# Cohomology of complete unordered flag manifolds

Lorenzo Guerra  Università di Roma Tor Vergata<EMAIL_ADDRESS>and
Santanil Jana  Mathematics Department, University of British Columbia
<EMAIL_ADDRESS>
###### Abstract.

We consider quotients of complete flag manifolds in $\mathbb{C}^{n}$ and
$\mathbb{R}^{n}$ by an action of the symmetric group on $n$ objects. We
compute their cohomology with field coefficients of any characteristic.
Specifically, we show that these topological spaces exhibit homological
stability and we provide a closed-form description of their stable cohomology
rings. We also describe a simple algorithmic procedure to determine their
unstable cohomology additively.

###### 2020 Mathematics Subject Classification:

Primary 14M15; Secondary 55N10, 55N91, 55P47, 55R20, 55R45, 55S12, 55T10,
20J06.

The first author acknowledges the MIUR Excellence Department Project awarded
to the Department of Mathematics, University of Rome Tor Vergata, CUP
E83C18000100006.

##  1\. Introduction

A complete flag over a field $\mathbb{F}$ is a nested sequence of
$\mathbb{F}$-linear subspaces of $\mathbb{F}^{n}$:

(1) $\\{0\\}=V_{0}\subset V_{1}\subset\cdots\subset V_{n}=\mathbb{F}^{n}$

such that $\mathrm{dim}_{\mathbb{F}}(V_{j})=j$, for all $j=1,2,\dots,n$. The
space of all complete flags of order $n$ over $\mathbb{F}$ forms an algebraic
variety called the full flag variety of order $n$ over $\mathbb{F}$, denoted
as $\operatorname{Fl}_{n}(\mathbb{F})$. In this paper, our focus is
exclusively on cases where $\mathbb{F}$ corresponds to the fields $\mathbb{R}$
and $\mathbb{C}$. Over these fields, $\operatorname{Fl}_{n}(\mathbb{F})$
endowed with the analytic topology is a topological manifold, that we call
real and complex flag manifold, respectively. The unitary group $U(n)$ (or the
orthogonal group $O(n)$) acts transitively on
$\operatorname{Fl}_{n}(\mathbb{C})$ (or $\operatorname{Fl}_{n}(\mathbb{R})$),
i.e. given any flag $\\{V_{j}\\}_{j=0}^{n}$ as in (1), there exists a $g\in
U(n)$ (or $O(n)$) which maps the flag (1) to the standard flag of
$\mathbb{C}^{n}$ (or $\mathbb{R}^{n}$):

$\\{0\\}\subset\mathbb{C}\\{e_{1}\\}\subset\cdots\subset\mathbb{C}\\{e_{1},\dots,e_{n}\\}=\mathbb{C}^{n},$

where $\\{e_{1},\dots,e_{n}\\}$ is the standard basis of $\mathbb{C}^{n}$.
Therefore, complex (or real) flags can be identified with the elements of
$U(n)$ (or $O(n)$), modulo the subgroup that leaves the standard flag fixed.
Therefore, the flag manifold $\operatorname{Fl}_{n}(\mathbb{C})$ (or
$\operatorname{Fl}_{n}(\mathbb{R})$) can be identified with $U(n)/T(n)$ (or
$O(n)/T(n)$), where $T(n)$ is the maximal torus in $U(n)$ (or $O(n)$).

Given a flag $\\{V_{j}\\}_{j=0}^{n}$ as in (1), there is an orthonormal basis
$\\{v_{1},\dots,v_{n}\\}$ of $\mathbb{C}^{n}$ (or $\mathbb{R}^{n}$) such that

$V_{j}=\mathbb{C}\\{v_{1},\dots,v_{j}\\}\text{ }(\text{or
}V_{j}=\mathbb{R}\\{v_{1},\dots,v_{j}\\}).$

An ordered orthonormal basis of $\mathbb{C}^{n}$ (or $\mathbb{R}^{n}$)
describes a complete flag. Any element in $\operatorname{Fl}_{n}(\mathbb{C})$
(or $\operatorname{Fl}_{n}(\mathbb{R})$) can be described as a set of $n$
ordered mutually orthogonal lines in $\mathbb{C}^{n}$ (or $\mathbb{R}^{n}$).
The symmetric group $\Sigma_{n}$ acts freely on
$\operatorname{Fl}_{n}(\mathbb{C})$ (or $\operatorname{Fl}_{n}(\mathbb{R})$)
by permuting the ordered basis elements. Note that the action of $\Sigma_{n}$
on $\operatorname{Fl}_{n}(\mathbb{C})$ (or
$\operatorname{Fl}_{n}(\mathbb{R})$) does not permute the subspaces $V_{j}$,
rather it permutes the basis elements $v_{j}$.

######  Definition 1.1.

The complex (or real) unordered flag manifold of order $n$ is defined as the
quotient $\operatorname{Fl}_{n}(\mathbb{C})/\Sigma_{n}$ (or
$\operatorname{Fl}_{n}(\mathbb{R})/\Sigma_{n}$) and denoted as
$\overline{\mathrm{Fl}}_{n}(\mathbb{C})$ (or
$\overline{\mathrm{Fl}}_{n}(\mathbb{R})$).

Let $N(n):=N_{U(n)}(T(n))$ be the normalizer of the maximal torus $T(n)$ in
$U(n)$. The Weyl group of the maximal torus is $N(n)/T(n)\cong\Sigma_{n}$. The
unordered flag manifold of order $n$ can also be described as

(2)
$\overline{\mathrm{Fl}}_{n}(\mathbb{C})=\operatorname{Fl}_{n}(\mathbb{C})/\Sigma_{n}\cong
U(n)/N(n).$

Similarly $\operatorname{B}_{n}:=N_{O(n)}(T(n))$ be the normalizer of the
maximal torus in $O(n)$. Then we can identify $O(n)/\operatorname{B}_{n}$ with
$\operatorname{Fl}_{n}(\mathbb{R})$. In §4.2 we use the alternating subgroup
$\operatorname{B}_{n}^{+}:=N_{SO(n)}(T(n)\cap SO(n))$ (so that
$\overline{\mathrm{Fl}}_{n}(\mathbb{R})\cong SO(n)/\operatorname{B}_{n}^{+}$)
for our computations.

In this paper, we delve into the study of the stable and unstable cohomology
of complex unordered flag manifolds. While complete flag manifolds have been
extensively studied in the fields of algebraic topology and geometry due to
their significance in Lie theory, unordered complete flag manifolds have
received less attention from algebraic topologists. Understanding the topology
of these unordered flag manifolds is not only intriguing in its own right, but
also carries important implications for emerging problems in algebraic
topology and convex geometry.

The cohomology of the unordered flag manifolds bears relevance to the
computation of the cohomology of the classifying spaces for commutativity. The
classifying space for commutativity $B_{com}G$ and the total space of the
associated principal $G$-bundle, $E_{com}G$, was described as a homotopy
colimit over a poset by Adem–Gómez [1]. This homotopy colimit diagram involves
the unordered flag manifold $\overline{\mathrm{Fl}}_{n}(\mathbb{C})$ (or
$\overline{\mathrm{Fl}}_{n}(\mathbb{R})$) when $G$ is $U(n)$ (or $O(n)$). The
cohomology of these spaces, particularly the $p$-torsion structure, has
implications for the study of spaces of homomorphism. The study of the
cohomology of unordered flag manifolds is also connected to the resolution of
a conjecture proposed by Atiyah [2]. However, providing a comprehensive
discussion of this conjecture is beyond the scope of the paper. For further
information and in-depth analysis, we refer readers to [2, §4].

The cohomology of unordered flag manifolds also has implications in the
estimation of the number of Auerbach bases of finite dimensional Banach
spaces. Let $\mathbb{X}$ be a $n$ dimensional complex (or real) Banach space
and $S_{\mathbb{X}}$ denote its unit sphere. A basis
$\mathcal{B}=\\{v_{1},\dots,v_{n}\\}$ of $\mathbb{X}$ is called an Auerbach
basis if $v_{i}\in S_{\mathbb{X}}$ and there is a basis ${v^{1},\dots,v^{n}}$
of the dual space $\mathbb{X}^{*}$ satisfying

$v^{i}(v_{j})=\delta_{ij},\quad\text{and}\quad v^{i}\in
S_{\mathbb{X}^{*}}\text{ for}\ i,j=1,2,\dots,n.$

Weber–Wojciechowski [20] provided an estimate of the number of Auerbach bases
of a finite-dimensional Banach space using topological methods. Here, one
identifies bases that differ only by permutation or multiplication by scalars
of absolute value one. In other words, two bases are said to be equivalent if
they lie in the same orbit of the action of $N(n)$ (or $\operatorname{B}_{n}$)
on $U(n)$ (or $O(n)$). In [20], the estimates of the Lusternick–Schnirelmann
category [7] $\mathrm{cat}({\overline{\mathrm{Fl}}}_{n}(\mathbb{R}))$ and
$\mathrm{rank}(H^{*}({\overline{\mathrm{Fl}}}_{n}(\mathbb{R})))$ along with
their complex counterparts were obtained using known results about the
cohomology of ordered flag manifolds. The estimates for
$\mathrm{rank}(H^{*}({\overline{\mathrm{Fl}}}_{n}(\mathbb{R})))$ and
$\mathrm{cat}({\overline{\mathrm{Fl}}}_{n}(\mathbb{C}))$ has been much-
improved by the computational results about the cohomology of the unordered
flag varieties for lower orders in [13].

Convention. In the rest of this paper, $\mathbb{F}$ will denote a field and
$p$ will denote a prime. The cohomological degree of a cohomology class
$\gamma$ will be denoted by $|\gamma|$.

Acknowledgements. The first author would like to thank Prof. Paolo Salvatore
for the helpful conversation and acknowledges the MIUR Excellence Department
Project awarded to the Department of Mathematics, University of Rome Tor
Vergata, CUP E83C18000100006.
The second author expresses their gratitude to Prof. Alejandro Adem for
engaging discussions that significantly contributed to the research presented
in this paper.

###### Contents

  1. 1 Introduction
  2. 2 Recollection of the cohomology of extended powers
  3. 3 Review of the cohomology of alternating subgroups of the hyperoctahedral groups
    1. 3.1 Mod $2$ cohomology of $\operatorname{B}_{n}^{+}$
    2. 3.2 Cohomology of $\operatorname{B}_{n}^{+}$ at odd primes
  4. 4 Stable cohomology of complete unordered flag varieties
    1. 4.1 The complex case
    2. 4.2 The real case
      1. 4.2.1 Mod $2$ cohomology
      2. 4.2.2 Cohomology at odd primes
      3. 4.2.3 Rational cohomology
    3. 4.3 Poincaré series of stable cohomology
      1. 4.3.1 Poincaré series of $H^{*}(BN(\infty);\mathbb{F}_{2})$
      2. 4.3.2 Poincaré series of $H^{*}(\overline{\mathrm{Fl}}_{\infty}(\mathbb{C});\mathbb{F}_{2})$
  5. 5 The spectral sequence computing $H^{*}(\overline{\operatorname{Fl}}_{n}(\mathbb{C});\mathbb{F})$ and $H^{*}(\overline{\operatorname{Fl}}_{n}(\mathbb{R});\mathbb{F})$
    1. 5.1 The complex case
    2. 5.2 The real case
  6. 6 Unstable cohomology of complete unordered flag varieties
    1. 6.1 Mod $2$ Cohomology of $\overline{\mathrm{Fl}}_{3}(\mathbb{C})$
    2. 6.2 Mod $2$ Cohomology of $\overline{\mathrm{Fl}}_{4}(\mathbb{C})$
    3. 6.3 Mod $2$ Cohomology of $\overline{\mathrm{Fl}}_{5}(\mathbb{C})$
    4. 6.4 Mod $p$ Cohomology of $\overline{\mathrm{Fl}}_{p}(\mathbb{C})$ for $p>2$
    5. 6.5 Mod $p$ cohomology of $\overline{\operatorname{Fl}}_{p}(\mathbb{R})$ for $p>2$

##  2\. Recollection of the cohomology of extended powers

Let $X$ be a Hausdorff compactly generated topological space. Define its
$n$-fold extended (symmetric) power $D_{n}(X_{+})$ as the homotopy quotient of
$X^{n}$ with respect to the action of the symmetric group $\Sigma_{n}$ that
permutes the cartesian factors. In particular, if $\\{*\\}$ is a space with
one point, $D_{n}(\\{*\\}_{+})=B(\Sigma_{n})$, the classifying space of the
symmetric group itself.

We are mostly interested in the cases where $\mathbb{F}=\mathbb{F}_{p}$ for
some prime number $p$ or $\mathbb{F}=\mathbb{Q}$. There will be some results
that are valid only for these particular choices; we will stress explicitly
this fact where it happens. Let $A_{X}=\bigoplus_{n,d}A_{X}^{n,d}$ be the
bigraded $\mathbb{F}$-vector space defined by
$A_{X}^{n,d}=H^{d}(D_{n}(X_{+}))$. We refer to the indices $n$ and $d$ as the
component and (cohomological) dimension, respectively. On $A_{X}$ there are
some structural morphisms, all natural in $X$, that provide it with a rich
algebraic structure:

  * •

the usual cup product, component by component, $\cdot\colon A_{X}\otimes
A_{X}\to A_{X}$.

  * •

a coproduct $\Delta\colon A_{X}\to A_{X}\otimes A_{X}$ induced by the maps
$p_{n,m}\colon D_{n}(X_{+})\times D_{m}(X_{+})\to D_{n+m}(X_{+})$ defined by
passing to quotients the following map:

$\displaystyle(p,(x_{1},\dots,x_{n}))\times(q,(x_{n+1},\dots,x_{n+m}))\in(E(\Sigma_{n})\times
X^{n})\times(E(\Sigma_{m})\times X^{m})$
$\displaystyle\mapsto(i_{n,m}(p,q),(x_{1},\dots,x_{n},x_{n+1},\dots,x_{n+m}))\in
E(\Sigma_{n+m})\times X^{n+m},$

where $i_{n,m}\colon E(\Sigma_{n})\times E(\Sigma_{m})\to E(\Sigma_{n+m})$ is
the $\Sigma_{n}\times\Sigma_{m}$-equivariant map induced by the standard
inclusion $\Sigma_{n}\times\Sigma_{m}\hookrightarrow\Sigma_{n+m}$.

  * •

since $p_{n,m}$ is homotopy equivalent to a finite covering, there is a
product $\odot\colon A_{X}\otimes A_{X}\to A_{X}$ corresponding to the
cohomological transfer maps of $p_{n,m}$.

######  Definition 2.1.

A commutative component bigraded Hopf ring is a bigraded vector space
$A=\bigoplus_{n,d}A^{n,d}$ with a coproduct $\Delta\colon A\to A\otimes A$,
two products $\odot,\cdot\colon A\otimes A\to A$, a unit
$\eta\colon\mathbb{F}\to A$, a counit $\varepsilon\colon A\to\mathbb{F}$, and
an antipode $S\colon A\to A$ such that

  * •

$\Delta,\odot,\eta,\varepsilon$ are bigraded, $\cdot$ is graded with respect
to $d$, and the products and the coproduct are graded commutative with respect
to $d$,

  * •

$(A,\Delta,\odot,\eta,\varepsilon)$ is a Hopf algebra,

  * •

$(A,\cdot,\Delta)$ is a bialgebra,

  * •

$\forall x\in A^{n,d},x^{\prime}\in A^{n^{\prime},d^{\prime}}\colon x\cdot
x^{\prime}=0$ if $n\not=n^{\prime}$,

  * •

and the following Hopf ring distributivity axiom holds for all $x,y,z\in A$,
where we use Sweedler’s notation $\Delta(x)=\sum x_{(1)}\otimes x_{(2)}$:

$x\cdot(y\odot z)=\sum(-1)^{d(y)d(x_{(2)})}(x_{(1)}\cdot y)\odot(x_{(2)}\cdot
z).$

######  Theorem 2.2 ([14, Corollary 2.31]).

$A_{X}$, with the morphisms defined above, the unit and counit that identify
$A_{X}^{(0,0)}$ with $\mathbb{F}$, and an antipode given by multiplication by
$\pm 1$ depending on the component, is a commutative bigraded component Hopf
ring.

In the case where $\mathbb{F}=\mathbb{F}_{p}$ or $\mathbb{F}=\mathbb{Q}$,
there are distinguished classes in $A_{X}$:

  * •

The first distinguished set of classes appears only if
$\mathbb{F}=\mathbb{F}_{p}$ for some prime $p$. The unique map $\pi\colon
X\to\\{*\\}$ induces an injective homomorphism $\pi^{*}\colon\bigoplus_{n\geq
0}H^{*}(\Sigma_{n};\mathbb{F}_{p})=A_{\\{*\\}}\to A_{X}$. By a slight abuse of
notation, we identify the classes $\gamma_{k,l}$ of Giusti–Salvatore–Sinha [8]
(if $p=2$) or the classes $\gamma_{k,l}$, $\alpha_{i,k}$ and $\beta_{i,j,l}$
of Guerra [10] (if $p>2$) with their image $\pi^{*}(\gamma_{k,l})$ under
$\pi^{*}$, thus realizing them as elements of $A_{X}$.

  * •

The second distinguished set of classes appears for all choices of
$\mathbb{F}$. For every even-dimensional cohomology class $\alpha\in H^{*}(X)$
and for all $n>0$, the twisted cross product
$\alpha_{[n]}=1\times_{\Sigma_{n}}\alpha^{\times n}\in
H^{*}(E(\Sigma_{n})\times_{\Sigma_{n}}X^{n})=H^{*}(D_{n}(X_{+}))$ is well-
defined. explicitly, the complex of singular chains of
$E(\Sigma_{n})\times_{\Sigma_{n}}X^{n}$ is quasi-isomorphic to
$W_{*}\otimes_{\Sigma_{n}}C_{*}(X)^{\otimes n}$, where $W_{*}$ is a free
resolution of $\mathbb{F}$ as a $\mathbb{F}[\Sigma_{n}]$-module. Let
$\varepsilon\colon W_{0}\to\mathbb{F}$ be the augmentation (representing the
unit in $H^{0}(\Sigma_{n})$) and $a$ be a cocycle representative of $\alpha$.
Then $\alpha_{[n]}$ is represented by the $\Sigma_{n}$-invariant cocycle
$\varepsilon\otimes a^{\otimes n}\colon W_{*}\otimes C_{*}(X)^{\otimes
n}\to\mathbb{F}$, which does not depend on the chosen representative. We also
define, by convention, $\alpha_{[0]}=1_{0}$, the unit of the $0$-th component
of $A_{X}$. If $\mathbb{F}=\mathbb{F}_{2}$, then $\varepsilon\otimes
a^{\otimes n}$ is $\Sigma_{n}$-invariant even if $a$ is an odd-dimensional
cochain; consequently, $\alpha_{[n]}$ is defined for all $\alpha\in
H^{*}(X;\mathbb{F}_{2})$, odd- and even-dimensional.

Guerra–Salvatore–Sinha proved that these distinguished classes suffice to
fully determine $A_{X}$ as a Hopf ring. We recall their results below.

######  Theorem 2.3 ([14, Theorem 2.37]).

Assume that $\mathbb{F}=\mathbb{F}_{2}$. As a commutative bigraded component
Hopf ring, $A_{X}$ is generated by the classes $\gamma_{k,l}$ (for $k,l\geq
1$) and $x_{n}$ (for $x\in H^{*}(X)$ and $n\geq 1$), with the following
relations:

  * •

the relations of [8, Theorem 1.2],

  * •

$x_{[n]}\cdot{x^{\prime}}_{[n]}=(x\cdot x^{\prime})_{[n]}$ for
$x,x^{\prime}\in H^{*}(X)$,

  * •

$\Delta(x_{[n]})=\sum_{i=0}^{n}x_{[i]}\otimes x_{[n-i]}$ for $x\in H^{*}(X)$,

  * •

$x_{[m]}\odot x_{[n]}=\binom{m+n}{n}x_{[m+n]}$ for $x\in H^{*}(X)$,

  * •

$(\lambda x)_{[n]}=\lambda^{n}x_{[n]}$ and
$(x+y)_{[n]}=\sum_{k=0}^{n}x_{[k]}\odot y_{[n-k]}$ for $x,y\in H^{*}(X)$ and
$\lambda\in\mathbb{F}$.

