# Cohomology rings of extended powers and of free infinite loop spaces

Lorenzo Guerra  Università di Roma Tor Vergata<EMAIL_ADDRESS>,
Paolo Salvatore  Università di Roma Tor Vergata<EMAIL_ADDRESS>and
Dev Sinha  Mathematics Department, University of Oregon<EMAIL_ADDRESS>
###### Abstract.

We calculate mod-$p$ cohomology of extended powers, and their group
completions which are free infinite loop spaces. We consider the cohomology of
all extended powers of a space together and identify a Hopf ring structure
with divided powers within which cup product structure is more readily
computable than on its own. We build on our previous calculations of
cohomology of symmetric groups, which are the cohomology of extended powers of
a point, the well-known calculation of homology, and new results on cohomology
of symmetric groups with coefficients in the sign representation. We then use
this framework to understand cohomology rings of related spaces such as
infinite extended powers and free infinite loop spaces.

###### 1991 Mathematics Subject Classification:

20J06, 20B30

The first two authors acknowledge the MIUR Excellence Department Project
awarded to the Department of Mathematics, University of Rome Tor Vergata, CUP
E83C18000100006.

##  1\. Introduction

Homotopy orbit spaces with respect to the symmetric group action on iterated
products, known as extended powers, play many key roles in algebraic topology.
Steenrod introduced them as a central character in the of study cohomology
operations [Ste53]. Extended powers of spectra play an essential role in
Nishida’s proof of his Nilpotence Theorem [Nis75]. By the Barratt-Priddy-
Quillen theorem [BP72], after group completion the infinite extended powers
functor provides a model for the free infinite loopspace functor $Q$. May and
his collaborators demonstrated that the stable version of extended powers not
only were essential in defining structured ring spectra but also provided
access to a wide range of phenomena which had previously been ad-hoc [BMMS86].
In addition to necessarily playing a role in derived algebra, modern
applications of extended powers and free infinite loop spaces range widely
from, for example, the calculus of functors [AM99] to the stable cohomology of
mapping class groups [Gal04].

We focus on the cohomology rings of these spaces, as this ring structure is
the most basic unstable homotopical structure. These rings have long been
implicitly understood through homology and coproduct calculations [CLM76]. But
calculations with such require application of Adem relations for Kudo-Araki-
Dyer-Lashof operations, so are difficult to work with. Here, as in our work on
symmetric groups and alternating groups [Gue17, GSS12, GS17], we find that
cohomology ring structure is best understood when coupled with a transfer (or
induction) product, first defined by Strickland and Turner [ST97]. Our
techniques in this setting for example have led to further development of the
Curtis-Wellington spectral sequence [Hun].

Our work is thus parallel to as well as building on the well-established
homology calculations. In [CLM76], the homology of extended powers all
together forms an algebra over the Dyer-Lashof algebra, which behaves
similarly to and in fact can be related to the cohomology rings of spaces over
the Steenrod Algebra [May70]. Our presentations capture cohomology ring
structure as part of a larger structure as well, ultimately presenting
cohomology as “universal component super-Hopf rigs with additive divided
powers” which then have small, if any, sets of relations. Prior to the
elaboration of such Hopf ring structures, calculations of mod-two cohomology
rings of classifying spaces of symmetric groups – the extended powers of a
point – had cumbersome descriptions [AM04, Fes02], and calculations were not
made at all for odd primes.

We develop these needed algebraic notions in Section 2, and we encourage the
reader to take a look at the statement of first main results, namely Theorems
2.34 through 2.38 to understand the goal of the algebraic work before going
through the details of that section. Such statements are remarkably simple,
given the complexity for example of individual component rings. In the odd
primary setting, such brevity is not possible without use of trivial and
twisted coefficients together. We follow these statements with a development
of additive bases in their algebraic frameworks. After extending well-known
results about homology to the twisted coefficient setting, as needed for our
cleanest statements, we prove the theorems about ground rings, namely Theorems
2.34 and 2.35, in Section 3. We then use those to prove our first main
theorems, namely Theorems 2.37 and 2.38, in Section 4. Our strategy is to use
the two products and divided powers structures to produce a set of classes in
cohomology which pairs perfectly with the standard basis in homology. This
perfect pairing both establishes the set of cohomology classes as a basis and
validates the algebraic presentation which governs them.

After treating the cohomology of extended powers with disjoint basepoints, we
calculate the cohomology of $DX$, $D_{\infty}X$, $CX$ (defined below) and $QX$
in Section 5. For $CX$ and $QX$ these reproduce Dung’s description of the
cohomology ring for $QX$ when $p=2$ and givie new results for odd primes.

While we capitalize on finding the right algebraic structures and rely on
standard techniques from algebraic topology and group cohomology as well as
previous homology calculations, geometry has inspired and guided us. We focus
on $C(X)$, the free $E_{\infty}$-space on $X$. We choose $E{\mathcal{S}}_{n}$
to be the (ordered) configuration space of $n$ points in $\mathbb{R}^{\infty}$

$E{\mathcal{S}}_{n}=\\{(x_{1},\dots,x_{n})\in(\mathbb{R}^{\infty})^{n}:x_{i}\not=x_{j}\forall
i\not=j\\},$

with an action of ${\mathcal{S}}_{n}$ given by permutation of coordinates.

######  Definition 1.1.

Let $X$ be a pointed topological space. Let $\phi_{i,n}\colon
E{\mathcal{S}}_{n}\to E{\mathcal{S}}_{n-1}$ the map that forgets the $i^{th}$
point of the configuration. Let $s_{i,n}\colon X^{n-1}\to X^{n}$ be the map
that adds the basepoint as the $i^{th}$ coordinate. Define $CX$, the space of
configurations in ${\mathbb{R}}^{\infty}$ with labels in $X$, as
${\bigsqcup_{n\geq
0}E{\mathcal{S}}_{n}\times_{{\mathcal{S}}_{n}}X^{n}}/{\sim}$, where $\sim$ is
the equivalence relation generated by $(\phi_{i,n}a,b)\sim(a,s_{i,n}b)$ for
all $a\in E{\mathcal{S}}_{n}$ and $b\in X^{n-1}$.

We consider the finite-dimensional versions of this model, with a limited
number of points in some ${\mathbb{R}}^{N}\subset{\mathbb{R}}^{\infty}$, in
which case if $X$ is a manifold then so is this quotient. In that setting, we
can use geometric chains and cochains [FMS22]. The homology story is well-
known through Kudo-Araki-Dyer-Lashof operations, which are manifestly
geometric. If $f:P\to X$ represents a mod-two geometric homology class $y$
then we can map $S^{i}\times_{C_{2}}(P\times P)$ to $CX$ by sending $v\times
p_{1}\times p_{2}$ to the configuration of two points at $v,-v$ with labels
$f(p_{1})$ and $f(p_{2})$. The resulting geometric homology class represents
the operation $q_{i}(y)$. On the cohomology side, the analogous geometric
cochain would be the submanifold of $CX$ where two points in the underlying
configuration share their first coordinate with labels both in some geometric
cochain $W\to X$. This geometry guided us in formulating our main structures,
with cup product corresponding to intersection as usual, transfer product a
type of “union”, and divided powers a “repeating”.

We see many possible investigations building on this work. Following Remark
2.39, it would be interesting and probably reflective of deeper structure to
fully understand divided powers operations on cohomology, as we only employ
them on a limited set of classes. The pairing between homology and cohomology
is not Kronecker, and is likely to be useful since the two settings differ in
which structures are most readily expressed. We have made partial progress on
computing Steenrod operations, a natural “self-serving” calculation to
consider. Calculating cup product more organically as we have done here could
help in extending these calculations to generalized cohomology theories, with
even understanding whether divided powers operations exist for the transfer
product an interesting first question. We suspect not, as the transfer product
is analogous to the Pontrjagin product on the homology of Eilenberg-MacLane
spaces, which has a divided powers structure [Car55, HHP+], but the homology
of other infinite loop spaces instead have Dyer-Lashof operations. Related to
this, there is possibility of further binding these structures through higher
algebra enhancements of transfer product, in which case all of this intricate
structure could likely be tied back to the homology of ${\mathbb{R}}P^{1}$ and
$\mathbb{C}P^{1}$.

###### Contents

  1. 1 Introduction
  2. 2 Hopf ring with additive divided powers structure on the cohomology of $DX$
    1. 2.1 Preliminaries
    2. 2.2 Component Hopf rings with additive divided powers
    3. 2.3 The algebraic extended powers functor
    4. 2.4 Divided powers on cohomology of symmetric groups with coefficients in a product series
    5. 2.5 Statement of first main results
    6. 2.6 Explicit construction of (relatively) universal component Hopf rings with divided powers
  3. 3 The homology and cohomology of symmetric groups with twisted coefficients
    1. 3.1 The homology of extended powers in terms of KADL operations
    2. 3.2 Cohomology of symmetric groups with coefficients in the sign representation mod-$p$
  4. 4 Cohomology rings of extended powers with basepoints
    1. 4.1 Additive basis for the cohomology of $DX_{+}$
    2. 4.2 Analyzing the pairing
  5. 5 Further cohomology ring calculations
    1. 5.1 General extended powers
    2. 5.2 Infinite extended powers
    3. 5.3 Free $E_{\infty}$ spaces
    4. 5.4 Free infinite loop spaces

##  2\. Hopf ring with additive divided powers structure on the cohomology of
$DX$

In this section we introduce the algebraic structure of component Hopf ring
with additive divided powers. We show in Corollary 2.31 that this structure
applies to the cohomology of extended powers, and then give an explicit
presentation of this cohomology in Theorem 2.35.

###  2.1. Preliminaries

Extended powers are our first main objects of study, and we assume throughout
that $X$ is Hausdorff and compactly generated, and that any basepoint has a
neighborhood deformation retract.

######  Definition 2.1.

If spaces $E$ and $Y$ are both ${\mathcal{S}}_{n}$-spaces, with action
preserving the base point of $Y$, then $E\rtimes_{{\mathcal{S}}_{n}}Y$ is
defined as the quotient $\left((E)_{+}\wedge Y\right)/{{\mathcal{S}}_{n}}$,
where $(E)_{+}$ is $E$ with a disjoint basepoint and ${\mathcal{S}}_{n}$ acts
diagonally.

Define the $n^{th}$ extended power of $X$ by
$D_{n}X=E{\mathcal{S}}_{n}\rtimes_{{\mathcal{S}}_{n}}X^{\wedge^{n}}$. Let
$DX=\bigvee_{n}(D_{n}X)$. Recall that if $X$ is a $G$-space we let $X_{hG}$
denote the homotopy quotient $EG\times_{G}X$, so that
${X^{n}}_{h{\mathcal{S}}_{n}}=D_{n}(X_{+})$. We will also consider this
unbased version of $D_{n}$, for which we use the symbol $\tilde{D}$ to
differentiate the two. Hence $\tilde{D}(X)=X_{h{\mathcal{S}}_{n}}$.

For $p=2$ let ${H_{EP}}^{*}(X):=\bigoplus_{n\geq
0}H^{*}(D_{n}X;\mathbb{F}_{2})$.

For odd primes let ${H_{EP}}^{*}(X):=\mathbb{F}_{p}\oplus\bigoplus_{n\geq
1}H^{*}(D_{n}X;\mathbb{F}_{p}\oplus sgn)$, where $\mathbb{F}_{p}$ denotes the
trivial representation and $sgn$ is the sign representation. These are bi- or
tri-graded by component, degree, and in the odd prime setting a mod-two
grading induced by an obvious grading on $\mathbb{F}_{p}\oplus sgn$.

Observe that $\tilde{D}(X)\cong D(X_{+})\cong C(X_{+})$.

We use both algebraic and geometric variants of the extended powers
topological construction, using the algebra to provide the main framework for
our arguments. Almost by definition, cohomology of $D_{n}X$ is the equivariant
cohomology of the $n$-fold tensor product of the cochains of $X$. But the
following basic computation shows one can consider only the cohomology of $X$
(see for example [May70], Lemma 1.1).

######  Proposition 2.2.

There are natural isomorphisms $H^{*}(D_{n}X;\mathbb{F}_{p})\cong
H^{*}({\mathcal{S}}_{n};{\widetilde{H}^{*}(X;\mathbb{F}_{p}))}^{\otimes{n}})$,
and similarly for homology, compatible with standard pairings.

Our main results present the cohomology rings, with $\mathbb{F}_{p}$
coefficients, of these three related spaces, namely $DX$, the extended power
of a topological space $X$, $CX$, the free $E_{\infty}$-space over $X$, and
$QX$, the free infinite-loop space over $X$. Because the homology of these
spaces, as co-algebras, is known and only depends on the homology of $X$
[CLM76], our first step is to develop the algebraic functors which will take
as their input the cohomology ring of $X$ and produce the cohomology rings of
these spaces.

###  2.2. Component Hopf rings with additive divided powers

We work primarily in a bigraded setting in a strong sense, namely products
vanish on elements which differ in the first grading.

######  Definition 2.3.

Let $(H,\Delta,\cdot)$ be a bialgebra over $R$. We say that $H$ is a bigraded
component bialgebra if
$H=\bigoplus_{n,d\in\mathbb{N}\times{\mathbb{N}}}H_{n,d}$, where

  * •

the product sends $H_{m,i}\otimes H_{m,j}$ to $H_{m,i+j}$

  * •

the product of $x$ and $y$ is zero if $x\in H_{m,i}$, $y\in H_{n,j}$ with
$n\not=m$

  * •

the coproduct is standardly bigraded, sending $H_{p,k}$ to
$\bigoplus_{n+m=p,k=i+j}H_{n,i}\otimes H_{m,j}$.

The first grading of an element is called its component, while the second is
called its dimension. For such $H$, we let $\overline{H}$ denote the direct
sum of $H_{n,d}$ for $n>0$ (and all $d\geq 0$). We require the connectedness
condition $A=\operatorname{im}(\eta)\bigoplus\overline{A}$, where $\eta\colon
R\to A$ is the image of the unit map.

Similarly, super-component bialgebra is a bialgebra graded over
$\mathbb{N}\times\frac{\mathbb{Z}}{2\mathbb{Z}}$ satisfying these same axioms.

Recall that a rig (ring without negatives) satisfies all ring axioms except
the existence of additive inverses.

######  Definition 2.4.

A Hopf rig $H$ is a rig object in the category of coalgebras, so that there
are two products, a “multiplication product” $\cdot$ and an “addition product”
$\odot$, both giving bialgebras with one coproduct $\Delta$, and a
distributivity axiom that $a\cdot(b\odot c)=\sum_{\Delta a=\sum
a^{\prime}\otimes a^{\prime\prime}}(a^{\prime}\cdot
b)\odot(a^{\prime\prime}\cdot c)$.

A bigraded component Hopf rig (respectively super-component Hopf rig) is one
for which $(H,\cdot,\Delta)$ is a bigraded component bialgebra (respectively
super-component bialgebra), and $\odot$ preserves both gradings. In this
setting, we modify the distributivity axiom above by introducing a coefficient
$(-1)^{|a^{\prime\prime}||b|}$ corresponding to the usual Koszul sign.

We require the structural morphisms to be graded (co)commutative with respect
to the dimension. When this commutativity condition is not necessarily
satisfied, we speak of a non-commutative bigraded component Hopf rig.

The Hopf rigs we study have antipodes, and are thus Hopf rings. But we will
not need the antipode structures, and only mention them in passing. Hopf rings
first appeared in topology in the study of homology of infinite loop spaces
which represent cohomology with product structure [Mil70, RW80, Wil00]. But
following Strickland and Turner [ST97], we have found them essential in
describing the cohomology of symmetric groups. An interesting related case of
Hopf rings arises when one introduces an induction or symmetrization product
on rings of symmetric polynomials. Readers who would like to see concrete
examples can see these treated in the second section of [GSS12].

We will show that the cohomology of $DX$ not only is a Hopf rig, but is also
endowed with divided powers operations.

######  Definition 2.5.

A divided powers algebra is a triple $(A,I,\gamma)$, where $A$ is an algebra,
$I\trianglelefteq A$ is a proper ideal, and
$\gamma=\left\\{\gamma_{n}\right\\}_{n\in\mathbb{N}}$ is a family of functions
from $I$ to $I$, also denoted $x^{[{n}]}:=\gamma_{n}(x)$, which satisfies the
following relations, whenever $x,y\in I$ and $\lambda\in A$:

  1. (1)

$x^{[{0}]}=1$, and $x^{[{1}]}=x$ ($0,1$ Cases)

  2. (2)

$\left(x+y\right)^{[{n}]}=\sum_{i=0}^{n}x^{[{i}]}y^{[{n-i}]}$ (Binomial)

  3. (3)

$\left(\lambda x\right)^{[{n}]}=\lambda^{n}x^{[{n}]}$ (Distributivity over
Multiplication)

  4. (4)

$x^{[{n}]}x^{[{m}]}=\left(\begin{array}[]{c}n+m\\\
n\end{array}\right)x^{[{n+m}]}$ (Law of Exponents)

  5. (5)

$(x^{[{m}]})^{[{n}]}=\frac{(nm)!}{(m!)^{n}n!}x^{[{nm}]}$. (Composition)

We also refer to such triples as divided powers structures, on the algebra
$A$. Introduced in [Car55], these have been extensively studied and applied to
various contexts, for example in [Rob65], from which we borrow notation, and
[Ber74, Haz78]. In algebraic topology, the homology of Eilenberg-MacLane
spaces are divided powers algebras [EML54, Tho57].

These relations imply that the formal series
$f(t)=\sum_{i=0}^{\infty}x^{[{i}]}t^{i}\in A[[t]]$ satisfies $f(0)=1$ and
$f(t+s)=f(t)f(s).$ The set $\operatorname{Exp}(A)$ of $f\in A[[t]]$ satisfying
these conditions constitutes an $R$-module. The left adjoint of
$\operatorname{Exp}$ viewed as a functor from $R$-algebras to $R$-modules is
the free divided powers functor $DP$, defined explicitly as follows.

######  Definition 2.6.

For an $R$-module $M$, we let $DP(M)$, the free divided powers algebra
generated by $M$, be generated as an algebra by the set
$\left\\{x^{[{n}]}\right\\}$, where $x\in M$, with relations (1)-(5) from
Definition 2.5 imposed. The ideal $I$ is given by the collection of
$x^{[{n}]}$ with $n\geq 1$ and divided powers maps are defined through the
Binomial, Scalar Multiplication and Composition relations of Definition 2.5.

If $M$ is graded, then naturally so are $DP(M)$ and the universal map. Over a
field, if $M$ is finite-dimensional, there is a standard additive basis of
$DP(M)$ which we can associate to a choice of basis
$\left\\{x_{1},\dots,x_{r}\right\\}$ for $M$, namely
$\left\\{\prod_{i=1}^{r}x_{i[n_{i}]}\right\\}_{n_{i}\in\mathbb{N}}$.

######  Definition 2.7.

For a divided powers algebra $(A,I,\gamma)$, the tensor product $A\otimes A$
has a natural divided powers structure $(A\otimes A,A\otimes I+I\otimes
A,\gamma_{\otimes})$ defined for $x\in I$ and $y\in A$ by $(x\otimes
y)^{[{n}]}=x^{[{n}]}\otimes y^{n}$ and $(y\otimes x)^{[{n}]}=y^{n}\otimes
x^{[{n}]}$.

A divided powers bialgebra is a divided powers structure
$(H,\overline{H},\gamma)$ on a bialgebra $H$, with $\overline{H}$ kernel of
the counit, such that the coproduct $\Delta\colon H\to H\otimes H$ is a
morphism of divided powers structures.

