^{\mathbf{Q}}$ .

3. Free Lie algebras and their derivation algebras

In this section, we review free Lie algebras and their derivation algebras. Furthermore, we consider a certain Lie subalgebra of the derivation algebra of the free Lie algebra.

For $n \geq 1$ , let $F_n$ be the free group of rank $n$ with basis $x_1, \ldots , x_n$ . Let $\Gamma _n(1) \supset \Gamma _n(2) \supset \cdots$ be the lower central series of $F_n$ defined by

\begin{equation*} \Gamma _n(1)\,:\!= F_n, \hspace {1em} \Gamma _n(k) \,:\!= [\Gamma _n(k-1),F_n], \hspace {1em} k \geq 2. \end{equation*}

We denote by $\mathcal{L}_n(k) \,:\!= \Gamma _n(k)/\Gamma _n(k+1)$ the $k$ -th graded quotient of the lower central series of $F_n$ , and by $\mathcal{L}_n \,:\!= {\bigoplus }_{k \geq 1} \mathcal{L}_n(k)$ the associated graded sum. For each $k \geq 1$ , we consider $\mathcal{L}_n(k)$ as an additive group. For example, for any $x, y, z \in F_n$ we have

\begin{equation*} [xy,z]=[x,z]+[y,z], \hspace {1em} [x^{-1},y]=-[x,y] \end{equation*}

in $\mathcal{L}_n(1)$ from (1) and (2). The graded sum $\mathcal{L}_n$ has the Lie algebra structure with the Lie bracket induced from the commutator bracket of $F_n$ . It is known that the Lie algebra $\mathcal{L}_n$ is isomorphic to the free Lie algebra generated by $\mathcal{L}_n(1)$ by a classical work of Magnus. (For details, see [Reference Magnus, Karras and Solitar24] and [Reference Reutenauer28] for example.) Thus, each of $\mathcal{L}_n(k)$ is a free abelian group of finite rank. Witt [Reference Witt39] gave a formula for the rank of $\mathcal{L}_n(k)$ as

(3) \begin{equation} r_n(k) \,:\!= \frac {1}{k} \sum _{d | k} \mu (d) n^{\frac {k}{d}} \end{equation}

where $\mu$ is the Möbius function, and $d$ runs over all positive divisors of $k$ . Hall [Reference Hall18] constructed an explicit basis of $\mathcal{L}_n(k)$ with basic commutators of weight $k$ . For example, basic commutators of weight less than five are listed below.

Here $(i,j) \gt (l,m)$ means the lexicographic order on $\mathbf{N} \times \mathbf{N}$ . (For details, see [Reference Hall19] and [Reference Reutenauer28] e.g.,.)

For simplicity, we denote by $H$ the abelianization $F_n^{\mathrm{ab}}=H_1(F_n,\mathbf{Z})$ of $F_n$ . The basis $x_1, \ldots , x_n$ of $F_n$ induces a basis of $H$ as a free abelian group. We also denote it by the same letters $x_1, \ldots , x_n$ by abuse of language. By fixing this basis of $H$ , we identify the automorphism group $\mathrm{Aut}\,H$ with the general linear group $\mathrm{GL}(n,\mathbf{Z})$ . Then $\mathrm{GL}(n,\mathbf{Z})$ naturally acts on each of $\mathcal{L}_n(k)$ . For example, as a $\mathrm{GL}(n,\mathbf{Z})$ -module, we have $\mathcal{L}_n(1) = H$ and $\mathcal{L}_n(2) = \wedge ^2 H$ . Furthermore, $\mathrm{GL}(n,\mathbf{Q})$ naturally acts on each of $\mathcal{L}_n^{\mathbf{Q}}(k)$ . The irreducible decompositions of $\mathcal{L}_n^{\mathbf{Q}}(k)$ for $1 \leq k \leq 4$ are given as follows.

Here $[\lambda ]$ means the irreducible polynomial representation of $\mathrm{GL}(n,\mathbf{Q})$ associated to the Young tableau $\lambda$ .

The universal enveloping algebra of $\mathcal{L}_n$ is the tensor algebra

\begin{equation*} T(H)\,:\!= \mathbf{Z} \oplus H \oplus H^{\otimes 2} \oplus \cdots \end{equation*}

of $H$ over $\mathbf{Z}$ . From Poincaré–Birkhoff–Witt’s theorem, the natural map $\iota\,:\,\mathcal{L}_n \rightarrow T(H)$ is injective. We denote by $\iota _k\,:\,\mathcal{L}_n(k) \rightarrow H^{\otimes k}$ the homomorphism of degree $k$ part of $\iota$ and consider $\mathcal{L}_n(k)$ as a $\mathrm{GL}(n,\mathbf{Z})$ -submodule $H^{\otimes k}$ through $\iota _k$ . Similarly, we consider $\mathcal{L}_n^{\mathbf{Q}}(k)$ as a $\mathrm{GL}(n,\mathbf{Q})$ -submodule $H_{\mathbf{Q}}^{\otimes k}$ through $\iota _k^{\mathbf{Q}}$ .

Next, we consider the derivation algebra of the free Lie algebra. We denote the Lie algebra of derivations of $\mathcal{L}_n$ by

\begin{equation*} \mathrm{Der}(\mathcal{L}_n) \,:\!= \{ f\,:\,\mathcal{L}_n \xrightarrow {\mathbf{Z}-\mathrm{linear}} \mathcal{L}_n \,|\, f([x,y]) = [f(x),y]+ [x,f(y)], \,\,\, x, y \in \mathcal{L}_n \}. \end{equation*}

The Lie algebra $\mathrm{Der}(\mathcal{L}_n)$ is a graded Lie algebra. For $k \geq 0$ , the degree $k$ part of $\mathrm{Der}(\mathcal{L}_n)$ is defined to be

\begin{equation*} \mathrm{Der}(\mathcal{L}_n)(k) \,:\!= \{ f \in \mathrm{Der}(\mathcal{L}_n) \,|\, f(x) \in \mathcal{L}_n(k+1), \,\,\, x \in H \}. \end{equation*}

For $k \geq 1$ , by considering the universality of the free Lie algebra, we indentify $\mathrm{Der}(\mathcal{L}_n)(k)$ with $\mathrm{Hom}_{\mathbf{Z}}(H,\mathcal{L}_n(k+1)) = H^* {\otimes }_{\mathbf{Z}} \mathcal{L}_n(k+1)$ as a $\mathrm{GL}(n,\mathbf{Z})$ -module. Let $x_1^*, \ldots , x_n^*$ be the dual basis of $x_1, \ldots , x_n \in H$ . Then $\mathrm{Der}(\mathcal{L}_n)(k)$ is generated by $x_i^* \otimes [x_{j_1}, x_{j_2}, \ldots , x_{j_{k+1}}]$ for all $1 \leq i, j_1, \ldots , j_{k+1} \leq n$ as a $\mathbf{Z}$ -module. Let $\mathrm{Der}^+(\mathcal{L}_n)$ be the graded Lie subalgebra of $\mathrm{Der}(\mathcal{L}_n)$ with positive degree. Similarly, we define the graded Lie algebra $\mathrm{Der}^+(\mathcal{L}_{n}^{\mathbf{Q}})$ . Then, we have $\mathrm{Der}^+(\mathcal{L}_{n}^{\mathbf{Q}}) = \mathrm{Der}^+(\mathcal{L}_n) \otimes _{\mathbf{Z}} \mathbf{Q}$ , and $\mathrm{Der}(\mathcal{L}_n^{\mathbf{Q}})(k)=H_{\mathbf{Q}}^* {\otimes }_{\mathbf{Q}} \mathcal{L}_n^{\mathbf{Q}}(k+1)$ for any $k \geq 1$ .

Definition 3.1. For any $k \geq 1$ , let $\mathfrak{p}_n(k)$ be the $\mathbf{Z}$ -submodule of $\mathrm{Der}(\mathcal{L}_n)(k)$ generated by elements

\begin{equation*} x_i^* \otimes [x_{j_1}, \ldots , x_{j_k}, x_i] \end{equation*}

for all $1 \leq i, j_1, \ldots , j_k \leq n$ . Let $\mathfrak{p}_n$ be the graded sum $\bigoplus _{k \geq 1} \mathfrak{p}_n(k)$ .

Alekseev and Torossian [Reference Alekseev and Torossian1] showed that $\mathfrak{p}_n$ is a Lie subalgebra of $\mathrm{Der}^+(\mathcal{L}_n)$ . (See also Satoh [Reference Satoh36].) We remark that $x_i^* \otimes [x_j, x_i]$ for any $1 \leq i \neq j \leq n$ form a basis of $\mathfrak{p}_n(1)$ , and $\mathfrak{p}_n(k)$ is isomorphic to $H \otimes _{\mathbf{Z}} \mathcal{L}_n(k)$ as an $\mathfrak{S}_n$ -module. Hence, we have $\mathrm{rank}_{\mathbf{Z}}(\mathfrak{p}_n(k)) = nr_n(k)$ for any $k \geq 2$ .

4. Johnson homomorphisms

Here we recall the Andreadakis–Johnson filtrations and the Johnson homomorphisms of the automorphism groups of free groups. For each $k \geq 1$ , the action of $\mathrm{Aut}\,F_n$ on the nilpotent quotient group $F_n/\Gamma _n(k+1)$ of $F_n$ induces the homomorphism

\begin{equation*} \mathrm{Aut}\,F_n \rightarrow \mathrm{Aut}(F_n/\Gamma _n(k+1)). \end{equation*}

We denote its kernel by $\mathcal{A}_n(k)$ . Then the groups $\mathcal{A}_n(k)$ define the descending filtration

\begin{equation*} \mathcal{A}_n(1) \supset \mathcal{A}_n(2) \supset \mathcal{A}_n(3) \supset \cdots \end{equation*}

of $\mathrm{Aut}\,F_n$ . This filtration is called the Andreadakis–Johnson filtration of $\mathrm{Aut}\,F_n$ , and the first term is called the IA-automorphism group of $F_n$ , denoted by $\mathrm{IA}_n$ . Historically, the Andreadakis–Johnson filtration was originally introduced by Andreadakis [Reference Andreadakis2] in the 1960s. He showed that

For each $k \geq 1$ , the group $\mathrm{Aut}\,F_n$ acts on $\mathcal{A}_n(k)$ by conjugation, and this action naturally induces the action of $\mathrm{GL}(n,\mathbf{Z})=\mathrm{Aut}\,F_n/\mathrm{IA}_n$ on the graded quotients $\mathrm{gr}^k (\mathcal{A}_n) \,:\!= \mathcal{A}_n(k)/\mathcal{A}_n(k+1)$ by Part (2) of Theorem 4.1.

In order to study the $\mathrm{GL}(n,\mathbf{Z})$ -module structure of ${\mathrm{gr}}^k (\mathcal{A}_n)$ , we consider the Johnson homomorphisms. For each $k \geq 1$ , define the homomorphism $\tilde {\tau }_k\,:\,\mathcal{A}_n(k) \rightarrow \mathrm{Hom}_{\mathbf{Z}}(H, {\mathcal{L}}_n(k+1))$ by

\begin{equation*} \sigma \hspace {0.3em} \mapsto \hspace {0.3em} (x \mapsto x^{-1} x^{\sigma }), \hspace {1em} x \in H. \end{equation*}

The kernel of $\tilde {\tau }_k$ is $\mathcal{A}_n(k+1)$ . Hence, it induces the injective homomorphism

\begin{equation*} \tau _k\,:\,\mathrm{gr}^k (\mathcal{A}_n) \hookrightarrow \mathrm{Hom}_{\mathbf{Z}}(H, \mathcal{L}_n(k+1)) = H^* \otimes _{\mathbf{Z}} \mathcal{L}_n(k+1). \end{equation*}

The homomorphisms $\tilde {\tau }_k$ and ${\tau }_{k}$ are called the $k$ -th Johnson homomorphisms of $\mathrm{Aut}\,F_n$ . Each of ${\tau }_{k}$ is $\mathrm{GL}(n,\mathbf{Z})$ -equivariant. The graded sum $\mathrm{gr}(\mathcal{A}_n) \,:\!= \bigoplus _{k \geq 1} \mathrm{gr}^k (\mathcal{A}_n)$ has the Lie algebra structure with the Lie bracket induced from the commutator bracket on $\mathrm{IA}_n$ . Furthermore, the graded sum $\tau \,:\!= \bigoplus _{k \geq 1} \tau _k\,:\,\mathrm{gr}(\mathcal{A}_n) \rightarrow \mathrm{Der}^+(\mathcal{L}_n)$ is a Lie algebra homomorphism.

Here we remark on the cokernels of the Johnson homomorphisms. For $k \geq 1$ , let ${\varphi }^{k}\,:\,H^* {\otimes }_{\mathbf{Z}} H^{\otimes (k+1)} \rightarrow H^{\otimes k}$ be the contraction map defined by

\begin{equation*} x_i^* \otimes x_{j_1} \otimes \cdots \otimes x_{j_{k+1}} \mapsto x_i^*(x_{j_1}) \, \cdot x_{j_2} \otimes \cdots \otimes \cdots \otimes x_{j_{k+1}}. \end{equation*}

For the natural embedding ${\iota }_{k+1}\,:\,\mathcal{L}_n(k+1) \rightarrow H^{\otimes (k+1)}$ , we obtain the $\mathrm{GL}(n,\mathbf{Z})$ -equivariant homomorphism

\begin{equation*} \Phi ^k = {\varphi }^{k} \circ ({id}_{H^*} \otimes {\iota }_{k+1})\,:\,H^* {\otimes }_{\mathbf{Z}} \mathcal{L}_n(k+1) \rightarrow H^{\otimes k}. \end{equation*}

We also call $\Phi ^k$ the contraction map of degree $k$ . From Lemma 3.1 in [Reference Satoh30], we have the following.