######  Theorem 2.4 ([14, Theorem 2.38], particular case).

Assume that $\mathbb{F}=\mathbb{F}_{p}$ with $p$ odd prime and that $H^{*}(X)$
is $0$ in odd degrees. Then, as a commutative bigraded component Hopf ring,
$A_{X}$ is generated by the classes $\gamma_{k,l}$ (for $k,l\geq 1$),
$\alpha_{i,k}$ (for $1\leq i\leq k$), $\beta_{i,j,k}$ (for $1\leq i<j\leq k$)
and $x_{[n]}$ (for $x\in H^{*}(X)$ and $n\geq 1$), with the following
relations:

  * •

the relations of [10, Theorem 2.7],

  * •

$x_{[n]}\cdot{x^{\prime}}_{[n]}=(x\cdot x^{\prime})_{[n]}$ for
$x,x^{\prime}\in H^{*}(X)$,

  * •

$\Delta(x_{[n]})=\sum_{i=0}^{n}x_{[i]}\otimes x_{[n-i]}$ for $x\in H^{*}(X)$,

  * •

$x_{[m]}\odot x_{[n]}=\binom{m+n}{n}x_{[m+n]}$ for $x\in H^{*}(X)$,

  * •

$(\lambda x)_{[n]}=\lambda^{n}x_{[n]}$ and
$(x+y)_{[n]}=\sum_{k=0}^{n}x_{[k]}\odot y_{[n-k]}$ for $x,y\in H^{*}(X)$ and
$\lambda\in\mathbb{F}$.

From this presentation, one can extract an additive basis for the cohomology
of $D_{n}(X_{+})$ under the hypotheses of Theorems 2.3 and 2.4.

######  Definition 2.5.

Let $\mathcal{B}$ be a graded basis of $H^{*}(X)$ as a $\mathbb{F}$-vector
space. A decorated gathered block in $A_{X}$ is a couple $(b,x)$, where $b$ is
a gathered block in $\bigoplus_{n\geq 0}H^{*}(\Sigma_{n})$, in the sense of
Giusti–Salvatore–Sinha [8, Definition 6.3] (if $\mathbb{F}=\mathbb{F}_{2}$) or
Guerra [10, page 964] (if $\mathbb{F}=\mathbb{F}_{p}$ with $p>2$), and
$x\in\mathcal{B}$ is a basis element. We call $x$ the decoration of the
gathered block.

A decorated Hopf monomial is a formal expression of the form
$b_{1}\odot\dots\odot b_{r}$, where $b_{1},\dots,b_{r}$ are decorated gathered
blocks, such that any two distinct elements among these $r$ gathered blocks
have not the same profile (in the sense of [8, Definition 6.3] (if
$\mathbb{F}=\mathbb{F}_{2}$) or [10, Definition 3.1] and the same decoration
concurrently.

######  Corollary 2.6 ([14, Proposition 4.5]).

Assume that $\mathbb{F}=\mathbb{F}_{p}$ with $p\geq 2$ and
$H^{d}(X;\mathbb{F}_{p})=0$ for $d$ odd. Let $\mathcal{B}$ be as in the
previous definition. Realize a decorated gathered block $(b,x)$, as an element
of $A_{X}$ by taking the cup product $\pi^{*}(b)\cdot x_{[n]}$, where $n$ is
the component of $b$. Realize a decorated Hopf monomial
$x=b_{1}\odot\dots\odot b_{r}$ as an element of $A_{X}$ by realizing the
constituent decorated gathered blocks as elements of $A_{X}$ and taking their
transfer product. Then the set $\mathcal{M}$ of decorated Hopf monomial is a
bigraded basis for $A_{X}$ as a $\mathbb{F}$-vector space.

Hopf ring distributivity and the relations of Theorems 2.3 and 2.4 are enough
to explicitly compute the products and the coproduct on decorated Hopf
monomials in $A_{X}$. We exemplify below the general procedure in a specific
case. The reader might also find the graphical algorithm described in [14, §4]
in terms of skyline diagrams more accessible.

######  Example 2.7.

Let $\mathcal{B}=\\{1,c,c^{2},\dots\\}$ be the graded basis of
$H^{*}(BU(1);\mathbb{F}_{2})$ as a $\mathbb{F}_{2}$-vector space as before.
Consider the following decorated Hopf monomials

$x=(\gamma_{1,2}^{2},c)\odot(\gamma_{1,1},c)\quad\text{and}\quad
y=(\gamma_{2,1},1)\odot(\gamma_{1,1},c).$

We will compute the cup product $x\cdot y$. First, we compute the coproduct of
$x$. Using the fact that the cup product and coproduct from a bialgebra, we
have

$\displaystyle\Delta((\gamma_{1,2}^{2},c))=\Delta(\gamma_{1,2}^{2}\cdot
c_{[4]})=\Delta(\gamma_{1,2})^{2}\cdot\Delta(c_{[4]})$ $\displaystyle\hskip
28.45274pt=(\gamma_{1,2}^{2}\otimes
1_{0}+\gamma_{1,1}^{2}\otimes\gamma_{1,1}^{2}+1_{0}\otimes\gamma_{1,2}^{2})\cdot(c_{[4]}\otimes
1_{0}+c_{[3]}\otimes c_{[1]}+\cdots+1_{0}\otimes c_{[4]})$
$\displaystyle\hskip 28.45274pt=\gamma_{1,2}^{2}\cdot c_{[4]}\otimes
1_{0}+\gamma_{1,1}^{2}\cdot c_{[2]}\otimes\gamma_{1,1}^{2}\cdot
c_{[2]}+1_{0}\otimes\gamma_{1,2}^{2}\cdot c_{[4]}$

and

$\displaystyle\Delta((\gamma_{1,1},c))$
$\displaystyle=\Delta(\gamma_{1,1})\cdot\Delta(c_{[2]})$
$\displaystyle=(\gamma_{1,1}\otimes
1_{0}+1_{0}\otimes\gamma_{1,1})\cdot(c_{[2]}\otimes 1_{0}+c_{[1]}\otimes
c_{[1]}+1_{0}\otimes c_{[2]})$ $\displaystyle=\gamma_{1,1}\cdot c_{[2]}\otimes
1_{0}+1_{0}\otimes\gamma_{1,1}\cdot c_{[2]}.$

Using the fact that the transfer product and the coproduct form a bialgebra,
we have

$\displaystyle\Delta(x)$
$\displaystyle=\Delta((\gamma_{1,2}^{2},c))\odot\Delta((\gamma_{1,1},c))$
$\displaystyle=x\otimes 1_{0}+(\gamma_{1,2}^{2}\cdot
c_{[4]})\otimes(\gamma_{1,1}\cdot c_{[2]})+((\gamma_{1,1}^{2}\cdot
c_{[2]})\odot(\gamma_{1,1}\cdot c_{[2]}))\otimes(\gamma_{1,1}^{2}\cdot
c_{[2]})$ $\displaystyle+(\gamma_{1,1}^{2}\cdot
c_{[2]})\otimes((\gamma_{1,1}^{2}\cdot c_{[2]})\odot(\gamma_{1,1}\cdot
c_{[2]}))+(\gamma_{1,1}\cdot c_{[2]})\otimes(\gamma_{1,2}^{2}\cdot
c_{[4]})+1_{0}\otimes x.$

Using the Hopf ring distributivity axiom, we have

$\displaystyle x\cdot y$
$\displaystyle=x\cdot((\gamma_{2,1},1)\odot(\gamma_{1,1},c))$
$\displaystyle=\sum_{\Delta x=\sum x^{\prime}\otimes
x^{\prime\prime}}(x^{\prime}\cdot\gamma_{2,1})\odot(x^{\prime\prime}\cdot\gamma_{1,1}\cdot
c_{[2]}).$

We observe that only addends $x^{\prime}\otimes x^{\prime\prime}$ in
$\Delta(x)$ that have the right components such that the cup products are non-
zero are $(\gamma_{1,2}^{2}\cdot c_{[4]})\otimes(\gamma_{1,1}\cdot c_{[2]})$
and $((\gamma_{1,1}^{2}\cdot c_{[2]})\odot(\gamma_{1,1}\cdot
c_{[2]}))\otimes(\gamma_{1,1}^{2}\cdot c_{[2]})$. Note that by Hopf ring
distributivity

$\gamma_{2,1}\cdot((\gamma_{1,1}^{2}\cdot c_{[2]})\odot(\gamma_{1,1}\cdot
c_{[2]}))=0.$

Therefore,

$\displaystyle x\cdot y$
$\displaystyle=(\gamma_{2,1}\cdot\gamma_{1,2}^{2}\cdot
c_{[4]})\odot(\gamma_{1,1}^{2}\cdot c_{[2]}^{2})$

The corresponding skyline diagram is described in Figure 1.

$\cdot$$=$$(c_{[4]}\gamma_{1,2}^{2})\odot(c_{[2]}\gamma_{1,1})$$(c_{[2]}\gamma_{1,1})\odot\gamma_{2,1}$$(c_{[4]}\gamma_{1,2}^{2}\gamma_{2,1})\odot(c_{[2]}^{2}\gamma_{1,1}^{2})$
Figure 1. Skyline diagram for the dot product of Hopf monomials $x$ and $y$
using Hopf ring distributivity

The rational cohomology of $D(X_{+})$ is well-known to experts. We slightly
enhance its usual description by incorporating the Hopf ring structure.

######  Proposition 2.8.

Let $\mathbb{F}=\mathbb{Q}$. Let $\mathcal{B}$ be a graded basis of the
cohomology of $X$. If $H^{*}(X;\mathbb{Q})$ is concentrated in even degrees,
then $A_{X}$ is the commutative bigraded component Hopf ring generated by
$x_{[n]}$ for $x\in\mathcal{B}$ with the following relations:

  * •

$x_{[n]}\cdot{x^{\prime}}_{[n]}=(x\cdot x^{\prime})_{[n]}$ for
$x,x^{\prime}\in H^{*}(X)$,

  * •

$\Delta(x_{[n]})=\sum_{i=0}^{n}x_{[i]}\otimes x_{[n-i]}$ for $x\in H^{*}(X)$,

  * •

$x_{[m]}\odot x_{[n]}=\binom{m+n}{n}x_{[m+n]}$ for $x\in H^{*}(X)$,

  * •

$(\lambda x)_{[n]}=\lambda^{n}x_{[n]}$ and
$(x+y)_{[n]}=\sum_{k=0}^{n}x_{[k]}\odot y_{[n-k]}$ for $x,y\in H^{*}(X)$ and
$\lambda\in\mathbb{F}$.

Moreover, the set $\mathcal{M}$ consisting of elements
$(x_{1})_{[n_{1}]}\odot\dots\odot(x_{k})_{[n_{k}]}$ up to permutation of
$\odot$-factors, where $x_{i}\not=x_{j}$ for $i\not=j$ and
$x_{i}\in\mathcal{B}$, form a bigraded basis for $A_{X}$.

###### Proof.

The proof that the relations hold and that $\mathcal{M}$ is a basis for the
commutative bigraded component Hopf ring with that presentation is the same as
in [14].

The rational cohomology of $D_{n}(X_{+})$ is isomorphic to the subspace of
invariants of ${H^{*}(X)}^{\otimes n}$ under the action of $\Sigma_{n}$. Under
this isomorphism, $(x_{1})_{[n_{1}]}\odot\dots\odot(x_{k})_{[n_{k}]}$
corresponds to the symmetrization by means of $(n_{1},\dots,n_{k})$-shuffles
of

$\underbrace{x_{1}\otimes\dots\otimes x_{1}}_{n_{1}\mbox{
times}}\otimes\underbrace{x_{2}\otimes\dots\otimes x_{2}}_{n_{2}\mbox{
times}}\otimes\dots\otimes x_{k}$

which constitute a basis for the subspace of invariants. ∎

We conclude this section by recalling some stable calculations. In the
following definitions, we fix a graded basis $\mathcal{B}$ for the cohomology
of $X$ containing the unit $1_{X}$.

######  Definition 2.9.

Let $\mathcal{M}$ be as in Corollary 2.6 or Proposition 2.8. An element
$x=b_{1}\odot\dots\odot b_{r}$ is pure if none of the $b_{i}$s is equal to the
unit class of a component $1_{n}\in H^{*}(D_{n}(X_{+}))$.

The stabilizer of $n+1$ in $\Sigma_{n+1}$ is isomorphic to $\Sigma_{n}$. This
provides an inclusion $i_{n}\colon\Sigma_{n}\hookrightarrow\Sigma_{n+1}$. If
we fix a basepoint $*\in X$, then the spaces $D(X_{+})$ exhibit homological
stability with respect to the stabilization maps

$\displaystyle j_{n}\colon(p,(x_{1},\dots,x_{n}))\in
E(\Sigma_{n})\times_{\Sigma_{n}}X^{n}=D_{n}(X_{+})$
$\displaystyle\mapsto(i_{n,1}(p),(x_{1},\dots,x_{n},*))\in
E(\Sigma_{n+1})\times_{\Sigma_{n+1}}X^{n+1}=D_{n+1}(X_{+}).$

$j_{n}$ depends on the basepoint, but if $X$ is path-connected, then any two
choices yield homotopic maps. This homologically stability property is
classically well-known and follows, for instance, from the calculations of [5,
Theorem 4.1]. Thus, under this connectedness hypothesis, the stable cohomology
of $D_{n}(X_{+})$ is given by
$A_{\infty}(X)=\varprojlim_{n}H^{*}(D_{n}(X_{+}))$ is well-defined and
coincides with $H^{*}(D_{n}(X_{+}))$ in low degrees.

The stabilization maps in cohomology have a section on the subspace generated
by pure Hopf monomials given by the transfer product with the units:

$1_{1}\odot\\_\colon H^{*}(D_{n}(X_{+}))\to H^{*}(D_{n+1}(X_{+}))$

Therefore, we can define the stabilization of a pure $x\in\mathcal{M}$ as the
unique class $x\odot 1_{\infty}$ restricting to $x\odot 1_{[n]}\in
H^{*}(D_{n+n(x)}(X_{+}))$ for all $n\in\mathbb{N}$.

######  Corollary 2.10 ([14, Lemma 6.5]).

If $\operatorname{char}(\mathbb{F})\not=2$, assume that $H^{*}(X)=0$ in odd
degrees. Then, the set $\\{x\odot 1_{\infty}:x\in\mathcal{M}\mbox{ pure}\\}$
is a graded basis for $A_{\infty}(X)$ as a $\mathbb{F}$-vector space.

######  Corollary 2.11 ([14, Theorem 6.8], particular case).

Assume that $H^{*}(X)$ is a polynomial algebra, with even-dimensional
generators if $\operatorname{char}(\mathbb{F})\not=2$. Let $\mathcal{B}$ be
the monomial basis of $H^{*}(X)$. If
$\operatorname{char}(\mathbb{F})=p\not=0$, then $A_{\infty}(X)$ is the free
graded commutative algebra generated by the classes $b\odot 1_{\infty}$, where
$b$ is a decorated gathered block satisfying the following conditions:

  * •

either the underlying (non-decorated) block of $b$ or the decoration of $b$ is
not a $p$-th power,

  * •

the width of $b$ is a power of $p$,

  * •

$b$ is different from the unit class.

If, instead, $\operatorname{char}(\mathbb{F})=0$, then $A_{\infty}(X)$ is a
polynomial algebra generated by classes $x_{[n]}\odot 1_{\infty}$ for all
$n\geq 1$ and $x$ ranging in a minimal set of algebra generators for
$H^{*}(X)$.

##  3\. Review of the cohomology of alternating subgroups of the
hyperoctahedral groups

The normalizer $N_{O(n)}(T(n))$ of the “torus” of diagonal matrices
$T(n)=\\{D\in O(n):D\mbox{ is diagonal}\\}$ is isomorphic to the wreath
product $\Sigma_{n}\wr(\mathbb{Z}/2\mathbb{Z})$. This can be realized as the
isometry group of a hyperoctahedron in $\mathbb{R}^{n}$. This makes
$N_{O(n)}(T(n))$ a reflection group that, under the classification of finite
Coxeter groups (see [16]), corresponds to the Dynkin diagram
$\operatorname{B}_{n}$ having the following form:

$s_{0}$$s_{1}$$s_{1}$$s_{n-1}$$4$$\dots$

The intersection $N_{O(n)}(T(n))\cap SO(n)$ is identified with the alternating
subgroup $\operatorname{B}_{n}^{+}$ of the Coxeter group
$\operatorname{B}_{n}$, defined as the kernel of the sign homomorphism
$\operatorname{sgn}_{\operatorname{B}_{n}}\colon\operatorname{B}_{n}\to
C_{2}=\\{-1,1\\}$ whose value on every reflection is $-1$.

###  3.1. Mod $2$ cohomology of $\operatorname{B}_{n}^{+}$

The mod $2$ cohomology of $\operatorname{B}_{n}$ has first been computed by
Guerra in [11], where the author shows that
$\bigoplus_{n}H^{*}(B\operatorname{B}_{n};\mathbb{F}_{2})$ is a Hopf ring.
Here we observe that this Hopf ring is isomorphic to
$A_{\mathbb{P}^{\infty}(\mathbb{R})}$ and that, consequently, his description
is a particular case of Theorem 2.3.

A similar structure exists on the mod $2$ cohomology of
$\operatorname{B}_{n}^{+}$. More precisely, on the direct sum
$A^{\prime}_{\operatorname{B}^{+}}=\bigoplus_{n,d\geq
0}H^{d}(B\operatorname{B}_{n}^{+};\mathbb{F}_{2})$ we consider structural
morphisms analogous to those defined in §2:

  * •

the usual cup product, component by component, $\cdot\colon
A^{\prime}_{\operatorname{B}^{+}}\otimes A^{\prime}_{\operatorname{B}^{+}}\to
A^{\prime}_{\operatorname{B}^{+}}$.

  * •

a coproduct $\Delta\colon A^{\prime}_{\operatorname{B}^{+}}\to
A^{\prime}_{\operatorname{B}^{+}}\otimes A^{\prime}_{\operatorname{B}^{+}}$
whose components are restriction maps associated with the inclusions of groups
$\operatorname{B}_{n}^{+}\times\operatorname{B}_{m}^{+}\to\operatorname{B}_{n+m}^{+}$.

  * •

a transfer product $\odot\colon A^{\prime}_{\operatorname{B}^{+}}\otimes
A^{\prime}_{\operatorname{B}^{+}}\to A^{\prime}_{\operatorname{B}^{+}}$ whose
components are transfer maps associated with the same inclusions.

######  Definition 3.1 (from [9]).

A commutative bigraded component almost-Hopf semiring is a bigraded vector
space $A=\bigoplus_{n,d}A^{n,d}$ with a coproduct $\Delta$, two products
$\odot,\cdot$, a unit $\eta\colon\mathbb{F}\to A$, and a counit
$\varepsilon\colon A\to\mathbb{F}$ satisfying all the properties of Definition
2.1, except those involving the antipode and $\Delta$ and $\odot$ forming a
bialgebra.