In the graded setting, we require divided powers operations to be compatible
with degrees, in the sense that $\deg(x^{[{k}]})=k\deg(x)$. In characteristic
not equal to two, elements of odd degree have zero squares and thus higher
powers, so in this setting $x^{[{k}]}=0$ if $\deg(x)$ is odd and $k\geq 2$.
This is also the case for the odd degree classes in the super-bigraded
component setting. By a classical theorem of Milnor and Moore, a graded Hopf
algebra $H$ that is bicommutative in the graded sense can always be split into
an even and odd part where the odd factor is an exterior algebra generated by
primitive elements in odd degree. A graded divided powers structure on $H$ is
thus equivalent to a divided powers structure in the standard sense on the
even part, and on the odd part the divided powers with exponent larger than
one are zero.

Admitting a divided powers structure is a very restrictive condition on a Hopf
algebra. Such algebras arise as dual Hopf algebras to free, primitively
generated Hopf algebras or more generally as duals of enveloping algebras of
Lie algebras [Sch67, And71].

Given a counital $R$-coalgebra $C$, let $\varepsilon\colon C\to R$ be the
counit and $\overline{C}=\ker(\varepsilon)$. The coproduct $\Delta\colon
C\rightarrow C\otimes C$ extends uniquely to $DP(\overline{C})\to
DP(\overline{C})\otimes DP(\overline{C})$, which by abuse of notation we still
denote $\Delta$. This is a coassociative and cocommutative coproduct, and
defines a bialgebra structure on $DP(\overline{C})$.

######  Definition 2.8.

Denote by $DP_{HA}$ the free divided powers Hopf algebra functor from
$R$-coalgebras to Hopf algebras with divided powers over $R$, with
$DP_{HA}(C)=DP(\overline{C})$ as an algebra, the coproduct induced by that of
$C$, and extended to morphisms by the universal property.

Then $DP_{HA}(C)$ is left adjoint to the forgetful functor from divided powers
bialgebras to coalgebras.

Our main structure we use comprises the two structures detailed above.

######  Definition 2.9.

A bigraded component Hopf ring with additive divided powers is a septad
$(A,\odot,\cdot,\Delta,\eta,\varepsilon,\\{\gamma_{n}\\})$ such that

  * •

$(A,\odot,\Delta,\\{\gamma_{n}\\})$ is a divided powers bigraded component
Hopf algebra

  * •

$(A,\odot,\cdot,\Delta)$ is a bigraded component Hopf rig

We also require that each component algebra $(A_{n},\cdot)$ is unital, and
that the unit of $A_{n}$ is the $n^{th}$ divided power of the unit of $A_{1}$.

A component super-Hopf ring with additive divided powers is a septad as above
where $A$ has an additional $\mathbb{Z}/2\mathbb{Z}$-grading $A=A_{even}\oplus
A_{odd}$, preserved by $\Delta$ and $\cdot$ and such that $A_{even}\odot
A_{odd}=0$, satisfying the following conditions:

  * •

$(A,\odot,\Delta,\\{\gamma_{n}\\})$ is a divided powers Hopf algebra

  * •

$(A,\cdot,\Delta)$ is a component super-algebra

  * •

the Hopf ring distributivity axiom holds in $A$

We will prove below that the Hopf rings of our interest have divided powers
operations for the addition product satisfying Definition 2.9. However, they
are not compatible with the multiplication product $\cdot$ in a simple way, as
we show in Remark 2.39.

###  2.3. The algebraic extended powers functor

Recall that our first main goal is to describe the algebraic functor which
takes the cohomology ring of a space $X$ and produces the cohomology of its
extended powers. In light of Proposition 2.2, this can be viewed in terms of
group cohomology. For optimal results, at odd primes we take cohomology with
coefficients which incorporate sign representation.

######  Definition 2.10.

Let $V$ be a representation of ${\mathcal{S}}_{n}$ over a field. Define the
cohomology of extended powers of a space $X$ with coefficients in $V$, denoted
$H^{*}(D_{n}X;V)$, to be the group cohomology
$H^{*}({\mathcal{S}}_{n};{\widetilde{H}^{*}(X)}^{\otimes{n}}\otimes V)$.

In our application $V$ will be either trivial when $p=2$ or for odd primes
will incorporate the sign representation. Clearly the notation is consistent
in the case of trivial coefficients by Proposition 2.2.

######  Definition 2.11.

By abuse, for any odd prime $p$ and any $n$ let $\rho$ be the representation
of ${\mathcal{S}}_{n}$ given by $\rho=\mathbb{F}_{p}\oplus sgn$, where
$\mathbb{F}_{p}$ denotes the trivial representation and $sgn$ denotes the sign
representation.

We grade $\rho$ by having the trivial representation in degree zero (even) and
the sign representation in degree one (odd). We use canonical isomorphisms
$\mathbb{F}_{p}\otimes V\cong V$ and $sgn\otimes sgn\cong\mathbb{F}_{p}$ to
define a graded algebra structure on $\rho$.

In our setting, the coefficients we are using have multiplicative structure
which is then reflected in cohomology.

######  Definition 2.12.

A product series of algebras over a field $k$ is a collection $\\{A_{n}\\}$
with $A_{n}$ $k[{\mathcal{S}}_{n}]$-algebra, with isomorphisms of
$k[{\mathcal{S}}_{i}\times{\mathcal{S}}_{j}]$-algebras $\chi_{i,j}\colon
A_{i+j}\to A_{i}\otimes_{k}A_{j}$ that are coherent in the sense that the
following two conditions are satisfied:

  1. (1)

for all $i,j,k>0$ the following diagram commutes in the category of
$k[{\mathcal{S}}_{i}\times{\mathcal{S}}_{j}\times{\mathcal{S}}_{k}]$-modules:

${A_{i+j+k}}$${A_{i}\otimes A_{j+k}}$${A_{i+j}\otimes A_{k}}$${A_{i}\otimes
A_{j}\otimes
A_{k}}$$\scriptstyle{\chi_{i,j+k}}$$\scriptstyle{\chi_{i+j,k}}$$\scriptstyle{\operatorname{id}\otimes\chi_{j,k}}$$\scriptstyle{\chi_{i,j}\otimes\operatorname{id}}$

  2. (2)

for all $n,m>0$, the following diagram commutes

${A_{n+m}}$${A_{n+m}}$${A_{n}\otimes A_{m}}$${A_{m}\otimes
A_{n},}$$\scriptstyle{\sigma_{n,m}}$$\scriptstyle{\chi_{n,m}}$$\scriptstyle{\chi_{m,n}}$$\scriptstyle{\tau}$

where $\tau$ exchanges the two factors and $\sigma_{n,m}\in\Sigma_{n+m}$ is
the permutation given by

$\sigma_{n,m}(i)=\left\\{\begin{array}[]{ll}m+i&\mbox{if }1\leq i\leq n\\\
i-n&\mbox{if }n+1\leq i\leq n+m\end{array}\right..$

A super-product series of algebras is a collection $\\{A_{n}\\}$ with $A_{n}$
$k[{\mathcal{S}}_{n}]$-module with the following additional structure:

  * •

a grading $A_{n}=A_{n,0}\oplus A_{n,1}$ of each $A_{n}$ over
$\mathbb{Z}/2\mathbb{Z}$,

  * •

a product $A_{n}\otimes A_{n}\to A_{n}$ that makes each $A_{n}$ a
$k[{\mathcal{S}}_{n}]$-super-algebra,

  * •

and $k[{\mathcal{S}}_{i}\times{\mathcal{S}}_{j}]$-module isomorphism
$\chi_{i,j,e}\colon A_{i+j,e}\to A_{i,e}\otimes_{k}A_{j,e}$ for all
$i,j\in\mathbb{N}$ and $e\in\mathbb{Z}/2\mathbb{Z}$, that are coherent in the
sense that the two conditions above are satisfied and such that
$\chi_{i,j,0}\oplus\chi_{i,j,1}\colon A_{i,j}\to A_{i}\otimes_{k}A_{j}$ are
super-algebra morphisms.

We note that a product series of algebras can be regarded as a super-product
series of algebras concentrated in $\mathbb{Z}/2\mathbb{Z}$-degree $0$.

The product series we use are built from a single algebra, ultimately the
cohomology ring of a space.

######  Definition 2.13.

Let $A$ be an algebra. Define $TA$ to be the product series of algebras with
${TA}_{n}=A^{\otimes{n}}$ with canonical restrictions. For odd primes define
$T_{\rho}A$ to be the sequence of modules with ${T_{\rho}A}_{n}=\rho\otimes
A^{\otimes n}$ and structure maps given by the identity map on $\rho$ tensored
with canonical restrictions.

######  Definition 2.14.

Let $\\{A_{n}\\}$ be a product series of algebras. Let $f\in
H^{*}({\mathcal{S}}_{n};A_{n})$ and $g\in H^{*}({\mathcal{S}}_{m};A_{m})$, so
$f$ is represented by a homomorphism from a resolution
$W_{*}^{{\mathcal{S}}_{n}}$ of $k$ over $k[{\mathcal{S}}_{n}]$ to a suspension
of $A_{n}$. Define the preliminary multiplicative structures on cohomology of
symmetric groups with coefficients in $\\{A_{n}\\}$ as follows.

  * •

A coproduct $\tilde{\Delta}\colon H^{*}({\mathcal{S}}_{i}+j;A_{i+j})\to
H^{*}({\mathcal{S}}_{i};A_{i})\otimes H^{*}({\mathcal{S}}_{j};A_{j})$ is
induced by the composition

${\hom_{{\mathcal{S}}_{i+j}}(W_{*}^{{\mathcal{S}}_{i+j}};A_{i+j})}$${\hom_{{\mathcal{S}}_{i}\times{\mathcal{S}}_{j}}(W_{*}^{{\mathcal{S}}_{i}}\otimes
W_{*}^{{\mathcal{S}}_{j}};A_{i+j})}$${\hom_{{\mathcal{S}}_{i}\times{\mathcal{S}}_{j}}(W_{*}^{{\mathcal{S}}_{i}}\otimes
W_{*}^{{\mathcal{S}}_{j}};A_{i}\otimes
A_{j})}$${\hom_{{\mathcal{S}}_{i}}(W_{*}^{{\mathcal{S}}_{i}};A_{i})\otimes\hom_{{\mathcal{S}}_{j}}(W_{*}^{{\mathcal{S}}_{j}};A_{j}),}$$\scriptstyle{\rho}$$\scriptstyle{\chi_{i,j}}$$\scriptstyle{\cong}$

where $\rho$ is the restriction to the resolution of a subgroup;

  * •

A transfer product $\tilde{\odot}\colon H^{*}({\mathcal{S}}_{n};A_{n})\otimes
H^{*}({\mathcal{S}}_{m};A_{m})\to H^{*}({\mathcal{S}}_{n+m};A_{n+m})$ is
induced by

${\hom_{{\mathcal{S}}_{n}}(W_{*}^{{\mathcal{S}}_{n}};A_{n})\otimes\hom_{{\mathcal{S}}_{m}}(W_{*}^{{\mathcal{S}}_{m}};A_{m})}$${\hom_{{\mathcal{S}}_{n}\times{\mathcal{S}}_{m}}(W_{*}^{{\mathcal{S}}_{n}}\otimes
W_{*}^{{\mathcal{S}}_{m}};A_{n}\otimes
A_{m})}$${\hom_{{\mathcal{S}}_{n}\times{\mathcal{S}}_{m}}(W_{*}^{{\mathcal{S}}_{n}}\otimes
W_{*}^{{\mathcal{S}}_{m}};A_{n+m})}$${\hom_{{\mathcal{S}}_{n+m}}(W_{*}^{{\mathcal{S}}_{n+m}};A_{n+m}),}$$\scriptstyle{\cong}$$\scriptstyle{\chi_{n,m}^{-1}}$$\scriptstyle{\operatorname{tr}}$

where $\operatorname{tr}$ denotes the usual transfer map;

  * •

a cup product $\cdot$ is defined in the standard way on each component, using
restriction along the diagonal on a resolution of $k$ over
$k[{\mathcal{S}}_{n}]$ and the product on $A_{n}$

${\hom_{{\mathcal{S}}_{n}}(W_{*}^{{\mathcal{S}}_{n}};A_{n})\otimes\hom_{{\mathcal{S}}_{n}}(W_{*}^{{\mathcal{S}}_{n}};A_{n})}$${\hom_{{\mathcal{S}}_{n}\times{\mathcal{S}}_{n}}(W_{*}^{{\mathcal{S}}_{n}}\otimes
W_{*}^{{\mathcal{S}}_{n}};A_{n}\otimes
A_{n})}$${\hom_{{\mathcal{S}}_{n}}(W_{*}^{{\mathcal{S}}_{n}};A_{n}\otimes
A_{n})}$${\hom_{{\mathcal{S}}_{n}}(W_{*}^{{\mathcal{S}}_{n}};A_{n}),}$$\scriptstyle{\cong}$

and is zero between distinct components;

More generally, if we start with a super-product series of algebras
$\\{A_{n}\\}$ with $A_{n}=\bigoplus_{e\in\mathbb{Z}/2\mathbb{Z}}A_{n,e}$, we
similarly define:

  * •

a coproduct $\tilde{\Delta}$ as the direct sum of the coproducts associated
with the addend product series $A_{*,e}$;

  * •

a transfer product that restrict to that defined above on each addend
$A_{*,e}$, and such that $f\tilde{\odot}g$ is zero if $f\in
H^{*}({\mathcal{S}}_{n};A_{n,e})$ and $g\in
H^{*}({\mathcal{S}}_{m};A_{m,e^{\prime}})$ with $e\not=e^{\prime}$;

  * •

if each $A_{n}$ is an algebra, a cup product $\cdot$ as above.

These structures are preliminary because of some additional signs which we
will include below.

######  Definition 2.15.

If $\\{A_{n}\\}$ is a product series of algebras or super-product series of
algebras, define $H^{*}({\mathcal{S}}_{\bullet};A_{\bullet})$ to be the direct
sum $\mathbb{F}_{p}\oplus\bigoplus_{n\geq 1}H^{*}({\mathcal{S}}_{n};A_{n})$,
obtained from $\bigoplus_{n\geq 0}H^{*}({\mathcal{S}}_{n};A_{n})$ by replacing
in component $0$ the representation
$\bigoplus_{e\in\mathbb{Z}/2\mathbb{Z}}A_{0,e}\cong\bigoplus_{e}\mathbb{F}_{p}$
via the epimorphism
$\sum_{e}\colon\bigoplus_{e}\mathbb{F}_{p}\to\mathbb{F}_{p}$, with coproduct
$\tilde{\Delta}$ and product $\tilde{\odot}$

######  Theorem 2.16.

$H^{*}({\mathcal{S}}_{\bullet};A_{\bullet})$ forms a non-commutative Hopf
algebra, with structures defined in Definition 2.14. Moreover,
$\left(H^{*}({\mathcal{S}}_{\bullet};A_{\bullet}),\tilde{\Delta},\cdot\right)$
is a bigraded component bialgebra.

###### Proof.

This theorem has been essentially proved by Strickland and Turner [ST97] for
generalized cohomology theories, and in particular for cohomology with
coefficients in the trivial representations. Their proof can be reinterpreted
diagrammatically in a group-theoretic setting as explained by
Giusti–Salvatore–Sinha [GSS12].In our case, the modification in component zero
is introduced only to ensure that the unit behaves correctly with both the
coproduct and the counit. With this exception, those diagrams also prove the
statement for coefficients in a general super-product series of algebras, with
the exception of those yielding the commutativity of $\tilde{\odot}$ and the
cocommutativity of $\tilde{\Delta}$, that involve conjugation by elements of
the symmetric groups.

The associativity of $\tilde{\Delta}$ and $\tilde{\odot}$ is shown using the
following commutative diagram of finite coverings:

${B({\mathcal{S}}_{n}\times{\mathcal{S}}_{m}\times{\mathcal{S}}_{l})}$${B({\mathcal{S}}_{n+m}\times{\mathcal{S}}_{l})}$${B({\mathcal{S}}_{n}\times{\mathcal{S}}_{m+l})}$${B({\mathcal{S}}_{n+m+l})}$

When taking cohomology with coefficient in the product series we obtain the
following commutative diagram:

${H^{*}({\mathcal{S}}_{n+m+l};A_{n+m+l})}$${H^{*}({\mathcal{S}}_{n+m}\times{\mathcal{S}}_{l};A_{n+m+l})}$${H^{*}({\mathcal{S}}_{n+m}\times{\mathcal{S}}_{l};A_{n+m}\otimes
A_{l})}$${H^{*}({\mathcal{S}}_{n+m};A_{n+m})\otimes
H^{*}({\mathcal{S}}_{l};A_{l})}$${H^{*}({\mathcal{S}}_{n}\times{\mathcal{S}}_{m+l};A_{n+m+l})}$${H^{*}({\mathcal{S}}_{n}\times{\mathcal{S}}_{m}\times{\mathcal{S}}_{l};A_{n+m+l})}$${H^{*}({\mathcal{S}}_{n}\times{\mathcal{S}}_{m}\times{\mathcal{S}}_{l};A_{n+m}\otimes
A_{l})}$${H^{*}({\mathcal{S}}_{n}\times{\mathcal{S}}_{m};A_{n+m})\otimes
H^{*}({\mathcal{S}}_{l};A_{l})}$${H^{*}({\mathcal{S}}_{n}\times{\mathcal{S}}_{m+l};A_{n}\otimes
A_{m+l})}$${H^{*}({\mathcal{S}}_{n}\times{\mathcal{S}}_{m}\times{\mathcal{S}}_{l};A_{n}\otimes
A_{m+l})}$${H^{*}({\mathcal{S}}_{n}\times{\mathcal{S}}_{m}\times{\mathcal{S}}_{l};A_{n}\otimes
A_{m}\otimes
A_{l})}$${H^{*}({\mathcal{S}}_{n}\otimes{\mathcal{S}}_{m};A_{n}\otimes
A_{m})\otimes
H^{*}({\mathcal{S}}_{l};A_{l})}$${H^{*}({\mathcal{S}}_{n};A_{n})\otimes
H^{*}({\mathcal{S}}_{m+l};A_{m+l})}$${H^{*}({\mathcal{S}}_{n};A_{n})\otimes
H^{*}({\mathcal{S}}_{m}\times{\mathcal{S}}_{l};A_{m}\otimes
A_{l})}$${H^{*}({\mathcal{S}}_{n};A_{n})\otimes
H^{*}({\mathcal{S}}_{m}\times{\mathcal{S}}_{l};A_{m}\otimes
A_{l})}$${H^{*}({\mathcal{S}}_{n};A_{n})\otimes
H^{*}({\mathcal{S}}_{m};A_{m})\otimes
H^{*}({\mathcal{S}}_{l};A_{l}).}$$\scriptstyle{\rho}$$\scriptstyle{\rho}$$\scriptstyle{\chi_{n+m,l}}$$\scriptstyle{\rho}$$\scriptstyle{\cong}$$\scriptstyle{\rho}$$\scriptstyle{\rho\otimes\operatorname{id}}$$\scriptstyle{\rho}$$\scriptstyle{\chi_{n,m+l}}$$\scriptstyle{\chi_{n+m,l}}$$\scriptstyle{\chi_{n,m+l}}$$\scriptstyle{\cong}$$\scriptstyle{\chi_{n,m}\otimes\operatorname{id}}$$\scriptstyle{\chi_{n,m}\otimes\operatorname{id}}$$\scriptstyle{\rho}$$\scriptstyle{\cong}$$\scriptstyle{\operatorname{id}\otimes\chi_{m,l}}$$\scriptstyle{\cong}$$\scriptstyle{\cong}$$\scriptstyle{\cong}$$\scriptstyle{\cong}$$\scriptstyle{\operatorname{id}\otimes\rho}$$\scriptstyle{\operatorname{id}\otimes\chi_{m,l}}$$\scriptstyle{\cong}$

In the diagram above $\rho$ denotes restriction maps, and the central square
commutes by the coherence condition $1$ of the definition of product series of
symmetric representation. The outer square looks as follows and encodes the
co-commutativity of $\tilde{\Delta}$.

