$\displaystyle\cong\pi_{*}(\mathrm{Lie}^{\pi}_{\mathbb{F}_{p},\mathbb{E}_{\infty}}(\Sigma^{j}H\mathbb{F}_{p}\oplus\Sigma^{k}H\mathbb{F}_{p}))\xrightarrow{\theta_{*}}\pi_{*}(A),$

for $p>2$.

More precisely, since the dual bar spectral sequence converging to
$\pi_{*}(\mathrm{Lie}^{\pi}_{\mathbb{F}_{p},\mathbb{E}_{\infty}}(\Sigma^{j}H\mathbb{F}_{p}\oplus\Sigma^{k}H\mathbb{F}_{p}))$
collapses on the $E^{2}$-page with no extension problems, there are unique
preimages of $x,y$ on the $E^{2}$-page, which we again call $x,y$ by abuse of
notations. The binary operation $[-,-]$ is then given at all primes $p$ by

$\displaystyle\Sigma^{-1}(\Sigma^{j}\mathbb{F}_{p}\oplus\Sigma^{k}\mathbb{F}_{p})\rightarrow\mathbb{F}_{p}\\{[x,y]\\}$
$\displaystyle\hookrightarrow\mathrm{wt}_{2}\big{(}\mathrm{Free}^{\mathrm{sLie}_{\mathbb{F}_{p}}}(\Sigma^{j}\mathbb{F}_{p}\oplus\Sigma^{k}\mathbb{F}_{p})\big{)}$
$\displaystyle\cong\pi_{*}\big{(}\mathbb{S}^{-1}\mathbin{\mathop{\otimes}\displaylimits^{h\Sigma_{2}}}(\Sigma^{j}H\mathbb{F}_{p}\oplus\Sigma^{k}H\mathbb{F}_{p})^{\mathbin{\mathop{\otimes}\displaylimits}2}\big{)}$
$\displaystyle\xrightarrow{(\iota_{2})_{*}}\mathrm{wt}_{2}[\pi_{*}(\mathrm{Lie}^{\pi}_{\mathbb{F}_{p},\mathbb{E}_{\infty}}(\Sigma^{j}H\mathbb{F}_{p}\oplus\Sigma^{k}H\mathbb{F}_{p}))]\xrightarrow{\theta_{*}}\pi_{*}(A)$

up to a nonzero scalar $c$. We fix a choice of the generator for the shifted
Lie bracket on the $E^{2}$-page of the dual bar spectral sequence so that
$c=1$, and by abuse of notation we use $[-,-]$ to denote both the binary
operation on $\pi_{*}(A)$ and the shifted Lie bracket on the $E^{2}$-page.

Translating to cohomological grading, we constructed a binary operation

$[-,-]:\overline{\mathrm{TAQ}}^{m}(A)\mathbin{\mathop{\otimes}\displaylimits}\overline{\mathrm{TAQ}}^{n}(A)\rightarrow\overline{\mathrm{TAQ}}^{m+n+1}(A)$

for any $\mathbb{E}^{\mathrm{nu}}_{\infty}$-$H\mathbb{F}_{p}$-algebra $A$.

First we show that this binary operation is indeed a shifted Lie bracket in a
general sense.

######  Proposition 5.9.

The binary operation $[-,-]$ in Construction 5.8 satisfies graded
commutativity $[x,y]=(-1)^{|x||y|}[y,x]$ and the graded Jacobi identity

$(-1)^{|x||z|}[x,[y,z]]+(-1)^{|x||y|}[y,[z,x]]+(-1)^{|y||z|}[z,[x,y]]=0.$

We shall see in the next proposition that $[x,x]=0$ for all $x$ at $p=2$ and
$[x,[x,x]]=0$ for all $x$ at $p=3$. Hence $[-,-]$ equips the homotopy groups
of any spectral partition Lie algebra with a shifted Lie algebra structure as
is the convention of this paper.

###### Proof.

Graded commutativity $[x,y]=(-1)^{|x||y|}[y,x]$ follows by construction, with
the sign coming from the induced action of the transposition
$(12)\in\Sigma_{2}$. To check the graded Jacobi identity of the shifted
bracket, we use an argument adapted from [AC20]. Let
$A=\Sigma^{j}H\mathbb{F}_{p}\oplus\Sigma^{k}H\mathbb{F}_{p}\oplus\Sigma^{l}H\mathbb{F}_{p}$.
The iteration $[[-,-],-]$ of the binary operation $[-,-]$ is given by the
weight 3 summand

$\displaystyle(\partial_{2}(\mathrm{Id})\mathbin{\mathop{\otimes}\displaylimits}H\mathbb{F}_{p})\mathbin{\mathop{\otimes}\displaylimits}\Big{(}\big{(}(\partial_{1}(\mathrm{Id})\mathbin{\mathop{\otimes}\displaylimits}H\mathbb{F}_{p})\mathbin{\mathop{\otimes}\displaylimits^{h\Sigma_{1}}}A\big{)}\mathbin{\mathop{\otimes}\displaylimits}\big{(}(\partial_{2}(\mathrm{Id})\mathbin{\mathop{\otimes}\displaylimits}H\mathbb{F}_{p})\mathbin{\mathop{\otimes}\displaylimits^{h\Sigma_{2}}}A^{\mathbin{\mathop{\otimes}\displaylimits}2}\big{)}\Big{)}$
$\displaystyle\xhookrightarrow{(\ref{liebracket})}$
$\displaystyle\mathrm{wt}_{3}\Big{[}(\partial_{2}(\mathrm{Id})\mathbin{\mathop{\otimes}\displaylimits}H\mathbb{F}_{p})\mathbin{\mathop{\otimes}\displaylimits^{h\Sigma_{2}}}\Big{(}\big{(}(\partial_{1}(\mathrm{Id})\mathbin{\mathop{\otimes}\displaylimits}H\mathbb{F}_{p})\mathbin{\mathop{\otimes}\displaylimits^{h\Sigma_{1}}}A\big{)}\oplus\big{(}(\partial_{2}(\mathrm{Id})\mathbin{\mathop{\otimes}\displaylimits}H\mathbb{F}_{p})\mathbin{\mathop{\otimes}\displaylimits^{h\Sigma_{2}}}A^{\mathbin{\mathop{\otimes}\displaylimits}2}\big{)}\Big{)}^{\mathbin{\mathop{\otimes}\displaylimits}2}\Big{]}$
$\displaystyle\rightarrow$
$\displaystyle(\partial_{3}(\mathrm{Id})\mathbin{\mathop{\otimes}\displaylimits}H\mathbb{F}_{p})\mathbin{\mathop{\otimes}\displaylimits^{h\Sigma_{3}}}A^{\mathbin{\mathop{\otimes}\displaylimits}3}$

of the monad composition
$\mathrm{Lie}^{\pi}_{\mathbb{F}_{p},\mathbb{E}_{\infty}}\circ\mathrm{Lie}^{\pi}_{\mathbb{F}_{p},\mathbb{E}_{\infty}}\rightarrow\mathrm{Lie}^{\pi}_{\mathbb{F}_{p},\mathbb{E}_{\infty}}$
applied to $A$. Since $\partial_{2}(\mathrm{Id})\simeq\mathbb{S}^{-1}$ and
$\partial_{1}(\mathrm{Id})\simeq\mathbb{S}$ both have trivial actions by the
symmetric groups, we deduce that the source of the above structure map is
equivalent to

$\partial_{2}(\mathrm{Id})\mathbin{\mathop{\otimes}\displaylimits}(\partial_{1}(\mathrm{Id})\mathbin{\mathop{\otimes}\displaylimits}\partial_{2}(\mathrm{Id}))\mathbin{\mathop{\otimes}\displaylimits}H\mathbb{F}_{p}\mathbin{\mathop{\otimes}\displaylimits}(A^{\mathbin{\mathop{\otimes}\displaylimits}3})^{h\Sigma_{3}}.$

Denote by $\nu$ the structure map
$\partial_{2}(\mathrm{Id})\mathbin{\mathop{\otimes}\displaylimits}(\partial_{1}(\mathrm{Id})\mathbin{\mathop{\otimes}\displaylimits}\partial_{2}(\mathrm{Id}))\rightarrow\partial_{3}(\mathrm{Id})$.
The graded Jacobi identity

$(-1)^{|x||z|}[x,[y,z]]+(-1)^{|x||y|}[y,[z,x]]+(-1)^{|y||z|}[z,[x,y]]=0$

is then equivalent to showing that $\nu+(\sigma)_{*}\nu+(\sigma)^{2}_{*}\nu$
is null-homotopic, where $(\sigma)_{*}$ is the induced action of the cyclic
permutation $(123)$. It was proved in [AC20, Proposition 5.2] that
$\nu+(\sigma)_{*}\nu+(\sigma)^{2}_{*}\nu$ is null-homotopic. ∎

Next we investigate the interaction between the shifted Lie bracket $[-,-]$
and the unary operation in Construction 5.1.

######  Proposition 5.10.

Given any $x,y\in\pi_{*}(A)$ and unary operation $\alpha$ of positive weight,
we have $[x,\alpha(y)]=0$ unless one of the following condition is satisfied:

(1) $p=2$ and $\alpha$ is an iteration of bottom unary operations on $y$. The
bottom unary operation on a class $y$ is given by $R^{-|y|+1}$ in Construction
5.1, which equips the shifted Lie bracket with a restriction map $x\mapsto
x^{[2]}$. In particular $[x,x]=0$ for all $x$.