Theorem 2.4 of [9] implies that $A^{\prime}_{\operatorname{B}^{+}}$, with the
morphisms above, is an almost-Hopf ring over $\mathbb{F}_{2}$. There is an
involution $\iota\colon A^{\prime}_{\operatorname{B}^{+}}\to
A^{\prime}_{\operatorname{B}^{+}}$ induced on the $n$-th component by
conjugation by a reflection in $\operatorname{B}_{n}$. Using $\iota$,
$A^{\prime}_{\operatorname{B}^{+}}$ can be extended to a bigger almost-Hopf
semiring $\widetilde{A^{\prime}_{\operatorname{B}^{+}}}$ that coincide with
$A^{\prime}_{\operatorname{B}^{+}}$ in positive component, and such that
$\widehat{A^{\prime}_{\operatorname{B}^{+}}}^{0,*}=\mathbb{F}_{2}\\{1^{+},1^{-}\\}$
concentrated in dimension $0$. The unit
$1_{0}\in{A^{\prime}_{\operatorname{B}^{+}}}^{0,0}$ is identified with
$1^{+}+1^{-}$. $1^{+}$ is the unit for the transfer product and $1^{-}\odot
x=\iota(x)$ for all $x\in A^{\prime}_{\operatorname{B}^{+}}$. The cup product
is extended on the $0$-th component by letting $1^{+}\cdot 1^{+}=1^{+}$,
$1^{-}\cdot 1^{-}=1^{-}$ and $1^{+}\cdot 1^{-}=1^{-}\cdot 1^{+}=0$. The
coproduct in $\widetilde{A^{\prime}_{\operatorname{B}^{+}}}$ is defined by

$\Delta(x)=1^{+}\otimes x+1^{-}\otimes x+x\otimes 1^{+}+x\otimes
1^{-}+\overline{\Delta}(x),$

where $\overline{\Delta}$ is the reduced coproduct in
$A^{\prime}_{\operatorname{B}^{+}}$. Proposition 4.5 of another paper from
these two authors [12], that will appear in the near future, guarantees that
the almost-Hopf semiring structure on
$\widetilde{A^{\prime}_{\operatorname{B}^{+}}}$ is well-defined. Clearly
knowledge of $\widetilde{A^{\prime}_{\operatorname{B}^{+}}}$ implies knowledge
of $A^{\prime}_{\operatorname{B}^{+}}$.

Moreover, in that same paper, a full presentation of
$\widetilde{A^{\prime}_{\operatorname{B}^{+}}}$ is determined:

######  Theorem 3.2 ([12, Theorem 12.6]).

Let $Q$ be the quotient Hopf ring of $A_{\mathbb{P}^{\infty}(\mathbb{R})}$
obtained by putting $\gamma_{k,l}=0$ if $k\geq 2$. Consider the bigraded
component bialgebra

$A=\bigoplus_{n\geq 0}\frac{Q^{n,*}}{(w,\gamma_{1,1}+w\odot
1_{1},\dots,\gamma_{1,1}\odot 1_{n-2}+w\odot 1_{n-1},\dots)}.$

Let $A^{0}$ be the Hopf ring obtained by adjoining a unit to $A$ and letting
$\odot$ of elements of $A$ be $0$.

$A^{0}$ is identified as a sub-Hopf semiring of
$\widetilde{A^{\prime}_{\operatorname{B}^{+}}}$. Moreover, as an almost-Hopf
semiring over $A^{0}$, $\widetilde{A^{\prime}_{\operatorname{B}^{+}}}$ is
generated by the two classes
$1^{\pm}\in\widehat{A^{\prime}_{\operatorname{B}^{+}}}^{0,0}$ and a family of
classes
$\\{\gamma_{k,l}^{+}\in\widehat{A^{\prime}_{\operatorname{B}^{+}}}^{l2^{k},l(2^{k}-1)}\\}_{k\geq
2,l\geq 1}$.

A complete set of relations is the following, where we denote
$\gamma_{k,l}^{+}\odot 1^{-}$ as $\gamma_{k,l}^{-}$ and where we add the apex
$0$ to emphasize when elements belong to $A$:

  1. (1)

$1^{+}+1^{-}=1^{0}$, the unit of the $0$-th component of $A$,

  2. (2)

$1^{+}$ is the Hopf ring unit of
$\widetilde{A^{\prime}_{\operatorname{B}^{+}}}$,

  3. (3)

$1^{-}\odot 1^{-}=1^{+}$,

  4. (4)

$\gamma_{k,l}^{+}\odot\gamma_{k,m}^{+}=\left(\begin{array}[]{c}l+m\\\
l\end{array}\right)\gamma_{k,l+m}$ for all $k\geq 2$, $l,m\geq 1$,

  5. (5)

$\gamma_{k,l}^{+}\cdot\gamma_{k^{\prime},l^{\prime}}^{-}=0$ unless
$k=k^{\prime}=2$,

  6. (6)

$\gamma_{2,m}^{+}\cdot\gamma_{2,m}^{-}=\left\\{\begin{array}[]{ll}(\gamma_{2,m}^{+})^{2}+(\gamma_{2,m}^{-})^{2}+(\gamma_{2,m-1}^{+})^{2}\odot(\gamma_{1,2}^{3})^{0}&\mbox{if
}m\mbox{ is odd}\\\
(\gamma_{2,m-1}^{+})^{2}\odot(\gamma_{1,2}^{3})^{0}&\mbox{if }m\mbox{ is
even}\end{array}\right.$ for all $m\geq 1$,

  7. (7)

for all Hopf monomials $x=b_{1}\odot\dots\odot
b_{r}\in\mathcal{Q}_{\mathbb{P}^{\infty}(\mathbb{R}),1}$,

$\gamma_{k,l}^{+}\cdot
x^{0}=\bigodot_{i=1}^{r}\left(\gamma_{k,\frac{n(b_{i})}{2^{k}}}^{+}\cdot(b_{i})^{0}\right),$

where $n(b_{i})$ is the component of $b_{i}$, and $\gamma_{k,l}$ is assumed to
be $0$ if $l$ is not an integer,

  8. (8)

$\Delta(x\odot y)=\Delta(x)\odot\rho_{+}\Delta(y)$ for all
$x,y\in\widetilde{A^{\prime}_{\operatorname{B}^{+}}}$,

  9. (9)

$\Delta(\gamma_{k,l}^{+})=\sum_{i=0}^{l}\left(\gamma_{k,i}^{+}\otimes\gamma_{k,l-i}^{+}+\gamma_{k,i}^{-}\otimes\gamma_{k,m-i}^{-}\right)$
for all $k\geq 2$, $l\geq 0$, with the convention that
$\gamma_{k,0}^{\pm}=1^{\pm}$.

There is also an explicit additive basis for
$\widetilde{A^{\prime}_{\operatorname{B}^{+}}}$.

######  Definition 3.3 (from [12]).

Let $\mathcal{B}=\\{1,w,w^{2},\dots\\}$ be the monomial basis of
$H^{*}(\mathbb{P}^{\infty}(\mathbb{R});\mathbb{F}_{2})=\mathbb{F}_{2}[w]$. Let
$\mathcal{M}$ be the associated decorated Hopf monomial basis of
$A_{\mathbb{P}^{\infty}(\mathbb{R})}$. We define $\mathcal{G}_{ann}$ as the
set of those $x=b_{1}\odot\dots\odot b_{r}\in\mathcal{M}$ whose constituent
gathered blocks $b_{i}$ all contain at least one factor $\gamma_{k,l}$ with
$k\geq 2$.

We also define $\mathcal{G}_{quot}$ as the set of those
$x=b_{1}\odot\dots\odot b_{r}\in\mathcal{M}$ satisfying one of the following
two conditions:

  * •

at least one constituent gathered block $b_{i}$ of $x$ is of the form
$(1_{n},w^{k})$, and the one with $k$ maximal (necessarily unique) satisfies
$n\geq 2$

  * •

no $b_{i}$ is of the form $(1_{n},w^{k})$ and, among the constituent gathered
blocks of the form $(\gamma_{1,n}^{l},w^{k})$ (if there are any), the one with
the couple $(k,l)$ maximal with respect to the lexicographic order
$\leq_{lex}$ on $\mathbb{N}\times\mathbb{N}$ satisfies $n\geq 2$ or $l=1$

For a gathered block $b$ containing at least one factor $\gamma_{k,l}$ with
$k\geq 2$,
$b=(w^{a_{0}})_{[m2^{n}]}\cdot\prod_{i=1}^{n}\gamma_{i,2^{n-i}m}^{a_{i}}$ for
some $a_{i}\geq 0$, with $n\geq 2$ and $a_{n}\not=0$. We write

$b^{+}=((w_{[m2^{n}]})^{a_{0}}\gamma_{1,m2^{n-1}}^{a_{1}})^{0}\sum_{\begin{subarray}{c}0\leq
k_{0}\leq a_{0}\\\ \dots\\\ 0\leq k_{n-1}\leq a_{n-1}\\\ 0\leq
k_{n}\leq\lfloor\frac{a_{n}}{2}\rfloor\end{subarray}}^{a_{n-1}}\prod_{i=2}^{n}\left(\begin{array}[]{c}a_{i}\\\
k_{i}\end{array}\right)(\gamma_{i,m2^{n-i}}^{-})^{k_{i}}(\gamma_{i,m2^{n-i}}^{+})^{a_{i}-k_{i}}.$

For an element $x=b_{1}\odot\dots\odot b_{r}\in\mathcal{G}_{ann}$, we let
$x^{+}=\bigodot_{i=1}^{r}b_{i}^{+}$. We also let $x^{-}=\iota(x^{+})$.

For an element $x\in\mathcal{G}_{quot}\setminus\mathcal{G}_{ann}$, we let
$x^{0}=\operatorname{res}(x)$, where $\operatorname{res}\colon
A_{\mathbb{P}^{\infty}(\mathbb{R})}\to A^{\prime}_{\operatorname{B}^{+}}$ is
the direct sum of the cohomological restriction maps
$\bigoplus_{n}\operatorname{res}^{\operatorname{B}_{n}}_{\operatorname{B}_{n}^{+}}$.

######  Theorem 3.4 ( Theorem 3.5 of [12]).

With reference to Definition 3.3, let

  * •

$\mathcal{M}_{0}=\\{x^{0}\\}_{x\in\mathcal{G}_{quot}\setminus\mathcal{G}_{ann}}$,

  * •

$\mathcal{M}_{+}=\\{x^{+}\\}_{x\in\mathcal{G}_{ann}}$,

  * •

and $\mathcal{M}_{-}=\\{x^{-}\\}_{x\in\mathcal{G}_{ann}}$.

The set
$\mathcal{M}_{charged}=\mathcal{M}_{0}\sqcup\mathcal{M}_{+}\sqcup\mathcal{M}_{-}$
is a bigraded basis of $\widetilde{A^{\prime}_{\operatorname{B}^{+}}}$ over
$\mathbb{F}_{2}$.

Moreover, $\operatorname{res}(x)=x^{+}+x^{-}$ for all $x\in\mathcal{G}_{ann}$.

Although the description above is more complicated than its analogs Theorem
2.3 and Corollary 2.6, we can still deduce the ring structure on the
components $H^{*}(B\operatorname{B}_{n}^{+};\mathbb{F}_{2})$. Since the
component of all elements of $\mathcal{G}_{ann}$ is a multiple of $4$, then
all classes belonging to the other components are restrictions of classes in
$A_{\mathbb{P}^{\infty}(\mathbb{R})}$. More precisely, we have the following
result.

######  Corollary 3.5 (of Theorem 3.4).

If $n\not\equiv 0\mod 4$, then the restriction map is surjective and induces
an isomorphism

$\frac{H^{*}(B\operatorname{B}_{n};\mathbb{F}_{2})}{(\gamma_{1,1}\odot
1_{n-2}+w\odot 1_{n-1})}\cong
H^{*}(B\operatorname{B}_{n}^{+};\mathbb{F}_{2}).$

In components that are multiples of $4$, the calculation of the cup product is
exemplified below.

######  Example 3.6.

We determine the cup product of the two charged Hopf monomials
$x=(\gamma_{2,2}^{3}\odot\gamma_{3,1}^{5}w_{[8]})^{+}$ and
$y=(\gamma_{2,1}\gamma_{1,2}\odot\gamma_{2,1}\odot\gamma_{3,1})^{+}$ in
$A^{\prime}_{\operatorname{B}^{+}}$. The first step is to compute the
coproduct of the charged gathered blocks appearing in $x$. We use the fact
that $\cdot$ and $\Delta$ form a bialgebra, the fact that cup products of
classes in different components are zero, and the relations (5), (6), (7) and
(8) of Theorem 3.2 to perform the computation. Explicitly

$\displaystyle\Delta((\gamma_{2,2}^{3})^{+})=\Delta((\gamma_{2,2}^{+})^{3}+(\gamma_{2,2}^{+})^{2}\cdot\gamma_{2,2}^{-})=\Delta((\gamma_{2,2}^{+}))^{3}+\Delta(\gamma_{2,2}^{+})\cdot\Delta(\gamma_{2,2}^{-})^{2}$
$\displaystyle=\Big{(}1^{+}\otimes(\gamma_{2,2}^{+})^{3}+1^{-}\otimes(\gamma_{2,2}^{+})^{3}+(\gamma_{2,1}^{+})^{3}\otimes(\gamma_{2,1}^{+})^{3}+(\gamma_{2,1}^{+})^{2}\gamma_{2,1}^{-}\otimes(\gamma_{2,1}^{+})^{2}\gamma_{2,1}^{-}$
$\displaystyle+\gamma_{2,1}^{+}(\gamma_{2,1}^{-})^{2}\otimes\gamma_{2,1}^{+}(\gamma_{2,1}^{-})^{2}+1^{+}(\gamma_{2,2}^{+})^{3}\otimes
1^{+}+(\gamma_{2,2}^{-})^{3}\otimes
1^{-}\Big{)}+\Big{(}1^{+}\otimes(\gamma_{2,2}^{+})^{2}\gamma_{2,2}^{-}$
$\displaystyle+1^{-}\otimes\gamma_{2,2}^{+}(\gamma_{2,2}^{-})^{2}+(\gamma_{2,1}^{+})^{3}\otimes(\gamma_{2,1}^{+})^{2}\gamma_{2,1}^{-}+(\gamma_{2,1}^{+})^{2}\gamma_{2,1}^{-}\otimes(\gamma_{2,1}^{+})^{3}$
$\displaystyle+\gamma_{2,1}^{+}(\gamma_{2,1}^{-})^{2}\otimes(\gamma_{2,1}^{-})^{3}+(\gamma_{2,1}^{-})^{3}\otimes\gamma_{2,1}^{+}(\gamma_{2,1}^{-})^{2}+(\gamma_{2,2}^{+})^{2}\gamma_{2,2}^{-}\otimes
1^{+}+\gamma_{2,2}^{+}(\gamma_{2,2}^{-})^{2}\otimes 1^{-}\Big{)}$
$\displaystyle=1^{+}\otimes(\gamma_{2,2}^{3})^{+}+1^{-}\otimes(\gamma_{2,2}^{3})^{+}+(\gamma_{2,2}^{3})^{+}\otimes
1^{+}+(\gamma_{2,2}^{3})^{-}\otimes
1^{-}+(\gamma_{2,1}^{3})^{+}\otimes(\gamma_{2,1}^{3})^{+}$
$\displaystyle+(\gamma_{2,1}^{3})^{-}\otimes(\gamma_{2,1}^{3})^{-}+(\gamma_{2,1}^{3})^{+}\otimes\gamma_{2,1}^{+}(\gamma_{2,1}^{-})^{2}+\gamma_{2,1}^{+}(\gamma_{2,1}^{-})^{2}\otimes(\gamma_{2,1}^{3})^{+}$
$\displaystyle+(\gamma_{2,1}^{3})^{-}\otimes(\gamma_{2,1}^{+})^{2}\gamma_{2,1}^{-}+(\gamma_{2,1}^{+})^{2}\gamma_{2,1}^{-}\otimes(\gamma_{2,1}^{-})^{3}.$

By relation (6) of Theorem 3.2,

$(\gamma_{2,1}^{+})^{2}\gamma_{2,1}^{-}=\left((\gamma_{2,1}^{-})^{2}+(\gamma_{1,2}^{3})^{0}+\gamma_{1,2}^{+}\gamma_{1,2}^{-}\right)\gamma_{2,1}^{-}=(\gamma_{2,1}^{3})^{-}+(\gamma_{2,1}\gamma_{1,2}^{3})^{-}$

and similarly for $\gamma_{2,1}^{+}(\gamma_{2,1}^{-})^{2}$. Substituting in
the expression for $\Delta((\gamma_{2,1}^{3})^{+})$ we obtain that

$\displaystyle\Delta((\gamma_{2,1}^{3})^{+})=1^{+}\otimes(\gamma_{2,2}^{3})^{+}+1^{-}\otimes(\gamma_{2,2}^{3})^{+}+(\gamma_{2,2}^{3})^{+}\otimes
1^{+}+(\gamma_{2,2}^{3})^{-}\otimes 1^{-}$
$\displaystyle+(\gamma_{2,1}^{3})^{+}\otimes(\gamma_{2,1}^{3})^{+}+(\gamma_{2,1}^{3})^{-}\otimes\gamma_{2,1}^{3})^{-}+(\gamma_{2,1}^{3})^{+}\otimes(\gamma_{2,1}\gamma_{1,2}^{3})^{-}$
$\displaystyle+(\gamma_{2,1}^{3})^{-}\otimes(\gamma_{2,1}\gamma_{1,2}^{3})^{+}+(\gamma_{2,1}\gamma_{1,2}^{3})^{+}\otimes(\gamma_{1,2}^{3})^{-}+(\gamma_{2,1}\gamma_{1,2}^{3})^{-}\otimes(\gamma_{2,1}^{3})^{+}.$

The reader is encouraged to read Remark 5.6 of [9] for the explanation of some
computational difficulties. A similar calculation shows that

$\displaystyle\Delta((\gamma_{3,1}^{5}w_{[8]})^{+})$
$\displaystyle=1^{+}\otimes(\gamma_{3,1}^{5}w_{[8]})^{+}+1^{-}\otimes(\gamma_{3,1}^{5}w_{[8]})^{-}$
$\displaystyle+(\gamma_{3,1}^{5}w_{[8]})^{+}\otimes
1^{+}+(\gamma_{3,1}^{5}w_{[8]})^{-}\otimes 1^{-}.$

As a second step, we exploit relations (3), (4) and (8) of Theorem 3.2 to
determine the coproduct of $x$:

$\displaystyle\Delta(x)=(\gamma_{3,1}^{5}w_{[8]})^{+}\otimes(\gamma_{2,2}^{3})^{+}+(\gamma_{3,1}^{5}w_{[8]})^{-}\otimes(\gamma_{2,2}^{3})^{+}+(\gamma_{2,2}^{3}\odot\gamma_{3,1}^{5}w_{[8]})^{+}\otimes
1^{+}$
$\displaystyle+(\gamma_{2,2}^{3}\odot\gamma_{3,1}^{5}w_{[8]})^{-}\otimes
1^{-}+(\gamma_{2,1}^{3}\odot\gamma_{3,1}^{5}w_{[8]})^{+}\otimes(\gamma_{2,1}^{3}\odot\gamma_{3,1}^{5}w_{[8]})^{+}$
$\displaystyle+(\gamma_{2,1}^{3}\odot\gamma_{3,1}^{5}w_{[8]})^{-}\otimes(\gamma_{2,1}^{3})^{-}+(\gamma_{2,1}^{3}\odot\gamma_{3,1}^{5}w_{[8]})^{+}\otimes(\gamma_{2,1}\gamma_{1,2}^{3})^{-}$
$\displaystyle+(\gamma_{2,1}^{3}\odot\gamma_{3,1}^{5}w_{[8]})^{-}\otimes(\gamma_{2,1}\gamma_{1,2}^{3})^{+}+(\gamma_{2,1}\gamma_{1,2}^{3}\odot\gamma_{3,1}^{5}w_{[8]})^{+}\otimes(\gamma_{1,2}^{3})^{-}$
$\displaystyle+(\gamma_{2,1}\gamma_{1,2}^{3}\odot\gamma_{3,1}^{5}w_{[8]})^{-}\otimes(\gamma_{2,1}^{3})^{+}+1^{+}\otimes(\gamma_{2,2}^{3}\odot\gamma_{3,1}^{5}w_{[8]})^{+}+1^{-}\otimes(\gamma_{2,2}^{3}\odot\gamma_{3,1}^{5}w_{[8]})^{+}$
$\displaystyle+(\gamma_{2,2}^{3})^{+}\otimes(\gamma_{3,1}^{5}w_{[8]})^{+}+(\gamma_{2,2}^{3})^{-}\otimes(\gamma_{3,1}^{5}w_{[8]})^{-}+(\gamma_{2,1}^{3})^{+}\otimes(\gamma_{2,1}^{3}\odot\gamma_{3,1}^{5}w_{[8]})^{+}$
$\displaystyle+(\gamma_{2,1}^{3})^{-}\otimes(\gamma_{2,1}^{3}\odot\gamma_{3,1}^{5}w_{[8]})^{-}+(\gamma_{2,1}^{3})^{+}\otimes(\gamma_{2,1}\gamma_{1,2}^{3}\odot\gamma_{3,1}^{5}w_{[8]})^{-}$
$\displaystyle+(\gamma_{2,1}^{3})^{-}\otimes(\gamma_{2,1}\gamma_{1,2}^{3}\odot\gamma_{3,1}^{5}w_{[8]})^{+}+(\gamma_{2,1}\gamma_{1,2}^{3})^{+}\otimes(\gamma_{1,2}^{3}\odot\gamma_{3,1}^{5}w_{[8]})^{-}$
$\displaystyle+(\gamma_{2,1}\gamma_{1,2}^{3})^{-}\otimes(\gamma_{2,1}^{3}\odot\gamma_{3,1}^{5}w_{[8]})^{+}$