${H^{*}({\mathcal{S}}_{n};A_{n})\otimes H^{*}({\mathcal{S}}_{m};A_{m})\otimes
H^{*}({\mathcal{S}}_{l};A_{l})}$${H^{*}({\mathcal{S}}_{n+m}\times{\mathcal{S}}_{l};A_{n+m})\cong
H^{*}({\mathcal{S}}_{n+m};A_{n+m})\otimes
H^{*}({\mathcal{S}}_{l};A_{l})}$${H^{*}({\mathcal{S}}_{n};A_{n})\otimes
H^{*}({\mathcal{S}}_{m+l};A_{m+l})}$${H^{*}({\mathcal{S}}_{n+m+l};A_{n+m+l}).}$$\scriptstyle{\tilde{\Delta}\otimes
1}$$\scriptstyle{1\otimes\tilde{\Delta}}$$\scriptstyle{\tilde{\Delta}}$$\scriptstyle{\tilde{\Delta}}$

Moreover, the representation considered in the diagram at the bottom are all
canonically identified with the restriction of the
${\mathcal{S}}_{n+m+l}$-representation $A_{n+m+l}$. Therefore, there is also
an induced diagram of transfer maps, obtained by replacing restrictions with
transfer maps in the commutative diagrams above:

${H^{*}({\mathcal{S}}_{n};A_{n})\otimes H^{*}({\mathcal{S}}_{m};A_{m})\otimes
H^{*}({\mathcal{S}}_{l};A_{l}))}$${H^{*}({\mathcal{S}}_{n+m}\times{\mathcal{S}}_{l};A_{n+m})\cong
H^{*}({\mathcal{S}}_{n+m};A_{n+m})\otimes
H^{*}({\mathcal{S}}_{l};A_{l})}$${H^{*}({\mathcal{S}}_{n};A_{n})\otimes
H^{*}({\mathcal{S}}_{m+l};A_{m+l})}$${H^{*}({\mathcal{S}}_{n+m+l};A_{n+m+l}).}$$\scriptstyle{\tilde{\odot}\otimes
1}$$\scriptstyle{1\otimes\tilde{\odot}}$$\scriptstyle{\tilde{\odot}}$$\scriptstyle{\tilde{\odot}}$

The fact that both $\tilde{\odot}$ and $\cdot$ are morphisms of coalgebras
with the preliminary coproduct is proved in a similar way using the following
commutative diagrams for all $n,m\in\mathbb{N}$, where $i$ denotes the
standard inclusions,
$\tau\colon{\mathcal{S}}_{b}\times{\mathcal{S}}_{c}\to{\mathcal{S}}_{c}\times{\mathcal{S}}_{b}$
the switching map, and $d$ the diagonal maps:

${\bigsqcup
B({\mathcal{S}}_{a}\times{\mathcal{S}}_{b}\times{\mathcal{S}}_{c}\times{\mathcal{S}}_{d})}$${B({\mathcal{S}}_{n}\times{\mathcal{S}}_{m})}$${\bigsqcup
B({\mathcal{S}}_{p}\times{\mathcal{S}}_{q})}$${B({\mathcal{S}}_{n+m})}$${B({\mathcal{S}}_{n}\times{\mathcal{S}}_{m})}$${B(({\mathcal{S}}_{n}\times{\mathcal{S}}_{m})^{2})}$${B({\mathcal{S}}_{n+m})}$${B({\mathcal{S}}_{n+m}^{2})}$$\scriptstyle{\bigsqcup
B(i_{a,b}\times i_{c,d})}$$\scriptstyle{\bigsqcup B(i_{a,c}\times
i_{b,d})\circ B(1\times\tau\times
1)}$$\scriptstyle{B(i_{n,m})}$$\scriptstyle{\bigsqcup
B(i_{p,q})}$$\scriptstyle{B(1\times\tau\times 1)\circ
B(d_{{\mathcal{S}}_{n}}\times
d_{{\mathcal{S}}_{m}})}$$\scriptstyle{i_{n,m}}$$\scriptstyle{i_{n,m}\times
i_{n,m}}$$\scriptstyle{d_{{\mathcal{S}}_{n+m}}}$

where the first unions are over all $p+q=n+m$ and the second union is over all
$a+b=n,c+d=m,a+c=p,b+d=q$.

It is well-known in general that $\cdot$ is associative. Since, under our
hypotheses, all the morphisms involved in the definition of the coproduct are
algebra maps with respect to the cup product, so is $\tilde{\Delta}$. ∎

In the cases of our interest, the statement of Theorem 2.16 can be improved to
show the commutativity of a twisted version of the coproduct and the products
and Hopf rig distributivity.

######  Definition 2.17.

Given a graded commutative algebra $A$ over the field $\mathbb{F}_{p}$, define
the algebraic extended powers functor  by
$EP(A)=\mathbb{F}_{p}\oplus\bigoplus_{n,d\geq
0}H^{d}({\mathcal{S}}_{n};{T_{\rho}A}_{n})$ for odd primes. Replace
$T_{\rho}A$ by $TA$ to obtain the definition when $p=2$. Given a pointed
topological space $X$, we define ${H_{EP}}^{*}(X)=EP(\tilde{H}(X))$.

We observe three different gradings on $EP(A)$. The first two are the
cohomological grading $d$, and component $n$, defined before. We also use the
decomposition ${T_{\rho}A}_{n}\cong A^{\otimes n}\oplus(sgn\otimes A^{\otimes
n})$ to induce a decomposition on $EP(A)$.

######  Definition 2.18.

Over $\mathbb{F}_{p}$ with $p$ odd, consider the decomposition

$EP(A)\cong\bigoplus_{n,d\geq 0}H^{d}({\mathcal{S}}_{n};A^{\otimes
n})\oplus\bigoplus_{n,d\geq 0}H^{d}({\mathcal{S}}_{n};sgn\otimes A^{\otimes
n}).$

We define sign degree $e$ of an element to be zero or even if it is in the
first summand of this decomposition of and to be one or odd if it is in the
second summand. Equivalently, we define $EP(A)^{d,n,e}$ to be the summand
$H^{d}({\mathcal{S}}_{n};A^{\otimes n})$ when $e=0$ or the summand
$H^{d}({\mathcal{S}}_{n};sgn\otimes A^{\otimes n})$ when $e=1$. We define the
total degree to be $t=ne+d$. As the sign degree, the total degree is only
defined modulo $2$.

By convention, if $p=2$, then the sign degree is always even/ zero. In order
to ultimately account for signs we need the following.

######  Definition 2.19.

A tri-graded component super-Hopf ring is a super-Hopf ring endowed with the
component grading $n$, the $\mathbb{Z}/2\mathbb{Z}$-grading $e$ arising from
the “super-” structure, and an additional grading $d$ over $\mathbb{N}$ where
$\Delta$ preserves all three gradings, $\odot$ preserves $n$ and $d$, $\cdot$
preserves $d$ and $e$ in each component, and the axioms of super-Hopf ring are
considered in the graded sense with respect to the total degree $t=ne+d$.

######  Definition 2.20.

Let $EP(A)$ be the extended powers of an algebra, as above. Let
$\tilde{\Delta}$ and $\tilde{\odot}$ be the maps constructed in Definition
2.14. We define:

  * •

For all $x$ such that, using Sweedler’s notation, $\tilde{\Delta}(x)=\sum
x_{(1)}\otimes x_{(2)}$, with $x_{(1)}$, $x_{(2)}$ all tri-homogeneous, the
modified coproduct is given by

$\Delta(x)=\sum(-1)^{d(x_{(2)})n(x_{(1)})e(x_{(1)})}x_{(1)}\otimes x_{(2)}.$

  * •

For all $x,y$ tri-homogeneous, the modified transfer product $x\odot
y=(-1)^{d(y)n(x)e(x)}x\tilde{\odot}y$.

We can finally state our main result of this section

######  Theorem 2.21.

Let $A$ be a graded commutative algebra of finite type over $\mathbb{F}_{p}$.
Then $EP(A)$, with the coproduct $\Delta$ and the products $\odot$ and
$\cdot$, is a super-component Hopf rig bigraded by the component $n$ and the
total degree $t$. If $p=2$, it is a component Hopf ring bigraded by the
component $n$ and the cohomological dimension $d$.

###### Proof.

Theorem 2.16 guarantees that $EP(A)$ is a non-commutative Hopf algebra with
the non-modified coproduct and transfer product $\tilde{\Delta}$ and
$\tilde{\odot}$. Clearly the additional signs do not disrupt the
(co)associativity of the transfer product $\odot$ and the coproduct $\Delta$,
that still form a Hopf algebra, and $\Delta$ and $\cdot$ still form a
component bialgebra. On the addend corresponding to the trivial
representation, the remaining Hopf rig axioms are proved diagrammatically as
in [GS17, Theorem 2.4]. On the addend corresponding to the sign
representation, one can adapt their proof by additionally keeping track of
coefficients. However, the cocommutativity of $\tilde{\Delta}$, the
commutativity of $\tilde{\odot}$ and the Hopf ring distributivity axiom fail
because, contrary to constant coefficients, inner automorphisms do not induce
the identity on cohomology with coefficients in non-trivial representations.
For instance, the diagram that would encode the commutativity of
$\tilde{\odot}$ and the cocommutativity of $\tilde{\Delta}$ is the following:

${{\mathcal{S}}_{n}\times{\mathcal{S}}_{m}}$${{\mathcal{S}}_{m}\times{\mathcal{S}}_{n}}$${{\mathcal{S}}_{n+m}}$${{\mathcal{S}}_{n+m},}$$\scriptstyle{\tau}$$\scriptstyle{\gamma}$

where $\tau$ is the obvious switching isomorphism and $\gamma$ is the
conjugation by a permutation $\sigma_{n,m}\in{\mathcal{S}}_{n+m}$ depending on
$n$ and $m$. In cohomology with trivial coefficients, such conjugation maps
induce the identity, but in cohomology with coefficients in
$\operatorname{sgn}$ they induce the multiplication by $(-1)^{nm}$. Therefore
$x\tilde{\odot}y=(-1)^{n(x)n(y)e(x)e(y)+d(x)d(y)}y\tilde{\odot}x$. The
additional signs introduced with the modified transfer product guarantee that
$x\odot y=(-1)^{t(x)t(y)}y\odot x$. The same argument shows that $\Delta$ is
cocommutative. By analyzing the corresponding diagram, one proves similarly
that $(EP(A),\Delta,\odot,\cdot)$ satisfy the graded version of Hopf ring
distributivity, with respect to the total degree. ∎

######  Remark 2.22.

Additively, $H^{*}({\mathcal{S}}_{n};\operatorname{sgn}\otimes A^{\otimes n})$
is isomorphic to $H^{*}({\mathcal{S}}_{n},\Sigma(A)^{\otimes n})$, where
$\Sigma$ is the suspension functor on graded vector spaces. Hence, for $p$
odd, $DP(A)\cong\bigoplus_{n\geq 0}H^{*}({\mathcal{S}}_{n};A^{\otimes
n})\oplus H^{*}({\mathcal{S}}_{n};\Sigma(A)^{\otimes n})$ as vector spaces.
The total degree on $EP(A)$ is defined in such a way that this isomorphism
becomes a degree-preserving linear map. Seen through this lense, the
correction signs appearing in the modified coproduct and transfer product
become less mysterious and arise from the Koszul sign conventions on
$Hom(W^{{\mathcal{S}}_{n}}_{*};\Sigma(A))$.

For $p=2$, ${H_{EP}}^{*}(X)$ differs from the cohomology of $DX$ by an
innocuous completion. We see something similar at odd primes if we take only
the $H^{*}(D_{n}X;\mathbb{F}_{p})$ summands. Our work ahead is to understand
the algebraic extended powers functor as a free functor to the category of
Hopf rings with additive divided powers, which we next develop.

######  Remark 2.23.

As we remarked in the first section, if $X$ is a manfiold then cochains of
finite dimensional approximations of $CX$ can be defined by submanifolds,
which informally we consider through “conditions” on the underlying points of
the configuration or on their labels in $X$.

Cup product as usual is given by intersecting, or in this case requiring that
two sets of conditions hold. Transfer product defines a condition on $n+m$
points by asking that a condition is satisfied on some subcollection of $n$
points and another is satisfied on $m$ points. The transfer product of a class
with itself is divisible by two because the condition is satisfied whether
some $n$ points or its complement is considered “first.” The divided powers
operation repeats a condition on $n$ points $k$ times to define a condition on
$nk$ points.

See Theorem 4.9 of [GSS12] where we discuss the representatives for Hopf ring
generators as subvarieties defined by $2^{n}$ points sharing a coordinate, and
[GS14] which gives a development through cellular models for one-point
compactifications of configuration spaces. A chain-level model for the
transfer product is given which would also work for the divided powers
operations. (Note that the cup product result claimed but not proven in [GS14]
is not correct.)

###  2.4. Divided powers on cohomology of symmetric groups with coefficients
in a product series

We now develop a divided powers structure on ${H_{EP}}^{*}(X)$, as a special
case of divided power structure on cohomology of symmetric groups with
coefficients in a product series of algebras. In order to define it we need
restriction and transfer maps for subgroups of symmetric groups which are
defined by partitions.

######  Definition 2.24.

A labeled multipartition $\pi$ of a set $S$ is the labeling of the leaves of a
rooted tree by the elements of $S$, possibly with additional labeling of
internal vertices. The subsets defined by considering all of the leaf labels
over a fixed internal vertex are called the blocks of the labeled
multipartition. The depth of a block is the number of edges between its
corresponding internal vertex and the root vertex.

Given a labeled multipartition $\pi$ of $\\{1,\dots,n\\}$ define its
automorphism group ${\mathcal{S}}_{\pi}$ to be the automorphisms of the tree
which preserve any additional labels of internal vertices. We identify
${\mathcal{S}}_{\pi}$ canonically with subgroup of ${\mathcal{S}}_{n}$ through
the action on leaves.

A labeled multipartition is determined up to additional labels by its blocks.
The additional labels of internal vertices are sometimes used to make such
structures more rigid, ruling out automorphisms between blocks. Since we only
use them in this way, we do not give explicit additional labels or name the
labeling set; we only indicate when labels are shared or they differ.

There are some basic constructions which give rise to the multipartitions and
thus automorphism subgroups we consider.

######  Definition 2.25.

If $\pi_{1},\dots,\pi_{k}$ are labeled multipartitions of $S_{1},\dots,S_{k}$
define the union $\pi_{1}\cup\dots\cup\pi_{k}$ to be the multipartition of
$\bigsqcup S_{i}$ in which the trees defining the $\pi_{i}$ are grafted to a
(new) root with $k$ edges, and the internal vertices corresponding to the
original roots are given distinct labels.

If ${\pi}$ is a labeled multipartition of $S$ define the multiple $m\cdot\pi$
to be the labeled multipartition of $\bigsqcup_{i=1}^{m}S$ in which $m$ copies
of the tree defining $\pi$ are grafted to a single root and given the same
label.

Let $\underline{n}$ denote the trivial labeled partition of $\\{1,\dots,n\\}$,
defined by a rooted tree with no internal vertices.

It is convenient to use the standard finite sets $\\{1,\dots,n\\}$ for all
labels, in which case we identify the abstract disjoint union
$\\{1,\dots,n_{1}\\}\sqcup\dots\sqcup\\{1,\dots,n_{k}\\}$ with the set
$\\{1,\dots,\sum n_{i}\\}$ through the standard “ordering on the page.” We
identify $\bigsqcup_{m}\\{1,\dots,n\\}\cong\\{1,\dots,mn\\}$ similarly.

The fact that the union of labeled multipartitions has distinct labels while
those of a multiple multipartition are repeated is a key distinction, in a
sense giving rise to our divided powers operations. In particular,
${\mathcal{S}}_{\underline{n}\cup\underline{m}}$ is isomorphic to
${\mathcal{S}}_{n}\times{\mathcal{S}}_{m}$, including when $n=m$, while
${\mathcal{S}}_{2\cdot\underline{n}}$ is isomorphic to
${\mathcal{S}}_{n}\wr{\mathcal{S}}_{2}$.

We consider standard set partitions as labeled multipartitions through trees
whose internal vertices all have depth one, and identical labels. Recall that
the set of partitions of a set is a poset under refinement and that this poset
has meet and join operators, which we denote $\wedge$ and $\vee$, that provide
it with a lattice structure. The following properties of ${\mathcal{S}}_{\pi}$
are straightforward.

######  Lemma 2.26.

Let $\pi=\\{S_{i}\\}$ and $\pi^{\prime}$ be partitions of $\\{1,\dots,n\\}$.

  1. (1)

${\mathcal{S}}_{m\cdot{\underline{k}}}$ is isomorphic to the wreath product
${\mathcal{S}}_{k}\wr{\mathcal{S}}_{m}$. More generally, if $m_{\pi}(i)$ is
the number of parts of $\pi$ whose cardinality is $i$, ${\mathcal{S}}_{\pi}$
is conjugate to
$\bigtimes_{i=0}^{n}{\mathcal{S}}_{i}\wr{\mathcal{S}}_{m_{\pi}(i)}$ in
${\mathcal{S}}_{n}$.

  2. (2)

For any $\sigma\in{\mathcal{S}}_{n}$, the conjugate
$\sigma{\mathcal{S}}_{\pi}\sigma^{-1}$ coincides with
${\mathcal{S}}_{\sigma\pi}$.

  3. (3)

The map $[\sigma]\mapsto\sigma\pi$ is a bijection between the cosets
${\mathcal{S}}_{n}/{\mathcal{S}}_{\pi}$ and the set of partitions that are
permutations of $\pi$.

Partition subgroups play key roles in defining the divided powers.

######  Definition 2.27.

Let $\\{A_{n}\\}$ be a product series of algebras. For $x\in
H^{*}({\mathcal{S}}_{n};A_{n})$ we define $x^{[{k}]}\in
H^{*}({\mathcal{S}}_{kn};A_{kn})$ to be

$x^{[{k}]}:=tr^{{\mathcal{S}}_{kn}}_{{\mathcal{S}}_{n}\wr{\mathcal{S}}_{k}}(x^{\otimes
k}),$

where we identify $A_{kn}$ with $A_{n}^{\otimes k}$ via the structural
isomorphism.

######  Proposition 2.28.

If $\\{A_{n}\\}$ is a product series of algebras then
$\bigoplus_{n}H^{*}({\mathcal{S}}_{n};A_{n})$ with cup and transfer product
from Definition 2.14, coproduct from Definition 2.20 and divided powers from
Definition 2.27 satisfies the axioms for divided powers structures.

Before proving this, we set aside a basic fact from group cohomology which we
repeatedly use, which follows from the standard fact that restriction followed
by transfer from a subgroup is multiplication by the index of a subgroup.

######  Lemma 2.29.

If $K\subset H\subset G$ are finite-index inclusions of groups then

$tr^{G}_{K}\circ res^{H}_{K}=[K:H]tr^{G}_{H}$

###### Proof of Proposition 2.28.

Let $x\in H^{*}({\mathcal{S}}_{n};A_{n})$ be represented by a homomorphism $f$
from a resolution of ${\mathcal{S}}_{n}$ to $A_{n}$, and similarly $y\in
H^{*}({\mathcal{S}}_{m};A_{m})$ be represented by $g$.