(2) $p>2$ and $y$ is in odd degree $\alpha$ is an iteration of bottom unary
operations on $y$. The bottom unary operation on an odd class $y$ is given by
$R^{(-|y|+1)/2}$ in Construction 5.1. In particular $[x,[x,x]]=0$ for all $x$
when $p=3$.

###### Proof.

We use an argument adapted from [Br17, Proposition 4.3.15]. Suppose that
$\alpha$ is a nonempty sequence of operations with weight $w$ divisible by
$p$, and $x:\Sigma^{j}H\mathbb{F}_{p}\rightarrow
A,y:\Sigma^{k}H\mathbb{F}_{p}\rightarrow A$ representing two homotopy classes
of a spectral partition Lie algebra $A$. The operation $[x,\alpha(y)]$ is
encoded by the weight $w+1$ part of the structure map
$\mathrm{Lie}^{\pi}_{\mathbb{F}_{p},\mathbb{E}_{\infty}}\circ\mathrm{Lie}^{\pi}_{\mathbb{F}_{p},\mathbb{E}_{\infty}}\rightarrow\mathrm{Lie}^{\pi}_{\mathbb{F}_{p},\mathbb{E}_{\infty}}$,
i.e.,

(6)
$\displaystyle\begin{split}\Sigma^{j+k+|\alpha|-1}H\mathbb{F}_{p}&\rightarrow
d_{2}\mathbin{\mathop{\otimes}\displaylimits}\Big{(}\big{(}d_{1}\mathbin{\mathop{\otimes}\displaylimits^{h\Sigma_{1}}}\Sigma^{j}H\mathbb{F}_{p}\big{)}\mathbin{\mathop{\otimes}\displaylimits}\big{(}d_{w}\mathbin{\mathop{\otimes}\displaylimits^{h\Sigma_{w}}}(\Sigma^{k}H\mathbb{F}_{p})^{\mathbin{\mathop{\otimes}\displaylimits}w}\big{)}\Big{)}\\\
&\xhookrightarrow{(\ref{liebracket})}\mathrm{wt}_{w+1}\Big{[}d_{2}\mathbin{\mathop{\otimes}\displaylimits^{h\Sigma_{2}}}\Big{(}\big{(}d_{1}\mathbin{\mathop{\otimes}\displaylimits^{h\Sigma_{1}}}\Sigma^{j}H\mathbb{F}_{p}\big{)}\oplus\big{(}d_{w}\mathbin{\mathop{\otimes}\displaylimits^{h\Sigma_{w}}}(\Sigma^{k}H\mathbb{F}_{p})^{\mathbin{\mathop{\otimes}\displaylimits}w}\big{)}\Big{)}^{\mathbin{\mathop{\otimes}\displaylimits}2}\Big{]}\\\
&\rightarrow
d_{w+1}\mathbin{\mathop{\otimes}\displaylimits^{h(\Sigma_{1}\times\Sigma_{w})}}(\Sigma^{j}H\mathbb{F}_{p}\otimes\Sigma^{kw}H\mathbb{F}_{p})\\\
&\hookrightarrow\mathrm{Lie}^{\pi}_{\mathbb{F}_{p},\mathbb{E}_{\infty}}(\Sigma^{j}H\mathbb{F}_{p}\oplus\Sigma^{k}H\mathbb{F}_{p})\rightarrow\mathrm{Lie}^{\pi}_{\mathbb{F}_{p},\mathbb{E}_{\infty}}(A)\rightarrow
A,\end{split}$

where we write $d_{n}$ for
$\partial_{n}(\mathrm{Id})\mathbin{\mathop{\otimes}\displaylimits}H\mathbb{F}_{p}$
for ease of notation. Note that the action obtained by restriction to
$\Sigma_{1}\times\Sigma_{w}\subset\Sigma_{w+1}$ on
$\partial_{w+1}(\mathrm{Id})$ is freely induced from the action of the trivial
subgroup on $\mathbb{S}^{-w}$. For any finite group $G$, we have
$(\mathrm{Ind}^{G}_{\\{e\\}}(X)\mathbin{\mathop{\otimes}\displaylimits}Y)^{hG}\simeq
X\mathbin{\mathop{\otimes}\displaylimits}Y$ for all $G$-spectra $Y$ by the
Wirthmüller isomorphism. Hence

$(\partial_{w+1}(\mathrm{Id})\mathbin{\mathop{\otimes}\displaylimits}H\mathbb{F}_{p})\mathbin{\mathop{\otimes}\displaylimits^{h(\Sigma_{1}\times\Sigma_{w})}}\Sigma^{j+kw}H\mathbb{F}_{p}\simeq\Sigma^{j+(k-1)w}H\mathbb{F}_{p}.$

In particular, the $\mathbb{F}_{p}$-module of weight $w+1$ operations on
spectral partition Lie algebras coming from the bracket of one weight one
operation and one weight $w$ operation is one-dimensional. Note that
$[[\cdots[[x,y],y]\cdots],y]$, where we take the bracket with $y$ exactly $w$
times, is a class of weight $w+1$ obtained by taking the bracket of $y$ with a
length $w$ bracket $[[\cdots[[x,y],y]\cdots],y]$ where we take the bracket
with $y$ exactly $w-1$ times in

$\displaystyle\mathrm{Free}^{\mathrm{sLie}^{\rho}_{\mathbb{F}_{p}}}\circ\mathrm{Free}^{\mathrm{sLie}^{\rho}_{\mathbb{F}_{p}}}(\Sigma^{j}\mathbb{F}_{p}\oplus\Sigma^{k}\mathbb{F}_{p})$
$\displaystyle\subset\mathrm{Free}^{\mathrm{sLie}^{\rho}_{\pazocal{R}^{!}}}\circ\mathrm{Free}^{\mathrm{sLie}^{\rho}_{\pazocal{R}^{!}}}(\Sigma^{j}\mathbb{F}_{p}\oplus\Sigma^{k}\mathbb{F}_{p})$
$\displaystyle\cong\pi_{*}(\mathrm{Lie}^{\pi}_{\mathbb{F}_{p},\mathbb{E}_{\infty}}\circ\mathrm{Lie}^{\pi}_{\mathbb{F}_{p},\mathbb{E}_{\infty}}(\Sigma^{j}H\mathbb{F}_{p}\oplus\Sigma^{k}H\mathbb{F}_{p}))\xrightarrow{\theta_{*}}\pi_{*}(A)$

when $p=2$ and

$\displaystyle\mathrm{Free}^{\mathrm{sLie}_{\mathbb{F}_{p}}}\circ\mathrm{Free}^{\mathrm{sLie}_{\mathbb{F}_{p}}}(\Sigma^{j}\mathbb{F}_{p}\oplus\Sigma^{k}\mathbb{F}_{p})$
$\displaystyle\subset\mathrm{Free}^{(\pazocal{R}^{\prime})^{!}}\mathrm{Free}^{\mathrm{sLie}_{\mathbb{F}_{p}}}\circ\mathrm{Free}^{(\pazocal{R}^{\prime})^{!}}\mathrm{Free}^{\mathrm{sLie}_{\mathbb{F}_{p}}}(\Sigma^{j}\mathbb{F}_{p}\oplus\Sigma^{k}\mathbb{F}_{p})$
$\displaystyle\cong\pi_{*}(\mathrm{Lie}^{\pi}_{\mathbb{F}_{p},\mathbb{E}_{\infty}}\circ\mathrm{Lie}^{\pi}_{\mathbb{F}_{p},\mathbb{E}_{\infty}}(\Sigma^{j}H\mathbb{F}_{p}\oplus\Sigma^{k}H\mathbb{F}_{p}))\xrightarrow{\theta_{*}}\pi_{*}(A)$

for $p>2$. In the free case this class is nonzero, so we obtain a generator
$\gamma$ of the $\mathbb{F}_{p}$-module of weight $w+1$ operations coming from
the bracket of one weight one operation and one weight $w$ operation.

If $\alpha$ represents an iteration of the self-bracket, then
$[x,\alpha(y)]=0$ by the Jacobi identity when $p\neq 3$. When $p=3$ and
$|x|=k$ is even, a degree count shows that $[x,[x,x]]=0$ since the weight 3
part of the $E^{2}$-page of the dual bar spectral sequence

$E^{2}\cong\mathrm{Free}^{(\pazocal{R}^{\prime})^{!}}\mathrm{Free}^{\mathrm{sLie}_{\mathbb{F}_{p}}}\circ\mathrm{Free}^{(\pazocal{R}^{\prime})^{!}}\mathrm{Free}^{\mathrm{sLie}_{\mathbb{F}_{p}}}(\Sigma^{k}\mathbb{F}_{p})\Rightarrow\pi_{*}(\mathrm{Lie}^{\pi}_{\mathbb{F}_{p},\mathbb{E}_{\infty}}\circ\mathrm{Lie}^{\pi}_{\mathbb{F}_{p},\mathbb{E}_{\infty}}(\Sigma^{k}H\mathbb{F}_{p}))$

has nothing in total degree $3k-2$.