Lemma 4.2. For any $1 \leq i, i_1, \ldots , i_{k} \leq n$ such that $i_1 \neq i$ , not required $i_1, \ldots , i_k$ are distinct, we have

\begin{align*} \begin{split} \Phi ^k(x_i^* \otimes & [x_i, x_{i_1}, \ldots , x_{i_{k}}]) \\ & = x_{i_1} \otimes \cdots \otimes x_{i_{k}} - \sum _{l=2}^k \delta _{i, i_l} [x_i, x_{i_1}, \ldots , x_{i_{l-1}}] \otimes x_{i_{l+1}} \otimes \cdots \otimes x_{i_k}. \end{split} \end{align*}

Let $\mathcal{C}_n(k)$ be the quotient module of $H^{\otimes k}$ by the action of the cyclic group $C_k$ of order $k$ on the components. Namely,

\begin{equation*} \mathcal{C}_n(k) \,:\!= H^{\otimes k} \big {/} \langle a_1 \otimes a_2 \otimes \cdots \otimes a_k - a_2 \otimes a_3 \otimes \cdots \otimes a_k \otimes a_1 \,|\, a_i \in H \rangle . \end{equation*}

Let $\varpi ^k\,:\,H^{\otimes k} \rightarrow \mathcal{C}_n(k)$ be the natural projection. For any $1 \leq j_1, \ldots , j_k \leq n$ , we write the image of $x_{j_1} \otimes \cdots \otimes x_{j_k} \in H^{\otimes k}$ by the map $\varpi ^k$ as $x_{j_1} x_{j_2} \cdots x_{j_k} \in \mathcal{C}_n(k)$ . Thus, we have $x_{j_1} x_{j_2} \cdots x_{j_k}=x_{j_2} \cdots x_{j_k} x_{j_1}$ in $\mathcal{C}_n(k)$ . By [Reference Reutenauer28], it is known that $\mathcal{C}_n(k)$ is a free abelian group, and the necklaces among $x_1, \ldots , x_n$ of length $k$ form a basis of $\mathcal{C}_n(k)$ . Hence, we have

\begin{equation*} \mathrm{rank}_{\mathbf{Z}}(\mathcal{C}_n(k)) = \frac {1}{k}\sum _{d|k}\varphi (d)n^{\frac {k}{d}} \end{equation*}

where $\varphi\,:\,\mathbf{N} \rightarrow \mathbf{N}$ is the Euler function, and $d$ runs over all positive divisors of $k$ .

Set $\mathrm{Tr}_k \,:\!= \varpi ^k \circ \Phi ^k\,:\,H^* {\otimes }_{\mathbf{Z}} \mathcal{L}_n(k+1) \rightarrow \mathcal{C}_n(k)$ . We call $\mathrm{Tr}^k$ the trace map of degree $k$ . By combining the results by Satoh [Reference Satoh30, Reference Satoh34] and Darne [Reference Darné12], we see that for any $k \geq 2$ and $n \geq k+2$ , the sequence

(4) \begin{equation} 0 \rightarrow \mathrm{gr}^k (\mathcal{A}_n) \xrightarrow {\tau _k} H^* {\otimes }_{\mathbf{Z}} \mathcal{L}_n(k+1) \xrightarrow {\mathrm{Tr}_k} \mathcal{C}_n(k) \rightarrow 0 \end{equation}

is a $\mathrm{GL}(n,\mathbf{Z})$ -equivariant exact sequence. Furthermore, Enomoto-Satoh [Reference Enomoto and Satoh14] gave a combinatorial description of the $\mathrm{GL}(n,\mathbf{Q})$ -irreducible decomposition of the stable rational cokernel $\mathcal{C}_n^{\mathbf{Q}}(k)$ of the Johnson homomorphism $\tau _{k,\mathbf{Q}}$ . For $1 \leq k \leq 5$ , the decompositions are given as follows.

Here we remark that for any $k \geq 2$ , the irreducible component $[k]$ , which is the symmetric product of $H_{\mathbf{Q}}$ of degree $k$ , appears in $\mathcal{C}_n^{\mathbf{Q}}(k)$ with multiplicity one. This result was originally obtained by Morita [Reference Morita27]. Enomoto-Satoh [Reference Enomoto and Satoh14] showed that $[1^k]$ for any odd $k \geq 3$ and $n \geq k+2$ and $[2,1^{k-2}]$ for even $k \geq 4$ and $n \geq k+2$ appear in $\mathcal{C}_n^{\mathbf{Q}}(k)$ with multiplicity one.

Here we remark the Andreadakis problem on $\mathrm{Aut}\,F_n$ . Let $\mathrm{IA}_n = \mathcal{A}_n'(1) \supset \mathcal{A}_n'(2) \supset \cdots$ be the lower central series of $\mathrm{IA}_n$ , Since the Andreadakis–Johnson filtration is a central filtration of $\mathrm{IA}_n$ , we see $\mathcal{A}_n'(k) \subset \mathcal{A}_n(k)$ for any $k \geq 1$ .

Andreadakis showed that the above conjecture is true for $n=2$ and any $k \geq 2$ and $n=3$ and $k \leq 3$ . It is known that $\mathcal{A}_n'(2) = \mathcal{A}_n(2)$ for any $n \geq 2$ due to Bachmuth [Reference Bachmuth4]. This result is also obtained from the fact that the first Johnson homomorphism is the abelianization of $\mathrm{IA}_n$ by independent works of Cohen-Pakianathan [Reference Cohen and Pakianathan9, Reference Cohen and Pakianathan10], Farb [Reference Farb15], and Kawazumi [Reference Kawazumi23]. Satoh [Reference Satoh37] showed that $\mathcal{A}_n'(3) = \mathcal{A}_n(3)$ for $n \geq 3$ . Bartholdi [Reference Bartholdi5, Reference Bartholdi6] showed that this conjecture is not true in general. In particular, he showed that $\mathcal{A}_3(4)/\mathcal{A}_3'(4) \cong (\mathbf{Z}/2\mathbf{Z})^{\oplus 14} \oplus (\mathbf{Z}/3\mathbf{Z})^{\oplus 9}$ and $\mathcal{A}_3(5)/\mathcal{A}_3'(5) \otimes _{\mathbf{Z}} \mathbf{Q} \cong \mathbf{Q}^{\oplus 3}$ . In the stable range, the Andreadakis conjecture is still open problem. We remark that Darné [Reference Darné12] gave affirmative answers for the Andreadakis problems restricted to certain subgroups of $\mathrm{Aut}\,F_n$ .

5. The basis-conjugating automorphism groups

Here we recall the basis-conjugating automorphisms of $F_n$ . Let $\mathrm{P}\Sigma _n$ be the subgroup of $\mathrm{Aut}\,F_n$ consisting of all basis-conjugating automorphisms of $F_n$ . The group $\mathrm{P}\Sigma _n$ is a subgroup of $\mathrm{IA}_n$ . For any $1 \leq i \neq j \leq n$ , let $K_{ij}$ be the basis-conjugating automorphism of $F_n$ defined by

\begin{align*} x_t \mapsto \begin{cases} {x_j}^{-1} x_i x_j & t=i, \\ x_t & t \neq i. \end{cases} \end{align*}

McCool gave a finite presentation for $\mathrm{P}\Sigma _n$ as follows.

We construct a group extension of $\mathrm{P}\Sigma _n$ by $\mathfrak{S}_n$ as follows. Let $B_n$ be Artin’s braid group of $n$ -strands. Artin [Reference Artin3] obtained the first finite presentation

\begin{align*} \begin{split} B_n & = \langle \,\, \sigma _1, \ldots , \sigma _{n-1} \,|\, \sigma _{i+1} \sigma _i \sigma _{i+1} \sigma _i^{-1} \sigma _{i+1}^{-1} \sigma _i^{-1} \,\,\, (1 \leq i \leq n-2), \\ & \hspace {7.8em} [\sigma _i, \sigma _j] \,\,\, (1 \leq i\lt j \leq n-1, \,\, |i-j| \geq 2) \,\,\rangle . \end{split} \end{align*}

Moreover, Artin constructed a faithful representation of $B_n$ into $\mathrm{Aut}\,F_n$ . For any $1 \leq i \leq n-1$ , the image of $\sigma _i$ by this representation is given by

\begin{align*} x_t \mapsto \begin{cases} x_i^{-1} x_{i+1} x_i & t=i, \\ x_i & t=i+1, \\ x_t & t \neq i, i+1. \end{cases} \end{align*}

We also denote this automorphism of $F_n$ by $\sigma _i$ and consider $B_n$ as a subgroup of $\mathrm{Aut}\,F_n$ . The subgroup $B_n \cap \mathrm{IA}_n$ of $B_n$ is the pure braid group of $n$ -strands and is denoted by $P_n$ . (For details, see [Reference Birman7] for example.)

According to [Reference Fenn, Rimányi and Rourke16], we call an element $\sigma \in \mathrm{Aut}\,F_n$ a braid-permutation automorphism of $F_n$ if there exists some $\mu \in \mathfrak{S}_n$ and some $a_i \in F_n$ for any $1 \leq i \leq n$ such that

\begin{equation*} x_i^{\sigma }=a_i^{-1} x_{\mu (i)} a_i \hspace {1em} (1 \leq i \leq n). \end{equation*}

Let $\mathrm{BP}_n$ be the subgroup of $\mathrm{Aut}\,F_n$ consisting of all braid-permutation automorphisms of $F_n$ . The group $\mathrm{BP}_n$ is called the braid-permutation automorphism group of $F_n$ . The subgroup $\mathrm{BP}_n \cap \mathrm{IA}_n$ of $\mathrm{BP}_n$ is the basis-conjugating automorphism group $\mathrm{P}\Sigma _n$ . For any $1 \leq i \leq n-1$ , let $s_i$ be the automorphism of $F_n$ which permutates $x_i$ and $x_{i+1}$ , and fixes the other generators of the basis of $F_n$ . Clearly, the subgroup of $\mathrm{Aut}\,F_n$ generated by $s_i$ for all $1 \leq i \leq n-1$ is isomorphic to $\mathfrak{S}_n$ . Fenn–Rimányi–Rourke gave a finite presentation for $\mathrm{BP}_n$ as follows.

Then we have the following commutative diagram whose three rows are group extensions, and whose all vertical maps are injective.

Here $\rho\,:\,\mathrm{Aut}\,F_n \rightarrow \mathrm{GL}(n,\mathbf{Z})$ is the homomorphism induced from the abelianization of $F_n$ .

For any $k \geq 1$ , set $\mathcal{P}_n(k)\,:\!= \mathrm{P}\Sigma _n \cap \mathcal{A}_n(k)$ . The filtration $\mathrm{P}\Sigma _n = \mathcal{P}_n(1) \supset \mathcal{P}_n(2) \supset \cdots$ is a descending central filtration of $\mathrm{P}\Sigma _n$ . Set $\mathrm{gr}^k(\mathcal{P}_n)\,:\!= \mathcal{P}_n(k)/\mathcal{P}_n(k+1)$ . In Proposition 3.5 in [Reference Satoh36], Satoh showed that for any $n \geq 2$ and $k \geq 1$ , the image of the restriction of $\tilde {\tau }_k\,:\,\mathcal{A}_n(k) \rightarrow H^* \otimes _{\mathbf{Z}} \mathcal{L}_n(k+1)$ to $\mathcal{P}_n(k)$ is contained in $\mathfrak{p}_n(k)$ .

Definition 5.3. We call the induced injective homomorphism

\begin{equation*} \tau _k^P\,:\,\mathrm{gr}^k(\mathcal{P}_n) \rightarrow \mathfrak{p}_n(k) \end{equation*}

from $\tilde {\tau }_k|_{\mathcal{P}_n(k)}$ the $k$ -th Johnson homomorphism of $\mathrm{P}\Sigma _n$ .

It is known that for each $k \geq 1$ , the Johnson homomorphism $\tau _k^P$ is $\mathrm{BP}_n/\mathrm{P}\Sigma _n = \mathfrak{S}_n$ -equvariant. and that $\tau _1^P$ induces the abelianization of $\mathrm{P}\Sigma _n$ . The graded sum $\mathrm{gr}(\mathcal{P}_n) \,:\!= \oplus _{k \geq 1} \mathrm{gr}^k(\mathcal{P}_n)$ is considered as the Lie subalgebra of $\mathrm{gr}(\mathcal{A}_n)$ , and the graded sum $\tau ^P \,:\!= \oplus _{k \geq 1} \tau _k^P\,:\,\mathrm{gr}(\mathcal{P}_n) \rightarrow \mathfrak{p}_n$ is a Lie algebra homomorphism.