The $0,1$-Cases axioms of Definition 2.5 are immediate. For the Binomial
axiom, we calculate $(x+y)^{[{r}]}$ by taking the transfer (induction) of
$(f+g)^{\otimes r}$, which is a sum over $i$ of shuffles of $f^{\otimes i}$
and $g^{\otimes(r-i)}$. But the transfer on such shuffles is exactly
$x^{[{i}]}\odot y^{[{r-i}]}$.

Checking the Exponent Axiom, $x^{[{m}]}\odot x^{[{r}]}$ and $x^{[{m+r}]}$ are
both images under transfer of $f^{\otimes m+r}$, but the former is induced up
from the subgroup
$({\mathcal{S}}_{m\cdot\underline{n}})\times({\mathcal{S}}_{r\cdot\underline{n}})$,
while the latter is induced up from ${\mathcal{S}}_{(m+r)\cdot\underline{n}}$.
By Lemma 2.29, the former is obtained from the latter by multiplication by the
quotient of the indices of these subgroups, namely $\binom{m+r}{m}$.

We next verify the Distributivity Axiom, which in the Hopf rig setting is that
$(x\odot y)^{[{k}]}=x^{\odot^{k}}\odot y^{[{k}]}$. By definition $(x\odot
y)^{\otimes^{k}}$ is a composite of induction maps of $(f\otimes
g)^{\otimes^{k}}$ from
${\mathcal{S}}_{k\cdot({\underline{n}\cup\underline{m}})}$ ultimately to
${\mathcal{S}}_{k(n+m)}$. Similarly, $x^{\odot^{k}}\odot y^{[{k}]}$ is
obtained by inducing $f^{\otimes^{k}}\otimes g^{\otimes^{k}}$ from
${\mathcal{S}}_{\cup_{k}\underline{n}\cup k\cdot\underline{m}}$. After
conjugating so that $(f\otimes g)^{\otimes^{k}}$ becomes
$f^{\otimes^{k}}\otimes g^{\otimes^{k}}$, both subgroups are then subgroups of
${\mathcal{S}}_{k\cdot\underline{n}\cup k\cdot\underline{m}}$, for which
$f^{\otimes^{k}}\otimes g^{\otimes^{k}}$ is invariant. Lemma 2.29 applies in
both cases, so that each result is the induction from
${\mathcal{S}}_{k\cdot\underline{n}\cup k\cdot\underline{m}}$ multiplied by
the index of the corresponding subgroup. The Distributivity axiom follows as
these two indices are the same.

We verify the Composition Axiom
$(x^{[{k}]})^{[{h}]}=\frac{(hk)!}{(k!)^{h}h!}x^{[{hk}]}$ similarly. By
definition $x^{[{hk}]}$ is given by inducing $f^{\otimes^{hk}}$ from
${\mathcal{S}}_{hk\cdot\underline{n}}$ to ${\mathcal{S}}_{nhk}$, while
$(x^{[{k}]})^{[{h}]}$ is the induction of the same cochain from
${\mathcal{S}}_{h\cdot(k\cdot\underline{n})}$. Since
${\mathcal{S}}_{h\cdot(k\cdot\underline{n})}\subset{\mathcal{S}}_{hk\cdot\underline{n}}$,
Lemma 2.29 applies to show that the differ by a multiplicative coefficient
equal to the index of this inclusion, which is exactly
$\frac{(hk)!}{(k!)^{h}h!}$. ∎

We next turn to compatibility of divided powers with coproduct. As our divided
powers structure are defined by transfers, and our coproduct is defined by
restriction, the Cartan-Eilenberg Double Coset Formula will be of use. Recall
that if $H$ and $K$ are subgroups of a finite group $G$ and $\mathcal{R}$ is a
set of representatives in $G$ for the set of double cosets $H\backslash G/K$,
then, on cohomology with coefficients in a given $G$-representation,

$\rho^{G}_{H}\circ\operatorname{tr}^{G}_{K}=\sum_{r\in\mathcal{R}}\operatorname{tr}^{H}_{H\cap
c_{r}K}\circ\rho^{c_{r}K}_{H\cap c_{r}K}\circ{c_{r}}^{*}$

where $c_{r}$ is the isomorphism of $K$ with $rKr^{-1}$ and $\rho$ and
$\operatorname{tr}$ denote restriction and transfer maps respectively. This is
proved for example as Theorem I.6.2 of [AM04].

######  Proposition 2.30.

If $\\{A_{n}\\}$ is a product series of algebras, then on
$\bigoplus_{n}H^{*}({\mathcal{S}}_{n};A_{n})$ the divided powers of Definition
2.27 commute with the coproduct of Definition 2.20.

###### Proof.

We start with a more explicit expression of divided powers of a coproduct. If
$x\in H^{*}({\mathcal{S}}_{n};A_{n})$, set $\Delta x=\sum x_{(i,n-i)}$, where
$x_{(i,n-i)}\in H^{*}({\mathcal{S}}_{i};A_{i})\otimes
H^{*}({\mathcal{S}}_{n-i};A_{n-i})$. By the Binomial Axiom, which was just
established in Proposition 2.28,

$\left(\Delta
x\right)^{[{k}]}=\sum_{j_{0}+\dots+j_{n}=k}\bigodot_{i=0}^{n}\left(x_{(i,n-i)}\right)^{[{j_{i}}]}.$

If $a+b=n$ the $(a,b)$ component of this consists of terms with $\sum
j_{i}\cdot i=a$. We show that $\left(\Delta(x^{[{k}]})\right)_{(a,b)}$ agrees
with this.

Let $x\in H^{*}({\mathcal{S}}_{n};A_{n})$ be represented by a homomorphism $f$
from a resolution of ${\mathcal{S}}_{n}$ to $A_{n}$. By definition
$(x^{[{k}]})_{(a,b)}$ will be represented by
$\rho^{{\mathcal{S}}_{nk}}_{{\mathcal{S}}_{\underline{a}\cup\underline{b}}}\circ\operatorname{tr}^{{\mathcal{S}}_{nk}}_{{\mathcal{S}}_{k\cdot\underline{n}}}(f^{\otimes
k})$. Applying the Cartan-Eilenberg Double Coset Formula, this coincides with

$\sum_{r\in\mathcal{R}}\operatorname{tr}_{{\mathcal{S}}_{\underline{a}\cup\underline{b}}\cap
c_{r}{\mathcal{S}}_{k\cdot\underline{n}}}^{{\mathcal{S}}_{\underline{a}\cup\underline{b}}}\circ\rho_{{\mathcal{S}}_{\underline{a}\cup\underline{b}}\cap
c_{r}{\mathcal{S}}_{k\cdot\underline{n}}}^{c_{r}{\mathcal{S}}_{k\cdot\underline{n}}}\circ
c_{r}^{\\#}(f^{\otimes^{k}}),$

where $\mathcal{R}$ is a set of representatives for the set of double cosets
$({\mathcal{S}}_{\underline{a}\cup\underline{b}})\backslash{\mathcal{S}}_{nk}/({\mathcal{S}}_{k\cdot\underline{n}})$.

To understand this double coset space, recall that the cosets of symmetric
groups modulo the automorphism groups ${\mathcal{S}}_{\pi}$ correspond to
partitions with the same shape as $\pi$. Thus
${\mathcal{S}}_{nk}/{\mathcal{S}}_{k\cdot\underline{n}}$ corresponds to
partitions of $\\{1,\dots,nk\\}$ with $k$ parts of cardinality $n$, while
${\mathcal{S}}_{\underline{a}\cup\underline{b}}\backslash{\mathcal{S}}_{nk}$
corresponds to partitions into two sets with cardinalities $a$ and $b$, or
equivalently bicolorings. Thus $\mathcal{R}$ corresponds to bicolored
partitions of $\\{1,\dots,nk\\}$ with $k$ parts of cardinality $n$.
Equivalently $\mathcal{R}$ is given by multipartitions governed by a tree with
$k$ edges attached to the root, and each of those with at most two edges
labeled $a$ and/or $b$, with a total of $n$ leaf edges over them. All labels
in the multipartition are identical. The intersections
${\mathcal{S}}_{\underline{a}\cup\underline{b}}\cap
c_{r}{\mathcal{S}}_{k\cdot\underline{n}}$ are the automorphisms of such
multipartitions. These are conjugate to
${\mathcal{S}}_{\cup_{i}m_{i}\cdot(\underline{i}\cup\underline{n-i})}$, where
$m_{i}$ is the number of times there are $i$ leaves in the group labeled by
$a$ within a $k$-block.

We then identify the restriction
$\rho_{{\mathcal{S}}_{\underline{a}\cup\underline{b}}\cap
c_{r}{\mathcal{S}}_{k\cdot\underline{n}}}^{c_{r}{\mathcal{S}}_{k\cdot\underline{n}}}\circ
c_{r}^{\\#}(f^{\otimes^{k}})$ with
$\bigotimes_{i=0}^{n}{f_{(i,n-i)}}^{\otimes^{m_{i}}}$, where ${f_{(i,n-i)}}$
represents $x_{(i,n-i)}$. To calculate
$\operatorname{tr}_{{\mathcal{S}}_{\underline{a}\cup\underline{b}}\cap
c_{r}{\mathcal{S}}_{k\cdot\underline{n}}}^{{\mathcal{S}}_{\underline{a}\cup\underline{b}}}$
applied to this class, we factor the inclusion of
${\mathcal{S}}_{\underline{a}\cup\underline{b}}\cap
c_{r}{\mathcal{S}}_{k\cdot\underline{n}}$ in
${\mathcal{S}}_{\underline{a}\cup\underline{b}}$. Since
${\mathcal{S}}_{\underline{a}\cup\underline{b}}\cap
c_{r}{\mathcal{S}}_{k\cdot\underline{n}}$ is conjugate to
${\mathcal{S}}_{\cup_{i}m_{i}\cdot(\underline{i}\cup\underline{n-i})}$ it is
contained in a conjugate of
${\mathcal{S}}_{\cup_{i}(\underline{im_{i}}\cup\underline{(n-i)m_{i}})}$. The
transfer map up to this subgroup is a product of transfer maps defining
divided powers, so $\bigotimes_{i=0}^{n}{f_{(i,n-i)}}^{\otimes^{m_{i}}}$ is
sent to $\bigotimes_{i=0}^{n}(f_{(i,n-i)})^{[{m_{i}}]}$. The transfer map from
${\mathcal{S}}_{\cup_{i}(\underline{im_{i}}\cup\underline{(n-i)m_{i}})}$ to
${\mathcal{S}}_{\underline{a}\cup\underline{b}}$ is the transfer product on
the tensor product of the cohomology with itself, so
$\bigotimes_{i=0}^{n}(f_{(i,n-i)})^{[{m_{i}}]}$ is sent to
$\bigodot_{i=0}^{n}(f_{(i,n-i)})^{[{m_{i}}]}$, which agrees with our
determination of $\left((\Delta x)^{[{k}]}\right)_{(a,b)}$ above. ∎

######  Corollary 2.31.

${H_{EP}}^{*}(X)$ is a bigraded component Hopf ring with additive divided
powers.

###  2.5. Statement of first main results

We state here our first main structure theorem, that we will prove in the
remaining sections.

We require two more definitions before stating our first main theorems.

######  Definition 2.32.

An element of a divided powers algebra of even degree when $p$ odd, or any
degree when $p=2$, is standard non-nilpotent if all $x^{[{m}]}$ are non-zero,
with $\Delta x^{[{m}]}=\sum_{i+j=m}x^{[{i}]}\otimes x^{[{j}]}$ and
$x^{[{i}]}\odot x^{[{j}]}=\binom{i+j}{i}x^{[{i+j}]}$.

######  Definition 2.33.

Let $\mathcal{C}$ be a category, $S$ a subcategory, and
$F:\mathcal{C}\to\mathcal{D}$ a functor. We say some object $x\in S$ is
universal among $S$ with respect to $F$ if its image under $F$ is initial in
the full subcategory generated by $F(\mathcal{C})$.

We first give an immediate reformulation of the calculations of
Giusti–Salvatore–Sinha utilizing divided powers structure.

######  Theorem 2.34 (from [GSS12]).

The mod-two cohomology of extended powers of $S^{0}$, namely
$H_{EP}^{*}(S^{0})\cong\bigoplus H^{*}(B{\mathcal{S}}_{n})$, is a component
Hopf rig with additive divided powers which contains standard non-nilpotent
classes $\gamma_{i}\in H^{2^{i}-1}(B{\mathcal{S}}_{2^{i}})$. It is universal
among such objects, with respect to the functor which forgets divided powers.

In other words, there are no relations required to understand
$H_{EP}^{*}(S^{0})$ as a Hopf ring other than those given in the axioms of
component Hopf rig with additive divided powers. This thus determines the
cohomology ring structure. In particular, the subrings generated by
$\gamma_{i}^{[{2^{j}}]}$ for $i+j=n$ are polynomial subrings of the cohomology
of $B{\mathcal{S}}_{n}$. But there are many relations for classes involving
the transfer product. At the moment, we do not fully understand the divided
powers structure – see Remark 2.39 below – so while we use divided powers to
generate classes we forget them in order to have a unique characterization.

The following is the odd primes counterpart of Theorem 2.34, that we will
prove in Section 3. For $p>2$, let
$T_{\rho}(H^{*}(X;\mathbb{F}_{p}))=\\{T_{\rho,n}(H^{*}(X;\mathbb{F}_{p}))\\}_{n\in\mathbb{N}}$
be the super-product series of algebras defined above. We consider the sign
degree $e$ that determines the super-structure and the degree $d$ induced by
the cohomological dimension of $H^{*}(X;\mathbb{F}_{p})$.

######  Theorem 2.35.

The mod-$p$ cohomology of extended powers of $S^{0}$, namely
$H_{EP}^{*}(S^{0})\cong\bigoplus H^{*}(B{\mathcal{S}}_{n};\rho)$, is a
trigraded component super-Hopf rig with additive divided powers which contains

  * •

standard non-nilpotent classes ${\gamma_{k}}$ in grading
$(2(p^{k}-1),p^{k},0)$,

  * •

primitive classes $\lambda_{k}$ in grading $(p^{k}-1,p^{k},1)$,

  * •

standard non-nilpotent classes ${\gamma^{\prime}_{k}}$ in grading
$(p^{k}-2,p^{k},1)$,

with cup product relations ${\lambda_{k}}^{2}={\gamma_{k}}$. It is universal
among such objects with respect to the functor which forgets divided powers.

######  Remark 2.36.

The sub-Hopf rig of ${H_{EP}}^{*}(S^{0})$ corresponding to the addend of even
sign degree has been calculated by Guerra [Gue17]. The generating classes
$\gamma_{k}$, $\alpha_{i,k}$ and $\beta_{i,j,k}$ appearing there can be
retrieved from our presentation as $\gamma_{k}$,
$(-1)^{\frac{p-1}{2}(k-i)}\lambda_{k}{\gamma^{\prime}}_{i}^{[{p^{k-i}}]}$ and
$(-1)^{\frac{p-1}{2}(j-i)}{\gamma^{\prime}}_{i}^{[{p^{k-i}}]}{\gamma^{\prime}}_{j}^{[{p^{k-i}}]}$,
respectively.

For any nonempty $X$, the cohomology of $D(X_{+})$ is an algebra over the
cohomology of $D(S^{0})$, through the projection map which sends $X$ to the
non-base point of $S^{0}$. This is injective, with any choice point in $X$
giving a splitting. We use this algebra structure for the two following
results, that will be proved in Section 4. For odd primes, we compute a bigger
Hopf ring with cohomology taken with coefficients in $\rho$, but one can
easily recover the ordinary mod $p$ cohomology of $\tilde{D}(X)$ as a Hopf
ring by extracting only the homogeneous part of even sign degree.

######  Theorem 2.37.

The mod-two cohomology of extended powers of $X_{+}$ is the universal
component super-Hopf rig with additive divided powers over the cohomology of
extended powers of $S^{0}$ which contains the cohomology classes of $X$ as
classes which are standard non-nilpotent. Relations are

$x^{[{n}]}\cdot y^{[{n}]}=(x\cdot y)^{[{n}]}\;\;{\text{f}or}\;\;x,y\in
H^{*}(X)\mbox{ and}$ $x^{[{nm}]}\cdot y^{[{n}]}=(x^{[{m}]}\cdot
y)^{[{n}]}\;\;{\text{f}or}\;\;x\in H^{*}(X),y\in H^{*}(D_{m}(S^{0})).$

It is universal among such objects with respect to the functor which forgets
divided powers.

######  Theorem 2.38.

For $p>2$, the mod-$p$ cohomology of extended powers of $X_{+}$ is a component
super-Hopf rig with additive divided powers over the cohomology of extended
powers of a point which contains two copies of the cohomology of $X$, which we
denote by $x_{e}$ degree $(d,1,e)$ for $e\in\\{0,1\\}$. These classes are
standard non-nilpotent if $d+e$ is even and primitive if $d+e$ is odd, with

$x_{e}^{[{n}]}\cdot y_{e^{\prime}}^{[{n}]}=(x\cdot y)_{e+e^{\prime}}^{[{n}]},$
$x_{e}^{[{nm}]}\cdot z^{[{n}]}=(x_{e}^{[{m}]}\cdot z)^{[{n}]}\mbox{ and}$
$x_{e}\odot y_{e^{\prime}}=0\mbox{ if }e\not=e^{\prime},$

if $x,y\in H^{*}(X;\mathbb{F}_{p})$, $z\in
H^{*}(D_{m}(S^{0});\mathbb{F}_{p})$, and $e,e^{\prime}\in\\{0,1\\}$, with the
sum $e+e^{\prime}$ understood modulo two. It is universal among such objects
with respect to the functor which forgets divided powers.

######  Remark 2.39.

Theorems 2.37 and 2.38 explicitly embed the cohomology of the symmetric groups
and the cohomology of $X$ in ${H_{EP}}^{*}(X)$. The relations provide a
compatibility identity between divided powers and cup product. We can
interpret this structure as a “bigraded component Hopf ring with divided
powers over the cohomology of symmetric groups generated by $H^{*}(X)$”. More
generally, one could define a “bigraded component Hopf ring with divided
powers” as a bigraded component Hopf ring with additive divided powers $A$
such that a similar compatibility condition between divided powers and cup
product hold for all $x,y\in A$ divided powers of primitive elements. However,
this simple identity does not extend to classes $x$ not arising from the
cohomology of $X$. For example, with $p=2$ and $X=\\{*\\}$,
${H_{EP}}^{*}(X_{+})$ becomes the ordinary mod $2$ cohomology of the symmetric
groups and the divided powers operations $\\_^{[{k}]}$ agree with the
cohomological transfer maps associated to
${\mathcal{S}}_{k}\wr{\mathcal{S}}_{n}\to{\mathcal{S}}_{nk}$. A procedure to
compute these transfer maps is described by Kechagias [Kec09]. Combining
Kechagias’s algorithm with the computations of the restriction to elementary
abelian subgroups achieved by Giusti–Salvatore–Sinha [GSS12] one obtains the
following equality

$(\gamma_{1})^{[{2}]}\cdot(\gamma_{1}^{2})^{[{2}]}=\gamma_{2}^{2}+(\gamma_{1}^{3})^{[{2}]}.$

As the right hand side is not simply $(\gamma_{1}^{3})^{[{2}]}$, the
relationship between the divided powers operations and the cup product in
${H_{EP}}^{*}(X)$ is not as expected and appears likely to be complicated in
general.