Suppose that $\alpha$ is not an iteration of the self-bracket. If $\alpha$ is
not an iteration of the bottom operation $R^{-|x|+1}$ on $x$ when $p=2$, or an
iteration of the bottom operation $R^{(-|x|+1)/2}$ on odd $x$ when $p>2$. Then
a comparison of topological degrees shows that $[x,\alpha(y)]$ has strictly
smaller topological degree than the generator
$\gamma=[[\cdots[[x,y],y]\cdots],y]$ of weight $w+1$ operations coming from
the bracket of one weight one operation and one weight $w$ operation.
Therefore it has to be zero.

If $p=2$ and $\alpha$ is the bottom operation $R^{-|y|+1}$ on $y$, then we
know that $[x,(Q^{-|y|})^{*}(y)]=[[x,y],y]$ on the $E^{\infty}\cong
E^{2}$-page of the dual bar spectral sequence converging to
$\pi_{*}(\mathrm{Lie}^{\pi}_{\mathbb{F}_{2},\mathbb{E}_{\infty}}(\Sigma^{j}H\mathbb{F}_{2}\oplus\Sigma^{k}H\mathbb{F}_{2})))$
by Proposition 4.9. This is the image of $[x,(Q^{-|y|})^{*}(y)]$ under the
summand

$\mathrm{Free}^{\mathrm{sLie}_{\mathbb{F}_{2}}}\circ\mathrm{Free}^{\mathrm{sLie}^{\rho}_{\pazocal{R}^{!}}}(\Sigma^{j}\mathbb{F}_{p}\oplus\Sigma^{k}\mathbb{F}_{p})\rightarrow\mathrm{Free}^{\mathrm{sLie}^{\rho}_{\pazocal{R}^{!}}}(\Sigma^{j}\mathbb{F}_{p}\oplus\Sigma^{k}\mathbb{F}_{p})$

of the map of $E^{2}$-pages of the dual bar spectral sequences for the monad
composition map

$\mathrm{Lie}^{\pi}_{\mathbb{F}_{p},\mathbb{E}_{\infty}}\circ\mathrm{Lie}^{\pi}_{\mathbb{F}_{p},\mathbb{E}_{\infty}}(\Sigma^{j}H\mathbb{F}_{p}\oplus\Sigma^{k}H\mathbb{F}_{p})\rightarrow\mathrm{Lie}^{\pi}_{\mathbb{F}_{p},\mathbb{E}_{\infty}}(\Sigma^{j}H\mathbb{F}_{p}\oplus\Sigma^{k}H\mathbb{F}_{p}).$

Passing to the $E^{\infty}$-pages of the dual bar spectral sequences of the
composition map, we deduce that $[x,R^{-|y|+1}(y)]=\gamma=[[x,y],y],$ i.e.,
the bottom operation serves as the restriction for the bracket. By induction
we conclude that if $\alpha$ is the $n$-th iteration of the restriction, then
$[x,\alpha(y)]=\gamma$ in weight $2n+1$. Note that there is only one unary
operation of degree $-|x|+1$ on any class $x$ up to a scalar, and the
restriction is such an unary operation. Hence we deduce that $[x,x]=0$ since
taking self-bracket is an additive operation while the restriction is not. ∎

When $p>2$, we expect to see a shifted restricted Lie algebra structure on the
$E^{\infty}$-page analogous to the case $p=2$ with the restriction given by
the bottom operation, as was noted in [BCN21, Remark 4.49] and by Basterra and
Mandell, cf. [Law20, Example 1.8.8].

######  Definition 5.11.

(cf. [Jac41], [Fre00]) A shifted restricted Lie algebra over $\mathbb{F}_{p}$,
denoted as a $\mathrm{sLie}^{\rho}_{\mathbb{F}_{p}}$-algebra, is a graded
$\mathbb{F}_{p}$-module $L=L_{\bullet}$ with a shifted Lie bracket
$L_{m}\mathbin{\mathop{\otimes}\displaylimits}L_{n}\rightarrow L_{m+n-1}$ and
a restriction map $x\mapsto x^{[p]}$ with $x^{[p]}\in L_{p|x|-p+1}$ whenever
$|x|$ is odd, satisfying the following identities:

  1. (1)

$(cx)^{[p]}=c^{p}x^{[p]}$ for all odd degree $x\in L$ and
$c\in\mathbb{F}_{p}$;

  2. (2)

$\mathrm{ad}(x^{[p]})=\mathrm{ad}(x)$ for all odd degree $x\in L$;

  3. (3)

For all odd degree $x,y\in L$,
$(x+y)^{[p]}=x^{[p]}+y^{[p]}+\sum_{i=1}^{p-1}\frac{s_{i}}{i}(x,y)$, where
$s_{i}$ is the coefficient of $t^{i-1}$ in the formal expression
$\mathrm{ad}(tx+y)^{p-1}(x)$.

Here $\mathrm{ad}(x)$ stands for the self-map $y\mapsto[y,x]$ on $L$.

######  Lemma 5.12.

If $j$ is odd, then
$\overline{\mathrm{TAQ}}^{-*}(\Sigma^{-j}H\mathbb{F}_{p})\cong\pi_{*}(\mathrm{Lie}^{\pi}_{\mathbb{F}_{p},\mathbb{E}_{\infty}}(\Sigma^{j}H\mathbb{F}_{p}))$
admits a restriction $x\mapsto x^{[p]}$ that coincides with the bottom
operation $R^{(-j+1)/2}$ on the generator $x$ in degree $j$ up to a unit
$\lambda_{j}$ that depends only on $j$. If $j$ is even, such a map does not
exist. In general the shifted Lie bracket on the homotopy groups of spectral
partition Lie algebras admits a restriction that on any odd degree class $x$
is represented by the bottom operation $R^{(-|x|+1)/2}$ up to a unit
$\lambda_{|x|}$.

###### Proof.

It suffices to check the cases of single generators and two generators. Since
the category of $H\mathbb{F}_{p}$-modules is equivalent to the derived
category of chain complexes over $\mathbb{F}_{p}$ ([Lur17, 7.1.1.16]), we can
think of
$\pi_{s+t}(\mathbb{D}|\mathrm{Bar}_{\bullet}(\mathrm{id},\mathbb{P},A)|)$ as
the homology of the chain complex

$C=C_{*}(\mathbb{D}|\mathrm{Bar}_{\bullet}(\mathrm{id},\mathbb{P},A)|)\cong\bigoplus_{n}(\mathrm{sLie}(n)\mathbin{\mathop{\otimes}\displaylimits}(A^{\vee})^{\mathbin{\mathop{\otimes}\displaylimits}n})^{h\Sigma_{n}}$

when $A$ is bounded by [BCN21, Remark 4.49], and its homotopy group is a
shifted restricted Lie algebra. Here $\mathrm{sLie}$ is the shifted Lie operad
in the dg-category, with $\mathrm{sLie}(r)$ concentrated in dimension $1-r$,
weight r. It remains to show that the restriction is nonzero and identify the
restriction on the generator $x$ of $\Sigma^{j}\mathbb{F}_{p}$.

There is a homotopy fixed points spectral sequence

$E^{2}_{s,t}=H^{s}\Big{(}\Sigma_{p},\pi_{t}\big{(}\mathrm{sLie}(p)\mathbin{\mathop{\otimes}\displaylimits}(\Sigma^{j}\mathbb{F}_{p})^{\mathbin{\mathop{\otimes}\displaylimits}p}\big{)}\Big{)}\Rightarrow\pi_{t-s}\Big{(}\big{(}\mathrm{sLie}(p)\mathbin{\mathop{\otimes}\displaylimits}(\Sigma^{j}H\mathbb{F}_{p})^{\mathbin{\mathop{\otimes}\displaylimits}p}\big{)}^{h\Sigma_{p}}\Big{)},$

where $\Sigma^{j}\mathbb{F}_{p}$ is considered as a one-dimensional chain
complex over $\mathbb{F}_{p}$ concentrated in homological degree $j$ with no
differentials. Since
$\pi_{t}(\mathrm{sLie}(p)\mathbin{\mathop{\otimes}\displaylimits}(\Sigma^{j}\mathbb{F}_{p})^{\mathbin{\mathop{\otimes}\displaylimits}p})=0$
unless $t=pj+1-p$, the $E^{2}$-page of the homotopy fixed points spectral is
concentrated on a single line $t=pj+1-p$. Hence the spectral sequence
collapses and there are no extension problems.