Let $\mathrm{P}\Sigma _n=\mathrm{P}\Sigma _n{(1)} \supset \mathrm{P}\Sigma _n{(2)} \supset \cdots$ be the lower central series of $\mathrm{P}\Sigma _n$ . We have $\mathrm{P}\Sigma _n{(k)} \subset \mathcal{P}_n(k)$ for any $k \geq 1$ . Satoh [Reference Satoh35] showed that $\mathrm{P}\Sigma _n{(k)} = \mathcal{P}_n(k)$ for $k \leq 3$ . In general, it is not known whether $\mathrm{P}\Sigma _n{(k)}$ coincides with $\mathcal{P}_n(k)$ or not for any $n \geq 3$ and $k \geq 4$ . Let $\mathrm{P}\Sigma _n^+$ be the subgroup of $\mathrm{P}\Sigma _n$ generated by $K_{ij}$ for all $1 \leq j \lt i \leq n$ . We remark that Darné [Reference Darné13] proved that $\mathrm{P}\Sigma _n^+ \cap \mathcal{A}_n(k)$ coincides with the $k$ -the subgroup of the lower central series of $\mathrm{P}\Sigma _n^+$ for any $k \geq 1$ .

6. Trace maps

Here we consider the trace maps and its variants to detect the cokernel of $\tau _k^P$ for any $k \geq 1$ . Let $\mathrm{Tr}_k^P\,:\,\mathfrak{p}_n(k) \rightarrow \mathcal{C}_n(k)$ be the restriction of $\mathrm{Tr}_k$ to $\mathfrak{p}_n(k)$ . From [Reference Satoh30] and Darné [Reference Darné12], we have $\mathrm{Im}(\tau _k) \subset \mathrm{Ker}(\mathrm{Tr}_k)$ for any $n \geq 2$ and $k \geq 2$ . From this we see $\mathrm{Im}(\tau _k^P) \subset \mathrm{Ker}(\mathrm{Tr}_k^P)$ . Hence, in order to study $\mathrm{Coker}(\tau _k^P)$ , it is interesting problem to determine the image of the trace map $\mathrm{Tr}_k^P$ .

It is easy to see that for any $k \geq 2$ , $\langle x_1^k, \ldots , x_n^k \rangle \cap \mathrm{Im}(\mathrm{Tr}_k^P) = \{ 0 \}$ . (See also Proposition 3.7 in [Reference Satoh36].) Set $\overline {\mathcal{C}}_n(k) \,:\!= \mathcal{C}_n(k)/\langle x_1^k, \ldots , x_n^k \rangle$ . By compositing $\mathrm{Tr}_k^P$ and the natural projection $\mathcal{C}_n(k) \rightarrow \overline {\mathcal{C}}_n(k)$ , we obtain the map $\overline {\mathrm{Tr}}_k^P\,:\,\mathfrak{p}_n(k) \rightarrow \overline {\mathcal{C}}_n(k)$ . As we will mention in Section 7, the map $\overline {\mathrm{Tr}}_k^P$ is surjective for $k \leq 3$ . Hence, we can describe the rational Johnson cokernel by using $\overline {\mathrm{Tr}}_k^P$ in these cases. In general, however, $\overline {\mathrm{Tr}}_k^P$ is not surjective, and it seems a difficult problem to determine its image. Here we consider some reduced versions of $\overline {\mathrm{Tr}}_k^P$ . In [Reference Satoh36], one of such reduced versions of $\mathrm{Tr}_k^P$ was studied. More precisely, consider the quotient $\mathfrak{S}_n$ -module $\overline {S}^k(H)\,:\!=S^k(H)/\langle x_1^k, \ldots , x_n^k \rangle$ and the natural projection $\lambda _k\,:\,\mathcal{C}_n(k) \rightarrow \overline {S}^k(H)$ . In Proposition 3.7 in [Reference Satoh36], Satoh showed that the map $\lambda _k \circ \mathrm{Tr}_k^P$ is surjective for $n \geq 2$ and $k \geq 2$ .

Here we introduce the other trace maps that detect different components other than $\overline {S}^k(H_{\mathbf{Q}})$ in $\mathrm{Coker}(\tau _{k,\mathbf{Q}}^P)$ .

For example, basis of $N(k)$ for $2 \leq k \leq 4$ are given as follows.

Set $\widetilde {\mathcal{C}}_n(k) \,:\!= \mathcal{C}_n(k)/N(k)$ . Let $\mu _k\,:\,\mathcal{C}_n(k) \rightarrow \widetilde {\mathcal{C}}_n(k)$ be the natural projection and set $\widetilde {\phantom{\,}\mathrm{Tr}\phantom{\,}}_{\!k}^{\!P}\,:\!= \mu _k \circ \mathrm{Tr}_k^P\,:\,\mathfrak{p}_n(k) \rightarrow \widetilde {\mathcal{C}}_n(k)$ . Then we have

\begin{equation*} \mathrm{Im}(\tau _k^P) \subset \mathrm{Ker}(\mathrm{Tr}_k^P) = \mathrm{Ker}(\overline {\mathrm{Tr}}_k^P) \subset \mathrm{Ker}(\widetilde {\phantom{\,}\mathrm{Tr}\phantom{\,}}_{\!k}^{\!P}) \subset \mathfrak{p}_n(k). \end{equation*}

Proof. As a $\mathbf{Z}$ -module, $\widetilde {\mathcal{C}}_n(k)$ is generated by elements $x_{j_1} x_{j_2} \cdots x_{j_k} \in \widetilde {\mathcal{C}}_n(k)$ such that there exists some $1 \leq l \leq k$ such that $j_l \neq j_m$ for all $m \neq l$ . By using Lemma 4.2, we have

\begin{align*} \begin{split} \widetilde {\phantom{\,}\mathrm{Tr}\phantom{\,}}_{\!k}^{\!P} & (x_{j_l}^* \otimes [x_{j_l}, x_{j_{l+1}}, \ldots , x_{j_k}, x_{j_1}, \ldots , x_{j_{l-1}}, x_{j_l}]) \\ & = x_{j_{l+1}} \cdots x_{j_k} x_{j_1} \cdots x_{j_l} - \mu _k \circ \varpi _k([x_{j_l}, x_{j_{l+1}}, \ldots , x_{j_k}, x_{j_1}, \ldots , x_{j_{l-1}}]). \end{split} \end{align*}

Since $\mathcal{L}_n(k) \subset \mathrm{Ker}(\varpi _k)$ , we see that

\begin{equation*} x_{j_1} x_{j_2} \cdots x_{j_k} = x_{j_{l+1}} \cdots x_{j_k} x_{j_1} \cdots x_{j_l} \in \mathrm{Im}(\widetilde {\phantom{\,}\mathrm{Tr}\phantom{\,}}_{\!k}^{\!P}). \end{equation*}

Thus, $\widetilde {\phantom{\,}\mathrm{Tr}\phantom{\,}}_{\!k}^{\!P}$ is surjective.

There is the natural surjective homomorphism $\widetilde {\mathcal{C}}_n(k) \rightarrow \overline {S}^k(H)$ . Hence, in the case where $k=2$ and $3$ , Proposition 3.7 in [Reference Satoh36] is a corollary of Theorem 6.2.

7. The Johnson cokernels in the case where $k \leq 4$

In this section, we determine the images and the cokernels of $\tau _k^P$ for $1 \leq k \leq 4$ by using the trace maps considered in Section 6. First, we recall some results in [Reference Satoh35].

For $k=1$ , the first Johnson homomorphism $\tau _1^P\,:\,\mathrm{gr}^1(\mathcal{P}_n) \rightarrow \mathfrak{p}_n(1)$ is an isomorphism, and $\mathrm{Im}(\tau _1^P)$ is a free abelian group of rank $n(n-1)$ with a basis $\{x_i^* \otimes [x_j, x_i] \,|\, 1 \leq i \neq j \leq n \}$ . From Lemma 2.3 in [Reference Satoh35], we see that $\mathrm{Im}(\tau _2^P)$ is a free abelian group of rank $\frac {n(n-1)^2}{2}$ with a basis

(5) \begin{equation} \begin{split} \{ x_i^* & \otimes [x_{j_1}, x_{j_2}, x_i] \,|\, 1 \leq j_2 \lt j_1 \leq n, \hspace {0.5em} i \neq j_1, j_2 \} \\ & \cup \{ x_i^* \otimes [x_i, x_j, x_i] - x_j^* \otimes [x_j, x_i, x_j] \,|\, 1 \leq i \neq j \leq n \} \end{split} \end{equation}

for $n \geq 3$ .

Proof. (1) By definition, we see $\overline {\mathrm{Tr}}_2^P=\widetilde {\mathrm{Tr}}_2^P$ . For any generator $x_i x_j$ ( $i \neq j$ ) of $\overline {\mathcal{C}}_n(2)$ , we have

\begin{equation*} \overline {\mathrm{Tr}}_2^P(x_i^* \otimes [x_i, x_j, x_i]) = x_jx_i=x_ix_j. \end{equation*}

Hence, $\overline {\mathrm{Tr}}_2^P$ is surjective.

(2) It suffices to show $\mathrm{Im}(\tau _2^P) \supset \mathrm{Ker}(\overline {\mathrm{Tr}}_2^P)$ . Take any

\begin{equation*} x \,:\!= \sum _{\substack {1 \leq i \leq n\\[1pt] 1 \leq j_2 \lt j_1 \leq n}} a_{j_1, j_2}^i x_i^* \otimes [x_{j_1}, x_{j_2}, x_i] \in \mathrm{Ker}(\overline {\mathrm{Tr}}_2^P). \end{equation*}

Since $\mathrm{Im}(\tau _2^P) \subset \mathrm{Ker}(\overline {\mathrm{Tr}}_2^P)$ , we show that we can add elements in the Johnson image to $x$ until we get zero. Namely, by adding elements in (5) to $x$ if necessary, we may assume

\begin{equation*} x=\sum _{1 \leq i \lt j \leq n} a_{i, j}^i x_i^* \otimes [x_i, x_j, x_i]. \end{equation*}

Then we obtain

\begin{equation*} 0=\overline {\mathrm{Tr}}_2^P(x)=\sum _{1 \leq i \lt j \leq n} a_{i, j}^i x_ix_j, \end{equation*}

and hence $a_{i, j}^i=0$ for any $i\lt j$ . Thus, $x=0$ .

Corollary 7.2. For $n \geq 3$ ,

\begin{equation*} 0 \rightarrow \mathrm{gr}^2(\mathcal{P}_n) \xrightarrow {\tau _2^P} \mathfrak{p}_n(2) \xrightarrow {\overline {\mathrm{Tr}}_2^P} \overline {\mathcal{C}}_n(2) \rightarrow 0 \end{equation*}

is $\mathfrak{S}_n$ -equivariant exact.

Next, we consider the case where $k=3$ . We prepare the following lemma.

Proof. We prove this lemma by induction on $k \geq 2$ . For $k=2$ , it is clear from (5). Assume $k \geq 3$ . By the inductive hypothesis, we see

\begin{equation*} x_i^* \otimes [x_{j_1}, \ldots , x_{j_k-1}, x_i] \in \mathrm{Im}(\tau _{k-1}^P). \end{equation*}

If we write $x_i^* \otimes [x_{j_1}, \ldots , x_{j_k-1}, x_i]$ as $\tau _{k-1}^P(\sigma )$ for $\sigma \in \mathcal{P}_n(k-1)$ , then we have

\begin{align*} \begin{split} \tau _k^P([\sigma , K_{i, j_k}]) &= [\tau _{k-1}^P(\sigma ), \tau _1^P(K_{i, j_k})] \\ &=[x_i^* \otimes [x_{j_1}, \ldots , x_{j_k-1}, x_i], x_i^* \otimes [x_i, x_{j_k}] ] \\ &=x_i^* \otimes [x_{j_1}, \ldots , x_{j_k-1}, [x_i, x_{j_k}] ] - x_i^* \otimes [x_{j_1}, \ldots , x_{j_k-1}, x_i, x_{j_k}] \\ &= - x_i^* \otimes [x_{j_1}, \ldots , x_{j_k}, x_i]. \end{split} \end{align*}

At the last equation, we use the Jacobi identity. Hence, the induction proceeds.

Proposition 7.4. For $n \geq 3$ , we have

1. (1) $\overline {\mathrm{Tr}}_3^P=\widetilde {\mathrm{Tr}}_3^P$ is surjective.

2. (2) $\mathrm{Im}(\tau _3^P)=\mathrm{Ker}(\overline {\mathrm{Tr}}_3^P)$ .

3. (3) $\mathrm{Im}(\tau _3^P)$ is a free abelian group with a basis

(6) \begin{equation} \begin{split} \{ x_i^* & \otimes [x_{j_1}, x_{j_2}, x_{j_3}, x_i] \,|\, j_1 \gt j_2 \leq j_3, \hspace {0.5em} i \neq j_1, j_2, j_3 \} \\ & \cup \{ x_i^* \otimes [x_i, x_j, x_i, x_i] - x_j^* \otimes [x_j, x_i, x_i, x_j] \,|\, 1 \leq i \neq j \leq n \} \\ & \cup \{ x_i^* \otimes [x_i, x_j, x_l, x_i] - x_j^* \otimes [x_j, x_l, x_i, x_j] \,|\, l \neq i \gt j \neq l \} \end{split} \end{equation}

Proof. (1) By definition, we see $\overline {\mathrm{Tr}}_3^P=\widetilde {\mathrm{Tr}}_3^P$ . $\overline {\mathcal{C}}_n(3)$ is generated by $x_ix_jx_l$ and $x_i^2x_j$ for all distinct $1 \leq i, j, l \leq n$ . Since

\begin{align*} \begin{split} \overline {\mathrm{Tr}}_3^P(x_i^* \otimes [x_i, x_j, x_l, x_i]) & = x_jx_lx_i=x_ix_jx_l, \\ \overline {\mathrm{Tr}}_3^P(x_i^* \otimes [x_i, x_j, x_i, x_i]) & = x_jx_i^2=x_i^2x_j,\end{split} \end{align*}

we see that $\overline {\mathrm{Tr}}_3^P$ is surjective.