######  Remark 2.40.

Recall from Remark 2.23 that Hopf ring with additive divided powers structures
all have simple geometric cochain models when using configuration space models
for extended powers of manifolds. Many of the basic classes in Theorems
2.34-2.38 have geometric cochain models as well. The $\gamma_{k}$ for
${\mathbb{F}}_{2}$ are represented by “$2^{k}$ points which share a
coordinate”. Formally one considers the manifold consisting of triples
$(x,X,Y)$ where $x\in{\mathbb{R}}$, $X$ is a configuration of $2^{k}$ distinct
unordered points in $x\times{\mathbb{R}}^{N-1}\subset{\mathbb{R}}^{N}$, and
$Y$ is a configuration of unordered points in the complement of $X$ in
${\mathbb{R}}^{N}$. This manifold maps properly to the configuration space,
generically an embedding but finite-to-one if some point in $Y$ is contained
in $x\times{\mathbb{R}}^{N-1}$ or some $2^{k}$ points in $Y$ contained in an
$x^{\prime}\times{\mathbb{R}}^{N-1}$. It thus defines a geometric cocycle,
which represents $\gamma_{k}$ as shown in Theorem 4.9 of [GSS12]. Similarly,
the $\gamma_{k}$ for odd primes are represented by the geometric cochain
defined by “$p^{k}$ points in $\mathbb{C}^{N}$ which share a complex
coordinate.” The cup products $\gamma_{k}\cdot x^{[{k}]}$ on the $k$th
extended power is represented by labeled points which share a coordinate and
are labeled by the representative of $x$. One must name which extended power
because the cup product of manifestations of these classes on higher extended
powers, given by transfer products with unit classes, can differ from this (as
some points in a configuration could share their coordinate while others share
their label).

We have not developed geometric representatives for the $\gamma_{k}^{\prime}$
and $\lambda_{k}$, or their even sign degree products as considered earlier in
[Gue17]. And as mentioned before, one must take care with geometry and cup
products since one would expect ${\gamma_{1}^{[{2}]}}^{3}$ to be represented
by the geometric cochain defined by four points which consist of two sets of
two points each of which share three coordinates. But it is represented by the
union of this along with a copy of the geometric cochain defined by four
points which share two coordinates.

###  2.6. Explicit construction of (relatively) universal component Hopf rings
with divided powers

We next better understand the structures elaborated in the main theorems just
stated, abstracting as follows.

######  Definition 2.41.

Let $A$ be a component super-Hopf ring with additive divided powers containing
$\rho$ as a subalgebra of $A_{1}$, its $\cdot$-algebra in component $1$, and
let $V$ be a connected graded algebra. Let $H_{alg}(A,V)$, which we generally
shorten to $H_{alg}$, be the super-Hopf ring with additive divided powers
satisfying the following five properties.

  1. (1)

There is a map $\pi\colon A\to H_{alg}$ that is both a super-Hopf ring map and
a morphism of divided powers structures preserving the $\cdot$-unit of each
component;

  2. (2)

There is a $\rho$-algebra homomorphism $\iota$ from $V\otimes\rho$ to the
subspace of elements of component $1$ inside $H_{alg}$, (that is, an algebra
morphism $\iota\colon V\to H_{alg}$ in component $1$ if $p=2$);

  3. (3)

For all $x,x^{\prime}\in V\otimes\rho$ both with even total degree
$\iota(x)^{[{n}]}\cdot\iota(x^{\prime})^{[{n}]}=\iota(xx^{\prime})^{[{n}]}$;

  4. (4)

For all $x\in V\otimes\rho$ with even total degree and for all $y\in A$ such
that $n(y)=m$,

$\iota(x)^{[{nm}]}\cdot\pi(y)^{[{n}]}=(\iota(x)^{[{m}]}\cdot\pi(y))^{[{n}]};$

  5. (5)

If $p>2$, $x^{[{n}]}\odot{x^{\prime}}^{[{m}]}=0$ if $x\in
V\otimes\mathbb{F}_{p}\subseteq V\otimes\rho$ and $x^{\prime}\in
V\otimes\operatorname{sgn}\subseteq V\otimes\rho$.

We wish to describe, in terms of additive bases of $A$ and $V$, a basis for
$H_{alg}$. In Section 4.1 will specialize it to our case of interest, where
$A=\mathbb{F}_{p}\oplus\bigoplus_{n\geq 1}H^{*}({\mathcal{S}}_{n};\rho)$ and
$V=H^{*}(X;\mathbb{F}_{p})$.

In the remainder of this section, we will assume that $p>2$, because the
treatment for $p=2$ is similar and much simpler. Let $V_{even}$ (respectively
$V_{odd}$) be the subspaces of $V\otimes\rho$ of even (respectively odd) total
degree.

As $A$, with the product $\odot$ and the coproduct $\Delta$ alone is a
bicommutative divided powers Hopf algebra, by a classical result [And71] it
must be the free divided powers Hopf algebra on its subspace of primitive
elements. That is, $A\cong DP(P(A))$. Since $A$ is a component super-Hopf
ring, $P(A)$ is a non-unital component super-algebra with $\cdot$, that
inherits a triple grading from $A$.

We consider the tensor product of algebras $V_{even}\otimes P(A)$, that we
tri-grade with the following rule: if $v\in V_{even}$ is homogeneous of degree
$d(v)$ and $y\in P(A)$ is tri-homogeneous of tri-degree $(d(y),n(y),e(y))$,
then $v\otimes y\in V\otimes P(A)$ is tri-homogeneous of tri-degree
$(d(v)n(y)+d(y),n(y),e(y)+d(v))$. Intuitively, we interpret a pure tensor
$v\otimes y\in V_{even}\otimes P(A)$ as the element
$\iota(v)^{[{n(y)}]}\cdot\pi(y)$. This justifies our choice of degrees. Recall
that the sign degree $e$ is defined only modulo $2$.

From a graded module $M$ we can construct an augmented non-unital coalgebra
$M^{pr}=\mathbb{F}_{p}\oplus M$ with the $\mathbb{F}_{p}$ addend in tri-degree
$(0,0,0)$ and a coproduct $\Delta(x)=1\otimes x+x\otimes 1$ for all $x\in M$.
This is the “primitive coalgebra extension” of the module $M$.

We then decompose $V_{even}\otimes P(A)$ as the direct sum $(V_{even}\otimes
P(A))_{+}\oplus(V_{even}\otimes P(A))_{-}$, where $(V_{even}\otimes P(A))_{+}$
is the subspace generated by elements $v\otimes x\in V_{even}\otimes P(A)$ in
which $e(v)=e(x)$ mod $2$ and $(V_{even}\otimes P(A))_{-}$ consists of such
with $e(v)\not=e(x)$ mod $2$). Then

$(H_{alg},\odot,\Delta)\cong{DP_{HA}((V_{even}\otimes P(A))_{+}^{pr})\oplus
DP_{HA}((V_{even}\otimes P(A))_{-}^{pr})}/{\sim},$

where $\sim$ is the equivalence relation that identifies the units of the two
addends. Additive bases for $V$ and $P(A)$ induce a basis for
$DP_{HA}((V_{even}\otimes P(A))_{\pm}^{pr})$, and consequently on $H_{alg}$,
consisting of elements of the form $\bigodot_{i}(v_{i}\otimes
y_{i})^{[{n_{i}}]}$ with $v_{i},y_{i}$ basis elements with $n_{i}=1$ if
$v_{i}\otimes y_{i}$ has odd total degree and $v_{i}\otimes y_{i}$ belong all
to $(V\otimes P(A))_{+}$ or all to $(V\otimes P(A))_{-}$.

For now, $H_{alg}$ is only a component divided powers super-Hopf algebra.
Under this isomorphism we next identify maps $\pi\colon A\to H_{alg}$ and
$\iota\colon V\otimes\rho\to H_{alg}$, as well as a second product
$\cdot\colon H_{alg}\otimes H_{alg}\to H_{alg}$ which provide it with a super-
Hopf ring structure.

First, we define $\pi\colon A\to H_{alg}$ as the unique divided powers super-
Hopf algebra morphism extending the linear map on primitives $\pi\colon y\in
P(A)\mapsto 1_{V}\otimes y\in V_{even}\otimes P(A)\subseteq P(H_{alg})$.
Second, let $\iota\colon V\otimes\rho\to H_{alg}$. If $v\in V_{even}$, then we
let $\iota(v)=v\otimes 1_{1}$, where $1_{1}\in A$ is the $\cdot$-product unit
of the $1$-component of $A$. If $v\in V_{odd}$, then let $s\in\rho$ be a
generator of the $\operatorname{sgn}$ addend. Since $vs\in V_{even}$, we can
define $\iota(v)$ as $\iota(vs)\otimes s$, where $s\in\rho\subseteq A_{1}$ is
considered as a primitive element of $A$.

We construct the second product $\cdot$ on $H_{alg}$ step-by-step. As always,
$\cdot$ between elements of different components will be zero. As an
intermediate step, we define the product $\iota(v)^{[{n}]}\cdot x$ for all
$v\in V_{even}$ and $x\in H_{alg}$ with $n(x)=n\geq 1$. The case $n=1$ reduces
to the product on $V\otimes\rho$, so we can assume that $n\geq 2$. If
$x=(w\otimes y)^{[{m}]}$ is a divided power of a primitive element $w\otimes
y\in V_{even}\otimes P(A)$, then we let $\iota(v)^{[{n}]}\cdot
x=(\iota(v)(w\otimes y))^{[{m}]}$. We can then extend it to general $x\in
H_{alg}$ by using Hopf ring distributivity, because $H_{alg}$ is generated
under $\odot$ by the divided powers of its primitive elements. Explicitly,
every element of $H_{alg}$ is a linear combination of products
$\bigodot_{i=1}^{r}x_{i}^{[{m_{i}}]}$ for some $x_{i}=(w_{i}\otimes y_{i})\in
V_{even}\otimes P(A)$ and $m_{i}\geq 1$. We let

$\iota(v)^{[{\sum_{i}n(x_{i})m_{i}}]}\cdot\bigodot_{i}x_{i}^{[{m_{i}}]}=\bigodot_{i}(\iota(v)(w_{i}\otimes
y_{i}))^{[{m_{i}}]}$

and we extend to all $H_{alg}$ by linearity.

######  Proposition 2.42.

There is a unique component super-Hopf ring structure on $H_{alg}$ such that
the multiplication by $\iota(v)^{[{n}]}$ for $v\in V_{even}$ has the form
defined above, and satisfying the five conditions of Definition 2.41.

###### Proof.

Let $v,w\in V_{even}$ and $y,z\in P(A)$ be tri-homogeneous elements. Note
that, by the definition of the multiplication by divided powers of elements of
$V_{even}$ above, we must have $v\otimes y=\iota(v)^{[{n(y)}]}\cdot\pi(y)$ and
$w\otimes z=\iota(w)^{[{n(z)}]}\cdot\pi(z)$. Hence, if $H_{alg}$ satisfies the
third and fourth conditions we must have

$\displaystyle(v\otimes y)^{[{a}]}\cdot(w\otimes z)^{[{b}]}$
$\displaystyle=(\iota(v)^{[{n(y)}]}\cdot\pi(y))^{[{a}]}\cdot(\iota(w)^{[{n(z)}]}\cdot\pi(z))^{[{b}]}$
$\displaystyle=(-1)^{t(y)d(w)n(z)ab}\iota(v)^{[{n(y)a}]}\cdot\iota(w)^{[{n(z)b}]}\pi(y)^{[{a}]}\pi(z)^{[{b}]}$
$\displaystyle=(-1)^{t(y)d(w)n(z)ab}\iota(vw)^{[{n(y)a}]}\cdot\pi(y)^{[{a}]}\cdot\pi(z)^{[{b}]}$

for all $a,b$ such that $n(y)a=n(z)b$. Moreover, if $\pi$ is a morphisms of
Hopf rings and of divided powers structure,
$\pi(y)^{[{a}]}\cdot\pi(z)^{[{b}]}=\pi(y^{[{a}]}\cdot z^{[{b}]})$. In
conclusion

$(v\otimes y)^{[{a}]}\cdot(w\otimes
z)^{[{b}]}=(-1)^{t(y)d(w)n(z)ab}\iota(vw)^{[{n(y)a}]}\cdot\pi(y^{[{a}]}\cdot
z^{[{b}]}).$

Consequently, the conditions above uniquely determine the values of $\cdot$ on
$\odot$-indecomposables. Therefore, if an extension of $\cdot$ exists, it is
necessarily unique.

To prove existence, we only need to check that the formula above provides a
well-defined bilinear product satisfying Hopf ring distributivity. This is
straightforward. For instance, one can fix bases of $A$ and $V$, define the
$\cdot$ product as above on the induced basis, extend it to all $H_{alg}$ by
bilinearity and directly check the axioms of Hopf rings using the basis. ∎

We now prove that $H_{alg}$ is our desired object.

######  Proposition 2.43.

$H_{alg}$ is universal among the super-Hopf rigs with additive divided powers
satisfying the five conditions of Definition 2.41.

###### Proof.

Let $B$ be a super-Hopf rig with additive divided powers with maps
$\pi_{B}\colon A\to B$, $\iota_{B}\colon V\otimes\rho\to B$ satisfying our
desired hypotheses. $DP_{HA}$ and $\\_^{pr}$ are left adjoints of the
forgetful functor from divided powers Hopf algebras to coalgbras and the
primitives $P\colon R\textit{-coalg}\to R\textit{-mod}$, respectively.
Therefore, a morphism of divided powers Hopf algebras $f\colon H_{alg}\to B$
such that $f\pi=\pi_{B}$ and $f\iota=\iota_{V}$ is uniquely determined by its
restriction to $V_{even}\otimes P(A)$. If, in addition, $f$ is a super-Hopf
rig morphism, then

$f(v\otimes
y)=f(\iota(v)^{[{n(y)}]}\cdot\pi(y))=\iota_{B}(v)^{[{n(y)}]}\cdot\pi_{B}(y)$

for all $v\in V_{even}$ and $y\in P(A)$. Therefore, such a morphism is unique
(if it exists).

To prove existence, we let $f\colon H_{alg}\to B$ be the divided powers super-
Hopf algebra morphism adjoint of the linear map $v\otimes y\in V_{even}\otimes
P(A)\mapsto\iota_{B}(v)\cdot\pi_{B}(y)\in P(B)$. We only need to check that
$f$ preserves the $\cdot$ product, and by Hopf ring distributivity it is
enough to prove this on $\odot$-indecomposables, that is divided powers of
elements of $P(H_{alg})=V_{even}\otimes P(A)$. By our construction of the
$\cdot$ product in $H_{alg}$, we immediately see that $f(\iota(v)^{[{a}]}\cdot
x)=\iota_{B}(v)^{[{a}]}\cdot f(x)$ for all $v\in V_{even}$ and $x\in H_{alg}$
and $a\geq 2$. Moreover, since $\pi_{B}$ is a super-Hopf rig morphism, we have
that

$\displaystyle f\left((v\otimes y)^{[{a}]}\cdot(w\otimes z)^{[{b}]}\right)$
$\displaystyle=(-1)^{t(y)d(w)n(z)ab}f\left(\iota(vw)^{[{n(y)a}]}\cdot\pi(y^{[{a}]}\cdot
z^{[{b}]})\right)$
$\displaystyle=(-1)^{t(y)d(w)n(z)ab}\iota_{B}(vw)^{[{n(y)a}]}\cdot\pi_{B}(y^{[{a}]}\cdot
z^{[{b}]})$
$\displaystyle=\left(\iota_{B}(v)^{[{n(y)}]}\cdot\pi_{B}(y)\right)^{[{a}]}\cdot\left(\iota_{B}(w)^{[{n(w)}]}\cdot\pi_{B}(z)\right)^{[{b}]}=f(v\otimes
y)\cdot f(w\otimes z).$

∎

Since we want to specialize to the case $A={H_{EP}}^{*}(S^{0})$ and
$V=H^{*}(X;\mathbb{F}_{p})$ and as shown in Remark 2.39 the relation between
cup product and divided powers in ${H_{EP}}^{*}(S^{0})$ is complicated, it is
preferable to rephrase the construction above in terms of indecomposables. In
a Hopf algebra with divided powers, the indecomposables are elements of the
form $x^{[{p^{k}}]}$ with $x$ primitive and $k\geq 0$. Consequently, as a
graded bicommutative Hopf algebra alone, $DP_{HA}{(V\otimes P(A)^{pr})}$ is
generated by

$\\{(v\otimes y)^{[{p^{k}}]}\\}_{v\in V,y\in P(A),k\geq 0}=V\otimes Q(A).$

We obtain an isomorphism of indecomposables $V_{even}\otimes Q(A)\cong
QH_{alg}$ given by $v\otimes y\mapsto v^{[{n(y)}]}\cdot y$ for all $v\in V$
and $y\in A$, and this realizes $H_{alg}$, under the product $\odot$ alone, as
the free graded commutative algebra generated by $V_{even}\otimes Q(A)$,
quotiented by the image of the Frobenius and the ideal generated by the
relation $(5)$. As a result, any pair of additive bases for $V$ and $Q(A)$
induce an additive basis for $H_{alg}$.

In summary, we have the following.

######  Lemma 2.44.

Let $e_{0}$ (respectively $e_{1}$) in $\rho$ be a non-zero element in the
constant representation (respectively sign representation) addend of $\rho$.
Let $Q(A)$ be the space of $\odot$-indecomposables of $A$. Let
$\mathcal{B}_{V}$ and $\mathcal{B}_{A}$ be totally ordered additive bases of
$V_{even}$ and $Q(A)$ respectively. Define $\mathcal{B}_{V}^{\rho}$ as the set
of elements $\\{v\otimes e_{i}\\}_{v\in\mathcal{B}_{V},i\in\\{0,1\\}}$. Order
$\mathcal{B}_{V}$ in any way (e.g. lexicographically). Let $\mathcal{B}$ be
the set of elements of the form
$\bigodot_{i=1}^{r}\left(\iota(v_{i})^{[{n(y_{i})}]}\cdot\pi(y_{i})\right)\in
H_{alg}$ with $v_{1}\leq\dots\leq v_{r}\in\mathcal{B}_{V}^{\rho}$,
$y_{1}\leq\dots\leq y_{r}\in\mathcal{B}_{A}$, such that:

  * •

the sign degree of all the $\odot$-factors is the same

  * •

the multiplicity of every $\odot$-factor of even total degree is at most
$p-1$,

  * •

and the multiplicity of every $\odot$-factor of odd total degree is at most
$1$.

Then $\mathcal{B}$ is a basis for $H_{alg}$ as a $\mathbb{F}_{p}$-vector
space.

##  3\. The homology and cohomology of symmetric groups with twisted
coefficients

Homology of extended powers and free infinite loop spaces are algebras over
the Kudo-Araki-Dyer-Lashof algebra, an algebra of homology operations that
constitute the basic building blocks of the homology of the extended powers of
a point – that is, the homology of symmetric groups. This structure is well-
understood by the work of Cohen–Lada–May [CLM76]. Our strategy in this paper
is show that our descriptions which build on cup product structure pair
perfectly with their descriptions. But for odd primes the cleanest
descriptions of cohomology require considering coefficients in $\rho$, the
$\mathbb{F}_{p}[{\mathcal{S}}_{n}]$-representation which is the sum of the
trivial and sign representations, introduced in the previous section.