Taking $s=0$, we deduce that

$\pi_{pn+1-p}\big{(}\mathrm{sLie}(p)\mathbin{\mathop{\otimes}\displaylimits}(\Sigma^{j}\mathbb{F}_{p})^{\mathbin{\mathop{\otimes}\displaylimits}p}\big{)}\cong
H^{0}\big{(}\Sigma_{p};\mathrm{sLie}(p)\mathbin{\mathop{\otimes}\displaylimits}(\Sigma^{j}\mathbb{F}_{p})^{\mathbin{\mathop{\otimes}\displaylimits}p}\big{)},$

where the right hand side has coefficients in $\mathrm{Mod}_{\mathbb{F}_{p}}$.
By [Fre00, Theorem 1.2.5], if $j$ is odd then

$H^{0}\big{(}\Sigma_{p};\mathrm{sLie}(p)\mathbin{\mathop{\otimes}\displaylimits}(\Sigma^{j}\mathbb{F}_{p})^{\mathbin{\mathop{\otimes}\displaylimits}p})\big{)}\cong\big{(}\mathrm{sLie}(p)\mathbin{\mathop{\otimes}\displaylimits}(\Sigma^{j}\mathbb{F}_{p})^{\mathbin{\mathop{\otimes}\displaylimits}p}\big{)}^{\Sigma_{p}}\cong\big{(}\Sigma^{-1}\mathrm{Lie}(p)\mathbin{\mathop{\otimes}\displaylimits}(\Sigma^{j+1}\mathbb{F}_{p})^{\mathbin{\mathop{\otimes}\displaylimits}p}\big{)}^{\Sigma_{p}}$

contains an element that serves as the restriction $x^{[p]}$ in the free
shifted restricted Lie algebra on $\Sigma^{j}\mathbb{F}_{p}$. Since $x^{[p]}$
is in topological degree $pj+1-p$ and the only element in the weight $p$ part
of
$\pi_{pj+1-p}(\mathrm{Lie}^{\pi}_{\mathbb{F}_{p},\mathbb{E}_{\infty}}(\Sigma^{j}H\mathbb{F}_{p}))$
comes from the bottom operation $(\beta Q^{(-j+1)/2})^{*}(x)$ up to a unit
$\lambda$ on the $E^{2}$-page of the dual bar spectral sequence, we conclude
that the restriction on $x$ is given by the image of $\lambda R^{(-j+1)/2}(x)$
in
$\pi_{pj+1-p}(\mathrm{Lie}^{\pi}_{\mathbb{F}_{p},\mathbb{E}_{\infty}}(\Sigma^{j}H\mathbb{F}_{p}))$.
This unit $\lambda$ is fixed for any class of a given degree $j$ by
functoriality of the restriction map and the bottom Dyer-Lashof operation on
an odd class $x$ as the $p$-fold Massey product on $x$.

The class $x$ does not admit a restriction when $j$ is even because
$\pi_{pj+1-p}(\mathrm{Lie}^{\pi}_{\mathbb{F}_{p},\mathbb{E}_{\infty}}(\Sigma^{j}H\mathbb{F}_{p}))=0$,
which is as expected for a shifted restricted Lie algebra.

Next we take $A=\Sigma^{j}H\mathbb{F}_{p}\oplus\Sigma^{k}H\mathbb{F}_{p}$,
with $j$ odd. There is a homotopy fixed points spectral sequence

$E^{2}_{s,t}=\bigoplus_{n}H^{s}\Big{(}\Sigma_{n},\pi_{t}\big{(}\mathrm{sLie}(n)\mathbin{\mathop{\otimes}\displaylimits}(\Sigma^{j}\mathbb{F}_{p}\oplus\Sigma^{k}\mathbb{F}_{p})^{\mathbin{\mathop{\otimes}\displaylimits}n}\big{)}\Big{)}\Rightarrow\bigoplus_{n}\pi_{t-s}\Big{(}\big{(}\mathrm{sLie}(n)\mathbin{\mathop{\otimes}\displaylimits}(\Sigma^{j}H\mathbb{F}_{p}\oplus\Sigma^{k}H\mathbb{F}_{p})^{\mathbin{\mathop{\otimes}\displaylimits}n}\big{)}^{h\Sigma_{n}}\Big{)}.$

On the line $s=0$, we have

$\bigoplus_{n}H^{0}\big{(}\Sigma_{n};\mathrm{sLie}(n)\mathbin{\mathop{\otimes}\displaylimits}(\Sigma^{j}\mathbb{F}_{p}\oplus\Sigma^{k}\mathbb{F}_{p})^{\mathbin{\mathop{\otimes}\displaylimits}n})\big{)}\cong\bigoplus_{n}\big{(}\mathrm{sLie}(n)\mathbin{\mathop{\otimes}\displaylimits}(\Sigma^{j}\mathbb{F}_{p}\oplus\Sigma^{k}\mathbb{F}_{p})^{\mathbin{\mathop{\otimes}\displaylimits}n}\big{)}^{\Sigma_{n}},$

which is the free shifted restricted Lie algebra on the
$\mathbb{F}_{p}$-module
$\Sigma^{j}\mathbb{F}_{p}\oplus\Sigma^{k}\mathbb{F}_{p}$ by [Fre00, Theorem
1.2.5]. A prior the bracket on the $E^{2}$-page of the homotopy fixed points
spectral sequence survives to a bracket on the $E^{\infty}$-page that agrees
with the shifted Lie bracket $[-,-]$ in Construction 5.8 up to a nonzero
scalar $c$. Hence we choose a generator for $\mathrm{sLie}(2)$ so that $c=1$,
and by abuse of notation we also denote the bracket on the $E^{2}$-page by
$[-,-]$. The $(p+1)$-th summand of the spectral sequence collapses on the
$E^{2}$-page with no extension problems, since the group cohomology of
$\Sigma_{n}$ with coefficients in $\mathrm{Mod}_{\mathbb{F}_{p}}$ is
concentrated in degree 0 when $n$ is coprime to $p$. From the computation on
one generator, we deduce that the restriction $x^{[p]}$ on the generator $x$
of $\Sigma^{j}\mathbb{F}_{p}$ on the $E^{2}$-page of the homotopy fixed points
spectral sequence survives to the element $\lambda_{j}R^{(-j+1)/2}(x)$ with
$\lambda_{j}$ the unit given in the first part of the proof. Furthermore, the
identity $[y,x^{[p]}]=[[\cdots[[y,x],x]\cdots],x]$ on the $E^{2}$-page
survives to an identity in
$\pi_{pj+k-p}(\mathrm{Lie}^{\pi}_{\mathbb{F}_{p},\mathbb{E}_{\infty}}(\Sigma^{j}H\mathbb{F}_{p}\oplus\Sigma^{k}H\mathbb{F}_{p}))$.

Similarly, suppose that $j$ and $k$ are both odd and let $x,y$ represent the
geneorator of $\Sigma^{j}\mathbb{F}_{p}$ and $\Sigma^{k}\mathbb{F}_{p}$ on the
$E^{2}$-page above. Then the identity
$(x+y)^{[p]}=x^{[p]}+y^{[p]}+\sum_{i=1}^{p-1}\frac{s_{i}}{i}(x,y)$ on the line
$s=0$ of the $E^{2}$-page survives to on identity
$(x+y)^{[p]}=\lambda_{j}R^{(-j+1)/2}(x)+\lambda_{k}R^{(-k+1)/2}(y)+\sum_{i=1}^{p-1}\frac{s_{i}}{i}(x,y)$
on the $E^{\infty}$-page via the collapse of the homotopy fixed points
spectral sequence, where $s_{i}$ is the coefficient of $t^{i-1}$ in the formal
expression $\mathrm{ad}(tx+y)^{p-1}(x)$. In particular, if $j=k$ then

$\lambda_{j}R^{(-j+1)/2}(x+y)=\lambda_{j}R^{(-j+1)/2}(x)+\lambda_{j}R^{(-j+1)/2}(y)+\sum_{i=1}^{p-1}\frac{s_{i}}{i}(x,y),$

so the bottom operation on an odd class is not additive in general even though
it lifts to an additive operation on the $E^{2}$-page of the dual bar spectral
sequence.

Finally, we want to show that the collection
$x^{[p]}:=\lambda_{j}R^{(-|x|+1)/2}(x)$ for $|x|=j$ odd and $\lambda_{j}$ a
unit depending only on $j$ defines a restriction map $(-)^{[p]}$ for the
shifted Lie bracket on the homotopy groups of spectral partition Lie algebras
by extending to linear sums of classes $x,y$ with $|x|=j\neq|y|=k$ odd via
$(x+y)^{[p]}:=\lambda_{j}R^{(-j+1)/2}(x)+\lambda_{k}R^{(-k+1)/2}(y)+\sum_{i=1}^{p-1}\frac{s_{i}}{i}(x,y)$.
The $p$th iteration of $[-,x]$ on a class $y$ is encoded by a summand in the
weight $p+1$ part of the iterated monad composition

$(\mathrm{Lie}^{\pi}_{\mathbb{F}_{p},\mathbb{E}_{\infty}})^{\circ
p}\rightarrow(\mathrm{Lie}^{\pi}_{\mathbb{F}_{p},\mathbb{E}_{\infty}})^{\circ
p-1}\rightarrow\cdots\rightarrow\mathrm{Lie}^{\pi}_{\mathbb{F}_{p},\mathbb{E}_{\infty}}$

applied to $H\Sigma^{j}\mathbb{F}_{p}\oplus H\Sigma^{k}\mathbb{F}_{p}$.
Explicitly, this summand is the $(p-1)$-th iteration of
$\partial_{2}(\mathrm{Id})\mathbin{\mathop{\otimes}\displaylimits}(\partial_{1}(\mathrm{Id})\mathbin{\mathop{\otimes}\displaylimits}(-))$
on $\partial_{2}(\mathrm{Id})$. Note that the last step
$\mathrm{Lie}^{\pi}_{\mathbb{F}_{p},\mathbb{E}_{\infty}}\circ\mathrm{Lie}^{\pi}_{\mathbb{F}_{p},\mathbb{E}_{\infty}}\rightarrow\mathrm{Lie}^{\pi}_{\mathbb{F}_{p},\mathbb{E}_{\infty}}$
of the above chain of compositions is