Thirdly, as
$y=(\gamma_{2,1}\gamma_{1,2})^{+}\odot(\gamma_{2,1}\odot\gamma_{3,1})^{+}$, we
can compute $x\cdot y$ from the previous coproduct calculations and Hopf ring
distributivity

$x\cdot
y=x\cdot\left((\gamma_{2,1}\gamma_{1,2})^{+}\odot(\gamma_{2,1}\odot\gamma_{3,1})^{+}\right)=\sum\left(x_{(1)}\cdot(\gamma_{2,1}\gamma_{1,2})^{+}\right)\odot\left(x_{(2)}\cdot(\gamma_{2,1}\odot\gamma_{3,1})^{+}\right).$

One observes that the component of $(\gamma_{2,1}w_{2})^{+}$ is $4$ and the
component of $(\gamma_{2,1}\odot\gamma_{3,1})^{+}$ is $12$. The addends in
$\Delta(x)$ in the correct components are
$(\gamma_{2,1}^{3})^{+}\otimes(\gamma_{2,1}^{3}\odot\gamma_{3,1}^{5}w_{[8]})^{+}$,
$(\gamma_{2,1}^{3})^{-}\otimes(\gamma_{2,1}^{3}\odot\gamma_{3,1}^{5}w_{[8]})^{-}$,
$(\gamma_{2,1}^{3})^{+}\otimes(\gamma_{2,1}\gamma_{1,2}^{3}\odot\gamma_{3,1}^{5}w_{[8]})^{-}$,
$(\gamma_{2,1}^{3})^{-}\otimes(\gamma_{2,1}\gamma_{1,2}^{3}\odot\gamma_{3,1}^{5}w_{[8]})^{+}$,
$(\gamma_{2,1}\gamma_{1,2}^{3})^{+}\otimes(\gamma_{1,2}^{3}\odot\gamma_{3,1}^{5}w_{[8]})^{-}$
and
$(\gamma_{2,1}\gamma_{1,2}^{3})^{-}\otimes(\gamma_{2,1}^{3}\odot\gamma_{3,1}^{5}w_{[8]})^{+}$.
For all these terms, $x_{(2)}$ is the transfer product of two primitive
elements. Therefore, the cup product
$x_{(2)}\cdot(\gamma_{2,1}\odot\gamma_{3,1})^{+}$ is easily determined by
applying Hopf ring distributivity again.

$\displaystyle x\cdot y$
$\displaystyle=(\gamma_{2,1}^{3})^{+}(\gamma_{2,1}\gamma_{1,2})^{+}\odot(\gamma_{2,1}^{3})^{+}\gamma_{2,1}^{+}\odot(\gamma_{3,1}^{2}w_{[8]})^{+}+(\gamma_{2,1}^{3})^{-}(\gamma_{2,1}\gamma_{1,2})^{+}$
$\displaystyle\odot(\gamma_{2,1}^{3})^{-}\gamma_{2,1}^{+}\odot(\gamma_{3,1}^{2}w_{[8]})^{+}+(\gamma_{2,1}^{3})^{+}(\gamma_{2,1}\gamma_{1,2})^{+}\odot(\gamma_{2,1}\gamma_{1,2}^{3})^{-}\gamma_{2,1}^{+}\odot(\gamma_{3,1}^{2}w_{[8]})^{+}$
$\displaystyle+(\gamma_{2,1}^{3})^{-}(\gamma_{2,1}\gamma_{1,2})^{+}\odot(\gamma_{2,1}\gamma_{1,2}^{3})^{+}\gamma_{2,1}^{+}\odot(\gamma_{3,1}^{2}w_{[8]})^{+}+(\gamma_{2,1}\gamma_{1,2}^{3})^{-}(\gamma_{2,1}\gamma_{1,2})^{+}$
$\displaystyle\odot(\gamma_{2,1}^{3})^{+}\gamma_{2,1}^{+}\odot(\gamma_{3,1}^{2}w_{[8]})^{+}+(\gamma_{2,1}\gamma_{1,2}^{3})^{+}(\gamma_{2,1}\gamma_{1,2})^{+}\odot(\gamma_{2,1}^{3})^{-}\gamma_{2,1}^{+}\odot(\gamma_{3,1}^{2}w_{[8]})^{+}.$

Finally, in order to write all the addends of $x\cdot y$ as linear
combinations of charged decorated Hopf monomials, we use our cup product
relations again. Explicitly, we observe that

$\displaystyle\gamma_{2,1}^{+}\cdot\gamma_{2,1}^{+}=(\gamma_{2,1}^{2})^{+},(\gamma_{2,1}^{3})^{+}\cdot\gamma_{2,1}^{+}=(\gamma_{2,1}^{4})^{+}+(\gamma_{2,1}^{4})^{-}+(\gamma_{2,1}^{2}\gamma_{1,3}^{3})^{+}+(\gamma_{1,2}^{6})^{0}$
and
$\displaystyle(\gamma_{2,1}^{3})^{-}\cdot\gamma_{2,1}^{+}=(\gamma_{2,1}^{4})^{-}+(\gamma_{2,1}^{2}\gamma_{1,2}^{3})^{-}.$

Substituting in the espression above for $x\cdot y$, many terms cancel out and
we obtain that $x\cdot
y=(\gamma_{2,1}^{4}\gamma_{1,2}\odot\gamma_{2,1}^{4}\odot\gamma_{3,1}^{2}w_{[8]})^{+}$.

###  3.2. Cohomology of $\operatorname{B}_{n}^{+}$ at odd primes

The cohomology of $\operatorname{B}_{n}^{+}$ modulo odd primes is much simpler
than its mod $2$ counterpart.

######  Theorem 3.7.

Let $n\in\mathbb{N}$ and let $p>2$ be a prime number. Then the projection
$\operatorname{B}_{n}^{+}\to\Sigma_{n}$ induces an isomorphism in mod $p$
cohomology. In particular,
$\bigoplus_{n}H^{*}(\operatorname{B}_{n}^{+};\mathbb{F}_{p})\cong A_{\\{*\\}}$
as Hopf rings.

###### Proof.

By Shapiro’s lemma $H^{*}(\operatorname{B}_{n}^{+};\mathbb{F}_{p})\cong
H^{*}(\operatorname{B}_{n};\operatorname{Ind}_{\operatorname{B}_{n}^{+}}^{\operatorname{B}_{n}}(\mathbb{F}_{p}))$,
where
$\operatorname{Ind}_{\operatorname{B}_{n}^{+}}^{\operatorname{B}_{n}}(\mathbb{F}_{p})$
is the induced representation of the constant representation $\mathbb{F}_{p}$.
$\operatorname{Ind}_{\operatorname{B}_{n}^{+}}^{\operatorname{B}_{n}}(\mathbb{F}_{p})\cong\mathbb{F}_{p}\oplus\operatorname{sgn}$,
where $\operatorname{sgn}$ is the mod $p$ sign representation of
$\operatorname{B}_{n}$, defined by $x.1=(-1)^{l(x)}$, where $l(x)$ is the
Coxeter length function.

By the Leray spectral sequence for the fibration
$B(\mathbb{Z}/2\mathbb{Z})^{n}\to B(\operatorname{B}_{n})\to B(\Sigma_{n})$,
in the formulation of Lyndon–Hochschild–Serre, there are spectral sequences

$\displaystyle E_{2}^{*,*}$
$\displaystyle=H^{*}(\Sigma_{n};{H^{*}(\mathbb{Z}/2\mathbb{Z};\mathbb{F}_{p})}^{\otimes
n})\Rightarrow H^{*}(\operatorname{B}_{n};\mathbb{F}_{p})$ and $\displaystyle
E_{2}^{*,*}$
$\displaystyle=H^{*}(\Sigma_{n};{H^{*}(\mathbb{Z}/2\mathbb{Z};\operatorname{sgn})}^{\otimes
n}\otimes\operatorname{sgn})\Rightarrow
H^{*}(\operatorname{B}_{n};\operatorname{sgn}).$

In the expression above, $\Sigma_{n}$ acts on
${H^{*}(\mathbb{Z}/2\mathbb{Z};\mathbb{F}_{p})}^{\otimes n}$ and
${H^{*}(\mathbb{Z}/2\mathbb{Z};\operatorname{sgn}}^{\otimes n})$ by permuting
the factors.

By a standard group-cohomological application of transfer maps, since the
order of $\mathbb{Z}/2\mathbb{Z}$ is coprime with $p$,
$H^{*}(\mathbb{Z}/2\mathbb{Z};\mathbb{F}_{p})\cong\mathbb{F}_{p}$ concentrated
in degree $0$ and $H^{*}(\mathbb{Z}/2\mathbb{Z};\operatorname{sgn})=0$.
Plugging these in the Lyndon–Hochschild–Serre spectral sequences above, we see
that the $E_{2}$ page of one of them is concentrated in the first row, while
the other is entirely zero. Therefore, they both collapse at the second page
and provide the desired isomorphism. ∎

We also record the following simple remark.

######  Corollary 3.8.

Let $p$ be an odd prime. Then $\operatorname{res}_{n}\colon
H^{*}(\operatorname{B}_{n};\mathbb{F}_{p})\to
H^{*}(\operatorname{B}_{n}^{+};\mathbb{F}_{p})$ is an isomorphism.

###### Proof.

The same Lyndon–Hochschild–Serre spectral sequence used in the proof of
Theorem 3.7 shows that $H^{*}(\operatorname{B}_{n};\mathbb{F}_{p})\cong
H^{*}(\Sigma_{n};\mathbb{F}_{p})$ via the projection map
$\operatorname{B}_{n}\to\Sigma_{n}$. ∎

##  4\. Stable cohomology of complete unordered flag varieties

In this section, we study the stable behavior of the cohomology of the
unordered flag manifolds $\\{\overline{\mathrm{Fl}}_{n}(\mathbb{C})\\}_{n}$
and $\\{\overline{\mathrm{Fl}}_{n}(\mathbb{R})\\}_{n}$. First, we recall the
following classical theorem about Borel spectral sequence, i.e. Serre spectral
sequence of fiber bundles $X\rightarrow X_{hG}\rightarrow BG$ associated with
the Borel construction (or homotopy quotient) $X_{hG}$. From now on we will be
using this theorem in many of our proofs and computations without explicitly
referring to it every time.

######  Theorem 4.1 ([18]).

Let $X\rightarrow X_{hG}\rightarrow BG$ be the Borel fibration associated to a
$G$-space $X$. There is a first quadrant spectral sequence of algebras
$E_{r}^{*,*},d^{r}\\}$, converging to $H^{*}(X_{hG};\mathbb{F}_{2})$ as an
algebra with

$E_{2}^{k,l}\cong H^{k}(BG;H^{l}(X)),$

where $H^{l}(X)$ is understood as a $G$-representation via the monodromy
action of $\pi_{1}(BG)$ onto the fiber. If $G$ acts trivially on the
cohomology $H^{*}(X)$ and field coefficients are used, then by Künneth’s
theorem

$E_{2}^{k,l}\cong H^{k}(BG)\otimes H^{l}(X).$

If $G$ acts freely on $X$, then $X_{hG}$ is known to be homotopy equivalent to
the strict quotient $X/G$.

######  Proposition 4.2.

For all $n\geq 1$, we have isomorphisms $BN(n)\cong D_{n}(BU(1)_{+})$ and
$B\operatorname{B}(n)\cong D_{n}(BO(1)_{+})$.

###### Proof.

We provide details for the complex case. The real case is similar. It is
enough to check that $D_{n}(BU(1))$ is the base space of a fibration with
fiber $N(n)$ and a contractible total space. Note that

$D_{n}(BU(1)_{+})=E\Sigma_{n}\times_{\Sigma_{n}}BU(1)^{n}\cong\mathrm{Conf}_{n}(\mathbb{R}^{\infty})\times_{\Sigma_{n}}(\mathbb{P}^{\infty}(\mathbb{C}))^{n}$

and consider the contractible space
$E=\mathrm{Conf}_{n}(\mathbb{R}^{\infty})\times(S^{\infty})^{n}$. There is an
action of $N(n)=\Sigma_{n}\ltimes U(1)^{n}$ on $E$ defined as follows:

  * •

$\Sigma_{n}$ acts diagonally on
$\mathrm{Conf}_{n}(\mathbb{R}^{\infty})\times(S^{\infty})^{n}$.

  * •

$U(1)^{n}$ acts on $(S^{\infty})^{n}$ factor by factor by identifying
$U(1)\cong S^{1}$.

This gives a well-defined action and a principal $N(n)$-bundle $E\rightarrow
D_{n}(BU(1))$. ∎

There are inclusions $U(n)\hookrightarrow U(n+1)$ and $O(n)\rightarrow O(n+1)$
given by

$M\longmapsto\begin{bmatrix}M&0\\\ 0&1\end{bmatrix}$

which restrict to inclusions $N(n)\hookrightarrow N(n+1)$ and
$\operatorname{B}(n)\hookrightarrow\operatorname{B}(n+1)$. Through the
isomorphism of Proposition 4.2, these inclusions correspond to the
stabilization maps in $D_{n}(BU(1)_{+})$ and $D_{n}(BO(1)_{+})$.

######  Corollary 4.3.

The sequence of spaces $\\{BN(n)\\}_{n}$ and $\\{B\operatorname{B}(n)\\}_{n}$
exhibit homological stability and if $N(\infty)=\varinjlim N(n)$ and
$\operatorname{B}(\infty)=\varinjlim\operatorname{B}(n)$, then
$H^{*}(BN(\infty);\mathbb{F}_{p})\cong A_{\infty}(BU(1))$ and
$H^{*}(B\operatorname{B}(\infty);\mathbb{F}_{p})\cong A_{\infty}(BU(1))$.

The above morphisms also determine quotient maps

$\displaystyle
i_{n}^{\mathbb{C}}\colon\frac{U(n)}{N(n)}\longrightarrow\frac{U(n+1)}{N(n+1)}$
$\displaystyle
i_{n}^{\mathbb{R}}\colon\frac{O(n)}{\operatorname{B}(n)}\longrightarrow\frac{O(n+1)}{\operatorname{B}(n+1)}$

and in cohomology

$\displaystyle(i_{n}^{\mathbb{C}})^{*}\colon
H^{*}\Big{(}\frac{U(n+1)}{N(n+1)};\mathbb{F}_{p}\Big{)}\longrightarrow
H^{*}\Big{(}\frac{U(n)}{N(n)};\mathbb{F}_{p}\Big{)}$
$\displaystyle(i_{n}^{\mathbb{R}})^{*}\colon
H^{*}\Big{(}\frac{O(n+1)}{\operatorname{B}(n+1)};\mathbb{F}_{p}\Big{)}\longrightarrow
H^{*}\Big{(}\frac{O(n)}{\operatorname{B}(n)};\mathbb{F}_{p}\Big{)}.$

######  Proposition 4.4.

The sequence of spaces $\\{U(n)/N(n)\\}_{n}$ and
$\\{O(n)/\operatorname{B}(n)\\}_{n}$ exhibit homological stability, i.e. for
all $d\geq 0$ there exists $N(d)$ such that for all $n\geq N(d)$

$\displaystyle(i_{n}^{\mathbb{C}})^{*}\colon
H^{d}\Big{(}\frac{U(n+1)}{N(n+1)};\mathbb{F}_{p}\Big{)}$
$\displaystyle\longrightarrow
H^{d}\Big{(}\frac{U(n)}{N(n)};\mathbb{F}_{p}\Big{)}$
$\displaystyle(i_{n}^{\mathbb{R}})^{*}\colon
H^{d}\Big{(}\frac{O(n+1)}{\operatorname{B}(n+1)};\mathbb{F}_{p}\Big{)}$
$\displaystyle\longrightarrow
H^{d}\Big{(}\frac{O(n)}{\operatorname{B}(n)};\mathbb{F}_{p}\Big{)}$

are isomorphisms.

###### Proof.

From the previous corollary $\\{BN(n)\\}_{n}$ exhibits homological stability.
Also, it is known classically that $\\{U(n)\\}_{n}$ exhibits homological
stability. Consider the Serre spectral sequence associated with the fiber
sequence

(3) $U(n)\longrightarrow\frac{U(n)}{N(n)}\longrightarrow BN(n).$

Since homological stability holds for both fiber and base it must hold at the
level of $E_{2}$-page. Since the spectral sequence converges, it must hold at
the $E_{\infty}$-page. The proof for $\\{O(n)/\operatorname{B}(n)\\}$ is
similar. ∎

Proposition 4.4 allows us to discuss the stable cohomology of $U(n)/N(n)$ and
$O(n)/\operatorname{B}(n)$. Let us denote the limit of
$\overline{\mathrm{Fl}}_{n}(\mathbb{C})$ and
$\overline{\mathrm{Fl}}_{n}(\mathbb{R})$ by

$\overline{\mathrm{Fl}}_{\infty}(\mathbb{C})=\varinjlim_{n}\overline{\mathrm{Fl}}_{n}(\mathbb{C})\text{
and
}\overline{\mathrm{Fl}}_{\infty}(\mathbb{R})=\varinjlim_{n}\overline{\mathrm{Fl}}_{n}(\mathbb{R})$

respectively. Then we can write

$\displaystyle
H^{*}(\overline{\mathrm{Fl}}_{\infty}(\mathbb{C});\mathbb{F}_{p})$
$\displaystyle=\varprojlim_{n}H^{*}(\overline{\mathrm{Fl}}_{n}(\mathbb{C});\mathbb{F}_{p})$
$\displaystyle
H^{*}(\overline{\mathrm{Fl}}_{\infty}(\mathbb{R});\mathbb{F}_{p})$
$\displaystyle=\varprojlim_{n}H^{*}(\overline{\mathrm{Fl}}_{n}(\mathbb{R});\mathbb{F}_{p}).$

Next, we will define a rank function on the basis of decorated Hopf monomials.
This rank function establishes a filtration on $A_{X}$, which holds
significant importance in our examination of the pullback of the Chern classes
under $f_{n}$.

######  Definition 4.5.

Let $X$ be a topological space, whose cohomology is concentrated in even
degrees if $p>2$. The rank of a decorated Hopf monomial in $A_{X}$ denoted as
$\operatorname{rk}$ is defined via the following rules:

  * •

The rank of a decorated gathered block $(b,a)$ is $0$ if $b$ contains at least
one of $\gamma_{k,l}$ (respectively $\alpha_{j,k}$, $\beta_{i,j,l}$, or
$\gamma_{k,l}$).