(2) It suffices to show $\mathrm{Im}(\tau _3^P) \supset \mathrm{Ker}(\overline {\mathrm{Tr}}_3^P)$ . Take any

\begin{equation*} x \,:\!= \sum _{\substack {1 \leq i \leq n\\[1pt] j_1\gt j_2 \leq j_3}} a_{j_1, j_2, j_3}^i x_i^* \otimes [x_{j_1}, x_{j_2}, x_{j_3}, x_i] \in \mathrm{Ker}(\overline {\mathrm{Tr}}_3^P). \end{equation*}

Observe

(7) \begin{equation} \begin{split} [x_{j_1}, x_{j_2}, x_i, x_i] &= -[x_i, [x_{j_1}, x_{j_2}], x_i] = -[x_i, x_{j_1}, x_{j_2}, x_i] + [x_i, x_{j_2}, x_{j_1}, x_i], \\ \tau _3^P([K_{ij},K_{ji}, K_{ji}]) &= x_i^* \otimes [x_i, x_j, x_i, x_i] - x_j^* \otimes [x_j, x_i, x_i, x_j] \end{split} \end{equation}

for any $1 \leq i, j, j_1, j_2 \leq n$ . Remark that

\begin{equation*} [x_j,x_i,x_j,x_i] - [x_j,x_i,x_i,x_j]=[x_j,x_i,[x_j,x_i]]=0 \end{equation*}

by the Jacobi identity. Since $\mathrm{Im}(\tau _3^P) \subset \mathrm{Ker}(\overline {\mathrm{Tr}}_3^P)$ , and from Lemma 7.3 and (7), we may assume

\begin{equation*} x=\sum _{\substack {1 \leq i, j, l \leq n\\[1pt] j, l \neq i}} a_{i, j, l}^i x_i^* \otimes [x_i, x_j, x_l, x_i]. \end{equation*}

Furthermore, by observing

\begin{equation*} \tau _3^P([[K_{ij},K_{il}], K_{ji}]) = x_j^* \otimes [x_j, x_l, x_i, x_j] - x_i^* \otimes [x_i, x_j, x_l, x_i] \end{equation*}

for any distinct $i, j$ and $l$ , we may also assume

\begin{equation*} x=\sum _{\substack {i\lt j, l\\[1pt] j \neq l}} a_{i, j, l}^i x_i^* \otimes [x_i, x_j, x_l, x_i] + \sum _{1 \leq i \neq j \leq n} a_{i, j, j}^i x_i^* \otimes [x_i, x_j, x_j, x_i]. \end{equation*}

Then we obtain

\begin{equation*} 0=\tau _3^P(x)=\sum _{\substack {i\lt j, l\\[1pt] j \neq l}} a_{i, j, l}^i x_ix_jx_l + \sum _{1 \leq i \neq j \leq n} a_{i, j, j}^i x_ix_j^2, \end{equation*}

and hence $a_{i, j, l}^i=0$ for any $i, j$ and $l$ . Thus, $x=0$ . (3) From Part (2), we see that $\mathrm{Im}(\tau _3^P)$ is generated by

\begin{align*} \begin{cases} x_i^* \otimes [x_{j_1}, x_{j_2}, x_{j_3}, x_i] \hspace {1em} & i \neq j_1, j_2, j_3, \\ x_i^* \otimes [x_i, x_j, x_i, x_i] - x_j^* \otimes [x_j, x_i, x_i, x_j] & i \neq j, \\ x_i^* \otimes [x_i, x_j, x_l, x_i] - x_j^* \otimes [x_j, x_l, x_i, x_j] & i \neq j \neq l \neq i \end{cases} \end{align*}

as a $\mathbf{Z}$ -module.

Corollary 7.5. For $n \geq 3$ ,

\begin{equation*} 0 \rightarrow \mathrm{gr}^3(\mathcal{P}_n) \xrightarrow {\tau _3^P} \mathfrak{p}_n(3) \xrightarrow {\overline {\mathrm{Tr}}_3^P} \overline {\mathcal{C}}_n(3) \rightarrow 0 \end{equation*}

is $\mathfrak{S}_n$ -equivariant exact.

In [Reference Enomoto and Satoh14], we showed that $\mathcal{C}_n^{\mathbf{Q}}(3)$ is isomorphic to $S^3 H_{\mathbf{Q}} \oplus \wedge ^3 H_{\mathbf{Q}}$ . From this fact and Corollary 7.5, we have the $\mathfrak{S}_n$ -equivariant exact sequence

\begin{equation*} 0 \rightarrow \mathrm{gr}^3(\mathcal{P}_n) \xrightarrow {\tau _3^P} \mathfrak{p}_n(3) \xrightarrow {} \overline {S}^3 H_{\mathbf{Q}} \oplus \wedge ^3 H_{\mathbf{Q}} \rightarrow 0 \end{equation*}

for $n \geq 3$ .

Finally, we consider the case where $k=4$ . At first, we prepare a few lemmas.

Lemma 7.6. For any $n \geq 3$ , the module $\mathfrak{p}_n(4)$ is generated by all

(8) \begin{equation} \begin{cases} x_i^* \otimes [x_{j_1}, x_{j_2}, x_{j_3}, x_{j_4}, x_i] \hspace {1em} & i \neq j_1, j_2, j_3, j_4, \\ x_i^* \otimes [x_i, x_{j_2}, x_{j_3}, x_{j_4}, x_i] & i \neq j_2, j_3, j_4, \\ x_i^* \otimes [x_i, x_{j_2}, x_{i}, x_{j_4}, x_i] & i \neq j_2, j_4, \\ x_i^* \otimes [x_i, x_{j_2}, x_{j_3}, x_i, x_i] & i \neq j_2, j_3, \\ x_i^* \otimes [x_i, x_{j_2}, x_i, x_i, x_i] & i \neq j_2 \\ \end{cases} \end{equation}

for $1 \leq i, j_1, j_2, j_3, j_4 \leq n$ as a $\mathbf{Z}$ -module.

Proof. By the definition, $\mathfrak{p}_n(4)$ is generated by elements $x_i^* \otimes [x_{j_1}, x_{j_2}, x_{j_3}, x_{j_4}, x_i]$ for all $1 \leq i, j_1, j_2, j_3, j_4 \leq n$ as a $\mathbf{Z}$ -module. Set $\omega \,:\!= x_i^* \otimes [x_{j_1}, x_{j_2}, x_{j_3}, x_{j_4}, x_i]$ . If $j_1, \ldots , j_4 \neq i$ , then $\omega$ is an element of first type. If $j_1, j_2 \neq i$ and $j_3=i$ , then by

\begin{align*} \begin{split} [x_{j_1}, x_{j_2}, x_i, x_{j_4}, x_i] &= -[x_i, [x_{j_1}, x_{j_2}], x_{j_4}, x_i] \\ & = -[x_i, x_{j_1}, x_{j_2}, x_{j_4}, x_i] + [x_i, x_{j_2}, x_{j_1}, x_{j_4}, x_i] \end{split} \end{align*}

from the Jacobi identity, we see that $\omega$ can be written as a sum of elements the form in (8). If $j_1, j_2, j_3 \neq i$ and $j_4=i$ , then we can show by an argument similar to that in case where $j_1, j_2 \neq i$ and $j_3=i$ . If $j_1=i$ and $j_2 \neq i$ , or if $j_2=i$ and $j_1 \neq i$ , then we see that $\omega$ is an element of the above form.

Proof. A basis of $\overline {\mathcal{C}}_n(4)$ is given by as follows:

• • $x_ax_bx_cx_d$ for any distinct $a, b, c, d$ such that $a\lt b, c, d$ ,

• • $x_a^2 x_bx_c$ for any distinct $a, b, c$ ,

• • $x_a x_b x_a x_c$ for any distinct $a, b, c$ such that $b\lt c$ ,

• • $x_a^2 x_b^2$ for any $a \lt b$ ,

• • $x_ax_bx_ax_b$ for any $a \lt b$ ,

• • $x_a^3 x_b$ for any $a \neq b$ .

On the other hand, the images of the generators of $\mathfrak{p}_n(4)$ shown in (8) by $\overline {\mathrm{Tr}}_4^P$ are given by

\begin{align*} \begin{cases} \overline {\mathrm{Tr}}_4^P(x_i^* \otimes [x_{j_1}, x_{j_2}, x_{j_3}, x_{j_4}, x_i])=0 \hspace {1em} & i \neq j_1, j_2, j_3, j_4, \\ \overline {\mathrm{Tr}}_4^P(x_i^* \otimes [x_i, x_{j_2}, x_{j_3}, x_{j_4}, x_i])= x_{j_2}x_{j_3}x_{j_4}x_i & i \neq j_2, j_3, j_4, \\ \overline {\mathrm{Tr}}_4^P(x_i^* \otimes [x_i, x_{j_2}, x_{i}, x_{j_4}, x_i])=2x_{j_2}x_ix_{j_4}x_i - x_{j_2}x_{j_4}x_i^2 & i \neq j_2, j_4, \\ \overline {\mathrm{Tr}}_4^P(x_i^* \otimes [x_i, x_{j_2}, x_{j_3}, x_i, x_i])=2x_{j_2}x_ix_{j_3}x_i - x_{j_2}x_{j_3}x_i^2 & i \neq j_2, j_3, \\ \overline {\mathrm{Tr}}_4^P(x_i^* \otimes [x_i, x_{j_2}, x_i, x_i, x_i])=x_{j_2}x_i^3 & i \neq j_2. \\ \end{cases} \end{align*}

Hence, the elements of the basis types of $x_ax_bx_cx_d$ , $x_a^2 x_bx_c$ , $x_a x_b x_a x_c$ , and $x_a^3 x_b$ are in $\mathrm{Im}(\overline {\mathrm{Tr}}_4^P)$ . Indeed, for example,

\begin{equation*} \overline {\mathrm{Tr}}_4^P(x_c^* \otimes [x_c, x_a, x_a, x_b, x_c])= x_a^2 x_bx_c. \end{equation*}

Therefore, we see that $\mathrm{Coker}(\overline {\mathrm{Tr}}_4^P)$ is isomorphic to the quotient module of the free $\mathbf{Z}$ -module with basis $x_a^2 x_b^2$ and $x_ax_bx_ax_b$ for all $a \lt b$ divided by the submodule generated by $2x_ax_bx_ax_b - x_a^2x_b^2$ for all $a \lt b$ . This completes the proof.

Proof. It suffices to show $\mathrm{Im}(\tau _4^P) \supset \mathrm{Ker}(\overline {\mathrm{Tr}}_4^P)$ . Take any $x \in \mathrm{Ker}(\overline {\mathrm{Tr}}_4^P)$ . For distinct $1 \leq i, j, l \leq n$ , set

\begin{equation*} D_1 \,:\!= x_i^* \otimes [x_j, x_l, x_i, x_i] + x_j^* \otimes [x_j, x_l, x_i, x_j] - x_j^* \otimes [x_j, x_i, x_l, x_j] \in \mathfrak{p}_n(3). \end{equation*}

Since $D_1 \in \mathrm{Ker}(\overline {\mathrm{Tr}}_3^P) = \mathrm{Im}(\tau _3^P)$ , there exists some $L_1 \in \mathcal{P}_n(3)$ such that $\tau _3^P(L_1)=D_1$ . From Lemma 7.3 and

(9) \begin{equation} \begin{split} \tau _4^P([K_{ij},K_{ji}, K_{ji}, K_{ji}]) &= x_i^* \otimes [x_i, x_j, x_i, x_i, x_i] - x_j^* \otimes [x_j, x_i, x_i, x_i, x_j], \\ \tau _4^P([K_{ij},K_{ji}, K_{ji}, K_{ij}]) &= x_i^* \otimes [x_i, x_j, x_j, x_i, x_i] - x_j^* \otimes [x_j, x_i, x_j, x_i, x_j], \\ \tau _4^P([K_{ji}, L_1]) &= x_i^* \otimes [x_i, x_j, x_l, x_i, x_i] - 2 x_j^* \otimes [x_j, x_i, x_l, x_i, x_j] \\ & \hspace {2em} + x_j^* \otimes [x_j, x_i, x_i, x_l, x_j], \\ \end{split} \end{equation}

for any distinct $1 \leq i, j, l \leq n$ , by reducing the generators shown in (8) by using these elements, we may assume

\begin{align*} \begin{split} x=\sum _{\substack {1 \leq i, j, l, m \leq n\\[1pt] j, l, m \neq i}} \alpha _{j, l, m}^i x_i^* \otimes [x_i, x_j, x_l, x_m, x_i] + \sum _{\substack {1 \leq i, j, m \leq n\\[1pt] j, m \neq i}} \beta _{j, m}^i x_i^* \otimes [x_i, x_j, x_i, x_m, x_i]. \end{split} \end{align*}

For distinct $1 \leq i, j, l \leq n$ , set

\begin{equation*} D_2 \,:\!= x_i^* \otimes [x_i, x_j, x_l, x_i] - x_l^* \otimes [x_l, x_i, x_j, x_l] \in \mathfrak{p}_n(3). \end{equation*}

Since $D_2 \in \mathrm{Ker}(\overline {\mathrm{Tr}}_3^P) = \mathrm{Im}(\tau _3^P)$ , there exists some $L_2 \in \mathcal{P}_n(3)$ such that $\tau _3^P(L_2)=D_2$ . Thus, we have

\begin{align*} \begin{split} \tau _4^P([L_2, K_{ji}]) &= x_i^* \otimes [x_i, x_j, x_i, x_l, x_i] - x_l^* \otimes [x_l, x_i, x_j, x_i, x_l] \\ & \hspace {2em} + x_l^* \otimes [x_l, x_i, x_i, x_j, x_l] - x_j^* \otimes [x_j, x_i, x_l, x_i, x_j]. \end{split} \end{align*}

Moreover, by the Jacobi identity, we have

\begin{equation*} [x_i, x_j, x_i, x_j, x_i] =- [x_i, x_j, [x_i,x_j], x_i] -[x_j,[x_i,x_j], x_i,x_i]=[x_i,x_j,x_j,x_i,x_i], \end{equation*}

and hence from the second equation of (9), we may assume $\beta _{j, m}^i=0$ if $j \neq m$ , or $j=m$ and $i\gt j$ .