$\displaystyle(\partial_{2}(\mathrm{Id})\mathbin{\mathop{\otimes}\displaylimits}H\mathbb{F}_{p})\mathbin{\mathop{\otimes}\displaylimits}\Big{(}\big{(}(\partial_{1}(\mathrm{Id})\mathbin{\mathop{\otimes}\displaylimits}H\mathbb{F}_{p})\mathbin{\mathop{\otimes}\displaylimits^{h\Sigma_{1}}}\Sigma^{j}H\mathbb{F}_{p}\big{)}\mathbin{\mathop{\otimes}\displaylimits}\big{(}(\partial_{p}(\mathrm{Id})\mathbin{\mathop{\otimes}\displaylimits}H\mathbb{F}_{p})\mathbin{\mathop{\otimes}\displaylimits^{h\Sigma_{p}}}(\Sigma^{k}H\mathbb{F}_{p})^{\mathbin{\mathop{\otimes}\displaylimits}p}\big{)}\Big{)}$
$\displaystyle\rightarrow$
$\displaystyle(\partial_{p+1}(\mathrm{Id})\mathbin{\mathop{\otimes}\displaylimits}H\mathbb{F}_{p})\mathbin{\mathop{\otimes}\displaylimits^{h(\Sigma_{1}\times\Sigma_{p})}}(\Sigma^{j}H\mathbb{F}_{p}\otimes\Sigma^{kp}H\mathbb{F}_{p})$

as in (6), which we showed to be one-dimensional in Proposition 5.10 with
$[[\cdots[[y,x],x]\cdots],x]$ where $x$ appears $p$ times a generator. The
monad composition induces a map of homotopy fixed points spectral sequences,
both of which collapse on the $E^{2}$-page at weight $p+1$ with no extension
problems. Hence we get a map that is the weight $p+1$ part of

$\bigoplus_{n}\Big{(}\mathrm{sLie}(n)\mathbin{\mathop{\otimes}\displaylimits}\big{(}\pi_{*}(\bigoplus_{m}\mathrm{sLie}(m)\mathbin{\mathop{\otimes}\displaylimits^{h\Sigma_{m}}}(\Sigma^{j}\mathbb{F}_{p}\oplus\Sigma^{k}\mathbb{F}_{p})^{\mathbin{\mathop{\otimes}\displaylimits}m})\big{)}^{\mathbin{\mathop{\otimes}\displaylimits}n}\Big{)}^{\Sigma_{n}}\rightarrow\bigoplus_{n}\Big{(}\mathrm{sLie}(n)\mathbin{\mathop{\otimes}\displaylimits}(\Sigma^{j}\mathbb{F}_{p}\oplus\Sigma^{k}\mathbb{F}_{p})^{\mathbin{\mathop{\otimes}\displaylimits}n}\Big{)}^{\Sigma_{n}}$

along the line $s=0$ on the $E^{2}$-pages. This map has as summand

$\displaystyle\mathrm{sLie}(2)\mathbin{\mathop{\otimes}\displaylimits^{\Sigma_{2}}}\Big{(}\big{(}\mathrm{sLie}(1)\mathbin{\mathop{\otimes}\displaylimits}(\Sigma^{j}\mathbb{F}_{p}\oplus\Sigma^{k}\mathbb{F}_{p})^{\mathbin{\mathop{\otimes}\displaylimits}1}\big{)}^{\Sigma_{1}}\oplus\big{(}\mathrm{sLie}(p)\mathbin{\mathop{\otimes}\displaylimits}(\Sigma^{j}\mathbb{F}_{p}\oplus\Sigma^{k}\mathbb{F}_{p})^{\mathbin{\mathop{\otimes}\displaylimits}p}\big{)}^{\Sigma_{p}}\Big{)}^{\mathbin{\mathop{\otimes}\displaylimits}2}$
$\displaystyle\rightarrow\big{(}\mathrm{sLie}(p+1)\mathbin{\mathop{\otimes}\displaylimits}(\Sigma^{j}\mathbb{F}_{p}\oplus\Sigma^{k}\mathbb{F}_{p})^{\mathbin{\mathop{\otimes}\displaylimits}p+1}\big{)}^{\Sigma_{p+1}},$

with further summand

$\displaystyle\phi:\mathrm{sLie}(2)\mathbin{\mathop{\otimes}\displaylimits}\Big{(}\Sigma^{j}\mathbb{F}_{p}\mathbin{\mathop{\otimes}\displaylimits}\big{(}\mathrm{sLie}(p)\mathbin{\mathop{\otimes}\displaylimits}(\Sigma^{k}\mathbb{F}_{p})^{\mathbin{\mathop{\otimes}\displaylimits}p}\big{)}^{\Sigma_{p}}\Big{)}\rightarrow\big{(}\mathrm{sLie}(p+1)\mathbin{\mathop{\otimes}\displaylimits}(\Sigma^{j}\mathbb{F}_{p}\oplus\Sigma^{k}\mathbb{F}_{p})^{\mathbin{\mathop{\otimes}\displaylimits}p+1}\big{)}^{\Sigma_{p+1}}.$

The map $\phi$ agrees with the construction 5.8 on the $E^{\infty}$-page,
i.e., it is the evaluation of the free shifted restricted Lie bracket. The
image of $[y,R^{(-|x|+1)/2}(x)]$ under $\phi$ is
$[y,x^{[p]}]=[[\cdots[[y,x],x]\cdots],x]$ up to a unit $\lambda_{|x|}$. Hence
on the $E^{\infty}$-page we have
$[y,\lambda_{|x|}R^{(-|x|+1)/2}(x)]=[[\cdots[[y,x],x]\cdots],x]$ as desired. ∎

To sum up, we have the following theorem.

######  Theorem 5.13.

The binary operation $[-,-]$ constructed above equips the homotopy groups of
any spectral partition Lie algebra $A$ with a shifted restricted Lie algebra
bracket

$[-,-]:\pi_{j}(A)\mathbin{\mathop{\otimes}\displaylimits}\pi_{k}(A)\rightarrow\pi_{j+k-1}(A)$

over $\mathbb{F}_{p}$. If $p=2$, for all $j$ and $x\in\pi_{j}(A)$ the
restriction $x^{[2]}$ is represented by the bottom operation $R^{-j+1}(x)$.
The restriction map on a sum of classes $x,y$ in degrees $j\neq k$ is given by

$(x+y)^{[2]}=\lambda_{j}R^{-j+1}(x)+\lambda_{k}R^{-k+1}(y)+[x,y].$

If $p>2$, for all odd $j$ and $x\in\pi_{j}(A)$ the restriction $x^{[p]}$ is
the bottom operation $R^{(-j+1)/2}(x)$ up to a unit $\lambda_{j}$. The
restriction map on a sum of classes $x,y$ in degrees $j\neq k$ are given by

$(x+y)^{[p]}=\lambda_{j}R^{(-j+1)/2}(x)+\lambda_{k}R^{(-k+1)/2}(y)+\sum_{i=1}^{p-1}\frac{s_{i}}{i}(x,y),$

where $s_{i}$ is the coefficient of $t^{i-1}$ in the formal expression
$\mathrm{ad}(tx+y)^{p-1}(x)$.

The bracket is compatible with the unary operations in Theorem 5.5 and 5.6 in
the sense that $[x,\alpha(y)]=0$ for $x,y\in\pi_{*}(A)$ and any unary
operations $\alpha$ of positive weight that is not an iteration of the
restriction map. Equivalently, for any
$\mathbb{E}^{\mathrm{nu}}_{\infty}$-$H\mathbb{F}_{p}$-algebra $A$, there is a
shifted Lie bracket with restriction

$[-,-]:\overline{\mathrm{TAQ}}^{j}(A)\mathbin{\mathop{\otimes}\displaylimits}\overline{\mathrm{TAQ}}^{k}(A)\rightarrow\overline{\mathrm{TAQ}}^{j+k+1}(A)$

satisfying the above conditions.

######  Remark 5.14.

The interaction between the unary operations and the shifted Lie bracket on
the homotopy groups of spectral partition Lie algebras differ from that on the
homology of spectral Lie algebras. It was shown in [AC20, Br17, Kja18] that on
the mod $p$ homology groups and the Lubin-Tate theory of spectral Lie
algebras, the bracket $[x,\alpha(y)]$ always vanishes if $\alpha$ is a unary
operation of positive weight. Note that unary operations on spectral Lie
algebras are all additive. In comparison, on the homotopy groups of spectral
partition Lie algebras the bracket $[x,\alpha(y)]$ does not necessarily vanish
when $\alpha$ is an iteration of the restriction map, i.e., a non-additive
unary operation. For instance, when $p=2$, the bottom non-vanishing operation
$\bar{Q}^{|x|}$ on a mod 2 homology class $x$ of a free spectral Lie algebra
is identified with the nonzero self-bracket $[x,x]$ by [AC20, Lemma 6.4].
Hence $[y,\bar{Q}^{|x|}(x)]=0$ by the Jacobi identity for all $x,y$. In
comparison, the bottom operation $R^{-|x|+1}$ on a class in the homotopy group
of a free spectral partition Lie algebra over $H\mathbb{F}_{2}$ represents the
restriction on $x$, so $[y,R^{-|x|+1}(x)]=[[y,x],x]$ is nonzero. Whereas self-
brackets always vanish in shifted restricted Lie algebras over
$\mathbb{F}_{2}$, cf. [Fre00, Remark 1.2.9].

###  5.3. Generation

Finally we put all the structures together to obtain the optimal target
category for the homotopy group of spectral partition Lie algebras, or
equivalently the reduced mod $p$ TAQ cohomology.