  * •

The rank of a decorated gathered block of the form $(1_{n},a)$ is $n|a|$.

  * •

$\operatorname{rk}(b_{1}\odot\cdots\odot
b_{r})=\sum_{i=1}^{r}\operatorname{rk}(b_{i})$.

We also define $\operatorname{rk}(x\otimes
y):=\operatorname{rk}(x)+\operatorname{rk}(y)$ in $A_{X}\otimes A_{X}$.

######  Definition 4.6.

We define the rank filtration $\mathcal{F}_{*}$ for $A_{X}$ by setting
$\mathcal{F}_{n}$ as the linear span of decorated Hopf monomials $x$ with
$\operatorname{rk}(x)\leq n$. Moreover, we define a rank filtration on
$A_{\infty}(X)$, by defining $\mathcal{F}_{n}$ as the linear span of
stabilized Hopf monomials $x\odot 1_{\infty}$ where $\operatorname{rk}(x)\leq
n$. With a slight abuse of notation, we also denote it as $\mathcal{F}_{*}$.

The rank filtration is exhaustive, increasing, and bounded from below.
Moreover, due to the Hopf ring distributivity law, it is multiplicative with
respect to the cup product, i.e.
$\mathcal{F}_{m}\cdot\mathcal{F}_{n}\subseteq\mathcal{F}_{m+n}$.

###  4.1. The complex case

The inclusions $N(n)\hookrightarrow U(n)$ induce maps $f_{n}\colon
BN(n)\rightarrow BU(n)$. We denote the limiting map by $f\colon
BN(\infty)\rightarrow BU(\infty)$. So,

$f^{*}\colon
H^{*}(BU(\infty);\mathbb{F}_{p})\cong\mathbb{F}_{p}[c_{1},c_{2},\dots]\longrightarrow
H^{*}(BN(\infty);\mathbb{F}_{p})\cong A_{\infty}(BU(1)).$

Let us recall some classical results from [19]. The vector bundle
corresponding to $\varphi\in[X,BU(n)]$ is given by the pullback
$\varphi^{*}(\eta_{n})$ of the universal vector bundle $\eta_{n}$ over
$BU(n)$. Taking $X=BU(k)\times BU(n-k)$, the bundle $\eta_{k}\oplus\eta_{n-k}$
is classified by the homotopy class of a map $\delta_{k,n-k}\colon BU(k)\times
BU(n-k)\rightarrow BU(n)$, induced by the direct sum of matrices. The induced
map in cohomology

$\Delta_{k,n-k}\colon H^{*}(BU(n))\longrightarrow H^{*}(BU(k))\otimes
H^{*}(B(n-k)).$

is associative and commutative up to isomorphism, and the direct sum of these
provides an associative and commutative coproduct

$\Delta\colon\bigoplus_{n\geq
0}H^{*}(BU(n))\longrightarrow\Big{(}\bigoplus_{n\geq
0}H^{*}(BU(n))\Big{)}\otimes\Big{(}\bigoplus_{n\geq 0}H^{*}(BU(n))\Big{)}.$

We can regard the morphism $\Delta$ as a coproduct and note that it commutes
with the cup product $\cdot$ as it is induced by homotopy classes of
topological maps. This makes $\bigoplus_{n\geq 0}H^{*}(BU(n))$ a bialgebra.
The stabilization maps of $BU(n)$ preserve $\Delta$, hence the limit
$H^{*}(BU(\infty))$ is also a bialgebra. In fact,
$\bigoplus_{n}f_{n}^{*}\colon\bigoplus_{n}H^{*}(BU(n))\to A_{BU(1)}$ and
$f^{*}\colon H^{*}(BU(\infty))\to{A}_{\infty}(BU(1))$ are bialgebra morphisms.

Chern classes of vector bundles over $X$ correspond to the pullbacks of the
universal Chern classes along its classifying map. Since,
$\delta_{k,n-k}^{*}(\eta_{n})$ is isomorphic to $\eta_{k}\oplus\eta_{n-k}$, we
have $\Delta(c_{k})=\sum_{i=0}^{k}c_{i}\otimes c_{k-i}$.

######  Lemma 4.7.

For $k\leq n$ let $c_{k}\in H^{2k}(BU(n);\mathbb{F}_{p})$ be the $k$-th
universal Chern class. Then $f_{n}^{*}(c_{k})=c_{[k]}\odot
1_{n-k}\mod\mathcal{F}_{2k-1}$.

###### Proof.

We prove the lemma by induction on $k$. Since the only two-dimensional classes
in $H^{*}(BN(n);\mathbb{F}_{p})$ are $c_{[1]}\odot 1_{n-1}$,
$\gamma_{1,1}^{2}\odot 1_{n-2}$, and $\gamma_{1,2}\odot 1_{n-4}$, we have

$f_{n}^{*}(c_{1})=\lambda c_{[1]}\odot 1_{n-1}+\mu\gamma_{1,1}^{2}\odot
1_{n-2}+\nu\gamma_{1,2}\odot 1_{n-4}$

for some $\lambda,\mu,\nu\in\mathbb{F}_{p}$. Since the restriction of $c_{1}$
to $H^{*}(BN(1);\mathbb{F}_{p})$ is $c_{[1]}$, we must have $\lambda=1$.
Hence, $f_{n}^{*}(c_{1})=c_{[1]}\odot 1_{n-1}+x$, with $x\in\mathcal{F}_{0}$.
We assume by the inductive hypothesis that the lemma is true for all $1<l<k$.
Consider the reduced coproduct map

$\overline{\Delta}\colon
H^{*}(BN(n);\mathbb{F}_{p})\longrightarrow\bigoplus_{l=1}^{n-1}H^{*}(BN(l);\mathbb{F}_{p})\otimes
H^{*}(BN(n-l);\mathbb{F}_{p}).$

Recall that $\Delta_{l,n-l}(c_{k})=\sum_{i=0}^{k}c_{i}\otimes c_{k-i}$ and the
maps $f_{n}$ induce coalgebra maps. Therefore we have

$\overline{\Delta}(f_{n}^{*}(c_{k}))=\sum_{l=1}^{n-1}\sum_{i=0}^{k}f_{l}^{*}(c_{i})\otimes
f_{n-l}^{*}(c_{k-i}).$

By the inductive hypothesis,

$\overline{\Delta}(f_{n}^{*}(c_{k}))=\sum_{l=1}^{n-1}\sum_{i=0}^{k}(c_{[i]}\odot
1_{l-i})\otimes(c_{[k-i]}\odot 1_{n-l-k+i})+x=\overline{\Delta}(c_{[k]}\odot
1_{n-k})+x$

where $x\in\mathcal{F}_{k-1}$. The reduced coproduct $\overline{\Delta}$
preserves rank, hence

$f_{n}^{*}(c_{k})=c_{[k]}\odot 1_{n-k}\mod\mathcal{F}_{2k-1}+y,$

where $y\in\operatorname{Prim}(A_{BU(1)})$. However, all the primitive
elements have rank $0$ except for $c_{[1]}^{j}$ ($j\geq 1$), which are in
component one. Hence, $f_{n}^{*}(c_{k})=c_{[k]}\odot
1_{n-k}\mod\mathcal{F}_{2k-1}$. ∎

To fully compute $f_{n}^{*}(c_{k})$ we can use an inductive argument using the
coproduct and Kochman’s formulas.

######  Theorem 4.8.

(Kochman’s formulae for $BU(\infty)$, [17]) Let

$Q_{*}^{r}\colon H^{d}(BU(\infty);\mathbb{F}_{2})\rightarrow
H^{d-r}(BU(\infty);\mathbb{F}_{2})$

be the dual of the $r$-th order Dyer-Lashof operation in homology. Then, for
all $r\leq k\leq n$, the following identity holds:

$Q_{*}^{2r}(c_{k})=\binom{r-1}{k-r-1}c_{k-r}.$

Moreover if $[n]\in
H_{0}(BU(\infty)\times\mathbb{Z};\mathbb{F}_{2})\cong\mathbb{Z}$ is the class
corresponding to $n\in\mathbb{Z}$, then

$Q^{2r}(1\otimes[1])=(c_{1}^{r})^{\vee}\otimes[2].$

Moreover, if $p>2$ is a prime number, the dual Dyer-Lashof operations

$Q^{r}_{*}\colon H^{d}(BU(\infty);\mathbb{F}_{p})\to
H^{d-2r(p-1)}(BU(\infty);\mathbb{F}_{p})$

satisfies

$\displaystyle Q_{*}^{r}(c_{k})=(-1)^{r+k}\left(\begin{array}[]{c}r-1\\\
pr-k\end{array}\right)c_{k-r(p-1)}$ and $\displaystyle
Q^{r}(1\otimes[1])=(c_{p-1}^{r})^{\vee}\otimes[p].$

As $H_{*}(BU(\infty)\times\mathbb{Z};\mathbb{F}_{p})$ is generated as an
algebra by the homology of its component corresponding to $0\in\mathbb{Z}$ and
$[1]$, Theorem 4.8 and the Cartan formula fully determine the Dyer-Lashof
operation on this Hopf algebra. The above identities are valid in the stable
cohomology ring $H^{*}(BU(\infty)\times\mathbb{Z};\mathbb{F}_{p})$. They also
determine the $Q_{*}^{r}$ unstably because the universal map from
$\coprod_{n\geq 0}BU(n)$ to its group completion $BU(\infty)\times\mathbb{Z}$
is injective in homology. It is worth noting that, if $p=2$,
$Q_{*}^{2r-1}=\beta Q_{*}^{2r}$, where $\beta$ is the mod $2$ Bockstein
homomorphism. Therefore, in this case knowledge of $Q_{*}^{2r}$ is sufficient
to determine $Q_{*}^{2r-1}$.

Let us denote the primitive elements in
$H^{*}(D_{n}(BU(1)_{+});\mathbb{F}_{p})$ as

$\mathrm{Prim}_{n}(A_{BU(1)})=\mathrm{Prim}(A_{BU(1)})\cap A_{BU(1)}^{n,*}.$

The module $\mathrm{Prim}_{n}(A_{BU(1)})$ injects into the $n$-th component of
the $\odot$-indecomposables $\mathrm{Indec}(A_{BU(1)})$ of $A_{BU(1)}$. The
following proposition can be deduced from [5].

######  Proposition 4.9.

Let $X$ be a topological space and let $\mathcal{B}$ be a homogeneous vector
space basis for $H^{*}(X;\mathbb{F}_{p})$. The module $\mathrm{Indec}(A_{X})$
is the linear dual of the subspace $M$ of $A_{X}$ with basis
$\\{Q_{*}^{I}(x^{\vee})\mid I\text{ admissible, }k\geq 0,x\in\mathcal{B}\\}$.
Moreover, for $m\geq 0$, the pairing between $\mathrm{Prim}_{2^{m}}(A_{X})$
and the subspace $M^{\prime}_{m}$ with basis $\\{Q_{*}^{I}(x^{\vee})\mid
I\text{ admissible, }l(I)=m,e(I)>0,k\geq 0,x\in\mathcal{B}\\}$ is perfect.
Here $l(I)$ and $e(I)$ are as defined in [5].

Using Proposition 4.9 with $X=BU(1)$ with the monomial basis on $c$, we will
be able to compute $f_{n}^{*}(c_{k})$ recursively as follows:

  1. (1)

From the proof of Lemma 4.7 we see that $f_{1}^{*}(c_{1})=c$ and the reduced
coproduct is given by

$\overline{\Delta}(f_{n}^{*}(c_{k}))=\sum_{l=1}^{n-1}\sum_{i=0}^{k}f_{l}^{*}(c_{i})\otimes
f_{n-l}^{*}(c_{k-i}).$

Hence, the knowledge of $f_{l}^{*}(c_{i})$ for $l<n$ is enough to recursively
compute $\overline{\Delta}(f_{n}^{*}(c_{k}))$. This gives us
$f_{n}^{*}(c_{k})$ up to primitives.

  2. (2)

In order to find the primitive part, we note that
$\mathrm{Prim}_{n}(A_{BU(1)})$ does not include any $\gamma_{k,l}$ unless $n$
is equal to $2^{m}$ for some $m\geq 0$. Then by applying Proposition 4.9, we
can determine the primitive part.

######  Lemma 4.10.

Let $k\geq 1$ and $\langle\cdot,\cdot\rangle\colon
H^{*}(BN(p^{k});\mathbb{F}_{p})\otimes
H_{*}(BN(p^{k});\mathbb{F}_{p})\rightarrow\mathbb{F}_{p}$ be evaluation
pairing between cohomology and homology of the space $BN(p^{k})$. Then, if
$p=2$,

$\big{\langle}f_{2^{k}}^{*}(c_{2^{k}-1}),Q^{2^{k}}Q^{2^{k-1}}\cdots
Q^{4}Q^{2}(1)\big{\rangle}=1,$

where $1\in H_{0}(BN(1);\mathbb{F}_{2})$. Similarly, if $p>2$,

$\big{\langle}f_{p^{k}}^{*}(c_{p^{k}-1}),Q^{p^{k-1}}Q^{p^{k-2}}\cdots
Q^{p}Q^{1}(1)\big{\rangle}=1,$

where $1\in H_{0}(BN(1);\mathbb{F}_{p})$.

###### Proof.

We prove the identity for $p=2$, as the argument for $p>2$ is entirely
similar. In the proof, we will use the notation $\langle\cdot,\cdot\rangle$
for the evaluation pairing between cohomology and homology in any topological
space, not just in $BN(2^{k})$. We also use the naturality of this pairing
with respect to continuous maps of spaces without further notice. since the
map $\coprod_{n}BN(n)\rightarrow\coprod_{n}BU(n)$ is an $E_{\infty}$ mapping,
it preserves the Dyer-Lashof operations in homology. Additionally, the map in
cohomology $H^{*}(BU(\infty);\mathbb{F}_{2})\rightarrow
H^{*}(BU(n);\mathbb{F}_{2})$ is surjective and the dual map
$H_{*}(BU(n);\mathbb{F}_{2})\rightarrow H_{*}(BU(\infty);\mathbb{F}_{2})$ is
injective. This allows us to perform computations in
$BU(\infty)\times\mathbb{Z}$. So, we prove the equivalent formula

$\big{\langle}c_{2^{k}-1}\otimes[2^{k}],Q^{2^{k}}Q^{2^{k-1}}\cdots
Q^{4}Q^{2}(1\otimes[1])\big{\rangle}=1,$

where $[1]\in H_{0}(BU(\infty)\times\mathbb{Z};\mathbb{F}_{2})$. We proceed by
induction on $k$. For $k=1$,

$\big{\langle}c_{1}\otimes[2],Q^{2}(1\otimes[1])\big{\rangle}=\big{\langle}c_{1}\otimes[2],c_{1}^{\vee}\otimes[2]\big{\rangle}=1.$

For $k>1$,

$\displaystyle\big{\langle}$ $\displaystyle
c_{2^{k}-1}\otimes[2^{k}],Q^{2^{k}}Q^{2^{k-1}}\cdots
Q^{4}Q^{2}(1\otimes[1])\big{\rangle}$
$\displaystyle\quad=\big{\langle}Q_{*}^{2^{k}}(c_{2^{k}-1}\otimes[2^{k}]),Q^{2^{k-1}}\cdots
Q^{4}Q^{2}(1\otimes[1])\big{\rangle}$
$\displaystyle\quad=\big{\langle}\binom{2^{k-1}-1}{2^{k-1}-2}c_{2^{k}-1-2^{k-1}}\otimes[2^{k-1}],Q^{2^{k-1}}\cdots
Q^{2}(1\otimes[1])\big{\rangle}\text{ (by Theorem\leavevmode\nobreak\
\ref{kof})}$
$\displaystyle\quad=\big{\langle}c_{2^{k-1}-1}\otimes[2^{k-1}],Q^{2^{k-1}}\cdots
Q^{4}Q^{2}(1\otimes[1])\big{\rangle}$

which is equal to $1$ by the inductive hypothesis. ∎

We will now derive the pullback formula for the Chern classes. We will state
the formula in terms of the total Chern class

$c_{*}=1+c_{1}+c_{2}+\cdots\in H^{*}(BU(\infty);\mathbb{F}).$

To obtain the pullback formula for a specific Chern class $c_{k}$, we only
need to consider the classes of cohomological dimension $2k$. We will
elaborate on this topic later in the section.

######  Theorem 4.11.

(Pullback Formula) Let $f\colon BN(\infty)\rightarrow BU(\infty)$ be the
limiting map induced by the inclusions $N(n)\hookrightarrow U(n)$ and
$\mathcal{X}=\\{\underline{a}=(a_{0},a_{1},a_{2},\dots)\in\mathbb{N}^{\mathbb{N}^{*}}\mid
a_{n}=0\text{ for }n\gg 0\\}$ be the set of infinite sequences of natural
numbers that have finite support i.e. that are eventually zero. Then, in
cohomology with coefficients in $\mathbb{F}$,

(4)
$f^{*}(c_{*})=\left\\{\begin{array}[]{ll}\sum_{\underline{a}\in\mathcal{X}}c_{[a_{0}]}\odot\big{(}\bigodot_{i,a_{i}\neq
0}\gamma_{i,a_{i}}^{2}\big{)}\odot 1_{\infty}&\mbox{if
}\operatorname{char}(\mathbb{F})=2\\\
\sum_{\underline{a}\in\mathcal{X}}c_{[a_{0}]}\odot\big{(}\bigodot_{i,a_{i}\neq
0}\gamma_{i,a_{i}}\big{)}\odot 1_{\infty}&\mbox{if
}\operatorname{char}(\mathbb{F})>p\\\ \sum_{n=0}^{\infty}c_{[n]}\odot
1_{\infty}&\mbox{if }\operatorname{char}(\mathbb{F})=0\\\ \end{array}\right..$

Note that requiring that every sequence in $\mathcal{X}$ has finite support
guarantees that the transfer product in the formula (4) is iterated a finite
number of times and well-defined.