Thus, this section is divided in two parts. In the first subsection we provide
a description of the homology of $\tilde{D}X$ with coefficients in $\rho$, in
terms of Kudo-Araki-Dyer-Lashof operations. We do not make any claim of
originality here, as we only review classical results by Cohen–Lada–May
[CLM76] for the trivial summand and their recent extension to sign-twisted
coefficients by Bernard [Ber]. In the second subsection we dualize these
results to obtain a description of $\bigoplus_{n\geq
0}H^{*}({\mathcal{S}}_{n};\rho)$ as a Hopf ring, and prove Theorem 2.35. While
the summand of sign degree zero is known by our previous work, namely [GSS12]
and [Gue17], the description of the sign degree one summand is new.

###  3.1. The homology of extended powers in terms of KADL operations

Since we work with field coefficients in a setting which is finite dimensional
in each grading, the homology of $DX$ with local coefficients given by the
${\mathcal{S}}_{n}$-representation $A_{n}$ of Definition 2.17 is isomorphic to
the bigraded linear dual of $\bigoplus_{n\geq
0}H^{*}(D_{n}(X);A_{n})={H_{EP}}^{*}(X)$. Hence it will be a Hopf co-ring,
endowed with two coproducts $\Delta_{\odot}$, $\Delta_{\cdot}$, and a product
$*$ dual to $\odot$, $\cdot$, and $\Delta$ respectively, satisfying all the
axioms of a Hopf ring with the directions of all morphisms reversed.

The product $*\colon H_{i}(D_{n}X)\otimes H_{j}(D_{m}X)\to H_{i+j}(D_{n+m}X)$.
also has a group-homology interpretation. Let $R_{{\mathcal{S}}_{n}}$ be a
resolution of $\mathbb{F}_{p}$ as a
$\mathbb{F}_{p}[{\mathcal{S}}_{n}]$-module, so that by Proposition 2.2 the
homology of
$R_{{\mathcal{S}}_{n}}\otimes_{{\mathcal{S}}_{n}}{\widetilde{H}_{*}(X)}^{\otimes{n}}$
is that of $D_{n}X$. The product $*$ is induced, up to sign, by the tensor
product of the map of resolutions $R_{{\mathcal{S}}_{n}}\otimes
R_{{\mathcal{S}}_{m}}\to R_{{\mathcal{S}}_{n+m}}$ induced by the inclusion
${\mathcal{S}}_{n}\times{\mathcal{S}}_{m}\hookrightarrow{\mathcal{S}}_{n+m}$
with the isomorphism
${\widetilde{H}_{*}(X)}^{\otimes{n}}\otimes{\widetilde{H}_{*}(X)}^{\otimes{m}}\cong{\widetilde{H}_{*}(X)}^{\otimes{n+m}}$.
Geometrically, the product on $D(X)$ and $C(X)$ can be defined through an
embedding
${\mathbb{R}}^{\infty}\coprod{\mathbb{R}}^{\infty}\to{\mathbb{R}}^{\infty}$,
through which one can take the image of the union of configurations with
labels in $X$. Since it arises from a homotopy commutative multiplication on
spaces we also call this product the Pontrjagin product.

Since the coproduct and the transfer product on cohomology form a bialgebra,
then for example if the evaluations of cohomology classes $a_{i}$ on homology
$x_{j}$ are Kronecker – that is, $\langle a_{i},x_{j}\rangle=\delta_{i,j}$ –
and the $a_{i}$ are primitive, then the evaluation
$\langle\bigodot_{i}a_{i},*_{i}x_{i}\rangle$ will be one. But if some classes
are repeated then coefficients are introduced, as for example
$\langle\bigodot_{n}a,x^{*n}\rangle=n!$. The divided powers operations in
cohomology “fill in” to produce duals in these cases, as $\langle
a^{[{n}]},x^{*n}\rangle$ is one, as it is given by the tensor product at the
chain and cochain level.

Just as the inclusions
${\mathcal{S}}_{n}\times{\mathcal{S}}_{m}\hookrightarrow{\mathcal{S}}_{n+m}$
give rise to the Pontrjagin product, the inclusions of wreath products
${\mathcal{S}}_{n}\wr{\mathcal{S}}_{k}\hookrightarrow{\mathcal{S}}_{nk}$ give
rise to the algebraic Kudo-Araki-Dyer-Lashof operations.

######  Definition 3.1 (compare [Ber]).

Use $R_{{\mathcal{S}}_{n}}\otimes(R_{{\mathcal{S}}_{k}})^{\otimes n}$ as a
resolution for
${\mathcal{S}}_{n}\wr{\mathcal{S}}_{k}\hookrightarrow{\mathcal{S}}_{nk}$. Let
$W_{*}$ be the standard minimal $\mathbb{F}_{p}$ resolution of $C_{p}$, the
cyclic group of order $p$ and $e_{i}$ a generator for $W_{i}$. To every chain
$e\in W_{*}$ we associate an operation $q(e)^{\\#}:R_{{\mathcal{S}}_{n}}\to
R_{{\mathcal{S}}_{pn}}$ as sending a chain $c$ the image of $e\otimes c^{p}$
under the map of resolutions induced by the inclusion of
$C_{p}\wr{\mathcal{S}}_{n}\hookrightarrow{\mathcal{S}}_{np}$.

We consider the linear morphisms

$q(e)\colon
H_{*}({\mathcal{S}}_{n};{\widetilde{H}_{*}(X)}^{\otimes{n}}\otimes\rho)\to
H_{*}({\mathcal{S}}_{pn};{\widetilde{H}_{*}(X)}^{\otimes{pn}}\otimes\rho)$

induced by the map

$c\otimes(x_{1}\otimes\cdots\otimes x_{n})\mapsto
q(e)^{\\#}(c)\otimes(x_{1}\otimes\cdots\otimes x_{n})^{\otimes p}.$

The Kudo-Araki-Dyer-Lashof (KADL) operations are linear morphisms

$q_{i}=q(e_{i})\colon
H_{m}({\mathcal{S}}_{n};{\widetilde{H}_{*}(X)}^{\otimes{n}}\otimes\rho)\to
H_{pm+i(p-1)}({\mathcal{S}}_{pn};{\widetilde{H}_{*}(X)}^{\otimes{pn}}\otimes\rho).$

Geometrically, $q_{i}$ on a homology class of $X$ (the $n=1$ case) when $i$ is
odd is represented in $CX$ by a family of configurations with labels where the
$p$ points in the configuration are on the $i$-sphere related by the action of
a $p$-th root of unity, with labels all in the same cycle on $X$.

For $p=2$ we consider $r$-tuples $(i_{1},\cdots,i_{r})$. For odd primes, let
$\beta$ be the Bockstein homomorphism. We associate to an $2r$-tuple
$I=(\varepsilon_{1},i_{1},\dots,\varepsilon_{r},i_{r})$, with
$\varepsilon_{k}\in\\{0,1\\}$ and all $i_{k}\geq 0$, the homology operation
$q_{I}=\beta^{\varepsilon_{1}}q_{i_{1}}\circ\dots\circ\beta^{\varepsilon_{r}}q_{i_{r}}$.
We say that $I$ is admissible if $i_{k}\leq i_{k+1}-\varepsilon_{k+1}$
(respectively $i_{k}\leq i_{k+1}$ when $p=2$) for all $1\leq k<r$ and, in case
$p>2$, $i_{k}\equiv i_{k+1}-\varepsilon_{k+1}\mod 2\,\forall 1\leq k<r$. We
say that $I$ is strongly admissible if it is admissible and $i_{1}>0$. This is
a reformulation of Bernard’s admissibility conditions, in Section 7 of [Ber],
with the lower indices notation.

######  Theorem 3.2 (Cohen–Lada–May [CLM76]).

The mod-$p$ homology of $CX$ is a free graded commutative algebra with respect
to the Pontrjagin product, generated by $q_{I}(x)$ where $x$ ranges over a
graded basis for the reduced homology of $X$ and $I$ ranges over strongly
admissible sequences, with the additional requirement that for $p$ odd $i_{r}$
and the homological degree of $x$ have the same parity, where
$I=(\varepsilon_{1},i_{1},\dots,\varepsilon_{r},i_{r})$

Similarly, the homology of $CX$ with twisted coefficients given by the mod $p$
sign representation for $p$ odd can be computed as an algebra using the
twisted versions of Dyer-Lashof operations.

######  Theorem 3.3 (Bernard [Ber]).

For $p$ odd, the homology of $\tilde{D}(X)=C(X_{+})$ with coefficients in
$\operatorname{sgn}$, the mod $p$ sign representation, is the free graded
commutative algebra with respect to the homology product generated by
$q_{I}(x)$, where $x$ ranges over a graded basis for the homology of $X$ and
$I$ ranges over strongly admissible sequences such that $i_{r}$ and the
homological degree of $x$ have different parity, with
$I=(\varepsilon_{1},i_{1},\dots,\varepsilon_{r},i_{r})$

######  Remark 3.4.

Admissible, but not strongly admissible, KADL operations on $CX$ can be
retrieved by the identity $Q_{0}(x)=x^{*p}$. Iterated non-admissible sequences
of KADL operations can be computed by means of Adem relations. Both in [CLM76]
and in [Ber], an “upper indices” notation is used, because homological degrees
behave better and Adem relations have a better form. We use a “lower indices”
notation, because it makes the argument for the computation of the dual module
and the construction of Hopf ring generators more transparent. The two
conventions differ only by coefficients and reindexing, and are essentially
equivalent. We refer to the two papers cited above for the precise relations
between the two notations.

######  Definition 3.5.

Let $p$ be an odd prime and let $k\geq 0$. Define $\mathcal{R}^{t}_{k}$ be the
$\mathbb{F}_{p}$-vector space spanned by the admissible KADL operations
$q_{I}$ where $I=(\varepsilon_{1},i_{1},\dots,\varepsilon_{k},i_{k})$ has
length $2k$. Let $\mathcal{R}^{t}=\bigoplus_{k\geq 0}\mathcal{R}^{t}_{k}$.

For $p=2$, let $\mathcal{R}_{k}$ be the $\mathbb{F}_{2}$-vector space spanned
by the admissible KADL operations $q_{I}$, where $I=(i_{1},\dots,i_{k})$ has
length $k$.

In light of our main application, we consider an admissible operation $q_{I}$
as acting on a $0$-dimensional class. In this case, the subspace
$\mathcal{R}_{k}\subseteq\mathcal{R}^{t}_{k}$ spanned by $q_{I}$ such that
$i_{k}$ is even corresponds to the classical untwisted KADL operations, and
the subspace $\mathcal{R}^{\prime}_{k}\subseteq\mathcal{R}^{t}_{k}$ spanned by
$q_{I}$ such that $i_{k}$ is odd corresponds to the twisted KADL operations.
Precisely, by sending $q_{I}$ to $q_{I}(\iota)$, for $\iota\neq
0\in\overline{H}_{0}(S^{0})$, we identify $\mathcal{R}_{k}$ as a subspace of
$H_{*}({\mathcal{S}}_{p^{k}};\mathbb{F}_{p})\subseteq
H_{*}(D(S^{0});\mathbb{F}_{p})$ and $\mathcal{R}^{\prime}_{k}$ as a subspace
of $H_{*}({\mathcal{S}}_{p^{k}};\operatorname{sgn})\subseteq
H_{*}(D(S^{0});\operatorname{sgn})$. The component of an operation
$q_{I}\in\mathcal{R}^{t}_{k}$ is $n(q_{I})=p^{k}$, and its homological degree
is
$d(q_{I})=\sum_{j=0}^{k-1}p^{j}\left(i_{j+1}(p-1)-\varepsilon_{j+1}\right)$.

As a consequence of Theorem 3.2,
$H_{*}(DS^{0};\mathbb{F}_{p})=H_{*}(CS^{0};\mathbb{F}_{p})$ has a basis given
by Pontryagin monomials in strongly admissible KADL operations in
$\mathcal{R}=\bigsqcup_{k}\mathcal{R}_{k}$. We denote this basis with
$\mathcal{B}$, and we call it Nakaoka basis. Similarly, by Theorem 3.3,
$H_{*}(DS^{0};\operatorname{sgn})$ has a basis given by $*$-monomials in
strongly admissible KADL operations in
$\mathcal{R}^{\prime}=\bigsqcup_{k}\mathcal{R}^{\prime}_{k}$, that we denote
with $\mathcal{B}_{\operatorname{sgn}}$. In both $\mathcal{B}$ and
$\mathcal{B}_{\operatorname{sgn}}$, operations with odd degree must not appear
twice in the same monomial.

For $p=2$, $\mathcal{R}=\bigsqcup_{k}\mathcal{R}_{k}$ embeds as a subspace of
$H_{*}(CS^{0};\mathbb{F}_{2})$ by sending $q_{I}$ to $q_{I}(\iota)$, whose
component is $n(q_{I})=2^{k}$ and whose homological degree is
$d(q_{I})=\sum_{j=0}^{k-1}2^{j}q_{j+1}$. Theorem 3.2 provides a Nakaoka
monomial basis as above.

###  3.2. Cohomology of symmetric groups with coefficients in the sign
representation mod-$p$

To complete the description of $H^{*}(DX;\mathbb{F}_{p})$ as a Hopf ring in
the odd primary case, we must first analyze the cohomology of the symmetric
groups with coefficients in the sign representation, mod $p$. When we consider
the cohomology of $X^{n}_{h{\mathcal{S}}_{n}}$, through say cellular chain
models, a $d$-cell in $X$ gives rise to a chain of dimension $dn$, and a
permutation $\sigma\in{\mathcal{S}}_{n}$ acts on this chain as multiplication
by $(-1)^{d\operatorname{sgn}(\sigma)}$. So for example
$H^{*+nd}(D_{n}(S^{d});\mathbb{F}_{p})$ has a summand isomorphic to
$H^{*}({\mathcal{S}}_{n};\operatorname{sgn}^{d})$. While additive structures
are isomorphic in the $d$ even and odd cases, the product structures here
differ significantly.

A first step in understanding a Hopf ring is through its
$\odot$-indecomposables, which by Hopf ring distributivity form a ring under
the $\cdot$-product. In the bigraded component setting, this ring is a
coproduct of the $\odot$-indecomposables on each component. For the cohomology
of symmetric groups these rings of $\odot$-indecomposables form the foundation
of our previous calculations [GSS12, Gue17], but the calculation of these
rings go back to the work of Cohen–Lada–May [CLM76], where they were called
$\mathcal{R}_{k}^{*}$.

Thus the focus of this section is Lemma 3.8, which consists of the calculation
of the dual algebra of $\mathcal{R}\bigoplus\mathcal{R}^{\prime}$. While this
computation for twisted operations is similar to the untwisted case, it has
not been computed previously. Our calculation here is a straightforward
generalization of arguments in Section I.3 of [CLM76], and as suggested in
that text we use lower index notation.

Let $\mathcal{R}_{k}^{*}$ and ${\mathcal{R}^{\prime}}_{k}^{*}$ be the spans of
linear duals $q_{I}^{\vee}$ of the operations $q_{I}$ of length $k$
respectively in $H^{*}({\mathcal{S}}_{p^{k}};\mathbb{F}_{p})$ and
$H^{*}({\mathcal{S}}_{p^{k}};\operatorname{sgn}\otimes\mathbb{F}_{p})$ with
respect to the Nakaoka bases. Because of the naturality of twisted KADL
operations, the proof of Theorem 4.13 in [GSS12] also shows that the subspace
of primitives with respect to the coproduct dual to the transfer product
$P(\bigoplus_{n\geq 0}H_{*}({\mathcal{S}}_{n};A_{n}))$ is
$\mathcal{R}^{t}=\mathcal{R}\bigoplus\mathcal{R}^{\prime}$. Because the
transfer product forms a bialgebra with the coproduct dual to $*$, the dual
classes $\\{q_{I}^{\vee}\\}$ for $I$ admissible are a basis for our Hopf ring
indecomposables. Moreover, each component of this module of
$\odot$-indecomposables is a graded commutative ring under cup product.

Recall that the coproduct $\Delta\colon H_{*}(C_{p})\to H_{*}(C_{p})\otimes
H_{*}(C_{p})$ dual to cup product in the homology of the cyclic group $C_{p}$
induces a unique coproduct $\psi$ on $\mathcal{R}^{t}$ such that
$\psi(q(e))=(q\otimes q)(\Delta(e))$. The coproduct $\psi$ has the form
$\psi(q_{I})=\sum_{i}\eta_{i}q_{J_{i}}\otimes q_{K_{i}}$, which encodes a
Cartan formula $q_{I}(x\otimes y)=\sum_{i}\eta_{i}q_{J_{i}}(x)\otimes
q_{K_{i}}(y)$. We refer to [CLM76] for the precise construction and related
calculations.

While the isomorphism $H^{*}({\mathcal{S}}_{n};\operatorname{sgn})\cong
H^{*}(D_{n}S^{1};\mathbb{F}_{p})$ guides us, we need to make a finer
distinction for this calculation. While the transfer product and the coproduct
are preserved by this isomorphism, the same does not happen for the cup
product. By letting an operation $q_{I}$ act on $[S^{1}]$ in
$H_{*}(DS^{1};\mathbb{F}_{p})$ and composing with that isomorphism, we can
nevertheless embed $\mathcal{R}^{\prime}_{k}$ into
$H_{*}({\mathcal{S}}_{p^{k}};\operatorname{sgn})$. Combining this with what we
know about the homology of the symmetric group with trivial coefficients, we
obtain a map $\kappa\colon\mathcal{R}\oplus\mathcal{R}^{\prime}\to
H_{*}({\mathcal{S}}_{n};\mathbb{F}_{p}\oplus\operatorname{sgn})$. By duality
this provides a map of $\mathbb{F}_{p}$-modules $\kappa^{*}\colon
H^{*}({\mathcal{S}}_{p^{k}};\mathbb{F}_{p}\oplus\operatorname{sgn})\to\mathcal{R}_{k}^{*}\oplus{\mathcal{R}^{\prime}_{k}}^{*}$.
We compare the products in those rings.

######  Proposition 3.6.

If $x\in H^{d}({\mathcal{S}}_{p^{k}};\operatorname{sgn}^{\varepsilon})$ and
$y\in
H^{d^{\prime}}({\mathcal{S}}_{p^{k}};\operatorname{sgn}^{\varepsilon^{\prime}})$,
then

$\psi^{*}(\kappa^{*}(x)\otimes\kappa^{*}(y))=(-1)^{\varepsilon(d^{\prime}+\frac{p-1}{2}k\varepsilon^{\prime})}\kappa^{*}(x\cdot
y).$

###### Proof.

Let $X$ be a space, and let $n=p^{k}$. Fix a graded basis ${B}$ of the
homology of $X$. We observe that the following diagram commutes, where $d$
denotes diagonal maps, $\tau$ the obvious shuffle map, and $q$ is the obvious
quotient map.

${\tilde{D}_{n}(X)}$${\tilde{D}_{n}(X)^{2}}$${(E({\mathcal{S}}_{n})\times
E({\mathcal{S}}_{n}))\times_{{\mathcal{S}}_{n}}(X^{n}\times
X^{n})}$${(E({\mathcal{S}}_{n})\times
E({\mathcal{S}}_{n}))\times_{({\mathcal{S}}_{n}\times{\mathcal{S}}_{n})}X^{2n}}$$\scriptstyle{d_{D_{n}(X)}}$$\scriptstyle{d_{E({\mathcal{S}}_{n})}\times_{{\mathcal{S}}_{n}}d_{X^{n}}}$$\scriptstyle{\tau}$$\scriptstyle{q}$

Consider $q_{I}(x)\in H_{*}(\tilde{D}_{n}(X))$, where $x\in H_{*}(X)$. Assume
that $\Delta(x)=\sum_{i}\lambda_{i}x_{i}^{\prime}\otimes x_{i}^{\prime\prime}$
in $H_{*}(X)$, for some $x_{i}^{\prime},x_{i}^{\prime\prime}\in{B}$. Write
$\psi(q_{I})=\sum_{l}\eta_{l}q_{J_{l}^{\prime}}\otimes
q_{J_{l}^{\prime\prime}}$. Let $\varphi$ be the coproduct in homology dual to
the cup product of
$H^{*}({\mathcal{S}}_{n};\mathbb{F}_{p}\oplus\operatorname{sgn})$.