######  Definition 5.15.

A $\pazocal{P}$-$\mathrm{sLie}^{\rho}$-algebra $L$ is a module over the power
ring $\pazocal{P}$, together with a shifted Lie bracket and a restriction
$(-)^{[p]}$, that satisfies the following conditions:

(1) If $p=2$, for all $j$ and $x\in\pi_{j}(A)$ the restriction $x^{[2]}$ is
given by the bottom operation $R^{-j+1}(x)$. The restriction map on a sum of
classes $x$ and $y$ is given by

$(x+y)^{[2]}=R^{-j+1}(x)+R^{-k+1}(y)+[x,y].$

If $p>2$, for all odd $j$ and $x\in\pi_{j}(A)$ the restriction $x^{[p]}$ is up
to a unit $\lambda_{j}$ the bottom operation $R^{(-j+1)/2}(x)$. The
restriction map on the sum of classes $x,y$ in degrees $j\neq k$ are given by

$(x+y)^{[p]}=\lambda_{j}R^{(-j+1)/2}(x)+\lambda_{k}R^{(-k+1)/2}(y)+\sum_{i=1}^{p-1}s_{i}(x,y),$

where $s_{i}$ is the coefficient of $t^{i-1}$ in the formal expression
$\mathrm{ad}(tx+y)^{p-1}(x)$;

(2) The bracket $[y,\alpha(x)]$ vanishes for any $x,y\in L$ and $\alpha$ a
unary operation of positive weight, unless $\alpha$ is an iteration of the
restriction map.

Denote by $\mathrm{sLie}^{\rho}_{\pazocal{P}}$ the category of
$\pazocal{P}$-$\mathrm{sLie}^{\rho}$-algebras.

Hence the homotopy group every spectral partition Lie algebra, or the reduced
TAQ cohomology of any $\mathbb{E}_{\infty}$-$H\mathbb{F}_{p}$-algebra, has the
structure of an $\mathrm{sLie}^{\rho}_{\pazocal{P}}$-algebra. The free
$\pazocal{P}$-$\mathrm{sLie}^{\rho}$-algebra functor
$\mathrm{Free}^{\mathrm{sLie}^{\rho}_{\pazocal{P}}}$ on a
$\mathbb{F}_{p}$-module $M$ can be computed as follows: first we take the free
shifted restricted Lie algebra over $\mathbb{F}_{p}$, then take the free
$\pazocal{P}$-module on
$\mathrm{Free}^{\mathrm{sLie}^{\rho}_{\mathbb{F}_{p}}}(M)$. If $p=2$ then we
define the bottom operation $R^{-|x|+1}(x)$ to be the restriction $x^{[2]}$;
if $p>2$, we identify the restriction $x\mapsto x^{[p]}$ with the bottom
operation $R^{(-|x|+1)/2}(x)$ up to a unit $\lambda_{|x|}$ for any odd degree
$x$. Finally we extend the shifted Lie bracket and the restriction map to the
quotient of
$\mathrm{Free}^{\pazocal{P}}\mathrm{Free}^{\mathrm{sLie}^{\rho}_{\mathbb{F}_{p}}}(M)$
by the above identification, subject to the conditions in Definition 5.16.

Note that when $p>2$, given an $\mathbb{F}_{p}$-module $M$ with basis
$\\{x_{1},\ldots,x_{k}\\}$, a basis for
$\mathrm{Free}^{\mathrm{sLie}^{\rho}_{\mathbb{F}_{p}}}(M)$ is given by

$\\{v\\}\cup\\{u,u^{[p]},(u^{[p]})^{[p]},\ldots\\},$

where $u$ ranges over shifted brackets represented by Lyndon words in letters
$x_{1},\ldots,x_{k}$ with odd degree, and $v$ those with even degree. (See,
for instance, [BKS05, section 2].) Proposition 4.9 and 4.15 immediately imply
the following.

######  Corollary 5.16.

The canonical map of $\pazocal{P}$-$\mathrm{sLie}^{\rho}$-algebras

$\alpha:\mathrm{Free}^{\mathrm{sLie}^{\rho}_{\pazocal{P}}}\pi_{*}(A)\rightarrow\pi_{*}(\mathrm{Lie}^{\pi}_{\mathbb{F}_{p},\mathbb{E}_{\infty}}(A))$

is an isomorphism when $A$ is any direct sum of shifts of $H\mathbb{F}_{p}$’s.

Hence we have identified the target category for the homotopy groups of
spectral partition Lie algebras and the reduced TAQ cohomology of any
$\mathbb{E}_{\infty}$-$H\mathbb{F}_{p}$-algebra.

##  6\. Operations on mod $p$ TAQ cohomology

In the last section, we record a computation of all natural operations on the
mod $p$ TAQ cohomology $\mathrm{TAQ}^{*}(-,\mathbb{S};H\mathbb{F}_{p})$ of
$\mathbb{E}_{\infty}$-$\mathbb{S}$-algebras, as well as determining their
relations. The results in this section are largely inspired by conversations
with Tyler Lawson.

Recall from [Law20, 1.8] that the mod $p$ TAQ homology of an
$\mathbb{E}_{\infty}$-$\mathbb{S}$-algebra $R$ can be computed by

$\mathrm{TAQ}_{*}(R,\mathbb{S};H\mathbb{F}_{p})\simeq\pi_{*}(|\mathrm{Bar}_{\bullet}(H\mathbb{F}_{p}\mathbin{\mathop{\otimes}\displaylimits}\mathrm{id},\mathbb{E}_{\infty},R)|).$

For any $\mathbb{E}^{\mathrm{nu}}_{\infty}$-$\mathbb{S}$-algebra $A$, the
reduced mod $p$ TAQ cohomology of an
$\mathbb{E}_{\infty}$-$\mathbb{S}$-algebra $\mathbb{S}\oplus A$ is the same as
the mod $p$ TAQ cohomology groups

$\overline{\mathrm{TAQ}}^{n}(A,\mathbb{S};H\mathbb{F}_{p}):=[\Sigma^{-n}|\mathrm{Bar}_{\bullet}(\mathrm{id},\mathbb{E}^{\mathrm{nu}}_{\infty},A)|,H\mathbb{F}_{p}]_{\mathrm{Sp}}.$

When $A$ is of finite type,
$\overline{\mathrm{TAQ}}^{*}(A,\mathbb{S};H\mathbb{F}_{p})\simeq\pi_{-*}(\mathbb{D}|\mathrm{Bar}_{\bullet}(H\mathbb{F}_{p}\mathbin{\mathop{\otimes}\displaylimits}\mathrm{id},\mathbb{E}^{\mathrm{nu}}_{\infty},A)|).$
Since all operations vanish on the unit of the mod $p$ TAQ cohomology except
for multiplication by units, we will again compute operations on
$\mathrm{TAQ}^{*}(-,\mathbb{S};H\mathbb{F}_{p})$ by throwing away the base
point and computing the dual bar spectral sequence on representing objects.

######  Corollary 6.1.

Unary operations on a degree $j$ cohomology class in the reduced mod $p$
$\mathbb{S}$-linear TAQ cohomology
$\overline{\mathrm{TAQ}}^{*}(A,\mathbb{S};H\mathbb{F}_{p})$ of
$\mathbb{E}^{\mathrm{nu}}_{\infty}$-$\mathbb{S}$-algebras $A$ are parametrized
by the free ${\pazocal{P}}$-$\mathrm{sLie}^{\rho}$-algebra
$\mathrm{Free}^{\mathrm{sLie}^{\rho}_{\pazocal{P}}}(\Sigma^{-j}\pazocal{A})$,
where $\pazocal{A}$ is the mod p Steenrod algebra with homological grading.

In general, for any tuple $(i_{1},\ldots i_{k})$, the $k$-ary cohomology
operations

$\prod^{k}_{i=1}\mathrm{TAQ}^{i_{l}}(-,\mathbb{S};H\mathbb{F}_{p})\rightarrow\mathrm{TAQ}^{m}(-,\mathbb{S};H\mathbb{F}_{p})$

away from the unit are parametrized by the homological degree $-m$ part of

$\mathrm{Free}^{\mathrm{sLie}^{\rho}_{\pazocal{P}}}(\Sigma^{-i_{1}}\pazocal{A}\oplus\cdots\oplus\Sigma^{-i_{k}}\pazocal{A}).$

###### Proof.

The representing objects for the mod $p$ TAQ cohomology functor
$\mathrm{TAQ}^{*}(-,\mathbb{S};H\mathbb{F}_{p})$ are the trivial square-zero
extensions
$\mathbb{S}\oplus\Sigma^{i_{1}}H\mathbb{F}_{p}\oplus\cdots\Sigma^{i_{n}}H\mathbb{F}_{p}$.
To compute the unary operations, we plug in the trivial algebras
$\mathrm{S}\oplus\Sigma^{j}H\mathbb{F}_{p}$. There is a base change formula

$\mathrm{TAQ}(R,\mathbb{S};H\mathbb{F}_{p})\mathbin{\mathop{\otimes}\displaylimits_{\mathbb{S}}}H\mathbb{F}_{p}\simeq\mathrm{TAQ}(R\mathbin{\mathop{\otimes}\displaylimits_{\mathbb{S}}}H\mathbb{F}_{p},H\mathbb{F}_{p};H\mathbb{F}_{p}),$

so unary operations on a degree $j$ cohomology class are parametrized by the
reduced mod $p$ TAQ cohomology
$\overline{\mathrm{TAQ}}^{*}(\Sigma^{j}H\mathbb{F}_{p}\mathbin{\mathop{\otimes}\displaylimits}H\mathbb{F}_{p},H\mathbb{F}_{p};H\mathbb{F}_{p})$.