###### Proof.

We first assume that $\operatorname{char}(\mathbb{F})=2$. Since $\mathbb{F}$
is a free module over $\mathbb{F}_{2}$, we can assume without loss of
generality that $\mathbb{F}=\mathbb{F}_{2}$. Let us denote by $x_{*}$ the
class

$x_{*}=\sum_{\underline{a}\in\mathcal{X}}c_{[a_{0}]}\odot\big{(}\bigodot_{i,a_{i}\neq
0}\gamma_{i,a_{i}}^{2}\big{)}\odot 1_{\infty}.$

We can write $x_{*}=\sum_{n\geq 0}x_{n}$, where $|x_{n}|=2n$, as
$\gamma_{i,a_{i}}^{2}$ and $c_{[a_{0}]}$ are always even dimensional
cohomology classes. We have that $\Delta x_{*}=x_{*}\otimes x_{*}$.
Extrapolating the $2n$-dimensional part, the coproduct of $x_{n}$ must be

$\Delta x_{n}=\sum_{j=0}^{n}x_{j}\otimes x_{n-j},$

similar to those satisfied by the pullbacks of the Chern classes

$\Delta f^{*}(c_{n})=\sum_{j=0}^{n}f^{*}(c_{j})\otimes f^{*}(c_{n-j}).$

When $m,n$ are finite positive numbers, the restriction of $c_{m}$ to
$H^{*}(BU(n);\mathbb{F}_{2})$ is $0$ for all $m>n$. Therefore, the restriction
of $f^{*}(c_{m})$ to $H^{*}(BN(n);\mathbb{F}_{2})$ is also $0$ for all $m>n$.
Claim 1. The restriction of $x_{m}$ to $H^{*}(BN(n);\mathbb{F}_{2})$ is $0$ if
$m>n$.
Proof of Claim 1. Recall that $|\gamma_{k,l}|=l(2^{k}-1)$ and $|c_{[k]}|=2k$.
Therefore the cohomological dimension of

$x_{\underline{a}}:=c_{[a_{0}]}\odot\big{(}\bigodot_{i,a_{i}\neq
0}\gamma_{i,a_{i}}^{2}\big{)}\odot 1_{\infty}$

corresponding to a sequence
$\underline{a}=(a_{0},a_{1},a_{2},\dots)\in\mathcal{X}$ is
$2a_{0}+\sum_{i=1}^{\infty}2a_{i}(2^{i}-1)$. The class $x_{m}$ is of the form
$\sum_{\underline{a}}x_{\underline{a}}$ for some sequences
$\underline{a}\in\mathcal{X}$. If the cohomological dimension of $x_{m}$ is
bigger that $2n$, i.e. $2m=2a_{0}+\sum_{i=1}^{\infty}2a_{i}(2^{i}-1)>2n$, then
$a_{0}+\sum_{i=1}^{\infty}a_{i}2^{i}>n$. Also, note that $x_{\underline{a}}$
is in the $(a_{0}+\sum_{i=1}^{\infty}a_{i}2^{i})$-th component as the width of
the skyline diagram corresponding to $x_{\underline{a}}$ is
$a_{0}+\sum_{i=1}^{\infty}a_{i}2^{i}$, which is bigger than $n$. Therefore the
restriction of $x_{\underline{a}}$ to $H^{*}(BN(n);\mathbb{F}_{2}$ is $0$.
$\blacksquare$
Assume that, $f^{*}(c_{*})\neq x_{*}$. Then there exists a minimal $m$ such
that $f^{*}(c_{m})\neq x_{m}$. By minimality and the coproduct formulas, we
have that $x_{m}-f^{*}(c_{m})$ is primitive of dimension $m$. The primitives
in $A_{\infty}(BU(1))$ are necessarily a linear combination of elements of the
form $b\odot 1_{\infty}$, where $b$ is a primitive gathered block in a
component equal to $2^{k}$ for some $k$. We will now determine the condition
on the gathered block $b$ that can appear in this linear combination.
Claim 2. For $x_{m}-f^{*}(c_{m})$ to be non-zero, we must have $m=2^{k}-1$ for
some $k\in\mathbb{N}$ and

$x_{2^{k}-1}-f^{*}(c_{2^{k}-1})=\lambda_{k}(\gamma_{k,1}^{2}\odot
1_{\infty}).$

Proof of Claim 2. By Claim 1, $x_{m}-f^{*}(c_{m})$ is in the kernel of the
restriction map
$H^{*}(BN(\infty);\mathbb{F}_{2})\xrightarrow{res}H^{*}(BN(n);\mathbb{F}_{2})$
for $n<m$. Note that
$\mathrm{ker}\big{(}H^{*}(BN(\infty);\mathbb{F}_{2})\xrightarrow{res}H^{*}(BN(m);\mathbb{F}_{2})\big{)}$
is generated by $y\odot 1_{\infty}$ for $y$ a full width Hopf monomial with
width strictly bigger than $m$. This rules out all primitive gathered blocks
in a component equal to $2^{k}$ for some $k<\lceil\log_{2}(m)\rceil$. It
follows from the calculation in [8] that only primitive gathered block in
component $2^{k}$ whose cohomological dimension is a multiple of $2$ and does
not exceed $2^{k+1}$ is $\gamma_{k,1}^{2}$ of dimension $2(2^{k}-1)$.
Therefore $m$ must be equal to $2^{k}-1$ for some $k\in\mathbb{N}$ and

(5) $x_{2^{k}-1}-f^{*}(c_{2^{k}-1})=\lambda_{k}(\gamma_{k,1}^{2}\odot
1_{\infty}),$

for some $\lambda_{k}\in\mathbb{F}_{2}$. $\blacksquare$
Claim 3. For all $k$, we have that $\lambda_{k}=0$.
Proof of Claim 3. We prove this by restricting (5) to
$H^{*}(BN(2^{k});\mathbb{F}_{2})$ and evaluate the restriction on the homology
class $Q^{2^{k}}Q^{2^{k-1}}\cdots Q^{4}Q^{2}(1)$, where $1\in
H_{0}(BN(1);\mathbb{F}_{2})$. Also note that $\gamma_{k,1}^{2}\odot
1_{\infty}$ restricts to $\gamma_{k,1}^{2}$ in
$H^{*}(BN(2^{k});\mathbb{F}_{2})$. Using a similar argument used to prove
Claim 1, we can show that the only addends in $x_{*}$ whose width is at most
$2^{k}$ and cohomological dimension is $2(2^{k}-1)$ are
$c_{[p^{k}-p^{l}]}\odot\gamma_{l,1}^{2}\odot 1_{\infty}$, with $1\leq l\leq
k$, and $c_{[2^{k}-1]}\odot 1_{\infty}$. Hence, the restriction of $x_{m}$ to
$H^{*}(BN(2^{k});\mathbb{F}_{2})$ is
$\sum_{l=1}^{k}c_{[p^{k}-p^{l}]}\odot\gamma_{l,1}^{2}+c_{[2^{k}-1]}\odot
1_{1}$ and

$f_{2^{k}}^{*}(c_{2^{k}-1})=(1-\lambda_{k})\gamma_{k,1}^{2}+\sum_{l=1}^{k-1}c_{[2^{k}-2^{l}]}\odot\gamma_{l,1}^{2}+c_{[2^{k}-1]}\odot
1_{1}.$

From the definition of $\gamma_{k,1}$, we have
$\langle\gamma_{k,1}^{2},Q^{2^{k}}Q^{2^{k-1}}\cdots Q^{4}Q^{2}(1)\rangle=1$
and from Lemma 4.10, we have $\langle
f_{2^{k}}^{*}(c_{2^{k}-1}),Q^{2^{k}}Q^{2^{k-1}}\cdots Q^{4}Q^{2}(1)\rangle=1$.
All the other addends pair trivially with $Q^{2^{k}}Q^{2^{k-1}}\cdots
Q^{4}Q^{2}(1)$. This implies $\lambda_{k}=0$. $\blacksquare$
Claim 3 implies that, $x_{m}-f^{*}(c_{m})=0$ for any $m\in\mathbb{N}$. This
contradicts our previous assumption that $x_{*}\neq f^{*}(c_{*})$. Therefore,
we conclude that $f^{*}(c_{*})$ must be equal to $x_{*}$ and the theorem is
proved.

If $\operatorname{char}(\mathbb{F})=p>2$, then the argument is essentially the
same with
$x_{*}=\sum_{\underline{a}\in\mathcal{X}}c_{[a_{0}]}\odot\big{(}\bigodot_{i,a_{i}\neq
0}\gamma_{i,a_{i}}\big{)}\odot 1_{\infty}$, but the analog of Claim 2 is
slightly more subtle. In order to prove that if $m$ is the minimal index such
that $x_{m}-f^{*}(c_{m})\not=0$ then $m=p^{k}-1$ and
$x_{p^{k}-1}-f^{*}(c_{p^{k}-1})=\lambda_{k}(\gamma_{k,1}\odot 1_{\infty})$,
one is lead to consider primitive gathered blocks in component $p^{k}$ whose
cohomological dimension does not excees $2p^{k}$. There are classes satisfying
this contraint, in dimension $2(p^{k}-p^{i}-1)$, $2(p^{k}-p^{i})-1$ for $1\leq
i\leq k$ and $2(p^{k}-1)$. The classes of dimension $2(p^{k}-p^{i})-1$ are
ruled out because $|x_{m}-f^{*}(c_{m})|$ is even, while $2(p^{k}-p^{i}-1)$ are
ruled out because their Bockstein is non-zero, while
$\beta(x_{p^{k}-1}-f^{*}(c_{p^{k}-1}))=0$. The only remaining primitive is
$\gamma_{k,1}$, of dimension $2(p^{k}-1)$.

If $\operatorname{char}(\mathbb{F})=0$, then Claims 2 and 3 are unnecessary,
because the only non-zero primitives lie in component $1$. ∎

Let us now recall the definition of a regular sequence. The following
definition and the subsequent lemma is a classical result [6].

######  Definition 4.12.

Let $A$ be a commutative algebra over a field $\mathbb{F}$. A sequence
$\\{a_{1},\dots,a_{n}\\}$ in $A$ is said to be a regular sequence if $a_{1}$
is not a zero divisor in $A$ and for all $2\leq i\leq n$, $a_{i}$ is not a
zero divisor in $A/(a_{1},\dots,a_{i-1})$.

If $\\{F_{i}\\}_{i\in I}$ is a filtration of $A$, with $I$ being a totally
ordered set, then we define the associated graded algebra of $A$ as

$\operatorname{gr}(A):=\bigoplus_{i\in I}F_{i}/F_{i-1}.$

######  Lemma 4.13 ([6]).

Let $A$ be an algebra over a field $\mathbb{F}$. Let $\mathcal{F}$ be a
multiplicative filtration of $A$ bounded below. For $i=1,2,\dots,n$ let
$x_{i}\in\mathcal{F}_{j_{i}}/\mathcal{F}_{j_{i}-1}\subseteq\operatorname{gr}_{\mathcal{F}}(A)$
be graded elements in the associated graded algebra lifting to
$\widetilde{x}_{i}\in\mathcal{F}_{j_{i}}\subseteq A$. If $\\{x_{1},\dots
x_{n}\\}$ is a regular sequence in $\operatorname{gr}_{\mathcal{F}}(A)$, then
$\\{\widetilde{x}_{1},\dots\widetilde{x}_{n}\\}$ is a regular sequence in $A$.

######  Lemma 4.14.

Let $A=H^{*}(BN(\infty);\mathbb{F}_{p})$, and $\mathcal{F}_{*}$ be the rank
filtration. The associated graded algebra is isomorphic to the polynomial
algebra

(6) $\operatorname{gr}_{\mathcal{F}}(A)=\mathcal{F}_{0}[c_{[1]}\odot
1_{\infty},c_{[2]}\odot 1_{\infty},\dots,c_{[k]}\odot 1_{\infty},\dots]$

###### Proof.

Every stabilized Hopf monomial $x=c_{[k_{1}]}^{m_{1}}\odot\cdots\odot
c_{[k_{l}]}^{m_{l}}\odot 1_{\infty}$ is uniquely determined by an eventually
zero sequence of non-negative integers of the form

$\underline{s}(x)=(\underbrace{m_{1},\dots,m_{1}}_{k_{1}\text{
times}},\dots,\underbrace{m_{l},\dots,m_{l}}_{k_{l}\text{ times}},0,0,\dots).$

Let $A^{\prime}$ be the subspace of $A$ generated by the linear span of the
Hopf monomials $c_{[k_{1}]}^{m_{1}}\odot\cdots\odot c_{[k_{l}]}^{m_{l}}\odot
1_{\infty}$ with $m_{1}>m_{2}>\cdots>m_{l}$ and $k_{1},\dots,k_{l}\geq 1$. By
Hopf ring distributivity, the product of any such two Hopf monomials is again
of this form and hence an element of $A^{\prime}$. This makes $A^{\prime}$ a
subalgebra of $A$. We define a total order on the set of these Hopf monomials
by letting $x<y$ if and only if $\underline{s}(x)<\underline{s}(y)$ in the
lexicographic order. The set of these monomials forms an ordered basis for
$A^{\prime}$ as a $\mathbb{F}_{p}$-vector space. Given an eventually zero
sequence of non-negative integers $s$, we denote the unique basis element
$x_{s}$ such that $\underline{s}(x_{s})=s$.
Claim. If $x=c_{[k_{1}]}^{m_{1}}\odot\cdots\odot c_{[k_{l}]}^{m_{l}}\odot
1_{\infty}$, then for $m\geq 1$ the leading term of $x\cdot(c_{[k]}\odot
1_{\infty})$ in the above ordered basis is
$x_{\underline{s}(x)+\underline{s}(c_{[k]}\odot 1_{\infty})}$.
Recall from Theorem 2.3 that $\Delta c_{[k]}=\sum_{j=0}^{k}c_{[j]}\otimes
c_{[k-j]}$. To prove this claim, we proceed by induction. If $l=1$, the Hopf
ring distributivity implies

$x\cdot(c_{[k]}\odot
1_{\infty})=\sum_{j=0}^{\mathrm{min}\\{k_{1},k\\}}c_{[j]}^{m_{1}+1}\odot
c_{[k_{1}-j]}^{m_{1}}\odot c_{[k-j]}\odot 1_{\infty}.$

The leading term is the one for which $j$ is maximal and this proves the claim
in the case when $l=1$.
To prove the general case, let $x^{\prime}=c_{[k_{2}]}^{m_{2}}\odot\cdots\odot
c_{[k_{l}]}^{m_{l}}\odot 1_{\infty}$. Again using the Hopf ring
distributivity, we have

$x\cdot(c_{[k]}\odot
1_{\infty})=\sum_{j=0}^{\mathrm{min}\\{k_{1},k\\}}c_{[j]}^{m_{1}+1}\odot
c_{[k_{1}-j]}^{m_{1}}\odot\big{(}x^{\prime}\cdot(c_{[k-j]}\odot
1_{\infty})\big{)}.$

The leading term is the one for which $j$ is maximal and this proves the
induction step.
As a consequence of the Claim $A^{\prime}=\mathbb{F}_{p}[c\odot
1_{\infty},\dots,c_{[n]}\odot 1_{\infty}]$ as an algebra. Therefore, to prove
the lemma it is enough to check that for all $x=\overline{x}\odot
1_{\infty}\in\mathcal{F}_{0}$ and for all $y=\overline{y}\odot 1_{\infty}\in
A^{\prime}$, the cup product $x\cdot y$ is equal to
$\overline{x}\odot\overline{y}\odot 1_{\infty}$ plus elements of lower rank.
Unpacking the Hopf ring distributivity $x\cdot y$ is obtained combinatorially
by splitting the columns of the skyline diagram of $\overline{y}$, and
stacking each column either onto a column of $\overline{x}$ with the same
width, or onto the $1_{\infty}$ part. Stacking a column of $\overline{y}$ onto
a column of $\overline{x}$ lowers rank. Therefore the term of maximal rank is
obtained by stacking $\overline{y}$ onto the $1_{\infty}$ part, which yields
$\overline{x}\odot\overline{y}\odot 1_{\infty}$. ∎

######  Theorem 4.15.

The sequence $\\{f^{*}(c_{1}),f^{*}(c_{2}),\dots\\}$ in $A_{\infty}(BU(1))$ is
a regular sequence.

###### Proof.

From Lemma 4.14, we have that the associated graded algebra
$\operatorname{gr}_{\mathcal{F}}(A)$ corresponding to the rank filtration
$\mathcal{F}_{*}$ of $A=H^{*}(BN(\infty);\mathbb{F}_{p})$ is isomorphic to
$\mathcal{F}_{0}[c_{[1]}\odot 1_{\infty},c_{[2]}\odot 1_{\infty},\dots]$.
Also, note that the sequence $\\{c_{[1]}\odot 1_{\infty},c_{[2]}\odot
1_{\infty},\dots\\}$ is a regular sequence in
$\operatorname{gr}_{\mathcal{F}}(A)$. From Lemma 4.7, we have
$f_{n}^{*}(c_{k})=c_{[k]}\odot 1_{n-k}$ modulo terms of lower rank and
therefore $c_{[k]}\odot 1_{\infty}\in\operatorname{gr}_{\mathcal{F}}(A)$ lifts
to $f^{*}(c_{k})\in A$. Hence by Lemma 4.13,
$\\{f^{*}(c_{1}),f^{*}(c_{2}),\dots\\}$ is a regular sequence in
$A_{\infty}(BU(1))$. ∎

As a direct consequence of Proposition 4.15, we have the following.

######  Corollary 4.16.

For all prime $p$, there are isomorphisms of algebras

$H^{*}\Big{(}\frac{U(\infty)}{N(\infty)};\mathbb{F}_{p}\Big{)}\cong\frac{A_{\infty}(BU(1))}{(f^{*}(c_{1}),f^{*}(c_{2}),\dots)}.$

###### Proof.

We consider the Serre spectral sequence associated with the fiber sequence

(7) $U(\infty)\longrightarrow\frac{U(\infty)}{N(\infty)}\longrightarrow
BN(\infty)$

and compare it with the Serre spectral sequence associated with the universal
fibration

(8) $U(\infty)\longrightarrow EU(\infty)\longrightarrow BU(\infty).$

Recall that $H^{*}(U(\infty);\mathbb{F}_{p})$ is the exterior algebra
generated by $z_{2i-1}$ for $i=1,2,\dots$ and $z_{2i-1}$ transgresses to
$c_{i}$ in the Serre spectral sequence associated with (8). Comparing the two
spectral sequences, we see that $z_{2i-1}$ transgresses to $f^{*}(c_{i})$ in
the Serre spectral sequence associated with (7). Hence, the differentials are
given as follows: for all $i\geq 1$, $d_{2i}\colon z_{2i-1}\mapsto
f^{*}(c_{i})$ and $d_{2i-1}\equiv 0$. By Theorem 4.15, the sequence
$\\{f^{*}(c_{1}),f^{*}(c_{2}),\dots\\}$ is a regular sequence in
$A_{\infty}(BU(1))$ and hence by [3, Theorem 11.22] is algebraically
independent over $\mathbb{F}_{2}$. Therefore, for all $i\geq 1$, $d_{2i}$ is
injective on the ideal generated by $z_{2i-1}$, otherwise, there will be
algebraic relations between the $f^{*}(c_{i})$’s. The $E_{\infty}$-page is
thus given by

$E_{\infty}^{*,*}=\frac{A_{\infty}(BU(1))}{(f^{*}(c_{1}),f^{*}(c_{2}),\dots)}$

and the result follows. ∎

######  Corollary 4.17.

$H^{*}(\overline{\mathrm{Fl}}_{\infty}(\mathbb{C});\mathbb{F}_{p})$ is the
free graded commutative algebra generated by the stabilization of decorated
gathered blocks $b\odot 1_{\infty}$ such that $\operatorname{rk}(b)=0$ and
satisfies the conditions of Corollary 2.11.

###### Proof.

By Lemma 4.14 and Corollary 4.16, the stable cohomology of the unordered flag
manifold is isomorphic to $\mathcal{F}_{0}$ the rank zero subalgebra of
$A_{\infty}(X)$. ∎

With rational coefficients, $A_{\infty}(BU(1))$ is the polynomial algebra
generated by $c_{[n]}\odot 1_{\infty}$ for $n\geq 1$. Therefore, our analysis
recovers the following well-known result.

######  Corollary 4.18.

$H^{*}(\overline{\mathrm{Fl}}_{\infty}(\mathbb{C});\mathbb{Q})\cong\mathbb{Q}$.

###  4.2. The real case

For the real case, we deal with the mod $2$, mod $p$ for $p>2$, and rational
cohomologies separately.

####  4.2.1. Mod $2$ cohomology

To compute the cohomology of $\operatorname{Fl}_{n}(\mathbb{R})$, we chose to
use the alternating subgroup $\operatorname{B}_{n}^{+}$ instead of
$\operatorname{B}_{n}$. Therefore, in order to compute the cohomology of the
limit $\operatorname{Fl}_{\infty}(\mathbb{R})$, we need to determine the
stable cohomology of $\operatorname{B}_{n}^{+}$.