Taking homology and keeping track of the image of $q_{I}(x)$ in upper path in
the diagram above act on, we obtain

$\begin{split}\tau_{*}\circ{d_{\tilde{D}_{n}(X)}}_{*}(q_{I}(x))&=\tau_{*}\left(\sum_{i,k}\lambda_{i}\eta_{l}q_{J_{l}^{\prime}}(x_{i}^{\prime})\otimes
q_{J_{l}^{\prime\prime}}(x_{i}^{\prime\prime})\right)\\\
&=\sum_{i,l}(-1)^{|q_{J_{l^{\prime\prime}}}||x_{i}^{\prime}|}\lambda_{i}\eta_{l}q_{J_{l}^{\prime}}\otimes
q_{J_{l}^{\prime\prime}}\otimes(x_{i}^{\prime})^{\otimes^{n}}\otimes(x_{i}^{\prime\prime})^{\otimes^{n}}.\end{split}$

Similarly, for the lower path, we have

$(d_{E({\mathcal{S}}_{n})}\times_{{\mathcal{S}}_{n}}d_{X^{n}})_{*}(q_{I}(x))=\sum_{i_{1},\dots,i_{n}}\prod_{j=1}^{n}\lambda_{i_{j}}(-1)^{\sum_{1\leq
j<l\leq
n}|x_{i_{j}}^{\prime\prime}||x_{i_{l}}^{\prime}|}{d_{E({\mathcal{S}}_{n})}}_{*}(q_{I})\otimes_{{\mathcal{S}}_{n}\times{\mathcal{S}}_{n}}\left(\bigotimes_{j=1}^{n}x_{i_{j}}^{\prime}\otimes\bigotimes_{j=1}^{n}x_{i_{j}}^{\prime\prime}\right).\\\
$

We can split this last summation as the sum over the set of $n$-tuples where
all $i_{k}$ are equal to each other (say $i$) and the sum over the set on
$n$-tuples which are not all equal. The first part can be rewritten as

$\sum_{i}\lambda_{i}(-1)^{\frac{n(n-1)}{2}|x_{i}^{\prime}||x_{i}^{\prime\prime}|}{d_{E({\mathcal{S}}_{n})}}_{*}(q_{I})\otimes({x_{i}^{\prime}}^{\otimes^{n}}\otimes{x_{i}^{\prime\prime}}^{\otimes^{n}}),$

which maps to
$\sum_{i}\lambda_{i}(-1)^{\frac{n(n-1)}{2}|x_{i}^{\prime}||x_{i}^{\prime\prime}|}\varphi(q_{I})\otimes{x_{i}^{\prime}}^{\otimes^{n}}\otimes{x_{i}^{\prime\prime}}^{\otimes^{n}}$
under $q$. We claim that the second part is zero. Recall that there is an
isomorphism $H_{*}((E({\mathcal{S}}_{n})\times
E({\mathcal{S}}_{n}))\times_{{\mathcal{S}}_{n}}X^{2n};\mathbb{F}_{p})\cong
H_{*}({\mathcal{S}}_{n};({H}_{*}(X;\mathbb{F}_{p})^{\otimes^{n}})^{\otimes^{2}})$.
The ${\mathcal{S}}_{n}$-subrepresentation of
$({H}_{*}(X;\mathbb{F}_{p})^{\otimes^{n}})^{\otimes^{2}}$ generated by
$(x_{i_{1}}^{\prime}\otimes\dots
x_{i_{n}}^{\prime})\otimes(x_{i_{1}}^{\prime\prime}\otimes\dots\otimes
x_{i_{n}}^{\prime\prime})$ with $i_{1},\dots,i_{n}$ not all equal is
isomorphic to the induced ${\mathcal{S}}_{n}$-representation of a
$G$-representation for a Young subgroup $G\lneq{\mathcal{S}}_{n}$. The
corresponding terms in the sum above amount to
$\operatorname{tr}_{G}^{{\mathcal{S}}_{n}}\rho_{G}^{{\mathcal{S}}_{n}}(q_{I})$,
which is multiplication by the index $[{\mathcal{S}}_{n}:G]$, which is zero
modulo $p$.

In conclusion,

$\sum_{i}\lambda_{i}(-1)^{|x_{i}^{\prime}||q_{J_{l}^{\prime\prime}}|}\psi(q_{I})\otimes(x_{i}^{\prime})^{\otimes^{n}}\otimes(x_{i}^{\prime\prime})^{\otimes^{n}}=\sum_{i}\lambda_{i}(-1)^{\frac{n(n-1)}{2}|x_{i}^{\prime}||x_{i}^{\prime\prime}|}\varphi(q_{I})\otimes(x_{i}^{\prime})^{\otimes^{n}}\otimes(x_{i}^{\prime\prime})^{\otimes^{n}}.$

Since $n=p^{k}$, $\frac{n(n-1)}{2}=\frac{p-1}{2}k\mod 2$. Hence, the equality
above is true for all spaces $X$ and for all classes $x\in
H_{*}(X;\mathbb{F}_{p})$ if and only if

$\sum_{l}\eta_{l}\kappa(q_{J_{l}}^{\prime})\otimes\kappa(q_{J_{l}}^{\prime\prime})=\sum_{l}\eta_{l}(-1)^{\varepsilon(|q_{I_{l}^{\prime\prime}}|+\frac{p-1}{2}\varepsilon^{\prime})}\varphi\kappa(q_{I})$

for $\varepsilon,\varepsilon^{\prime}\in\\{0,1\\}$. Dualizing this we obtain
the desired formula. ∎

To summarize, the coproduct dual to the cup product, when restricted on
$\mathcal{R}^{t}$, has the form

$\varphi(q_{I})=\sum_{J+K=I}(-1)^{\sum_{l=1}^{n}\left(\frac{p-1}{2}j_{l}k_{l}+\delta_{l}j_{l}\right)}q_{J}\otimes
q_{K},$

where $J=(\varepsilon_{1},j_{1},\dots,j_{n})$ and
$K=(\delta_{1},k_{1},\dots,k_{n})$.

The difficulty with this formula is that the sum is over all ways to decompose
the sequence $I$, not just admissible sequences. Adem relations are needed to
make calculations. The dual algebra is in the end manageable, though to this
day (to our knowledge) even in the trivial coefficient setting the pairing
between the Nakaoka basis on homology and the standard basis on cohomology
from analysis such as we give below is not known.

To calculate the product dual to $\varphi$ in the sign representation setting,
we continue to follow the trivial coefficient treatment of [CLM76, Section
I.3]. We consider sequences of operations with minimal entries.

Let $S\subseteq\\{1,\dots,k\\}$ and define $I_{S}[k]$ and $I^{\prime}_{S}[k]$
respectively as the sets of the admissible $2k$-tuples
$I=(\varepsilon_{1},i_{1},\dots,\varepsilon_{k},i_{k})$ such that $i_{j}$ is
equal to $|S\cap\\{1,\dots,k-j\\}|$ (respectively
$(|S\cap\\{1,\dots,k-j\\}|+1)$) mod $2$ and $\varepsilon_{j}=1$ if and only if
$k+1-j\in S$. Note that $\\{q_{I}:I\in I_{S}[k]\\}\subseteq\mathcal{R}_{k}$
and $\\{q_{I}:I\in I^{\prime}_{S}[k]\\}\subseteq\mathcal{R}^{\prime}_{k}$.

For all $k\in\mathbb{N}$ and $S\subseteq\\{1,\dots,k\\}$, we can define a
partial order $\leq$ on $I_{S}[k]$ and $I^{\prime}_{S}[k]$ by comparison of
all entries. This partially ordered set possesses all meets, obtained by
taking the minimum on each entry. In particular, $I_{S}[k]$ and
$I^{\prime}_{S}[k]$ have a minimal element, that we denote $L_{S,k}$ and
$L^{\prime}_{S,k}$ respectively.

Although we will not need this, it is straightforward to write down the
elements $L_{S,k}$ and $L^{\prime}_{S,k}$ explicitly. For instance,
$L_{\\{1,2\\},3}=(0,0,1,1,1,2)$. Up to switching to upper-index notation, the
$(2k)$-tuples $L_{S,k}$ coincide with those defined in Section I.3 of [CLM76].

The following is straightforward combinatorics, following the same argument as
Lemmas I.3.3 and I.3.4 in [CLM76].

######  Lemma 3.7.

For all $j,k\in\mathbb{N}$ such that $1\leq j\leq k$, let

$I_{j,k}=(\underbrace{0,\dots,0,}_{2(k-j)}\underbrace{0,2,\dots,0,2}_{2j})\in
I_{\phi}[k].$

The weighted sum $(n_{1},\dots,n_{k})\mapsto
L^{\prime}_{S,k}+\sum_{j=1}^{k}n_{j}I_{j,k}$ is a bijections from
$\mathbb{N}^{k}\to I^{\prime}_{S}[k]$ for all subset $S$ of $\\{1,\dots,k\\}$.

We can now compute the algebra dual to $(\mathcal{R}^{t},\varphi)$.

######  Lemma 3.8.

Assume that $p$ is odd. We consider the following dual classes in
$(\mathcal{R}^{t})^{\vee}$ with respect to the admissible sequences basis:

  * •

$\xi_{j,k}=q_{I_{j,k}}^{\vee}$ for $1\leq j\leq k$

  * •

$\xi^{\prime}_{k}=q_{L^{\prime}_{\phi,k}}^{\vee}$ for $k\geq 1$

  * •

$\tau^{\prime}_{j,k}=q_{L^{\prime}_{\\{j\\},k}}^{\vee}$ for $1\leq j\leq k$

As a $\mathbb{F}_{p}[\xi_{1,k},\dots,\xi_{k,k}]$-module,
$(\mathcal{R}^{t}_{k})^{\vee}$ is isomorphic to
$\mathbb{F}_{p}[\xi_{1,k},\dots,\xi_{k,k}]\otimes(M_{k}\bigoplus
M^{\prime}_{k})$, where

  * •

$M_{k}$ is the $\mathbb{F}_{p}$-vector space with basis
$\\{q_{L_{S,k}}^{\vee}\\}_{S\subseteq\\{1,\dots,k\\}}$, as in Cohen–Lada–May,
and

  * •

$M^{\prime}_{k}$ is the $\mathbb{F}_{p}$-vector space with basis
$\\{q_{L^{\prime}_{S,k}}^{\vee}\\}_{S\subseteq\\{1,\dots,k\\}}$.

As a graded commutative algebra, $(\mathcal{R}^{t})^{\vee}$ is generated by
$\xi_{j,k}$ with $1\leq j\leq k$, $\xi^{\prime}_{k}$ and $\tau^{\prime}_{j,k}$
with $1\leq j\leq k$, under the relations

${\xi^{\prime}_{k}}^{2}=\xi_{k,k}.$

###### Proof.

Theorem I.3.7 in [CLM76] states that $\mathcal{R}_{k}^{\vee}$ is additively
isomorphic to $\mathbb{F}_{p}[\xi_{1,k},\dots,\xi_{k,k}]\otimes M_{k}$. The
exact argument used to prove that result, with Lemma 3.7 in place of Lemma
I.3.4 of [CLM76], yields an additive isomorphism
${\mathcal{R}^{\prime}_{k}}^{\vee}\cong\mathbb{F}_{p}[q_{I_{j,k}}^{\vee}]_{1\leq
j\leq k}\otimes M^{\prime}_{k}$. In particular,
$\mathbb{F}_{p}[\xi_{1,k},\dots,\xi_{k,k}]$ is the dual of the quotient
coalgebra $\mathcal{R}_{\varnothing}[k]$. Let $+$ denote component-wise
addition of $2k$-tuples. One can check that

$\displaystyle L_{S,k}$ $\displaystyle=\left\\{\begin{array}[]{ll}\sum_{s\in
S}L^{\prime}_{\\{s\\},k}&\mbox{if }|S|\mbox{ is even}\\\ \sum_{s\in
S}L^{\prime}_{\\{s\\},k}+L^{\prime}_{\phi,k}&\mbox{if }|S|\mbox{ is
odd}\end{array}\right.$ $\displaystyle L^{\prime}_{S,k}$
$\displaystyle=\left\\{\begin{array}[]{ll}\sum_{s\in
S}L^{\prime}_{\\{s\\},k}+L^{\prime}_{\phi,k}&\mbox{if }|S|\mbox{ is even}\\\
\sum_{s\in S}L^{\prime}_{\\{s\\},k}&\mbox{if }|S|\mbox{ is
odd}\end{array}\right..$

This is true because the function $\\_+L^{\prime}_{\\{\ell\\},k}\colon
I_{S}[k]\to I^{\prime}_{S\cup\\{\ell\\}}[k]$ (respectively
$\\_+L^{\prime}_{\\{\ell\\},k}\colon I^{\prime}_{S}[k]\to
I_{S\cup\\{\ell\\}}[k]$) is an order-preserving bijection for all
$S\subseteq\\{1,\dots,\ell-1\\}$ of even (respectively odd) cardinality, thus
it must preserve minimal elements. Since application of Adem relations to a
sequence $q_{I}$ produce elements $q_{J}$ with $J>I$ and
$L_{S,k},L^{\prime}_{S,k}$ are minimal, the argument in the proof of [CLM76,
I.3.7] can be used to prove that the pairing between products of the form
$\prod_{s\in S}\tau^{\prime}_{s,k}$ (if $|S|$ is even) or
$\xi^{\prime}_{k}\prod_{s\in S}\tau^{\prime}_{s,k}$ (if $|S|$ is odd) and the
$q_{L^{\prime}_{S,k}}$ is perfect. This implies by Lemma 3.7 that
$\xi^{\prime}_{k}$ and $\tau^{\prime}_{j,k}$ with $1\leq j\leq k$ generate
$(\mathcal{R}^{t}_{k})^{\vee}$ as an
$\mathbb{F}_{p}[\xi_{1,k},\dots,\xi_{k,k}]$-algebra. We can check the
relations ${\xi^{\prime}_{k}}^{2}=\xi_{k,k}$ between the generators also
following [CLM76]. Explicitly, since
$I_{k,k}=L^{\prime}_{\phi,k}+L^{\prime}_{\phi,k}$,
$\langle{\xi_{k}^{\prime}}^{2},q_{I_{k,k}}\rangle=1$. Tracking degrees, the
only possible summand of ${\xi^{\prime}_{k}}^{2}$ is $q_{I_{k,k}}^{\vee}$,
hence the relation holds. By comparing dimensions, we see that these provide a
presentation of $(\mathcal{R}^{t}_{k})^{\vee}$ as a graded commutative
algebra. ∎

######  Remark 3.9.

We recover the elements $\tau_{j,k}$ and $\sigma_{i,j,k}$ defined at page 28
in Cohen, Lada and May’s book by the identities
$\tau_{j,k}=\pm\xi^{\prime}_{k}\tau^{\prime}_{j,k}$ and
$\sigma_{i,j,k}=\pm\tau^{\prime}_{i,k}\tau^{\prime}_{j,k}$. These are
indecomposables in $\mathcal{R}_{k}^{\vee}$, but not in
$(\mathcal{R}_{k}^{t})^{\vee}$.

We now provide an additive basis for the twisted cohomology of the symmetric
groups. In the following definition, gathered blocks are elements of the
universal Hopf ring of the statement of Theorem 2.35. As mentioned above, the
product of a collection of $\gamma_{S_{i}}^{[{p^{n_{i}}}]}$ only depends on
the union of the $S_{i}$ as well as the list, with multiplicity, of the
$n_{i}$.

######  Definition 3.10.

Let $S=\\{s_{1}<s_{2}\dots<s_{r}\\}\subset\\{1,\ldots,k\\}$ and let
$D=\\{n_{i}\\}_{i=1}^{k}$ be a finite sequence of non-negative integers of
length $k$. If $|S|$ is odd let $m=1$, but if it is even let $m\geq 1$.

Set

$\Gamma_{S,D,m}=\prod_{i=1}^{k}\left(\gamma_{\varnothing,i}^{[p^{k-i}m]}\right)^{n_{i}}\prod_{j=1}^{r}(\gamma^{\prime}_{s_{j}})^{[{mp^{k-s_{j}}}]}{\lambda_{k}}^{\varepsilon(r)},$

where $\varepsilon(r)$ is $0$ if $r$ is even and $1$ if $r$ is odd

Similarly, we define

$\Gamma^{\prime}_{S,D,m}=\prod_{i=1}^{k}\gamma_{\varnothing,i}^{[p^{k-i}m]}\prod_{j=1}^{r}(\gamma^{\prime}_{s_{j}})^{[{mp^{j-s_{j}}}]}{\lambda_{k}}^{1-\varepsilon(r)},$

for $S\subseteq\\{1,...,k\\}$, with $S$ and $n_{i}$ as above but now $|S|=1$
when $m=1$.

We call $\Gamma_{S,D,m}$ and $\Gamma^{\prime}_{S,D,m}$ gathered blocks with
profile $(S,D)$ in $\bigoplus_{n}H^{*}({\mathcal{S}}_{n};\mathbb{F}_{p})$ or
$\bigoplus_{n}H^{*}({\mathcal{S}}_{n};\operatorname{sgn})$ respectively.

We define our preferred Hopf monomials in ${H_{EP}}^{*}(S^{0})$ as a $\odot$
product of divided powers of primitive gathered blocks with pairwise different
profiles, all belonging to
$\bigoplus_{n}H^{*}({\mathcal{S}}_{n};\mathbb{F}_{p})$ or all belonging to
$\bigoplus_{n}H^{*}({\mathcal{S}}_{n};\operatorname{sgn})$

We observe that $\Gamma_{S,D}$ is primitive if and only if $k\in S$, $n_{k}>0$
or $r$ is odd, while $\Gamma^{\prime}_{S,D}$ is primitive if and only if $k\in
S$, $n_{k}>0$ or $r$ is even.

These gathered blocks will be the building blocks of Hopf monomial bases for
the cohomology we compute, and can be assembled into “skyline diagrams.” The
reader is encouraged to compare this with the notion of gathered block in the
classical mod $p$ cohomology of the symmetric groups, as defined in Guerra’s
paper [Gue17] at page 964.

######  Lemma 3.11.

The universal Hopf ring described by the statement of Theorem 2.35 is spanned
as an $\mathbb{F}_{p}$-vector space by Hopf monomials.

###### Proof.

By unraveling the definition of the universal object, we directly see that a
complete set of relations for the considered component super-Hopf rig between
the generators $\gamma_{k}^{[{n}]}$, ${\gamma^{\prime}}_{i}^{[{n}]}$,
$\lambda_{k}$ is the following:

  * •

$(\lambda_{k})^{2}=\gamma_{k}$

  * •

the $\odot$ product of a class in $H^{*}({\mathcal{S}}_{n};\mathbb{F}_{p})$
and a class in $H^{*}({\mathcal{S}}_{n};\operatorname{sgn})$ is $0$.