It follows from the limiting case of Proposition 4.9 and 4.15 that the dual
bar spectral sequence takes the form

$E^{2}_{s,t}=\pi_{s}(\mathrm{Bar}_{\bullet}(\mathrm{id},\mathrm{Poly}_{\pazocal{R}},\pi_{*}(\Sigma^{j}H\mathbb{F}_{p}\mathbin{\mathop{\otimes}\displaylimits}H\mathbb{F}_{p}))^{\vee})_{t}\Rightarrow\overline{\mathrm{TAQ}}^{-s-t}(\Sigma^{j}H\mathbb{F}_{p}\mathbin{\mathop{\otimes}\displaylimits}H\mathbb{F}_{p},H\mathbb{F}_{p};H\mathbb{F}_{p})$

and collapses on the $E^{2}$-page. Hence we deduce that

$E^{\infty}_{s,t}\cong
E^{2}_{s,t}\cong\mathrm{Free}^{\mathrm{sLie}^{\rho}_{R^{!}}}(\Sigma^{-j}\pazocal{A})\cong\mathrm{Free}^{\mathrm{sLie}^{\rho}_{\pazocal{P}}}(\Sigma^{-j}\pazocal{A}),\,\,p=2,$
$E^{\infty}_{s,t}\cong
E^{2}_{s,t}\cong\mathrm{Free}^{(\pazocal{R}^{\prime})^{!}}\mathrm{Free}^{\mathrm{sLie}_{\mathbb{F}_{p}}}(\Sigma^{-j}\pazocal{A})\cong\mathrm{Free}^{\mathrm{sLie}^{\rho}_{\pazocal{P}}}(\Sigma^{-j}\pazocal{A}),\,\,p>2$

The computation for $k>1$ is similar. ∎

The Steenrod operations commute with the bracket via the usual Cartan formula

$Sq^{a}[x,y]=\sum_{i}[Sq^{i}(x),Sq^{a-i}(y)],\,\,\mathrm{for\ }p=2,$
$P^{a}[x,y]=\sum_{i}[P^{i}(x),P^{a-i}(y)],\,\,\,\,\beta
P^{a}[x,y]=\sum_{i}([\beta P^{i}(x),P^{a-i}(y)]+[P^{i}(x),\beta P^{a-i}(y)])$

for $p>2$.

Finally we deduce the relations between the Steenrod operations and the unary
$\mathbb{F}_{p}$-linear TAQ cohomology operations.

######  Proposition 6.2.

The Steenrod operations commute with unary $\mathbb{F}_{p}$-linear TAQ
cohomology operations $R^{i}$ via the Nishida relations on mod $p$ cohomology,
i.e.,

$Sq^{a}R^{-|x|+1}(x)=\sum\binom{j-c}{a-2c}R^{a+j+1-c}Sq^{c}(x)+\sum_{l<k,l+k=a}[Sq^{l}(x),Sq^{k}(x)],$
$Sq^{a}R^{b}(x)=\sum\binom{b-1-c}{a-2c}R^{a+b-c}Sq^{c}(x),\,\,\,\,b>-|x|+1$

for $p=2$. For $p>2$ we have

$\displaystyle P^{n}\beta R^{j}(x)=$
$\displaystyle(-1)^{n-i}\sum_{i}\binom{(j-i)(p-1)}{n-pi}\beta
R^{n+j-i}P^{i}(x)$
$\displaystyle+(-1)^{n-i}\sum_{i}\binom{(j-i)(p-1)-1}{n-pi-1}R^{n+j-i}\beta
P^{i}(x),$
$P^{n}R^{j}(x)=(-1)^{n-i}\sum_{i}\binom{(j-i)(p-1)-1}{n-pi}R^{n+j-i}P^{i}(x)$

for all $2j>-|x|+1$, as well as

$\displaystyle P^{n}R^{j}(x)=$
$\displaystyle(-1)^{n-i}\sum_{i}\binom{(j-i)(p-1)-1}{n-pi}R^{n+j-i}P^{i}(x)$
$\displaystyle+\frac{1}{\lambda_{|x|}}\sum_{I,\sigma\in\Sigma_{p},\sigma(1)=1}[[\cdots[[P^{i_{\sigma(1)}}(x),P^{i_{\sigma(2)}}(x)],P^{i_{\sigma(3)}}(x)]\cdots],P^{i_{\sigma(p)}}(x)]$

when the degree of $x$ is odd and $2j=-|x|+1$, where the bracket term sums
over all nondecreasing sequences $I=(0\leq i_{1}\leq i_{2}\leq\ldots\leq
i_{p})$ with $i_{1}+i_{2}+\cdots+i_{p}=n$ for $p>2$.

Recall that $\lambda_{|x|}$ is the unit by which bottom operation on an odd
degree class $x$ differs from the restriction $x^{[p]}$ on $x$, cf. Lemma
5.12.

######  Remark 6.3.

Note that the commuting relations between the Steenrod operations and the TAQ
cohomology operations $R^{i}$ coincide with the Adem relations for Steenrod
algebras, thereby reinforcing the heuristics that the operations $R^{i}$ are
extended Steenrod operations.

###### Proof of Proposition 6.2.

Since the operations $R^{i}=(Q^{i-1})^{*}$ come from the linear dual of the
Dyer-Lashof operations $Q^{i-1}$, the Steenrod operations commute with $R^{i}$
via the Nishida relations on cohomology. When $p=2$ the relations are worked
out explicitly, for example, by Miller in [Mil16]. The Nishida relations for
applying a Steenrod operation to the bottom operation on $x$ involves an extra
bracket term because the bottom operation is the restriction on $x$.

For $p>2$, the Nishida relations on cohomology can be read off from Theorem 3
and its corollary in Nishida’s original paper [Nis68]:

$P^{n}\beta R^{j}=(-1)^{n-i}\sum_{i}\binom{(j-i)(p-1)}{n-pi}\beta
R^{n+j-i}P^{i}+(-1)^{n-i}\sum_{i}\binom{(j-i)(p-1)-1}{n-pi-1}R^{n+j-i}\beta
P^{i},$
$P^{n}R^{j}=(-1)^{n-i}\sum_{i}\binom{(j-i)(p-1)-1}{n-pi}R^{n+j-i}P^{i}.$

Analogous to the case $p=2$, when $x$ is a class in odd degree, the Nishida
relations for the steenrod action on the bottom class $P^{n}R^{(-|x|+1)/2}(x)$
involve extra bracket terms since $\lambda_{|x|}R^{(-|x|+1)/2}(x)$ is the
restriction on $x$.

In order to determine the extra bracket terms, we need an explicit expression
for the restriction map on an odd class. In the setting of unshifted graded
$\mathbb{F}_{p}$-modules, this is worked out by Fresse in [Fre00, Remark
1.2.8]. Note that there is an embedding of the Lie operad into the associative
operad Assoc. Furthermore, there is an identity

(7) $\sum_{\sigma\in\Sigma_{p}}X_{\sigma(1)}\cdots
X_{\sigma(p)}=\sum_{\sigma\in\Sigma_{p},\sigma(1)=1}\langle\langle\cdots\langle\langle
X_{\sigma(1)},X_{\sigma(2)}\rangle,X_{\sigma(3)}\rangle\cdots\rangle,X_{\sigma(p)}\rangle$

in the associative operad, where $\langle x,y\rangle=xy-yx$ is the commutator.
For $x\in V$ in even degree, the $p$th power on $x$ is given by

$\sum_{\sigma\in\Sigma_{p}}X_{\sigma(1)}\cdots
X_{\sigma(p)}\mathbin{\mathop{\otimes}\displaylimits}x^{\mathbin{\mathop{\otimes}\displaylimits}p}\in(\mathrm{Assoc}(p)\mathbin{\mathop{\otimes}\displaylimits}V^{\mathbin{\mathop{\otimes}\displaylimits}p})^{\Sigma_{p}}\cong(\mathrm{Assoc}(p)\mathbin{\mathop{\otimes}\displaylimits}V^{\mathbin{\mathop{\otimes}\displaylimits}p})_{\Sigma_{p}}.$

Using the identity (7), we can pull back the $p$th power on $x$ along the
embedding

$(\mathrm{Lie}(p)\mathbin{\mathop{\otimes}\displaylimits}V^{\mathbin{\mathop{\otimes}\displaylimits}p})^{\Sigma_{p}}\hookrightarrow(\mathrm{Assoc}(p)\mathbin{\mathop{\otimes}\displaylimits}V^{\mathbin{\mathop{\otimes}\displaylimits}p})^{\Sigma_{p}}.$

The resulting element is the restriction on $x$ in the free restricted Lie
algebra on $V$, i.e.,

(8)
$x^{[p]}=\Big{(}\sum_{\sigma\in\Sigma_{p},\sigma(1)=1}[[\cdots[[X_{\sigma(1)},X_{\sigma(2)}],X_{\sigma(3)}]\cdots],X_{\sigma(p)}]\Big{)}\mathbin{\mathop{\otimes}\displaylimits}x^{\mathbin{\mathop{\otimes}\displaylimits}p}\in(\mathrm{Lie}(p)\mathbin{\mathop{\otimes}\displaylimits}V^{\mathbin{\mathop{\otimes}\displaylimits}p})^{\Sigma_{p}}.$