On the other hand, consider the following equations.

(10) \begin{equation} \tau _4^P([[K_{ac},K_{ad}], [K_{ca},K_{cb}]]) = x_a^* \otimes [x_a, x_b, x_c, x_d, x_a] - x_c^* \otimes [x_c, x_d, x_a, x_b, x_c] \end{equation}

for $1 \leq a, b, c, d \leq n$ such that $c \neq d$ and $b \neq c$ .

(11) \begin{equation} \tau _4^P([K_{ab}, K_{ac}, K_{ad}, K_{ba}]) = x_a^* \otimes [x_a, x_b, x_c, x_d, x_a] - x_b^* \otimes [x_b, x_c, x_d, x_a, x_b] \end{equation}

for $1 \leq a, b, c, d \leq n$ such that $b \neq c$ .

(12) \begin{equation} \tau _4^P([K_{ad}, [K_{da}, K_{db}, K_{dc}]]) = x_i^* \otimes [x_a, x_b, x_c, x_d, x_a] - x_d^* \otimes [x_d, x_a, x_b, x_c, x_d] \end{equation}

for $1 \leq a, b, c, d \leq n$ such that $b, c \neq d$ . Then from (12) for the case where $b=c$ , we see that we may assume $\alpha _{j,l,l}^i=0$ . Similarly, from (10) for the case where $b=d$ , we see that we may assume $\alpha _{j,l,j}^i=0$ if $i\gt j$ . Furthermore, for distinct $i, j, l, m$ , by using (10), (11) and (12), we may assume $\alpha _{j, l, m}^i=0$ for the case other than $i\lt j, l, m$ . Therefore, $x$ can be written as

\begin{align*} \begin{split} x &=\sum _{\substack {i\lt j, l, m \\[1pt] j, l, m\,:\,\text{distinct}}} \alpha _{j, l, m}^i x_i^* \otimes [x_i, x_j, x_l, x_m, x_i] + \sum _{\substack {i\lt l \\[1pt] j \neq i, l}} \alpha _{j, l, j}^i x_i^* \otimes [x_i, x_j, x_l, x_j, x_i] \\ & + \sum _{i, j, l\,:\,\text{distinct}} \alpha _{j, j, l}^i x_i^* \otimes [x_i, x_j, x_j, x_l, x_i] + \sum _{i \neq j} \alpha _{j, j, j}^i x_i^* \otimes [x_i, x_j, x_j, x_j, x_i] \\ &+ \sum _{i\lt j} \beta _{j, j}^i x_i^* \otimes [x_i, x_j, x_i, x_j, x_i]. \end{split} \end{align*}

By observing

\begin{align*} \begin{split} 0=\overline {\mathrm{Tr}}_4^P(x) &=\sum _{\substack {i\lt j, l, m \\[1pt] j, l, m\,:\,\text{distinct}}} \alpha _{j, l, m}^i x_j x_l x_m x_i + \sum _{\substack {i\lt l \\[1pt] j \neq i, l}} \alpha _{j, l, j}^i x_j x_l x_j x_i \\ & + \sum _{i, j, l\,:\,\text{distinct}} \alpha _{j, j, l}^i x_j^2 x_l x_i + \sum _{i \neq j} \alpha _{j, j, j}^i x_j^3 x_i \\ &+ \sum _{i\lt j} \beta _{j, j}^i (2x_i x_j x_i x_j - x_i^2x_j^2), \end{split} \end{align*}

we obtain that all of the coefficients in the above equation is $0$ , and hence $x=0$ .

Corollary 7.9. For $n \geq 3$ ,

\begin{equation*} 0 \rightarrow \mathrm{gr}^4(\mathcal{P}_n) \xrightarrow {\tau _4^P} \mathfrak{p}_n(4) \xrightarrow {\overline {\mathrm{Tr}}_4^P} \overline {\mathcal{C}}_n(4) \rightarrow Q \rightarrow 0 \end{equation*}

is $\mathfrak{S}_n$ -equivariant exact. In particular, $\mathrm{Im}(\tau _4^P)$ is a free abelian group of rank $\frac {1}{4}n^2(n-1)^2(n+1)$ .

We remark that for any $1 \leq a \lt b \leq n$ , since

\begin{equation*} 0 \neq 2x_ax_bx_ax_b - x_a^2x_b^2 \in \mathrm{Im}(\mathrm{Tr}_4^P) \cap N(4) \end{equation*}

from the above calculation, we see $\mathrm{Ker}(\overline {\mathrm{Tr}}_4^P) \subsetneq \mathrm{Ker}(\widetilde {\mathrm{Tr}}_4^P)$ . In general, we have the following.

Proof. For any distinct $1 \leq i, j \leq n$ , consider an element $x_i^* \otimes [x_i, x_j, x_i, x_j, \ldots , x_j, x_i] \in \mathfrak{p}_n(k)$ . Then we have

\begin{equation*} \mathrm{Tr}_k^P(x_i^* \otimes [x_i, x_j, x_i, x_j, \ldots , x_j, x_i]) = 2x_jx_ix_j^{k-3}x_i - x_i^2x_j^{k-2} \in N(k). \end{equation*}

This shows $\mathrm{Ker}(\overline {\mathrm{Tr}}_k^P) \subsetneq \mathrm{Ker}(\widetilde {\phantom{\,}\mathrm{Tr}\phantom{\,}}_{\!k}^{\!P})$ .

On the other hand, for $k=4$ , the $\mathfrak{S}_n$ -irreducible component in $\mathrm{Coker}(\tau _4^P)$ which cannot detect $\widetilde {\mathrm{Tr}}_4^P$ is the module

\begin{equation*} A \,:\!= \langle x_a^* \otimes [x_a,x_b,x_a,x_b,x_a] \,|\, 1 \leq a\lt b \leq n \rangle . \end{equation*}

This component can be detected as follows. Let $J$ be the quotient module of $\wedge ^2 H \otimes \wedge ^2 H$ by the submodule generated by

• • $(v \wedge w) \otimes (x \wedge y) - (x \wedge y) \otimes (v \wedge w)$ ,

• • $(v \wedge w) \otimes (x \wedge y) - (x \wedge w) \otimes (v \wedge y) - (v \wedge x) \otimes (w \wedge y)$

for any $v, w, x, y \in H$ . Then $J$ is a free abelian group of rank $n^2(n^2-1)/12$ . As a $\mathrm{GL}(n,\mathbf{Q})$ -module, $J_{\mathbf{Q}}$ is irreducible and isomorphic to $[2,2]$ . In [Reference Satoh33], we construct a trace map which detect $J_{\mathbf{Q}}$ in the cokernel of $\tau _{4,\mathbf{Q}}$ . For $i=1, 2$ , let $f_i\,:\,H^{\otimes 4} \rightarrow J$ be the projection defined by

\begin{align*} f_i(v \otimes w \otimes x \otimes y) = \begin{cases} (v \wedge x) \cdot (w \wedge y), \hspace {1em} & i=1, \\ (v \wedge y) \cdot (w \wedge x), \hspace {1em} & i=2. \\ \end{cases} \end{align*}

Set

\begin{equation*} \mathrm{Tr}_J \,:\!= f_1 \circ \Phi ^4 - 2 (f_2 \circ \Phi ^4)\,:\,H^* \otimes _{\mathbf{Z}} H^{\otimes 5} \rightarrow J. \end{equation*}

By Proposition 4.2 in [Reference Satoh33] and Darné’s result in [Reference Darné12], we have $\mathrm{Im}(\tau _4) \subset \mathrm{Ker}(\mathrm{Tr}_J)$ . Denote by $\mathrm{Tr}_J^P\,:\,\mathfrak{p}_n(4) \rightarrow J$ the restriction of $\mathrm{Tr}_J$ to $\mathfrak{p}_n(4)$ . Since we see

\begin{equation*} \mathrm{Tr}_J(x_a^* \otimes [x_a,x_b,x_a,x_b,x_a]) = 5(x_a \wedge x_b) \cdot (x_a \wedge x_b) \neq 0, \end{equation*}

we see that $\mathrm{Tr}_J$ detects $A$ .

Here we give the list of the rank of the free abelian groups $\mathrm{gr}^k(\mathcal{P}_n)$ , $\mathfrak{p}_n(k)$ , $\mathcal{C}_n(k)$ and $\mathrm{Coker}(\overline {\mathrm{Tr}}_k^P)$ for $n \geq 3$ and $1 \leq k \leq 4$ .

8. Observations of the Johnson images and cokernels for $k \geq 5$

In this section, we give some observations of the dimensions of the Johnson images and cokernels for the case of $k \geq 5$ based on some direct computer calculations. We will use the following notations. For an $\alpha =(\alpha _1,\alpha _2, \ldots ,\alpha _n) \in \mathbb{Z}_{\ge 0}^n$ such that $\alpha _1+ \alpha _2+ \cdots +\alpha _n=k$ , let $p(\alpha )$ be the set of all permutations of $(1^{\alpha _1},2^{\alpha _2}, \cdots ,n^{\alpha _n})$ . We denote $\mathcal{L}_n(k,\alpha )$ by the subspace of $\mathcal{L}_n(k)$ generated by all left-normed elements $[x_{i_1},x_{i_2}, \ldots ,x_{i_k}]$ such that $(i_1,i_2, \ldots ,i_k) \in p(\alpha )$ . Notice that $\alpha _j=\sharp \{1 \leq u \leq k \ | \ i_u=j\} \ (1 \leq j \leq n)$ . We also note that $\dim _{\mathbf{Q}}\mathcal{L}_n^{\mathbf{Q}}(k,\alpha )=0$ for $\alpha =(k)$ . Similarly, let $\overline {\mathcal{C}}_n(k,\alpha )$ be the subspace of $\overline {\mathcal{C}}_n(k)$ generated by all cosets (necklace) of $x_{i_1}x_{i_2} \cdots x_{i_k}$ for $(i_1,i_2, \ldots ,i_k) \in p(\alpha )$ .

For an element $[x_{i_1}, \ldots ,x_{i_k}] \in \mathcal{L}_n(k,\alpha )$ , through the natural embedding $\mathcal{L}_n(k)$ into $H^{\otimes k}$ , we consider $[x_{i_1}, \ldots ,x_{i_k}] \in H^{\otimes k}$ , and take the expression $[x_{i_1}, \ldots ,x_{i_k}]=\sum _{j=1}^{n}x_j \otimes b_j(i_1,\ldots ,i_k)$ , where $b_j(i_1,\ldots ,i_k) \in H^{\otimes {k-1}}$ . Since $\varpi ^k([x_{i_1}, \ldots ,x_{i_k}])=0$ , we have

\begin{align*} &\overline {\mathrm{Tr}}_k^P(x_i^* \otimes [x_{i_1}, \ldots ,x_{i_k},x_i])= \overline {\mathrm{Tr}}_k^P(x_i^* \otimes ([x_{i_1}, \ldots ,x_{i_k}] \otimes x_i-x_i \otimes [x_{i_1}, \ldots ,x_{i_k}]))\\ & =\overline {\varpi }^k(b_i(i_1, \ldots ,i_k) \otimes x_i-[x_{i_1},\ldots ,x_{i_k}]) =\overline {\varpi }^k(b_i(i_1, \ldots ,i_k) \otimes x_i) \in \overline {\mathcal{C}}_n(k,\alpha ), \end{align*}

where $\overline {\varpi }^k\,:\,H^{\otimes {k}} \to \overline {\mathcal{C}}_n(k)$ is the composition of $\varpi ^k$ and the natural projection $\mathcal{C}_n(k) \to \overline {\mathcal{C}}_n(k)$ . Moreover,

\begin{equation*} c_\alpha =\dim _{\mathbf{Q}} \left \langle \overline {\varpi }^k(b_i(i_1, \ldots ,i_k) \otimes x_i) \ | \ 1 \leq i \leq n,(i_1,\ldots ,i_k) \in p(\alpha ) \right \rangle _{\mathbf{Q}} \end{equation*}

is invariant under the left (variable permutation) action of $\mathfrak{S}_n$ on $\alpha$ . Thus, we may consider the case where $\alpha _1 \geq \alpha _2 \geq \cdots \geq \alpha _n$ . We obtain the following proposition.