######  Corollary 4.19.

The sequence of spaces $\\{B\operatorname{B}_{n}^{+}\\}_{n\in\mathbb{N}}$
exhibits homological stability with stabilization maps
$B\operatorname{B}_{n}^{+}\to B\operatorname{B}_{n+1}^{+}$ induced by the
standard group inclusions
$\operatorname{B}_{n}^{+}\hookrightarrow\operatorname{B}_{n+1}^{+}$. Moreover,
for all primes $p$, the stable cohomology ring is

$H^{*}(B\operatorname{B}_{\infty}^{+};\mathbb{F}_{p})\cong\varinjlim_{n}H^{*}(B\operatorname{B}_{n}^{+};\mathbb{F}_{p})\cong\left\\{\begin{array}[]{ll}\frac{A_{\infty}(\mathbb{P}^{\infty}(\mathbb{R}))}{(e_{\infty})}&\mbox{if
}p=2\\\ A_{\infty}(\mathbb{P}^{\infty}(\mathbb{R}))&\mbox{if
}p>2\end{array}\right.,$

where $e_{\infty}=w\odot 1_{\infty}+\gamma_{1,1}\odot 1_{\infty}$.

###### Proof.

If $p>2$, the result follows immediately from Theorem 3.7 and the homological
stability of $B(\operatorname{B}_{n})$.

For $p=2$, we observe that, with the notation of Definition 3.3, every element
of $\mathcal{G}_{ann}\cap H^{*}(\operatorname{B}_{n};\mathbb{F}_{2})$ has
degree at least $3\lfloor n/4\rfloor$, which goes to $\infty$ as $n\to\infty$.

We deduce that, for all $d\geq 0$ and for $n$ large enough, in the mod $2$
cohomological Gysin sequence of $B\operatorname{B}_{n}^{+}\to
B\operatorname{B}_{n}$ the connecting homomorphisms is injective in degree
$d-1$. Therefore there is a short exact sequence

$0\longrightarrow
H^{d-1}(\operatorname{B}_{n};\mathbb{F}_{2})\stackrel{{\scriptstyle
e_{n}}}{{\longrightarrow}}H^{d}(\operatorname{B}_{n};\mathbb{F}_{2})\longrightarrow
H^{d}(\operatorname{B}_{n}^{+};\mathbb{F}_{2})\longrightarrow 0.$

The stabilization maps induce homomorphisms of short exact sequences, hence
the homological stability of $B\operatorname{B}_{n}$ implies that of
$B\operatorname{B}_{n}^{+}$. Moreover, the stable cohomology of
$B\operatorname{B}_{n}^{+}$ is the quotient of the stable cohomology of
$B\operatorname{B}_{n}$ by the limit of the classes $e_{n}$, which is
$e_{\infty}$. ∎

######  Definition 4.20.

Let $p$ be a prime number. We define
${A}_{\infty}^{\prime}(\operatorname{B}^{+})$ as the ring
$H^{*}(\operatorname{B}_{\infty}^{+};\mathbb{F}_{p})$ computed in the previous
corollary.

The inclusions $\operatorname{B}_{n}^{+}\hookrightarrow SO(n)$ induce maps
$g_{n}\colon B\operatorname{B}_{n}^{+}\rightarrow BSO(n)$. We call the
limiting map $g\colon B\operatorname{B}_{\infty}^{+}\rightarrow BSO(\infty)$.
Similarly, we have maps $h_{n}\colon B\operatorname{B}_{n}\rightarrow BO(n)$
and $h\colon B\operatorname{B}_{\infty}\to BO(\infty)$

Note that, in analogy with the complex case, the homomorphism
$\delta_{n,m}\colon SO(n)\times SO(m)\to SO(n+m)$ given by the direct sum of
matrices induces a coassociative and cocommutative coproduct $\Delta$ on
$\bigoplus_{n\geq 0}H^{*}(BSO(n);\mathbb{F}_{p})$. Since it is compatible with
cup product, this object is a bialgebra. The stabilization maps of $BSO(n)$
preserve $\Delta$ and hence the limit $H^{*}(BSO(\infty))$ is also a
bialgebra. The morphisms
$\bigoplus_{n}g_{n}^{*}\colon\bigoplus_{n}H^{*}(BSO(n);\mathbb{F}_{p})\to
A^{\prime}_{\operatorname{B}^{+}}$ and $g^{*}\colon
H^{*}(BSO(\infty))\to{A}_{\infty}^{\prime}(\operatorname{B}^{+})$ are
bialgebra morphisms.

Similarly to the complex case, there is a universal oriented real vector
bundle $\eta_{n}$ on $BSO(n)$. Pullbacks of $\eta_{n}$ along (homotopy classes
of) maps $X\to BSO(n)$ classify the isomorphism classes of oriented real
vector bundles over $X$. Stiefel-Whitney classes of such a bundle correspond
to pullbacks of the universal Stiefel-Whitney classes along its classifying
map. Since $\delta_{n,m}^{*}(\eta_{n+m})$ is isomorphic to
$\eta_{n}\oplus\eta_{m}$, the formula for characteristic classes of direct
sums imply that

$\forall k\in\mathbb{N}:\quad\Delta(w_{k})=\sum_{n+m=k}w_{n}\otimes w_{m}.$

Similarly, $\bigoplus_{n}H^{*}(BO(n);\mathbb{F}_{p})$ and
$H^{*}(BO(\infty);\mathbb{F}_{p})$ are bialgebras, and

$\displaystyle\bigoplus_{n}h_{n}^{*}\colon\bigoplus_{n}H^{*}(BO(n);\mathbb{F}_{p})\longrightarrow
A_{\mathbb{P}^{\infty}(\mathbb{R})}$ and $\displaystyle h^{*}\colon
H^{*}(BO(\infty);\mathbb{F}_{p})\longrightarrow
A_{\infty}(\mathbb{P}^{\infty}(\mathbb{R}))$

are bialgebra homomorphisms. The same formula for the coproduct of universal
characteristic classes holds in $BO(n)$.

Let us define a charged version of the rank filtration when $p=2$.

######  Definition 4.21.

Assume that $p=2$. The rank of a charged Hopf monomial in
$A^{\prime}_{\operatorname{B}^{+}}$ denoted as $\operatorname{rk}$ is defined
as the rank of the corresponding non-charged decorated Hopf monomial in
$A_{\mathbb{P}^{\infty}(\mathbb{R})}$. We also define
$\operatorname{rk}(x\otimes y):=\operatorname{rk}(x)+\operatorname{rk}(y)$ in
$A^{\prime}_{\operatorname{B}^{+}}\otimes A^{\prime}_{\operatorname{B}^{+}}$.

######  Definition 4.22.

Assume that $p=2$ We define the rank filtration $\mathcal{F}_{*}$ for
$A^{\prime}_{\operatorname{B}^{+}}$ by setting $\mathcal{F}_{n}$ as the linear
span of charged Hopf monomials $x$ with $\operatorname{rk}(x)\leq n$.
Moreover, we define a rank filtration on
${A}_{\infty}^{\prime}(\operatorname{B}^{+})$, by defining $\mathcal{F}_{n}$
as the linear span of stabilized Hopf monomials $x\odot 1_{\infty}$ where
$\operatorname{rk}(x)\leq n$. With a slight abuse of notation, we also denote
it as $\mathcal{F}_{*}$.

The rank filtration is exhaustive, increasing, and bounded from below.
Moreover, due to the almost-Hopf ring distributivity law, it is multiplicative
with respect to the cup product, i.e.
$\mathcal{F}_{m}\cdot\mathcal{F}_{n}\subseteq\mathcal{F}_{m+n}$.

We can now compute the pullbacks $g^{*}(w_{k})$ of Stiefel-Whitney classes
with essentially the same argument used for Chern classes in the previous
subsection.

######  Lemma 4.23.

  1. (1)

For $1\leq k\leq n$ let $w_{k}\in H^{k}(BO(n);\mathbb{F}_{2})$ be the $k$-th
universal Stiefel-Whitney class. Then $h_{n}^{*}(w_{k})=w_{[k]}\odot
1_{n-k}\mod\mathcal{F}_{k-1}$.

  2. (2)

For $2\leq k\leq n$, let $w_{k}\in H^{k}(BSO(n);\mathbb{F}_{2})$ be the $k$-th
universal Stiefel-Whitney class. Then $g_{n}^{*}(w_{k})=(w_{[k]}\odot
1_{n-k})^{0}\mod\mathcal{F}_{k-1}$.

###### Proof.

The proof of (1) follows from the same argument used for the proof of Lemma
4.7, by replacing Chern classes with Stiefel-Whitney classes.

There is a commutative diagram

${B\operatorname{B}_{n}^{+}}$${BSO(n)}$${B\operatorname{B}_{n}}$${BO(n).}$$\scriptstyle{g_{n}}$$\scriptstyle{h_{n}}$

The left vertical map induces $\operatorname{res}_{n}$ in cohomology, while
pullback along the right vertical map sends $w_{k}$ to $w_{k}$ if $k\geq 2$
and $w_{1}$ to $0$. (2) follows from the induced commutative diagram in
cohomology, by observing that $\operatorname{res}_{n}$ preserves the rank
filtration. ∎

There is a version of Kochman’s formulae for $BO(\infty)$.

######  Theorem 4.24 (Kochman’s formulae for $BO$, [17]).

Let $Q_{*}^{r}\colon H^{d}(BO(\infty);\mathbb{F}_{2})\to
H^{d-r}(BO(\infty);\mathbb{F}_{2})$ be the dual of the $r$-th order Dyer-
Lashof operation in homology. Then, for all $r\leq k$, the following equality
holds:

$Q_{*}^{r}(w_{k})=\left(\begin{array}[]{c}r-1\\\
k-r-1\end{array}\right)w_{k-r}.$

Moreover, if $[n]\in
H^{0}(BO(\infty)\times\mathbb{Z};\mathbb{F}_{2})\cong\mathbb{Z}$ corresponds
to $n\in\mathbb{Z}$, then

$Q^{r}([1])=(w_{1}^{r})^{\vee}*[2].$

Using this theorem, Lemma 4.23 and Proposition 4.9 with $X=BO(1)$ and the
monomial basis on $w$, we can prove the pullback formula for $h^{*}(w_{k})$.
The proof is very similar to the complex case: we use Lemma 4.7 to reduce to
primitives, and compute the primitive part of the pullback by applying
Proposition 4.9 and Kochman’s formulae via Lemma 4.25. For this reason, we
only provide the statements and we omit proofs.

######  Lemma 4.25.

Let $k\geq 1$ and $\langle\cdot,\cdot\rangle\colon
H^{*}(B\operatorname{B}_{2^{k}};\mathbb{F}_{2})\otimes
H_{*}(B\operatorname{B}_{2^{k}};\mathbb{F}_{2})\rightarrow\mathbb{F}_{2}$ be
evaluation pairing between cohomology and homology of the space
$B\operatorname{B}_{2^{k}}$. Then,

$\big{\langle}h_{2^{k}}^{*}(w_{2^{k}-1}),Q^{2^{k}}Q^{2^{k-1}}\cdots
Q^{4}Q^{2}(1)\big{\rangle}=1,$

where $1\in H_{0}(B\operatorname{B}_{1};\mathbb{F}_{2})$.

We will again state the pullback formula in terms of the total Stiefel-Whitney
class

$w_{*}=1+w_{1}+w_{2}+\cdots\in H^{*}(BO(\infty);\mathbb{F}).$

To obtain the pullback formula for a specific Stiefel class $w_{k}$, we only
need to consider the classes of cohomological dimension $k$.

######  Theorem 4.26.

(Pullback Formula) Let
$\mathcal{X}=\\{\underline{a}=(a_{0},a_{1},a_{2},\dots)\in\mathbb{N}^{\mathbb{N}^{*}}\mid
a_{n}=0\text{ for }n\gg 0\\}$ be the set of infinite sequences of natural
numbers that have finite support i.e. that are eventually zero. Then,

(9)
$h^{*}(w_{*})=\sum_{\underline{a}\in\mathcal{X}}w_{[a_{0}]}\odot\big{(}\bigodot_{i,a_{i}\neq
0}\gamma_{i,a_{i}}\big{)}\odot 1_{\infty}.$

Note that requiring that every sequence in $\mathcal{X}$ has finite support
guarantees that the transfer product in the formula (9) is iterated a finite
number of times and well-defined.

Modulo $2$, the real counterpart of Lemma 4.14 is proved with essentially the
same argument.

######  Lemma 4.27.

le $A=H^{*}(B\operatorname{B}_{\infty};\mathbb{F}_{2})$ and $\mathcal{F}_{*}$
be the rank filtration. The associated graded algebra is isomorphic to the
polynomial algebra

$\operatorname{gr}_{\mathcal{F}}(A)=\mathcal{F}_{0}[w_{[1]}\odot
1_{\infty},w_{[2]}\odot 1_{\infty},\dots,w_{[k]}\odot 1_{\infty},\dots].$

Consequently, the description of the mod $2$ stable cohomology of unordered
real flag manifold is similar to the complex one.

######  Theorem 4.28.

Let $h^{*}\colon H^{*}(BO(\infty);\mathbb{F}_{2})\to
H^{*}(B\operatorname{B}_{\infty};\mathbb{F}_{2})$ be the map induced by $h$ in
cohomology. Then the sequence $\\{h^{*}(w_{1}),\dots,h^{*}(w_{k}),\dots\\}$ is
a regular sequence in $H^{*}(B\operatorname{B}_{\infty};\mathbb{F}_{2})$.

###### Proof.

The proof is analogous to that of Theorem 4.15. ∎

######  Corollary 4.29.

There is an isomorphism of algebras

$H^{*}(\overline{\operatorname{Fl}}_{\infty}(\mathbb{R});\mathbb{F}_{2})\cong\frac{A_{\infty}(BO(1))}{(h^{*}(w_{1}),h^{*}(w_{2}),\dots)}.$

###### Proof.

We compare the Serre spectral sequences associated with the fiber sequences

$\displaystyle
SO(\infty)\longrightarrow\frac{SO(\infty)}{\operatorname{B}_{\infty}^{+}}\longrightarrow
B\operatorname{B}_{\infty}^{+}$ and $\displaystyle SO(\infty)\longrightarrow
ESO(\infty)\longrightarrow BSO(\infty).$

Recall that $H^{*}(SO(\infty);\mathbb{F}_{2})$ is the exterior algebra
generated by classes $a_{i-1}$ for $i=2,3,\dots$ and $a_{i-1}$ transgresses to
$w_{i}$ in the spectral sequence associated to the bottom fibration.
Therefore, $d_{i}(a_{i-1})=g^{*}(w_{i})$ in the Serre spectral sequence
associated to the top fibration. By Corollary 4.19, the mod $2$ cohomology of
$\operatorname{B}_{\infty}^{+}$ is isomorphic to
$A_{\infty}(BO(1))/(h^{*}(w_{1}))$. As $\operatorname{res}_{\infty}\circ h=g$,
under this isomorphism $g^{*}(w_{k})$ is identified with the image of
$h^{*}(w_{k})$ in the quotient. By Theorem 4.28
$\\{h^{*}(w_{1}),h^{*}(w_{2}),\dots\\}$ is a regular sequence, and hence we
must have for all $i\geq 1$, $d_{i}$ injective on the ideal generated by
$a_{i-1}$. ∎

######  Corollary 4.30.

$H^{*}(\overline{\operatorname{Fl}}_{\infty}(\mathbb{R});\mathbb{F}_{2})$ is
the free commutative algebra generated by the stabilization of decorated
gathered blocks $b\odot 1_{\infty}$ in $A_{\infty}(BO(1))$ such that
$\operatorname{rk}(b)=0$.

####  4.2.2. Cohomology at odd primes

With coefficients in $\mathbb{F}_{p}$, $p>2$, the cohomology of
$\mathbb{P}^{\infty}(\mathbb{R})$ is trivial. Thus, every Hopf monomial in
$A_{\mathbb{P}^{\infty}(\mathbb{R})}$ has rank $0$ and the rank filtration
becomes useless. Consequently, we must fully exploit the mod $p$ pullback
formula to perform our stable computations.

We recall that there are Pontrjagin classes $\mathcal{p}_{k}\in
H^{4k}(BSO(\infty);\mathbb{Z})$ that are pullbacks of even Chern classes via
the complexification map $cpx\colon BO(\infty)\to BU(\infty)$. We state our
pullback formula for these by means of the total Pontrjagin class

$\mathcal{p}_{*}=\sum_{n=0}^{\infty}\mathcal{p}_{n}=1+\mathcal{p}_{1}+\mathcal{p}_{2}+\dots\in
H^{*}(BO(\infty)),$

where we let, by convention, $\mathcal{p}_{0}=1$.

######  Proposition 4.31.

Let
$\mathcal{Y}=\\{\underline{a}=(a_{1},a_{2},\dots)\in\mathbb{N}^{\mathbb{N}^{*}}\mid
a_{n}=0\text{ for }n\gg 0\\}$ be the set of infinite sequences of natural
numbers that have finite support i.e. that are eventually zero. Then,

(10)
$g^{*}(\mathcal{p}_{*})=\sum_{\underline{a}\in\mathcal{Y}}\big{(}\bigodot_{i,a_{i}\neq
0}\gamma_{i,a_{i}}\big{)}\odot 1_{\infty}.$

Note that requiring that every sequence in $\mathcal{Y}$ has finite support
guarantees that the transfer product in the formula (10) is iterated a finite
number of times and well-defined.

###### Proof.

We first prove that the desired identity holds for $h^{*}(\mathcal{p}_{*})\in
H^{*}(B\operatorname{B}_{\infty};\mathbb{F}_{p})$. There is a commutative
diagram

${D_{\infty}(\\{*\\}_{+})}$${D_{\infty}(BO(1)_{+})}$${BO(\infty)}$${BN(\infty)}$${BU(\infty)}$$\scriptstyle{j_{1}}$$\scriptstyle{j_{2}}$$\scriptstyle{h}$$\scriptstyle{cpx}$$\scriptstyle{f}$

where $j_{1}$ and $j_{2}$ are the obvious inclusions. By passing to cohomology
we deduce that

$j_{1}^{*}h^{*}(\mathcal{p}_{k})=j_{1}^{*}h^{*}cpx^{*}(c_{2k})=j_{2}^{*}f^{*}(c_{k}).$

By functoriality, $j_{2}^{*}$ is the stabilization of the Hopf ring morphism
$A_{BU(1)}\to A_{\\{*\\}}$ induced by the inclusion $\\{*\\}\to BU(1)$. In
particular, $j_{2}^{*}(\gamma_{k,l})=\gamma_{k,l}$ and $j_{2}^{*}(c_{[l]})=0$
for all $k,l\geq 1$. As $j_{1}^{*}$ is an isomorphism in mod $p$ cohomology,
the result follows from Theorem 4.11.

To pass from $h^{*}(\mathcal{p}_{*})$ to $g^{*}(\mathcal{p}_{*})$, it is
enough to recall that the restriction map
$H^{*}(\operatorname{B}_{\infty};\mathbb{F}_{p})\to
H^{*}(\operatorname{B}_{\infty}^{+};\mathbb{F}_{p})$ is an isomoprhism by
Corollary 3.8. ∎

######  Corollary 4.32.

Let $p>2$ be an odd prime. Then, in mod $p$ cohomology, the following
statements hold:

  1. (1)

$g^{*}(\mathcal{p}_{k})=0$ if $k$ is not a multiple of $(p-1)/2$

  2. (2)

$\\{g^{*}(\mathcal{p}_{\frac{p-1}{2}}),g^{*}(\mathcal{p}_{2\frac{p-1}{2}}),g^{*}(\mathcal{p}_{3\frac{p-1}{2}}),\dots\\}$
is a regular sequence in
$H^{*}(B\operatorname{B}_{\infty}^{+};\mathbb{F}_{p})$.

###### Proof.

Statement $1$ is immediate from Lemma 4.31, because the right-hand side of the
formula for $g^{*}(\mathcal{p}_{*})$ only have terms in degrees that are
multiple of $2(p-1)$.