Since we are working with shifted graded $\mathbb{F}_{p}$-modules, the
commutator in the shifted graded associative algebra is $\langle
x,y\rangle=xy-(-1)^{(|y|-1)(|x|-1)}yx$. If $x,y$ are both in odd degrees, then
$\langle x,y\rangle=xy-yx$. Hence the identity (7) pulls back to the
restriction map (8) on an odd class $x$ in the free shifted graded restricted
Lie algebra over $\mathbb{F}_{p}$. Now we apply the Steenrod operation $P^{n}$
to the $p$th power on $x$ and use the Cartan formula. Note that the Steenrod
operations $P^{a}$ raises degree by an even number, so none of the signs are
altered. Pulling back to the free shifted restricted Lie algebra, we deduce
that the bracket terms in the Nishida relation for $P^{n}R^{(-|x|+1)/2}(x)$
consists of terms

$\sum_{\sigma\in\Sigma_{p},\sigma(1)=1}[[\cdots[[P^{i_{\sigma(1)}}(x),P^{i_{\sigma(2)}}(x)],P^{i_{\sigma(3)}}(x)]\cdots],P^{i_{\sigma(p)}}(x)]$

for all nondecreasing sequences $0\leq i_{1}\leq i_{2}\leq\ldots\leq i_{p}$
with $i_{1}+i_{2}+\cdots+i_{p}=n$. ∎

## References

  * [AC20] Antolín Camarena, O. (2020). The mod 2 homology of free spectral Lie algebras. Transactions of the American Mathematical Society, 373(9), 6301-6319.
  * [AB21] Arone, G. Z., & Brantner, D. L. B. (2021). The action of Young subgroups on the partition complex. Publications mathématiques de l’IHÉS, 133(1), 47-156.
  * [ADL13] Arone, G. Z., Dwyer, W. G., & Lesh, K. (2013). Bredon homology of partition complexes. arXiv preprint arXiv:1306.0056.
  * [AM99] Arone, G. & Mahowald, M. (1999). The Goodwillie tower of the identity functor and the unstable periodic homotopy of spheres. Inventiones mathematicae, 135(3), 743-788.
  * [Bas99] Basterra, M. (1999). André–Quillen cohomology of commutative S-algebras. Journal of Pure and Applied Algebra, 144(2), 111-143.
  * [Beh12] Behrens, M. (2012). The Goodwillie tower and the EHP sequence (Vol. 218, No. 1026). American Mathematical Soc..
  * [Bou68] Bousfield, A. K. (1968). Operations on derived functors of non-additive functors. Brandeis Univ.. Available at https://math.mit.edu/~hrm/manuscripts/bousfield-operations.pdf.
  * [BC70] Bousfield, A. K., & Curtis, E. B. (1970). A spectral sequence for the homotopy of nice spaces. Transactions of the American Mathematical Society, 151(2), 457-479.
  * [BKS05] Bryant, R. M., Kovács, L. G., & Stöhr, R. (2005). Subalgebras of free restricted Lie algebras. Bulletin of the Australian Mathematical Society, 72(1), 147-156.
  * [BMMS86] Bruner, R. R., May, J. P., McClure, J. E., & Steinberger, M. (2006). H ring spectra and their applications (Vol. 1176). Springer.
  * [BHK19] Brantner, L., Hahn, & J., Knudsen, B. (2019). The Lubin-Tate Theory of Configuration Spaces: I. arXiv preprint arXiv:1908.11321.
  * [Br17] Brantner, D. L. B. (2017). The Lubin-Tate theory of spectral Lie algebras (Doctoral dissertation). https://people.maths.ox.ac.uk/brantner/brantnerthesis.pdf.
  * [BCN21] Brantner, L., Campos, R., & Nuiten, J. (2021). PD Operads and Explicit Partition Lie Algebras. arXiv preprint arXiv:2104.03870.
  * [BM19] Brantner, L., & Mathew, A. (2019). Deformation theory and partition Lie algebras. arXiv preprint arXiv:1904.07352.
  * [Car58] Cartan, H. (1958). Séminaire Henri Cartan: ann. 7 1954/1955; Algèbres d’Eilenberg-Maclane et homotopie. Secrétariat mathématique.
  * [Chi05] Ching, M. (2005). Bar constructions for topological operads and the Goodwillie derivatives of the identity. Geometry & Topology, 9(2), 833-934.
  * [CH19] Ching, M., & Harper, J. E. (2019). Derived Koszul duality and TQ-homology completion of structured ring spectra. Advances in Mathematics, 341, 118-187.
  * [CLM76] Cohen, F. R., Lada, T. J., & May, P. J. (2007). The homology of iterated loop spaces (Vol. 533). Springer.
  * [DL62] Dyer, E., & Lashof, R. K. (1962). Homology of iterated loop spaces. American Journal of Mathematics, 84(1), 35-88.
  * [Dwy80] Dwyer, W. G. (1980). Homotopy operations for simplicial commutative algebras. Transactions of the American Mathematical Society, 260(2), 421-435.
  * [FG12] Francis, J., & Gaitsgory, D. (2012). Chiral koszul duality. Selecta Mathematica, 18(1), 27-87.
  * [Fre00] Fresse, B. (2000). On the homotopy of simplicial algebras over an operad. Transactions of the American Mathematical Society, 352(9), 4113-4141.
  * [Fre04] Fresse, B. (2004). Koszul duality of operads and homology of partition posets. In Homotopy theory: relations with algebraic geometry, group cohomology and algebraic K-theory, Contemp. Math. 346, Amer. Math. Soc. pp. 112-215.
  * [GK94] Ginzburg, V., & Kapranov, M. (1994). Koszul duality for operads. Duke mathematical journal, 76(1), 203-272.
  * [Goe90] Goerss, P. G. (1990). On the André-Quillen cohomology of commutative $\mathbb{F}_{2}$-algebras. Astérisque, (1).
  * [Jac41] Jacobson, N. (1941). Restricted Lie algebras of characteristic p. Transactions of the American Mathematical Society, 50(1), 15-25.
  * [Joh95] Johnson, B. (1995). The derivatives of homotopy theory. Transactions of the American Mathematical Society, 347(4), 1295-1321.
  * [JN14] Johnson, N. & Noel, J. (2014). Lifting homotopy T-algebra maps to strict maps. Advances in Mathematics, 264, 593-645.
  * [KA56] Kudo, T., & Araki, S. (1956). TOPOLOGY OF Hn-SPACES AND H-SQUARING OPERATIONS. Memoirs of the Faculty of Science, Kyushu University. Series A, Mathematics, 10(2), 85-120.
  * [Kja18] Kjaer, J. J. (2018). On the odd primary homology of free algebras over the spectral Lie operad. Journal of Homotopy and Related Structures, 13(3), 581-597.
  * [KL83] Kraines, D., & Lada, T. (1983). The cohomology of the Dyer-Lashof algebra. In Proceedings of the Northwestern Homotopy Theory Conference (pp. 145-152).
  * [Kri93] Kriz,Igor. (1993). Towers of $E_{\infty}$ ring spectra with an application to $BP$. Preprint.
  * [Kuh85] Kuhn, N. J. (1985). Chevalley group theory and the transfer in the homology of symmetric groups. Topology, 24(3), 247-264.
  * [Law20] Lawson, T. (2020). $E_{n}$-spectra and Dyer-Lashof operations. In  Handbook of Homotopy Theory (pp. 793-849). Chapman and Hall/CRC.
  * [Lur17] Lurie, J. (2017). Higher algebra.
  * [May72] May, J. P. (1972). The geometry of iterated loop spaces. Lecture Notes in Math., Vol. 271, SpringerVerlag, Berlin New York, 1972.
  * [McC01] McCleary, J. (2001). A user’s guide to spectral sequences (No. 58). Cambridge University Press.
  * [Mil16] Miller, H.R. Nishida relations and Singer construction. https://klein.mit.edu/~hrm/papers/nishida3.pdf.
  * [MM65] Milnor, J. W., & Moore, J. C. (1965). On the structure of Hopf algebras. Annals of Mathematics, 211-264.
  * [Nak57] Nakaoka, M. (1957). Cohomology mod p of the p-fold symmetric products of spheres. Journal of the Mathematical Society of Japan, 9(4), 417-427.
  * [Nak58] Nakaoka, M. (1958). Cohomology mod $p$ of symmetric products of spheres. Journal of the Institute of Polytechnics, Osaka City University. Series A: Mathematics, 9(1), 1-18.
  * [Nis68] Nishida, G. (1968). Cohomology operations in iterated loop spaces. Proceedings of the Japan Academy, 44(3), 104-109.
  * [Pri70] Priddy, S. B. (1970). Koszul resolutions. Transactions of the American Mathematical Society, 152(1), 39-60.
  * [Pri73] Priddy, S. (1973). Mod-p right derived functor algebras of the symmetric algebra functor. Journal of Pure and Applied Algebra, 3(4), 337-356.
  * [Zh21] Zhang, A. Y. (2021). André-Quillen homology of spectral Lie algebras, with application to mod $p$ homology of labeled configuration spaces. arXiv preprint arXiv:2110.08428.