Proposition 8.1. The dimension of the image of $\overline {\mathrm{Tr}}_{k,\mathbf{Q}}^P$ is given by the following formula;

(13) \begin{align} \dim _{\mathbf{Q}}\mathrm{Im}(\overline {\mathrm{Tr}}_{k,\mathbf{Q}}^P)= \sum _{\substack { \alpha _1 \ge \alpha _2 \ge \cdots \ge \alpha _n \\ \alpha _1+\alpha _2+ \cdots +\alpha _n=k}}|\mathfrak{S}_n \cdot \alpha | \cdot c_\alpha , \end{align}

where $|\mathfrak{S}_n \cdot \alpha |$ is the order of $\mathfrak{S}_n$ -orbit of $\alpha$ .

When $\alpha$ contains $1$ , we can detect all cosets in $\overline {\mathcal{C}}_n(k,\alpha )$ as in the image of $\overline {\mathrm{Tr}}_k^P$ by the surjectivity of $\widetilde {\phantom{\,}\mathrm{Tr}\phantom{\,}}_{\!k}^{\!P}$ from Theorem 6.2. In this case we have $c_\alpha =\dim _{\mathbf{Q}}\overline {\mathcal{C}}_n^{\mathbf{Q}}(k,\alpha )=\dim _{\mathbf{Q}}\mathcal{L}_n^{\mathbf{Q}}(k,\alpha )$ , since in this case all necklaces in $\overline {\mathcal{C}}_n(k,\alpha )$ have the period $k$ (i.e., primitive). Therefore, in order to compute the dimension of $\mathrm{Im}(\overline {\mathrm{Tr}}_{k,\mathbf{Q}}^P)$ , it suffices to consider the case where all positive parts of $\alpha$ are greater than or equal to $2$ .

For $5 \leq k \leq 9$ , we obtain the following tables of $c_\alpha$ and the difference $d_\alpha \,:\!=c_\alpha -\dim _{\mathbf{Q}}\mathcal{L}_n^{\mathbf{Q}}(k,\alpha )$ by computer calculations.

\begin{align*} \begin{array}{cccc} k & \alpha & c_\alpha & d_\alpha \\ \hline \hline 5 & (3,2) & 1 & 0 \\ \hline 6 & (4,2) & 2 & 0 \\ & (3,3) & 3 & 0 \\ & (2^3) & 15 & 1 \\ \hline 7 & (5,2) & 3 & 0 \\ & (4,3) & 5 & 0 \\ & (3,2^2) & 30 & 0 \\ \end{array} \quad \quad \begin{array}{cccc} k & \alpha & c_\alpha & d_\alpha \\ \hline \hline 8 & (6,2) & 2 & -1 \\ & (5,3) & 6 & -1 \\ & (4,4) & 7 & -1 \\ & (4,2^2) & 52 & 1 \\ & (3^2,2) & 69 & -1 \\ & (2^4) & 316 & 4\\ & \quad & & \end{array} \quad \quad \begin{array}{cccc} k & \alpha & c_\alpha & d_\alpha \\ \hline \hline 9 & (7,2) & 4 & 0 \\ & (6,3) & 9 & 0 \\ & (5,4) & 14 & 0 \\ & (5,2^2) & 84 & 0 \\ & (4,3,2) & 140 & 0 \\ & (3^3) & 188 & 2 \\ & (3,2^3) & 840 & 0 \end{array} \end{align*}

Using (13), for $n \ge 3$ , we have

\begin{align*} &\dim _{\mathbf{Q}}\mathrm{Im}(\overline {\mathrm{Tr}}_{5,\mathbf{Q}}^P)=r_n(5), \quad \dim _{\mathbf{Q}}\mathrm{Im}(\overline {\mathrm{Tr}}_{6,\mathbf{Q}}^P)=r_n(6)+\tbinom {n}{3}, \quad \dim _{\mathbf{Q}}\mathrm{Im}(\overline {\mathrm{Tr}}_{7,\mathbf{Q}}^P)=r_n(7), \\ &\dim _{\mathbf{Q}}\mathrm{Im}(\overline {\mathrm{Tr}}_{8,\mathbf{Q}}^P)=r_n(8)-2n(n-1)-\tbinom {n}{2}+4\tbinom {n}{4} \delta _{n \geq 4}, \\ &\dim _{\mathbf{Q}}\mathrm{Im}(\overline {\mathrm{Tr}}_{9,\mathbf{Q}}^P)=r_n(9)+2\tbinom {n}{3}, \end{align*}

where we recall that $r_n(k)$ is the rank of $\mathcal{L}_n(k)$ . For example, when $k=5$ , the suitable $\alpha$ is only $(3,2)$ . Moreover, $c_{(3,2)}=\dim _{\mathbf{Q}}\mathcal{L}^{\mathbf{Q}}_n(5,(3,2))$ . Therefore, $\dim _{\mathbf{Q}}\overline {C}_n^{\mathbf{Q}}(5,\alpha )=\dim _{\mathbf{Q}}\mathcal{L}_n^{\mathbf{Q}}(5,\alpha )$ for all $\alpha$ . Hence, we have $\dim _{\mathbf{Q}}\mathrm{Im}(\overline {\mathrm{Tr}}_{5,\mathbf{Q}}^P)=r_n(5)$ by (13). When $k=6$ , we have $\dim _{\mathbf{Q}}\overline {C}_n^{\mathbf{Q}}(6,(2,2,2))=\dim _{\mathbf{Q}}{}\mathcal{L}_n^{\mathbf{Q}}(6,(2,2,2))+1$ . Thus, we obtain $\dim _{\mathbf{Q}}\mathrm{Im}(\overline {\mathrm{Tr}}_{6,\mathbf{Q}}^P)=r_n(6)+\binom {n}{3}$ by (13) since $|\mathfrak{S}_n \cdot (2,2,2)|=\binom {n}{3}$ .

Let us proceed to the observations of the Johnson images. By fixing $n$ , we can directly calculate the dimension of the submodule

\begin{equation*} \mathrm{Im}(\tau _k^P)_1 \,:\!= \langle [K_{i_1j_1},K_{i_2j_2}, \ldots ,K_{i_kj_k}] \ | \ 1 \leq i_1,i_2, \ldots i_k, j_1, \ldots ,j_k \leq n \rangle \subset \mathrm{Im}(\tau _k^P) \subset \mathfrak{p}_n(k) \end{equation*}

generated by the degree $1$ part by using a computer. We obtain the following proposition for the gaps of $\mathrm{Im}(\tau _k^P)_1$ and $\mathrm{Ker}(\overline {\mathrm{Tr}}_{k,\mathbf{Q}}^P)$ .

Proof. (1) In the case of $n=2$ , we have $\mathcal{P}_2(k)=\mathcal{A}_2(k)$ for any $k \geq 1$ , and see that the filtration $\mathcal{P}_2(1) \supset \mathcal{P}_2(2) \supset \cdots$ is the lower central series of the inner automorphism group $\mathrm{Inn}\,F_2$ . Hence, $\mathrm{Im}(\tau _k^P) \cong \mathcal{L}_2(k)$ for any $k \geq 1$ . As in the proof of Corollary 8.2, we have $r_2(11)-\dim _{\mathbf{Q}}\mathrm{Im}(\overline {\mathrm{Tr}}_{11,\mathbf{Q}}^{P})\gt 0$ by considering the cases of $\alpha =(8,3),(7,4)$ . Since $\dim _{\mathbf{Q}}\mathrm{Ker}(\overline {\mathrm{Tr}}_{k,\mathbf{Q}}^P)=n r_n(k)- \dim _{\mathbf{Q}}\mathrm{Im}(\overline {\mathrm{Tr}}_{k,\mathbf{Q}}^P)$ , we obtain $r_2(11)=\mathrm{dim}_{\mathbf{Q}} \mathcal{L}^{\mathbf{Q}}_2(11) \lt \mathrm{dim}_{\mathbf{Q}} \mathrm{Ker}(\overline {\mathrm{Tr}}_{11,\mathbf{Q}}^P)$ . Therefore $\mathrm{Im}(\tau _{11,\mathbf{Q}}^P)_1 \subsetneq \mathrm{Ker}(\overline {\mathrm{Tr}}_{11,\mathbf{Q}}^P)$ .

(2) In the case of $n=3$ , we have the following table on the dimensions of $\mathrm{Im}(\tau _{k,\mathbf{Q}}^P)_1$ and the kernels of $\overline {\mathrm{Tr}}_{k,\mathbf{Q}}^P$ by computer calculations.

\begin{align*} \begin{array}{c||cccccccc} k &\,\,\, 1 &\,\,\, 2 &\,\,\, 3 &\,\,\, 4 &\,\,\, 5 &\,\,\, 6 &\,\,\, 7 &\,\,\, 8 \\ \dim _{\mathbf{Q}}\mathrm{Im}(\tau _{k,\mathbf{Q}}^P)_1 &\,\,\, 6 &\,\,\, 6 &\,\,\, 16 &\,\,\, 36 &\,\,\, 96 &\,\,\, 231 &\,\,\, 618 &\,\,\, 1596 \\ \dim _{\mathbf{Q}}\mathrm{Ker}(\overline {\mathrm{Tr}}_{k,\mathbf{Q}}^P) &\,\,\, 6 &\,\,\, 6 &\,\,\, 16 &\,\,\, 36 &\,\,\, 96 &\,\,\, 231 &\,\,\, 624 &\,\,\, 1635 \end{array} \end{align*}

We find a gap at the incusion $\mathrm{Im}(\tau _{k,\mathbf{Q}}^P)_1 \subsetneq \mathrm{Ker}(\overline {\mathrm{Tr}}_{k,\mathbf{Q}}^P)$ for $k=7$ . By computer calculations, we can also see that this gap appears only the subspace for $(j_1,\ldots ,j_7) \in \mathfrak{S}_3 \cdot (1^3,2^2,3^2)$ . Moreover, we can compute the character of the module $\mathrm{Ker}(\overline {\mathrm{Tr}}_{7,\mathbf{Q}}^P)/\mathrm{Im}(\tau _{7,\mathbf{Q}}^P)_1$ directly. From this, it turns out that it is isomorphic to the regular representation of $\mathfrak{S}_3$ .

In general, it seems to be difficult to determine whether or not $\mathrm{Im}(\tau _k^P)=\mathrm{Ker}(\overline {\mathrm{Tr}}_k^P)$ . Here we give a partial answer to this problem. For any $\alpha =(\alpha _1,\alpha _2, \ldots ,\alpha _n) \in \mathbb{Z}_{\ge 0}^n$ such that $\alpha _1+ \alpha _2+ \cdots +\alpha _n=k$ , denote by $\mathfrak{p}_n(k,\alpha )$ the subspace of $\mathfrak{p}_n(k)$ generated by all $x_i^* \otimes [x_{i_1},x_{i_2}, \ldots ,x_{i_k}, x_i]$ such that $(i_1,i_2, \ldots , i_k) \in p(\alpha )$ .

Proof. For any $X \in \mathrm{Ker}(\overline {\mathrm{Tr}}_k^P|_{\mathfrak{p}_n(k,\alpha )})$ , set

\begin{equation*} X = \sum a_{i, i_1, \ldots , i_k} x_i^* \otimes [x_{i_1},x_{i_2}, \ldots , x_{i_k}, x_i] \end{equation*}

for some $a_{i, i_1, \ldots , i_k} \in \mathbf{Z}$ .

Step 1. We show that $x_i^* \otimes [x_{i_1}, \ldots , x_{i_k}, x_i] \in \mathrm{Im}(\tau _k^P)$ if $i_1, \ldots , i_k \neq i$ by the induction on $k \geq 1$ . For $k =1$ , it is trivial since $\tau _1^P$ is surjective. Assume $k \geq 2$ . By the inductive hypothesis, there exist some $\sigma \in \mathcal{P}_n(k-1)$ and $\tau \in \mathcal{P}_n(1)$ such that

\begin{equation*} \tau _{k-1}^P(\sigma ) = x_i^* \otimes [x_{i_1},x_{i_2}, \ldots , x_{i_{k-1}}, x_i], \hspace {1em} \tau _{1}^P(\tau ) = x_i^* \otimes [x_{i_k}, x_i]. \end{equation*}

Then we have $\tau _k^P([\sigma , \tau ])=x_i^* \otimes [x_{i_1},x_{i_2}, \ldots , x_{i_k}, x_i]$ , and hence the induction proceeds. Since $\mathrm{Im}(\tau _k^P) \subset \mathrm{Ker}(\overline {\mathrm{Tr}}_k^P)$ , this shows that we may assume that $X$ is a linear combination of $x_i^* \otimes [x_{i_1}, \ldots , x_{i_k}, x_i]$ such that there exists some $1 \leq l \leq k$ such that $i_l=i$ . Furthermore, if we use the Jacobi identity repeatedly, we have

\begin{align*} \begin{split} x_i^* & \otimes [x_{i_1}, \ldots x_{i_{l-1}}, x_i, x_{i_{l+1}}, \ldots , x_{i_k}, x_i] \\ & =- x_i^* \otimes [x_i, [x_{i_1}, \ldots x_{i_{l-1}}], x_{i_{l+1}}, \ldots , x_{i_k}, x_i], \\ & =- x_i^* \otimes [x_i, [x_{i_1}, \ldots x_{i_{l-2}}], x_{i_{l-1}}, x_{i_{l+1}}, \ldots , x_{i_k}, x_i] + x_i^* \otimes [x_i, x_{i_{l-1}}, [x_{i_1}, \ldots x_{i_{l-2}}], x_{i_{l+1}}, \ldots , x_{i_k}, x_i] \\ & = \cdots . \end{split} \end{align*}

Thus, we may assume that $X$ is a linear combination of elements of type $x_i^* \otimes [x_i, x_{i_2}, \ldots , x_{i_k}, x_i]$ .