  * •

$\Delta(\lambda_{k})=1\otimes\lambda_{k}+\lambda_{k}\otimes 1$

  * •

$\Delta(\gamma_{k}^{[{n}]})=\sum_{i}\gamma_{k}^{[{i}]}\otimes\gamma_{k}^{[{n-i}]}$

  * •

$\Delta({\gamma^{\prime}}_{k}^{[{n}]})=\sum_{i}{\gamma^{\prime}}_{k}^{[{i}]}\otimes{\gamma^{\prime}}_{k}^{[{n-i}]}$

  * •

$\gamma_{k}^{[{n}]}\odot\gamma_{k}^{[{m}]}=\left(\begin{array}[]{c}n+m\\\
n\end{array}\right)\gamma_{k}^{[{m+n}]}\\\ $

  * •

${\gamma^{\prime}}_{k}^{[{n}]}\odot{\gamma^{\prime}}_{k}^{[{m}]}=\left(\begin{array}[]{c}n+m\\\
n\end{array}\right){\gamma^{\prime}}_{k}^{[{m+n}]}\\\ $

  * •

the $\odot$ product of elements with different sign degree is $0$

Since our Hopf rig is generated by the classes above, monomials in both
products ($\cdot$ and $\odot$) generate it as a vector space. By Hopf ring
distributivity we can reduce to $\odot$-products of $\cdot$-monomials of
generators. Moreover, we can further restrict to considering $\cdot$-monomials
of generators all belonging to the same component and we can discard monomial
in which $\lambda_{k}$ appears at least twice thanks to the relation
$\lambda_{k}^{2}=\gamma_{k}$. All the relations above preserve profile of such
$\cdot$-monomials and can be used to equate any product which has the same
profile as a gathered block. Hence, Hopf monomials generate our Hopf ring as a
vector space. ∎

We can now prove the main result of this section.

###### Proof of Theorem 2.35.

Given a subset $S\subset\\{1,\dots,k\\}$, define
$\gamma^{\prime}_{S,k}=q_{L^{\prime}_{S,k}}^{\vee}$, where linear duals are
taken with respect to the Nakaoka monomial basis. Similarly, define
$\gamma_{S,k}=q_{L_{S,k}}^{\vee}$. We also define:

  * •

$\lambda_{k}=\gamma^{\prime}_{\varnothing,k}$

  * •

$\gamma_{j,m}=(q_{I_{j,j}}^{*m})^{\vee}$

  * •

$\gamma_{k}=\gamma_{k,1}$

  * •

$\gamma^{\prime}_{i,m}=(q_{L_{\\{i\\},i}}^{*m})^{\vee}$

  * •

$\gamma^{\prime}_{k}=\gamma^{\prime}_{k,1}$

  * •

$\gamma_{k}=\gamma_{\varnothing,k}$

Since $q_{0}$ is the $p^{th}$ power in the mod $p$ homology of $D(S^{0})$ with
coefficients in $\rho$, then $\gamma_{j,p^{k-j}}=(q_{I_{j,k}})^{\vee}$ and
$\gamma^{\prime}_{i,p^{k-i}}=\gamma^{\prime}_{\\{i\\},k}$.

We use Proposition 3.6 and Lemma 3.8 to deduce our cup product relations. The
structure of $\mathcal{R}_{k}^{\vee}$ is described by Theorem I.3.7 in
[CLM76]. In this reference, the product relations involve dividing by
$\gamma_{\varnothing,k}$, but they are equivalent to the identities stated in
[Gue17], of whose our relations (as far as $\mathcal{R}_{k}^{\vee}$ is
concerned) are straightforward reformulations that use lower indexes. For the
cohomology with coefficients in the sign representation, a similar argument
using Lemma 3.8 yields our description.

We first observe that $\gamma_{j,k}$, $\gamma^{\prime}_{i,k}$ and
$\lambda_{k}$ are liftings of the generators $\xi_{j,k}$,
$\tau^{\prime}_{i,k}$ and $\xi^{\prime}_{k}$ of
$(\mathcal{R}^{t}_{k})^{\vee}$.

We first prove by induction on $m\geq 1$ that
${\gamma^{\prime}}_{k}^{[{m}]}=\gamma^{\prime}_{k,m}$. The base of the
induction ($m=1$) is obvious, so we assume $m>1$. Since the product $*$ in
homology is linear dual to the coproduct $\Delta$ in ${H_{EP}}^{*}(S^{0})$, a
straightforward calculation by induction on $i$ yields the coproduct formula:

$\Delta(\gamma^{\prime}_{k,m})=\sum_{i+j=m}\gamma^{\prime}_{k,i}\otimes\gamma^{\prime}_{k,j}$

Similarly, using the compatibility between coproduct and divided powers, we
deduce by induction that

$\Delta({\gamma^{\prime}}_{k}^{[m]})=\sum_{j=0}^{m}{\gamma^{\prime}}_{k}^{[j]}\otimes{\gamma^{\prime}}_{k}^{[m-j]}.$

By combining these two coproduct identities with the induction hypothesis we
deduce that the difference between ${\gamma^{\prime}_{k}}^{[{m}]}$ and
$\gamma^{\prime}_{k,m}$ must be primitive. If $m$ is not a power of $p$, then
there are no $\odot$-indecomposables in the right component. If $m=p^{l}$, the
primitives are determined by Lemma 3.8 and there is none in the correct
dimension. In all cases, this difference must be zero. This shows that
$\gamma^{\prime}_{k}$ is standard non-nilpotent. A similar argument shows that
$\gamma_{k}^{[{m}]}=\gamma_{k,m}$ and $\gamma_{k}$ is standard non-nilpotent.

It is obvious from its definition that $\lambda_{k}$ is primitive. The cup-
product relation $(\lambda_{k})^{2}=\gamma_{k}$ holds because both sides are
primitive, hence $\odot$-indecomposables, and because this identity holds in
$(\mathcal{R}^{t})^{\vee}$. Then our cup-product relations are a reformulation
of those in $(\mathcal{R}^{t}_{k})^{\vee}$.

Let $H$ be the universal Hopf rig described in the statement of the Theorem
2.35. Since we have checked that all the required relations hold in
${H_{EP}}^{*}(S^{0})$, there must be a unique Hopf rig map $\pi\colon
H\to{H_{EP}}^{*}(S^{0})$ compatible with our choice of generators. As the ring
of $\odot$-indecomposables in $\bigoplus_{n\geq
0}H^{*}({\mathcal{S}}_{n};\rho)$ is the linear dual of $\mathcal{R}^{t}$,
which is generated by the images of $\gamma_{i}^{[{p^{k}}]}$,
${\gamma^{\prime}}_{i}^{[{p^{k}}]}$ and $\lambda_{k}$ by Lemma 3.8,
${H_{EP}}^{*}(S^{0})$ is generated by the classes defined at the beginning of
this proof and, consequently, $\pi$ is surjective. Hence, by Lemma 3.11, the
images of Hopf monomials generate ${H_{EP}}^{*}(S^{0})$ as a
$\mathbb{F}_{p}$-vector space. To prove that $\pi$ is an isomorphism, it is
enough to check that they form a basis, which can be done by comparing
dimension degree-wise with the Nakaoka basis in homology, as done in [Gue17,
Theorem 2.7]. ∎

As a byproduct of this proof we directly obtain the following.

######  Corollary 3.12.

The set of Hopf monomials in ${H_{EP}}^{*}(S^{0})$, defined in Definition
3.10, is a tri-graded additive basis for ${H_{EP}}^{*}(S^{0})$ as a
$\mathbb{F}_{p}$-vector space.

##  4\. Cohomology rings of extended powers with basepoints

We are now ready to establish our main calculations. In Section 2 we developed
the algebraic framework which governs the cohomology rings of extended powers,
when considered together. Such a framework demands twisted coefficients, so in
Section 3 we extended well-known results about homology to that setting. We
now show that the cohomology generated within the framework of component Hopf
rings with divided powers pairs perfectly with homology. We develop an
additive basis for the former before proving the pairing result.

###  4.1. Additive basis for the cohomology of $DX_{+}$

We describe explicit additive bases of the super-Hopf ring with additive
divided powers given Theorems 2.37 and 2.38. We use these to pair with
homology. Our basis is useful to perform concrete calculations, because we
describe an algorithm to compute the two products and the coproduct in terms
of it.

We primarily combine results from Section 2.6 with Lemma 3.11. We apply Lemma
2.44 when $A={H_{EP}}^{*}(S^{0})$ and $V=H^{*}(X;\mathbb{F}_{p})$, so that a
basis for $Q(A)$ is given by gathered blocks of component equal to a power of
$p$ by Corollary 3.12. The basis provided by Lemma 2.44 consists of such
gathered blocks, with the extra datum of a class in the cohomology of $X$.
Informally, by the Hopf ring relations in $A$, the transfer product of two
such gathered blocks with the same profile is again another gathered block.
Therefore, we can merge $\odot$-factors with the same profile and the same
extra cohomology class of $X$ and remove the constraint on the component. We
formalize this as follows.

######  Definition 4.1.

Let $X$ be a topological space and let $\mathcal{B}$ be a graded additive
basis for $H^{*}(X;\mathbb{F}_{p})$. A decorated gathered block on
$\mathcal{B}$ is a pair $(x,b)$, where $b$ is a gathered block in
$\bigoplus_{n\geq 0}H^{*}({\mathcal{S}}_{n};A_{n})$ and $x\in\mathcal{B}$. The
profile of the decorated gathered block $(x,b)$ is the profile of $b$, and its
decoration is $x$. Its _sign degree_ is $0$ if the degree of $x$ and the sign
degree of $b$ have the same parity, is $1$ if they have different parity. A
decorated Hopf monomial on $\mathcal{B}$ is a formal expression of the form
$b_{1}\odot\dots\odot b_{r}$, where $b_{1},\dots,b_{r}$ are decorated gathered
blocks on $\mathcal{B}$, up to permutation of the $b_{i}$, where no two
$b_{i}$s have the same profile and decoration, and where all the $b_{i}$s have
the same sign degree.

Our previous results assemble as follows.

######  Proposition 4.2.

Let $X$ be a topological space and let $\mathcal{B}$ be a graded additive
basis for $H^{*}(X;\mathbb{F}_{p})$. Let $A=\bigoplus_{n\geq
0}H^{*}({\mathcal{S}}_{n};A_{n})$ and let $V=H^{*}(X;\mathbb{F}_{p})$. Let
$H_{alg}$ be the universal object among Hopf ring with additive divided powers
satisfying the conditions of the statement of Theorem 2.38. Let $\mathcal{M}$
be the set of decorated Hopf monomials on $\mathcal{B}$. We map $\mathcal{M}$
to $H_{alg}$ by realizing a decorated gathered block $(x,b)$ as
$\iota(x)^{[{n(b)}]}\cdot\pi(b)$, and a decorated Hopf monomial as the
$\odot$-product in $H_{alg}$ of the constituent gathered blocks. Then
$\mathcal{M}$ is an additive basis for $H_{alg}$ as a $\mathbb{F}_{p}$-vector
space.

We can give a graphical description of our additive basis using a mild
generalization of Giusti, Salvatore, and Sinha’s skyline diagrams. Moreover,
all our structural morphisms can be understood graphically. We first consider
undecorated skyline diagrams, which correspond to classes in
${H_{EP}}^{*}(S^{0})$.

First, we recall some definitions from [GSS12] and [Gue17].

######  Definition 4.3.

A skyline diagram is a display of columns placed one next to the other
horizontally. Each column is comprised of rectangular boxes with the same
width stacked one on top of the other. The possible dimensions of these boxes
depend on a prime $p$:

  * •

If $p=2$, a column can be made of any number of boxes of width $n2^{k}$ and
height $1-2^{-k}$, for some $k,n\in\mathbb{N}$, $n\geq 1$.

  * •

If $p>2$, a column can be made of any number of hollow rectangles of width
$np^{k}$ and height $2(1-p^{-k})$ and at most one solid box of width $np^{k}$
and height dependent on a subset $S\subseteq\\{1,\dots,k\\}$. To simplify
notation, we let $|S|=2a+\varepsilon$, with $a\in\mathbb{N}$ and
$\varepsilon\in\\{0,1\\}$. For any choice of $k$ and $S$, we have two possible
heights: $n[2(a+\varepsilon-\sum_{s\in S}p^{-s})-\varepsilon p^{-k}]$ and
$n[2(a-\sum_{s\in S}p^{-s})-(1-\varepsilon)p^{-k}+1]$.

If a box has height given by the first formula, we say that it is of even
type, while we say that it is of odd type otherwise. Boxes of even type are
always defined if the subset has even cardinality $|S|=2a$, while they are
allowed for $|S|=2a+1$ only if $n=1$. Boxes of odd type are always defined if
the subset has odd cardinality $|S|=2a+1$, while they are allowed for $|S|=2a$
only if $n=1$.

We say that a column is of even type if its solid box is of even type or is
absent, and we say that it is of odd type if its solid box is of odd type.

We require that a skyline diagram does not contain two columns made of
fundamental rectangles with matching heights, and that its constituent columns
are all of the even type or all of odd type. We do not keep track of the order
in which columns are placed or in which boxes are stacked inside each column,
so two skyline diagrams that differ by a permutation of their columns are
considered the same.

Each fundamental box should be interpreted as a class in $\bigoplus_{n\geq
0}H^{*}({\mathcal{S}}_{n};\mathbb{F}_{p})$ or $\bigoplus_{n\geq
0}H^{*}({\mathcal{S}}_{n};\operatorname{sgn})$, in the component corresponding
to its width. Precisely:

  * •

if $p=2$, the rectangle of dimensions $n2^{k}$ and $1-2^{-k}$ corresponds to
${\gamma_{k}}^{[{n}]}$

  * •

if $p>2$, we regard the hollow rectangle whose width is $np^{k}$ and whose
height is $2(1-p^{-k})$ as ${\gamma_{k}}^{[{n}]}$

  * •

if $p>2$, the solid box of even (respectively odd) type with width $np^{k}$
associated to the subset $S$ corresponds to the gathered block
$\Gamma_{S,\underline{0},n}$ (respectively
$\Gamma^{\prime}_{S,\underline{0},n}$), as defined in Definition 3.10.

A column is understood as the cup product of the constituent fundamental
boxes, and a general skyline diagram is interpreted as the transfer product of
its columns. With these correspondences, the elements of the additive bases
for $\bigoplus_{n\geq 0}H^{*}(\mathcal{S}_{n};\mathbb{F}_{2})$ and
$\bigoplus_{n\geq 0}H^{*}(\mathcal{S}_{n};\mathbb{F}_{p})$ ($p>2$) obtained in
the papers mentioned above are described as those skyline diagrams (with
columns of even type if $p>2$). For example, in Figure 1 we depict the skyline
diagrams corresponding to $\gamma_{1}^{3}\odot\gamma_{2}\in
H^{*}({\mathcal{S}}_{6};\mathbb{F}_{2})$ and
${\gamma_{1}}^{[{3}]}\gamma_{\\{1\\},2}\odot\gamma_{1}^{2}\in
H^{*}({\mathcal{S}}_{12};\mathbb{F}_{3})$.

It is sometimes useful to consider vertical dashed lines dividing some
fundamental rectangles into equal parts. Explicitly, boxes corresponding to
${\gamma_{k}}^{[{n}]}$ are divided into $n$ parts of width $p^{k}$ and, if
$p>2$, even-dimensional solid boxes of even type and odd-dimensional solid
boxes of odd type corresponding to a subset $S$ are divided into parts of
width $p^{\max(S)}$.

$\gamma_{1}^{3}\odot\gamma_{2}$${\gamma_{1}}^{[{3}]}{\gamma^{\prime}_{1}}^{[{3}]}\lambda_{2}\odot\gamma_{1}^{2}$
Figure 1. Examples of skyline diagrams in
$H^{*}({\mathcal{S}}_{6};\mathbb{F}_{2})$ (top) and
$H^{*}({\mathcal{S}}_{12};\mathbb{F}_{3})$ (bottom)

######  Proposition 4.4.

Hopf monomials in ${H_{EP}}^{*}(S^{0})$ are in bijective correspondence with
skyline diagrams as defined above

###### Proof.

We already noted that solid blocks correspond to gathered blocks
$\Gamma_{S,D,n}$ or $\Gamma^{\prime}_{S,D,n}$ and hollow rectangles correspond
to $\gamma_{k}^{[{n}]}$. Hence, the set of all possible columns is in
bijective correspondence with the set of gathered blocks, with taking the cup
product of the classes corresponding to the stacking of constituent boxes.
Similarly, a skyline diagram give rise to a Hopf monomial in
${H_{EP}}^{*}(S^{0})$ by taking the transfer product of the gathered blocks
corresponding to its constituent columns.

To define this precisely, one needs to fix an ordering of the boxes and the
columns for each skyline diagram, because cup product and transfer product are
only commutative up to sign. With that subtlety, it is straightforward to
check directly from the definitions that this is a bijection. ∎

We next describe our generalization of skyline diagrams for
${H_{EP}}^{*}(X_{+})$.

######  Definition 4.5.

Let $p$ be a prime number and $V$ be a graded algebra over $\mathbb{F}_{p}$. A
column decorated in $V$ is a column diagram as in Definition 4.3, together
with an element $x\in V$. Graphically, we depict the decoration of the column
as an additional rectangle labeled $x$ with width equal to its width and
height equal to the degree of $x$, placed beneath that column. When $p>2$, we
say that a decorated column is of even type if the type of underlying un-
decorated column and the dimension of the decoration are both even or both
odd. We say that it is of odd type otherwise.

A skyline diagram decorated in $V$ is a diagram constituted by columns
decorated in $V$ placed one next to each other horizontally, in which there
are not two columns made of blocks of corresponding heights and having the
same decoration and in which the constituent columns are all of even type or
all of odd type.

If $\mathcal{B}$ is an additive basis for $V$ as a graded
$\mathbb{F}_{p}$-vector space, then we define a skyline diagram decorated in
$\mathcal{B}$ as a skyline diagram decorated in $V$ whose decorations are
elements of that basis.

We now assume $V=H^{*}(X;\mathbb{F}_{p})$.

######  Proposition 4.6.

Let $p$ be a prime number and let $X$ be a topological space and $\mathcal{B}$
be a basis for $H^{*}(X;\mathbb{F}_{p})$. There is a bijective correspondence
between the set of skyline diagrams decorated in $\mathcal{B}$ and Hopf
monomials with respect to the given basis in ${H_{EP}}^{*}(X_{+})$ as in
Definition 4.1. Stacking boxes on top of each other corresponds to taking the
cup product of the associated cohomology classes (restricted to the relevant
component) and placing columns next to each other horizontally corresponds to
taking their transfer product.

###### Proof.

If $c$ is a column with decoration $x$, we associate with it the decorated
gathered block $\Gamma(c)\cdot x^{[{w}]}$, where $w$ is the width of the
column and $\Gamma(c)$ is the gathered block in ${H_{EP}}^{*}(S^{0})$
corresponding to $c$. We then associate to a decorated skyline diagram the
transfer product of the decorated gathered blocks corresponding to its
constituent decorated columns. It is easy to deduce from Proposition 4.4 that
this is a bijection. ∎

By taking only decorated skyline diagrams whose columns are all of even type,
we obtain a basis for $\bigoplus_{n\geq
0}H^{*}(\tilde{D}_{n}(X);\mathbb{F}_{p})$.

Our next aim is to describe a way to compute the structural morphisms via
skyline diagrams. The rough graphical interpretation of the cup product of
gathered blocks is “splitting and piling the corresponding columns one on top
of the other.” However, if $p>2$, this procedure does not always yield an
“allowable” column. Hence, we must use our cup product relations in
$H^{*}(\mathcal{S}_{n};\mathbb{F}_{p})$ or