Step 2. We consider elements $x_i^* \otimes [x_i, x_{i_2}, \ldots , x_{i_k}, x_i]$ such that $i_p=i$ for some $2 \leq p \leq k$ . By the hypothesis of the proposition for $\alpha$ , there exists some $2 \leq l \leq k$ such that $i_l$ appears only one time in $i, i_2, \ldots , i_k$ . We see $i_l \neq i$ . We show that $x_i^* \otimes [x_i, x_{i_2}, \ldots , x_{i_k}, x_i]$ is a linear combination of elements of type

\begin{equation*} x_{i_l}^* \otimes [x_{i_l}, x_{j_2}, \ldots , x_{j_k}, x_{i_l}] \hspace {1em} (j_2, \ldots , j_k \neq i_l) \end{equation*}

modulo $\mathrm{Im}(\tau _k^P)$ by the induction on $k \geq 2$ . For $k=2$ , we consider $x_i^* \otimes [x_i, x_{i_2}, x_i]$ for $i_2 \neq i$ . We see that the claim hold in this case by

\begin{equation*} \tau _2^P([K_{i_2 i}, K_{i i_2}]) = x_{i_2}^* \otimes [x_{i_2}, x_i, x_{i_2}] - x_i^* \otimes [x_i, x_{i_2}, x_i]. \end{equation*}

Assume $k \geq 3$ . First, we consider the case where $i_{l+1}, \ldots , i_k \neq i$ . From the argument is Step 1, we see

\begin{align*} \begin{split} \mathrm{Im}(\tau _k^P) \ni & [x_i^* \otimes [x_{i_l}, x_{i_{l+1}}, \ldots , x_{i_k}, x_i], x_{i_l}^* \otimes [x_i, x_{i_2}, \ldots , x_{i_{l-1}}, x_{i_l}]] \\ & = x_i^* \otimes [x_i, x_{i_2}, \ldots , x_{i_k}, x_i] - x_{i_l}^* \otimes [x_{i_l}, x_{i_{l+1}}, \ldots , x_{i_k}, x_i, x_{i_2}, \ldots , x_{i_l}] \\ & \hspace {0.5em} - \sum _{\substack {2 \leq m \leq l\\[1pt] i_m=i}} x_{i_l}^* \otimes [x_{i}, x_{i_2}, \ldots , x_{i_{m-1}}, [x_{i_l}, \ldots , x_{i_k}, x_i], x_{i_{m+1}}, \ldots , x_{i_l}], \end{split} \end{align*}

and see that the claim holds by using the Jacobi identity. Next we consider the case where $i_m=i$ for some $l+1 \leq m \leq k$ . By the inductive hypothesis, an element

\begin{equation*} x_i^* \otimes [x_{i_l}, x_{i_{l+1}}, \ldots , x_{i_k}, x_i] = - x_i^* \otimes [x_i, [x_{i_l}, \ldots , x_{i_{m-1}}], x_{i_{m+1}}, \ldots , x_{i_k}, x_i] \end{equation*}

can be written as a linear combination of elements of type $x_{i_l}^* \otimes [x_{i_l}, x_{j_2}, \ldots , x_{j_{k-l+1}}, x_{i_l}]$ for $j_2, \ldots , j_{k-l+1} \neq i_l$ modulo $\mathrm{Im}(\tau _{k-l+1}^P)$ . On the other hand, we see

\begin{align*} \begin{split} [x_{i_l}^* \otimes [x_{i_l}, & x_{j_2}, \ldots , x_{j_{k-l+1}}, x_{i_l}], x_{i_l}^* \otimes [x_i, x_{i_2}, \ldots , x_{i_{l-1}}, x_{i_l}]] \\ & = x_{i_l}^* \otimes [x_i, x_{i_2}, \ldots , x_{i_{l-1}}, x_{i_l}, x_{j_2}, \ldots , x_{j_{k-l+1}}, x_{i_l}] \\ & \hspace {1em} + x_{i_l}^* \otimes [x_{i_l}, x_{j_2}, \ldots , x_{j_{k-l+1}}, [x_i, x_{i_2}, \ldots , x_{i_{l-1}}, x_{i_l}]] \\ & \hspace {1em} - x_{i_l}^* \otimes [x_i, x_{i_2}, \ldots , x_{i_{l-1}}, [x_{i_l}, x_{j_2}, \ldots , x_{j_{k-l+1}}, x_{i_l}]] \\ & = x_{i_l}^* \otimes [x_i, x_{i_2}, \ldots , x_{i_{l-1}}, x_{i_l}, x_{j_2}, \ldots , x_{j_{k-l+1}}, x_{i_l}] \\ & \hspace {1em} + x_{i_l}^* \otimes [x_{i_l}, x_{j_2}, \ldots , x_{j_{k-l+1}}, [x_i, x_{i_2}, \ldots , x_{i_{l-1}}], x_{i_l}]. \end{split} \end{align*}

Therefore, by the same arguments as the previous case, we see that the claim holds by using the Jacobi identity. Thus, the induction proceeds. we may assume that $X$ is a linear combination of elements of type $x_i^* \otimes [x_i, x_{i_2}, \ldots , x_{i_k}, x_i]$ for $i_2, \ldots , i_k \neq i$ .

Step 3. Consider an element $x_i^* \otimes [x_i, x_{i_2}, \ldots , x_{i_k}, x_i]$ for $i_2, \ldots , i_k \neq i$ . If there exists another $2 \leq l \leq k$ such that $i_l$ appears only one time in $i, i_2, \ldots , i_k$ , from the argument in Step 1, we have some $\sigma \in \mathcal{P}_n(k-l+1)$ and $\tau \in \mathcal{P}_n(l)$ such that

\begin{equation*} \tau _{k-l+1}^P(\sigma ) = x_i^* \otimes [x_{i_l},x_{i_{l+1}}, \ldots , x_{i_k}, x_i], \hspace {1em} \tau _{l-1}^P(\tau ) = x_{i_l}^* \otimes [x_i, x_{i_2}, \ldots , x_{i_l}], \end{equation*}

and

\begin{equation*} \tau _k^P([\sigma ,\tau ]) = x_i^* \otimes [x_i, x_{i_2}, \ldots , x_{i_k}, x_i] - x_{i_l}^* \otimes [x_{i_l}, x_{i_{l+1}}, \ldots , x_{i_k}, x_i, x_{i_2}, \ldots , x_{i_l}]. \end{equation*}

This shows that we may assume that $X$ is a linear combination of elements of type $x_i^* \otimes [x_i, x_{i_2}, \ldots , x_{i_k}, x_i]$ for $i_2, \ldots , i_k \neq i$ such that $i$ is the minimum element in $i, i_2, \ldots , i_k$ , which appears only one time in $i, i_2, \ldots , i_k$ . Set

\begin{equation*} X = {\sum }' a_{i, i_2, \ldots , i_k} x_i^* \otimes [x_i, x_{i_2}, \ldots , x_{i_k}, x_i] \end{equation*}

where the sum runs over all $(i, i_2, \ldots , i_k)$ such that $i_2, \ldots , i_k \neq i$ and $i$ is the minimum element in $i, i_2, \ldots , i_k$ , which appears only one time in $i, i_2, \ldots , i_k$ . Then we have

\begin{equation*} \overline {\mathrm{Tr}}_k^P(X) = {\sum }' a_{i, i_2, \ldots , i_k} x_i x_{i_2} \cdots x_{i_k} \in \overline {\mathcal{C}}_n(k). \end{equation*}

Since elements $x_i x_{i_2} \cdots x_{i_k}$ are linearly independent in $\overline {\mathcal{C}}_n(k)$ , we obtain $a_{i, i_2, \ldots , i_k}=0$ for all $(i, i_2, \ldots , i_k)$ . This completes the proof of Theorem 8.5.

9. The Andreadakis conjecture for $\mathcal{P}_n(4)$

In this section, as an application of Proposition 7.4, we show that $\mathcal{P}_n(4)=\mathrm{P}\Sigma _n(4)$ . First, we give the strategy to show it, which is the same way as the proof of $\mathcal{P}_n(3)=\mathrm{P}\Sigma _n(3)$ given in [Reference Satoh35]. Since the natural map

\begin{equation*} \nu\,:\,\mathrm{P}\Sigma _n(3)/\mathrm{P}\Sigma _n(4) \rightarrow \mathrm{gr}^3(\mathcal{P}_n) \end{equation*}

is surjective, and $\mathrm{P}\Sigma _n(4) \subset \mathcal{P}_n(4)$ , if the injectivity of $\nu$ is proved then $\nu$ is an isomorphism, and hence we obtain $\mathrm{P}\Sigma _n(4) = \mathcal{P}_n(4)$ . Since $\mathrm{gr}^3(\mathcal{P}_n)$ is a free abelian group of rank $n(n-1)^2(n+1)/3$ , see the table at the end of Section 7, if we show that

\begin{align*} \text{``}\mathrm{P}\Sigma _n(3)/\mathrm{P}\Sigma _n(4)\, \textrm {is generated by}\, n(n-1)^2(n+1)/3\, \textrm {elements as a}\ \mathbf{Z}-\textrm {module}\text{''} \end{align*}

then there exists a section $\mathrm{gr}^3(\mathcal{P}_n) \rightarrow \mathrm{P}\Sigma _n(3)/\mathrm{P}\Sigma _n(4)$ which is the inverse of $\nu$ , and hence $\nu$ is injective. Thus, we show the following.

Proof. First, we count the number of the elements. The number of elements of (E1) and (E2) is equal to the total number of choices of $i \in \{ 1, \ldots , n\}$ times the total number of choices of $j, l, m \in \{ 1 \ldots,{} i-1, i+1, \ldots , n\}$ such that $j\gt l \leq m$ . It is $n \times 2 \binom {(n-1)+1}{3}$ . The number of elements of (E3) is $\binom {n}{3} \times 4$ . The number of elements of (E4) is $n(n-1)$ . Thus, the total number of the elements is equal to $n(n-1)^2{}(n+1)/3$ .

From (5), we see that $\mathrm{P}\Sigma _n(2)/\mathrm{P}\Sigma _n(3)=\mathrm{gr}^2(\mathcal{P}_n)$ is generated by

\begin{align*} \begin{split} \{ [K_{ij}, K_{il}] \,|\, 1 \leq l \lt j \leq n, \hspace {0.5em} i \neq j, l \} \cup \{ [K_{ij}, K_{ji}] \,|\, 1 \leq i \neq j \leq n \}. \end{split} \end{align*}

Hence, $\mathrm{P}\Sigma _n(3)/\mathrm{P}\Sigma _n(4)$ is generated by

\begin{align*} \begin{cases} [K_{ij}, K_{il}, K_{pq}] \hspace {1em} & i \neq l\lt j \neq i, \\ [K_{ij}, K_{ji}, K_{pq}] & i \neq j \end{cases} \end{align*}

for any $p \neq q$ . We reduce these generators.

(I) The generators type of $X\,:\!=[K_{ij}, K_{il}, K_{pq}]$ .

Set $N\,:\!=\sharp \{ i, j, l, p, q \}$ . If $N=5$ , then $X=0$ since as an automorphism of $F_n$ , $[K_{ij}, K_{il}, K_{pq}]=1$ in $\mathrm{Aut}\,F_n$ . Consider the case where $N=4$ . In this case, $p \in \{ i, j, l \}$ or $q \in \{i, j, l \}$ . If $p=i$ and $l\lt q$ , then $X$ is of type (E1). If $p=i$ and $l\gt q$ , by using the Jacobi identity, we see

\begin{equation*} [K_{ij}, K_{il}, K_{iq}] = -[K_{il}, K_{iq}, K_{ij}] +[K_{ij}, K_{iq}, K_{il}]. \end{equation*}

If $p=j$ , by the Jacobi idenity and $[K_{ij}, K_{jq}]=-[K_{ij}, K_{iq}]$ from the relation (P3), we have

\begin{equation*} [K_{ij}, K_{il}, K_{jq}] = -[K_{il}, K_{jq}, K_{ij}] -[K_{jq}, K_{ij}, K_{il}] = [K_{ij}, K_{jq}, K_{il}] = -[K_{ij}, K_{iq}, K_{il}]. \end{equation*}

This is the same case as $p=i$ . Hence, in each of the above cases, $X$ can be written as a sum of elements type of (E1). So, we can remove $X$ from the generating set in this case. Similarly, we can reduce the generators in other cases. For simplicity, we give a list as follows.

Here (E3)’ means

\begin{align*} (\textbf{E3})\text{'}\,:\, [K_{ij}, K_{il}, K_{ji}]\ \textrm{for any distinct}\ 1 \leq i, j, l \leq n. \end{align*}

We show that any element type of (E3)’ can be written as that of (E3) later.

(II) The generators type of $Y\,:\!=[K_{ij}, K_{ji}, K_{pq}]$ .

Set $N'\,:\!=\sharp \{ i, j, p, q \}$ . By arguments similar to those in (I), we have the following.

Finally, we consider (E3)’. For any distinct $1 \leq i, j, l \leq n$ , we have

where $\stackrel {\text{J}}{=}$ and $\stackrel {\textbf {(P3)}}{=}$ mean the equlities induced from the Jacobi identity and the relation (P3), respectively. Hence for any $1 \leq l\lt j\lt i \leq n$ , we can remove $[K_{ij}, K_{il}, K_{ji}]$ and $[K_{il}, K_{ij}, K_{li}]$ from the generating set.

Therefore, we obtain the required result. This completes the proof of Theorem 9.1.

Hence, we obtain the following.

10. Remarks on $H_2(\mathrm{P}\Sigma _n, \mathbf{Z})$

In this section, as applications of our results above, we give some remarks on the second homology and the second cohomology groups of $\mathrm{P}\Sigma _n$ . All of the results in this subsection were alreday obtained by Brownstein and Lee [Reference Brownstein and Lee8], and Jensen, McCammond and Meiyer [Reference Jensen, McCammond and Meier21]. The method is the same as that in the study of the integral second homology group of the lower-triangular IA-automorphism group of $F_n$ in [Reference Satoh31].

We prepare some notation. Let $F$ be the free group of rank $n(n-1)$ with a basis $K_{ij}$ for all $1 \leq i \neq j \leq n$ . Let $\pi\,:\,F \rightarrow \mathrm{P}\Sigma _n$ be the standard surjective homomorphism, and set $R\,:\!=\mathrm{Ker}(\pi )$ . Then we have the group extension

(14) \begin{equation} 1 \rightarrow R \rightarrow F \xrightarrow {\pi } \mathrm{P}\Sigma _n \rightarrow 1. \end{equation}

From Theorem 5.1, since the abelianization of $\mathrm{P}\Sigma _n$ is a free abelian group generated by (the coset classes of) all $K_{ij}$ s, we see that $\pi$ induces an isomorphism

\begin{equation*} \pi _*\,:\,H_1(F,\mathbf{Z}) \rightarrow H_1(\mathrm{P}\Sigma _n,\mathbf{Z}). \end{equation*}

From the homological five-term exact sequence of (14), we have

\begin{align*} \begin{split} 0=H_2(F,\mathbf{Z}) \rightarrow H_2(\mathrm{P}\Sigma _n,\mathbf{Z}) \rightarrow H_1(R,\mathbf{Z})_F \rightarrow H_1(F,\mathbf{Z}) \xrightarrow {\pi _*} H_1(\mathrm{P}\Sigma _n,\mathbf{Z}) \rightarrow 0, \end{split} \end{align*}

and hence

\begin{equation*} H_2(\mathrm{P}\Sigma _n,\mathbf{Z}) \cong H_1(R,\mathbf{Z})_F. \end{equation*}

Let $F=\Gamma _F(1) \supset \Gamma _F(2) \supset \cdots$ be the lower central series of $F$ , and set $\mathcal{L}_F(k) \,:\!= \Gamma _F(k)/\Gamma _F(k+1)$ for each $k \geq 1$ . Let $\{ R_k \}_{k \geq 1}$ be the descending filtration of $R$ defined by $R_k \,:\!= R \cap \Gamma _F(k)$ for each $k \geq 1$ . We have $R_k=R$ for $k=1$ and $2$ since $R \subset [F,F]$ . For each $k \geq 1$ , let $\pi _k\,:\,\mathcal{L}_F(k) \rightarrow \mathrm{P}\Sigma _n(k)/\mathrm{P}\Sigma _n(k+1)$ be the homomorphism induced from $\pi\,:\,F \rightarrow \mathrm{P}\Sigma _n$ . By observing $R_k/R_{k+1} \cong (R_k \, \Gamma _F(k+1))/\Gamma _F(k+1)$ , we obtain an exact sequence

(15) \begin{equation} 0 \rightarrow R_k/R_{k+1} \rightarrow \mathcal{L}_F(k) \xrightarrow {\pi _k} \mathrm{P}\Sigma _n(k)/\mathrm{P}\Sigma _n(k+1) \rightarrow 0. \end{equation}

For each $k \geq 2$ . The natural projection $R \rightarrow R/R_{k+1}$ induces the surjective homomorphism $\psi _k\,:\,H_1(R,\mathbf{Z}) \rightarrow H_1(R/R_{k+1},\mathbf{Z})$ . By considering the right action of $F$ on $R$ , defined by

\begin{equation*} r \cdot x \,:\!= x^{-1} r x, \hspace {1em} r \in R, \,\,\, x \in F, \end{equation*}

we see that $\psi _k$ is $F$ -equivariant. Hence, it induces the surjective homomorphism

\begin{equation*} H_1(R,\mathbf{Z})_{F} \rightarrow H_1(R/R_{k+1},\mathbf{Z})_{F}, \end{equation*}

which is also denoted by $\psi _k$ . For $k=2$ , $H_1(R/R_{3},\mathbf{Z})_{F} = R/R_{3}$ since $F$ acts on $R/R_{3}$ trivially.

Proof. Observe that $\mathrm{P}\Sigma _n(2)/\mathrm{P}\Sigma _n(3)=\mathrm{gr}^2(\mathcal{P}_n)$ is a free abelian group of rank $n(n-1)^2/2$ as mentioned above. (See the table at the end of Section 7.) By considering (15) for $k=2$ , we see

\begin{align*} \begin{split} \mathrm{rank}_{\mathbf{Z}}(R/R_{3}) & = \mathrm{rank}_{\mathbf{Z}}(\mathcal{L}_F(2)) - \mathrm{rank}_{\mathbf{Z}}(\mathrm{gr}^2(\mathcal{P}_n)) \\ & = \frac {1}{2}n(n-1)(n(n-1)-1) - \frac {1}{2}n(n-1)^2 \\ & = \frac {1}{2}n^2(n-1)(n-2). \end{split} \end{align*}

This means $H_2(\mathrm{P}\Sigma _n,\mathbf{Z})$ contains a free abelian group of rank $n^2(n-1)(n-2)/2$ . On the other hand, $H_1(R,\mathbf{Z})_F \cong R/[F,R]$ is generated by (P1), (P2) and (P3) as a $\mathbf{Z}$ -module. Since the total number of elements of (P1), (P2) and (P3) is $n^2(n-1)(n-2)/2$ , we see that $\psi _2\,:\,R/[F,R] = H_1(R,\mathbf{Z})_F \rightarrow H_1(R/R_3,\mathbf{Z})_{F}=R/R_3$ is an isomorphism, and obtain the required result.

By the universal coefficient theorem, we have

\begin{equation*} H^2(\mathrm{P}\Sigma _n,\mathbf{Z}) \cong \mathbf{Z}^{\frac {1}{2}n^2(n-1)(n-2)}. \end{equation*}

11. Twisted first cohomology groups of $\mathrm{BP}_n$

From the group extension

\begin{equation*} 1 \rightarrow \mathrm{P}\Sigma _n \rightarrow \mathrm{BP}_n \rightarrow \mathfrak{S}_n \rightarrow 1, \end{equation*}

the symmetric group $\mathfrak{S}_n$ acts on the integral homology groups $H_p(\mathrm{P}\Sigma _n, \mathbf{Z})$ for $p \geq 1$ . We consider $H_p(\mathrm{P}\Sigma _n, \mathbf{Z})$ as a $\mathrm{BP}_n$ -module through the homomorphism $\mathrm{BP}_n \rightarrow \mathfrak{S}_n$ .

Note that $\mathfrak{p}_n^{\mathbf{Q}}(1)\cong H_{\mathbf{Q}} \otimes \mathcal{L}_n^{\mathbf{Q}}(1) \cong ((n) \oplus (n-1,1)) \otimes (n-1,1)$ as an $\mathfrak{S}_n$ -module. We can decompose the tensor product representation $(n-1,1) \otimes (n-1,1) \cong (n) \oplus (n-1,1) \oplus (n-2,2) \oplus (n-2,1^2)$ . (See the example in [[Reference Tokuyama38],1.8].) Therefore, as an $\mathfrak{S}_n$ -equivariant $\mathbf{Q}$ -vector space, we have

\begin{equation*} H_1(\mathrm{P}\Sigma _n, \mathbf{Q}) =\mathfrak{p}_n^{\mathbf{Q}}(1) \cong (n) \oplus 2(n-1,1) \oplus (n-2,1^2) \oplus (n-2,2) \end{equation*}

for $n \geq 4$ . In this section, we consider the first cohomology groups of $\mathrm{BP}_n$ with coefficients in $(n)$ and $(n-1,1)$ .

First, the irreducible $\mathfrak{S}_n$ -module $(n)$ . This is isomorphic to the trivial module $\mathbf{Q}$ . From Theorem 5.2 by Fenn-Rimányi-Rourke, we see

\begin{equation*} H_1(\mathrm{BP}_n, \mathbf{Z}) = \mathbf{Z} \oplus \mathbf{Z}/2\mathbf{Z}. \end{equation*}

The free part is generated by (the coset class of) $\sigma _1$ , and the torsion part is generated by (the coset class of) $s_1$ . Thus, by the universal coefficient theorem, we obtain $H^1(\mathrm{BP}_n, \mathbf{Z}) \cong \mathbf{Z}$ , and $H^1(\mathrm{BP}_n, \mathbf{Q}) \cong \mathbf{Q}$ .

Next, we consider the standard $\mathfrak{S}_n$ -representation $(n-1,1)$ . Recall the standard $\mathrm{GL}(n,\mathbf{Z})$ -representation $H$ . Let $\boldsymbol{e}_1, \ldots , \boldsymbol{e}_n \in H$ be the standard basis of $H$ . For any element $x=x_1\boldsymbol{e}_1 + \cdots + x_n \boldsymbol{e}_n \in H$ , we write $x=(x_1, \ldots , x_n)$ for simplicity. As an $\mathfrak{S}_n$ -representation, the module $H$ is decomposed into the direct sum of the two irreducible representations

\begin{align*} \begin{split} V &\,:\!=\{ (x_1, \ldots , x_n) \in H \,|\, x_1+ \cdots + x_n=0 \}, \\ W &\,:\!=\{ (x, \ldots , x) \in H \,|\, x \in \mathbf{Z} \}. \end{split} \end{align*}

The representation $W$ is the trivial representation, and $V \otimes _{\mathbf{Z}} \mathbf{Q} \cong (n-1,1)$ . We compute the first cohomology group of $\mathrm{BP}_n$ with coefficients in $V$ . Here we consider $\boldsymbol{e}_1-\boldsymbol{e}_n, \ldots , \boldsymbol{e}_{n-1}-\boldsymbol{e}_n$ as a basis of $V$ , and write any element $v=v_1(\boldsymbol{e}_1-\boldsymbol{e}_n) + \cdots + v_{n-1}(\boldsymbol{e}_{n-1}-\boldsymbol{e}_n) \in V$ as $v=(v_1, \ldots , v_{n-1})$ for simplicity.

Here we give the outline of the computation. Let $Z^1(\mathrm{BP}_n, V)$ be the $\mathbf{Z}$ -module of all crossed homomorphisms from $\mathrm{BP}_n$ to $V$ , and $B^1(\mathrm{BP}_n, V)$ be the $\mathbf{Z}$ -module of all principal crossed homomorphisms from $\mathrm{BP}_n$ to $V$ . Then we have $H^1(\mathrm{BP}_n, V) \cong Z^1(\mathrm{BP}_n, V)/B^1(\mathrm{BP}_n, V)$ . We compute $Z^1(\mathrm{BP}_n, V)$ by using the presentation of $\mathrm{BP}_n$ in Theorem 5.2. Let $F$ be the free group with a basis $\sigma _i$ and $s_i$ for $1 \leq i \leq n-1$ , and $\varphi\,:\,F \rightarrow \mathrm{BP}_n$ the standard projection. Then the kernel $R$ of $\varphi$ is the normal closure of all of the relators coming from the relations in Theorem 5.2. Considering the five-term exact sequence of the Lyndon–Hochshild–Serre spectral sequence of the group extension

\begin{equation*} 1 \rightarrow R \rightarrow F \rightarrow \mathrm{BP}_n \rightarrow 1, \end{equation*}

we obtain the exact sequence

\begin{equation*} 0 \rightarrow H^1(\mathrm{BP}_n, V) \rightarrow H^1(F, V) \rightarrow H^1(R, V)^F. \end{equation*}

By observing this sequence at the cocycle level, we also obtain the exact sequence

(16) \begin{equation} 0 \rightarrow Z^1(\mathrm{BP}_n,V) \rightarrow Z^1(F,V) \xrightarrow {\iota ^*} Z^1(R,V) \end{equation}

where $\iota ^*$ is the map induced from the inclusion $\iota\,:\,R \hookrightarrow F$ . Thus, we consider $Z^1(\mathrm{BP}_n,V)$ as a subgroup of $Z^1(F,V)$ which are killed by $\iota ^*$ . In other words, $Z^1(\mathrm{BP}_n,V)$ consists of all elements of $Z^1(F,V)$ which preserve all of the relations in Theorem 5.2. By the universality of the free group $F$ , it is easily seen that a crossed homomorphism $f\,:\, F \rightarrow V$ is uniquely determined by the images of the basis $\sigma _i$ and $s_i$ for $1 \leq i \leq n-1$ , and hence $Z^1(F,V)$ is isomorphic to $V^{\oplus 2(n-1)}$ .