To prove the second claim, we first define an alternative rank function on
stabilized Hopf monomials in
$H^{*}(B\operatorname{B}_{\infty}^{+};\mathbb{F}_{p})$, which is identified
with $H^{*}(B(\Sigma_{\infty});\mathbb{F}_{p})=A_{\infty}(\\{*\\})$ by Theorem
3.7. Given a gathered block $b$, we define

$\operatorname{rk}(b)=\left\\{\begin{array}[]{ll}nm&\mbox{if
}b=\gamma_{1,n}^{m}\\\ 0&\mbox{if }b\not=\gamma_{1,n}^{m}\forall
n,m\in\mathbb{N}\end{array}\right..$

Given a Hopf monomial $x=b_{1}\odot\dots\odot b_{r}$, we define
$\operatorname{rk}(x)=\sum_{i=1}^{r}\operatorname{rk}(b_{i})$. Finally, for
all stabilized Hopf monomials $x\odot 1_{\infty}$, with $x$ pure, we let
$\operatorname{rk}(x\odot 1_{\infty})=\operatorname{rk}(x)$. This produces an
increasing filtration $\mathcal{F}=\\{\mathcal{F}_{n}\\}_{n\geq 0}$ such that
$\mathcal{F}_{n}$ is the subspace linearly spanned by basis element of rank at
most $n$.

The filtration $\mathcal{F}_{n}$ is multiplicative, and the same arguments
used in our proof of Lemma 4.14 shows that there is an isomorphism

$\operatorname{gr}_{\mathcal{F}}(A_{\infty}(\\{*\\}))\cong\mathcal{F}_{0}[\gamma_{1,1}\odot
1_{\infty},\dots,\gamma_{1,n}\odot 1_{\infty},\dots].$

As an immediate consequence of Proposition 4.31,
$g^{*}(\mathcal{p}_{k\frac{p-1}{2}})=\gamma_{1,k}\odot 1_{\infty}$ modulo
$\mathcal{F}_{k-1}$ and that the sequence $(\gamma_{1,1}\odot
1_{\infty},\dots,\gamma_{1,n}\odot 1_{\infty},\dots)$ is a regular sequence in
$\operatorname{gr}_{\mathcal{F}}(A_{\infty}(\\{*\\}))$ by the same argument
used in the proof of Theorem 4.15. ∎

######  Corollary 4.33.

If $p>2$ is an odd prime, the quotient algebra with mod $p$ coefficients

$\frac{A_{\infty}(\\{*\\})}{(g^{*}(\mathcal{p}_{1}),g^{*}(\mathcal{p}_{2}),\dots)}$

is the free graded commutative algebra generated by stabilized gathered blocks
$b\odot 1_{\infty}$ satisfying the conditions of Corollary 2.11 such that
$\operatorname{rk}(b)=0$.

Finally, we are ready to complete the computation of the stable mod $p$
cohomology of $O(\infty)/B(\infty)=SO(\infty)/\operatorname{B}_{\infty}^{+}$.

######  Theorem 4.34.

Let $p>2$ be an odd prime. Then there is an isomorphism of
$\mathbb{F}_{p}$-algebras

$H^{*}\left(\frac{O(\infty)}{B(\infty)};\mathbb{F}_{p}\right)\cong\Lambda(\\{a_{4n-1}:\frac{p-1}{2}\not|n\\})\otimes\frac{A_{\infty}(\\{*\\})}{(g^{*}(\mathcal{p}_{1}),g^{*}(\mathcal{p}_{2}),\dots)},$

where $\Lambda$ is the exterior algebra functor, and the classes $a_{k}$ are
indexed by their degree.

###### Proof.

We consider the cohomological Serre spectral sequence associated with the
fiber sequence
$SO(\infty)\to\overline{\operatorname{Fl}}_{\infty}(\mathbb{R})\to
B\operatorname{B}_{\infty}^{+}$. Recall that
$H^{*}(SO(\infty);\mathbb{F}_{p})$ is the exterior algebra generated by
classes $a_{3},a_{7},\dots,a_{4k-1},\dots$. Note that the action of the
fundamental group of the base on the fiber is homotopically trivial, so the
$E_{2}^{*,*}$ page involves cohomology with constant coefficients. By
comparing it with the fibration $SO(\infty)\to ESO(\infty)\to BSO(\infty)$ we
deduce, as in the complex case, that $a_{4k-1}$ is transgressive and
transgresses to $g^{*}(\mathcal{p}_{k})$. This completely determines the
differentials on each page.

In the cohomological Serre spectral sequence of the bottom fibration, the
class $a_{4n-1}\in H^{*}(SO(\infty))$ is transgressive and transgresses to the
Pontryagin class $\mathcal{p}_{n}$. By comparing the Serre spectral sequences
of the two fibrations above, we see that, for the top fibration, $a_{4n-1}$ is
also transgressive and transgresses to $g^{*}(\mathcal{p}_{n})$. This spectral
sequence is multiplicative with respect to the cup product, hence the remark
above completely determines all its differentials.

By Corollary 4.32, the pullbacks of Pontryagin classes
$g^{*}(\mathcal{p}_{n})$ for $\frac{p-1}{2}|n$ form a regular sequence in the
cohomology of the base and that the differentials of the remaining classes are
$0$. Consequently, one can check inductively that the classes $a_{4n-1}$ with
$\frac{p-1}{2}\not|n$ survive to the limit page and the differential $d_{4n}$
with $\frac{p-1}{2}|n$ is injective on the ideal generated by $a_{4n-1}$.
Therefore

$E_{\infty}^{*,*}\cong\Lambda(\\{a_{4n-1}:\frac{p-1}{2}\not|n\\})\otimes\frac{A_{\infty}(\\{*\\})}{(g^{*}(p_{1}),g^{*}(p_{2}),\dots)}.$

Since we are using field coefficients,
$H^{*}(\overline{\operatorname{Fl}}_{\infty}(\mathbb{R});\mathbb{F}_{p})\cong
E_{\infty}^{*,*}$ as a graded vector space. A priori, this isomorphic might
not preserve the product. In this particular case, however, the generators
$a_{4k-1}$, being odd-dimensional, must actually span an exterior algebra
inside
$H^{*}(\overline{\operatorname{Fl}}_{\infty}(\mathbb{R});\mathbb{F}_{p})$, and
thus this is an isomorphism of algebras. ∎

####  4.2.3. Rational cohomology

The rational cohomology of unordered flag manifolds is known. However, we
stress that we can retrieve it as a simple consequence of our machinery.

######  Proposition 4.35.

$H^{*}(\overline{\operatorname{Fl}}_{\infty}(\mathbb{R});\mathbb{Q})\cong
H^{*}(SO(\infty);\mathbb{Q})\cong\Lambda(\\{a_{4k-1}:k\in\mathbb{N}\\}).$

###### Proof.

The rational cohomology of the finite group $\operatorname{B}_{\infty}^{+}$ is
trivial by Serre’s theorem and the action of $\operatorname{B}_{\infty}^{+}$
on $SO(\infty)$ is homotopicallly trivial. Hence, the rational Serre spectral
sequence of the fiber sequence
$SO(\infty)\to\overline{\operatorname{Fl}}_{\infty}(\mathbb{R})\to
B\operatorname{B}_{\infty}^{+}$ collapses at the second page, which reduces to
the cohomology of the fiber. ∎

###  4.3. Poincaré series of stable cohomology

We now determine the Poincaré series formula for stable cohomology,
specifically
$H^{*}(\overline{\mathrm{Fl}}_{\infty}(\mathbb{C});\mathbb{F}_{2})$. For this,
we need to obtain the Poincaré series of the stabilization
$D_{\infty}(BU(1)_{+})\cong BN(\infty)$.

####  4.3.1. Poincaré series of $H^{*}(BN(\infty);\mathbb{F}_{2})$

To obtain $H^{*}(BN(\infty);\mathbb{F}_{2})$, we take the stabilization of
full-width Hopf monomials in $\bigoplus_{n\geq
0}H^{*}(D_{n}(BU(1)_{+});\mathbb{F}_{2})$. This means we exclude Hopf
monomials that have a $\odot$-factor of $1_{m}$ in their expression. Let us
denote the bigraded Poincaré series of $\bigoplus_{n\geq
0}H^{*}(D_{n}(BU(1)_{+});\mathbb{F}_{2})$ by $\Pi(t,s)$. To obtain the
Poincaré series $\Pi_{stab}(t)$ of $H^{*}(BN(\infty);\mathbb{F}_{2})$, we need
to

  * •

forget the second degree (component),

  * •

keep only terms corresponding to full-width Hopf monomials.

Therefore $\Pi_{stab}(t)=[(1-s)\cdot\Pi(t,s)]\mid_{s=1}$.

####  4.3.2. Poincaré series of
$H^{*}(\overline{\mathrm{Fl}}_{\infty}(\mathbb{C});\mathbb{F}_{2})$

We know from Theorem 4.15 that the sequence
$\big{(}f^{*}(c_{1}),f^{*}(c_{2}),\dots\big{)}$ is regular in
$H^{*}(BN(\infty);\mathbb{F}_{2})$ and from Corollary 4.16

$H^{*}(\overline{\mathrm{Fl}}_{\infty}(\mathbb{C});\mathbb{F}_{2})\cong\frac{H^{*}(BN(\infty);\mathbb{F}_{2})}{\big{(}f^{*}(c_{1}),f^{*}(c_{2}),\dots\big{)}}.$

From the additivity of Poincaré series and the short exact sequence of the
graded vector spaces

$\displaystyle
0\longrightarrow\frac{H^{*}(BN(\infty);\mathbb{F}_{2})}{\big{(}f^{*}(c_{1}),\dots,f^{*}(c_{n})\big{)}}[2(n+1)]$
$\displaystyle\longrightarrow\frac{H^{*}(BN(\infty);\mathbb{F}_{2})}{\big{(}f^{*}(c_{1}),\dots,f^{*}(c_{n})\big{)}}$
$\displaystyle\longrightarrow\frac{H^{*}(BN(\infty);\mathbb{F}_{2})}{\big{(}f^{*}(c_{1}),\dots,f^{*}(c_{n+1})\big{)}}\longrightarrow
0$

one can inductively compute the Poincaré series of
$\frac{H^{*}(BN(\infty);\mathbb{F}_{2})}{\big{(}f^{*}(c_{1}),\dots,f^{*}(c_{n})\big{)}}$
and pass to the limit to get the Poincaré series of
$H^{*}(\overline{\mathrm{Fl}}_{\infty}(\mathbb{C});\mathbb{F}_{2})$:

$\Pi_{\frac{U(\infty)}{N(\infty)}}(t)=\Pi_{stab}(t)\cdot\big{(}1-\sum_{n=1}^{\infty}t^{2n}\big{)}.$

It still remains to compute $\Pi(t,s)$. This can be done by first determining
the bigraded Poincaré series of the primitive component and subsequently, by
using the Hopf algebra structure, extending the result to obtain the Poincaré
series $\Pi(t,s)$.

##  5\. The spectral sequence computing
$H^{*}(\overline{\operatorname{Fl}}_{n}(\mathbb{C});\mathbb{F})$ and
$H^{*}(\overline{\operatorname{Fl}}_{n}(\mathbb{R});\mathbb{F})$

###  5.1. The complex case

In this section, we prepare the ground for the general spectral sequence
argument that we will use to compute
$H^{*}(\overline{\mathrm{Fl}}_{n}(\mathbb{C});\mathbb{F}_{p})$. For $G=N(n)$
and $X=U(n)$ in Theorem 4.1, we obtain the fiber sequence

(11) $U(n)\longrightarrow\overline{\mathrm{Fl}}_{n}(\mathbb{C})\longrightarrow
BN(n)$

The $E_{2}$-page of the spectral sequence associated with (11) is given by

$H^{*}(BN(n);H^{*}(U(n);\mathbb{F}_{p}))\cong
H^{*}(D_{n}(BU(1)_{+});\mathbb{F}_{p})\otimes H^{*}(U(n);\mathbb{F}_{p}).$

Recall from [15, Proposition 3D.4] that

(12) $H^{*}(U(n);\mathbb{F}_{p})\cong\Lambda_{\mathbb{F}_{p}}[z_{2k-1}\mid
k=1,2,\dots,n],$

where $z_{2k-1}\in H^{2k-1}(U(n);\mathbb{F}_{p})$. We will express the
differentials in the spectral sequence associated with (11) using the
pullbacks of the Chern classes $c_{k}$ under the map $f_{n}\colon
D_{n}(BU(1)_{+})\rightarrow BU(n)$.

######  Lemma 5.1.

In the spectral sequence associated with the fiber sequence (11) the
$z_{2k-1}$ transgresses to $f_{n}^{*}(c_{k})$.

###### Proof.

Recall that a map between the fiber sequences induces a map between the
corresponding spectral sequences which commute with the differentials. The
lemma follows from this by comparing the Serre spectral sequence associated
with the fiber sequence
$U(n)\rightarrow\overline{\mathrm{Fl}}_{n}(\mathbb{C})\rightarrow BN(n)$ and
the universal fibration $U(n)\rightarrow EU(n)\rightarrow BU(n)$. ∎

The pullback formulas in Theorem 4.11 have a limitation in that they are
expressed using the total Chern class $c_{*}$ and hence do not immediately
give us $f_{n}^{*}(c_{k})$ for specific $n$ and $k$, which is needed for the
unstable calculations. However, this issue can be addressed by imposing
certain constraints on the sequences that contribute to the sum for
$f_{n}^{*}(c_{k})$ in the original formula. The following proposition outlines
the conditions that need to be met.

######  Proposition 5.2.

The following statements are true:

  1. (1)

Let $\mathcal{X}_{n}$ denote the set of sequences
$\underline{a}=(a_{0},a_{1},a_{2},\dots)\in\mathcal{X}$ such that $\sum_{i\geq
0}a_{i}2^{i}\leq n$ and $\mathcal{X}_{n,k}$ denote the set of sequences
$\underline{a}\in\mathcal{X}_{n}$ such that $a_{0}+\sum_{i\geq
1}a_{i}(2^{i}-1)=k$. Then,

$f_{n}^{*}(c_{k})=\sum_{\underline{a}\in\mathcal{X}_{n,k}}c_{[a_{0}]}\odot\big{(}\bigodot_{i,a_{i}\neq
0}\gamma_{i,a_{i}}^{2}\big{)}\odot 1_{m},$

where $m\in\mathbb{N}$ is such that the addend is in component $n$.

  2. (2)

Let $p$ be an odd prime. Let $\mathcal{Y}_{n}$ denote the set of sequences
$\underline{a}=(a_{1},a_{2},\dots)\in\mathcal{Y}$ such that $\sum_{i\geq
1}a_{i}p^{i}\leq n$ and $\mathcal{Y}_{n,k}$ denote the set of sequences
$\underline{a}\in\mathcal{Y}_{n}$ such that $\sum_{i\geq 1}a_{i}(p^{i}-1)=2k$.
Then, in mod $p$ cohomology,

$f_{n}^{*}(c_{k})=\sum_{\underline{a}\in\mathcal{Y}_{n,k}}c_{[a_{0}]}\odot\big{(}\bigodot_{i,a_{i}\neq
0}\gamma_{i,a_{i}}\big{)}\odot 1_{m},$

where $m\in\mathbb{N}$ is such that the addend is in component $n$.

###### Proof.

The proof is straightforward. Recall that the component to which
$\gamma_{i,a_{i}}^{2}$ belongs is $a_{i}2^{i}$ and the component to which
$c_{[a_{0}]}$ belongs is $a_{0}$. Therefore, the component to which
$c_{[a_{0}]}\odot\big{(}\bigodot_{i,a_{i}\neq 0}\gamma_{i,a_{i}}^{2}\big{)}$
belongs is $a_{0}+\sum_{i\geq 1}a_{i}2^{i}=\sum_{i\geq 0}a_{i}2^{i}$. For the
addend to be in $\mathrm{Im}(f_{n}^{*})\subset H^{*}(BN(n);\mathbb{F}_{2})$,
this value must be less than or equal to $n$. Moreover the cohomological
dimension of $\gamma_{i,a_{i}}^{2}$ is $2a_{i}(2^{i}-1)$ and the cohomological
dimension of $c_{[a_{0}]}$ is $2a_{0}$. Note that the pullback $f_{n}^{*}$
preserves the cohomological dimension. So, a sequence
$\underline{a}\in\mathcal{X}_{n}$ is an addend in $f_{n}^{*}(c_{k})$ if and
only if $2a_{0}+\sum_{i\geq 1}2a_{i}(2^{i}-1)=2k$. ∎

The following example demonstrates the use of this pullback formula for $n=5$.

######  Example 5.3.

First, we compute $\mathcal{X}_{5,k}$ for $1\leq k\leq 5$.

$\displaystyle\mathcal{X}_{5,1}$
$\displaystyle=\\{(1,0,0,\dots),(0,1,0,\dots)\\},$
$\displaystyle\mathcal{X}_{5,2}$
$\displaystyle=\\{(2,0,0,\dots),(1,1,0,\dots),(0,2,0,\dots)\\},$
$\displaystyle\mathcal{X}_{5,3}$
$\displaystyle=\\{(3,0,0,\dots),(2,1,0,\dots),(1,2,0,\dots),(0,0,1,0,\dots)\\},$
$\displaystyle\mathcal{X}_{5,4}$
$\displaystyle=\\{(4,0,0,\dots),(3,1,0,\dots),(1,0,1,0,\dots)\\},$
$\displaystyle\mathcal{X}_{5,5}$ $\displaystyle=\\{(5,0,0,\dots)\\}.$

Hence by Proposition 5.2 we have

$\displaystyle f_{5}^{*}(c_{1})$ $\displaystyle=c\odot
1_{4}+\gamma_{1,1}^{2}\odot 1_{3},$ $\displaystyle f_{5}^{*}(c_{2})$
$\displaystyle=c_{[2]}\odot 1_{3}+c\odot\gamma_{1,1}^{2}\odot
1_{2}+\gamma_{1,2}^{2}\odot 1_{1},$ $\displaystyle f_{5}^{*}(c_{3})$
$\displaystyle=c_{[3]}\odot 1_{2}+c_{[2]}\odot\gamma_{1,1}^{2}\odot
1_{1}+c\odot\gamma_{1,2}^{2}+\gamma_{2,1}^{2}\odot 1_{1},$ $\displaystyle
f_{5}^{*}(c_{4})$ $\displaystyle=c_{[4]}\odot
1_{1}+c_{[3]}\odot\gamma_{1,1}^{2}+c\odot\gamma_{2,1}^{2},$ $\displaystyle
f_{5}^{*}(c_{5})$ $\displaystyle=c_{[5]}.$

As $\overline{\mathrm{Fl}}_{1}\cong\\{*\\}$ is just a point,
$H^{*}(\overline{\mathrm{Fl}}_{1};\mathbb{F}_{2})\cong\mathbb{F}_{2}$ is
trivial. For $n=2$, we have an isomorphism
$\overline{\mathrm{Fl}}_{2}(\mathbb{C})\cong\mathbb{P}^{2}(\mathbb{R})$. For
$n\geq 3$, we will be using the Serre spectral sequence $E$ associated with
the fiber sequence
$U(n)\rightarrow\overline{\mathrm{Fl}}_{n}(\mathbb{C})\rightarrow BN(n)$ to
determine the cohomology of $\overline{\mathrm{Fl}}_{n}(\mathbb{C})$. The
$E_{2}$-page is given by
$E_{2}^{k,l}=H^{k}(BN(n);H^{l}(U(n);\mathbb{F}_{2}))$. Recall the
transgressive differentials $d_{2k}$ described in Lemma 5.1 as follows

$d_{2k}:z_{2k-1}\longmapsto f_{n}^{*}(c_{k}).$

The differential $d_{2k}$ can be fully described from the multiplicative
structure of the spectral sequence. Also, note that odd differentials are
zero. The pullbacks $f_{n}^{*}(c_{k})$ are computed using the formula from
Proposition 5.2.

On the $E_{2}$-page of the spectral sequence, the $l=0$ row is given by
$H^{*}(BN(n);\mathbb{F}_{2})$ and the higher rows are
$H^{*}(BN(n);\mathbb{F}_{2})$ multiplied by the product of some finite
combination of the $z_{2k-1}$’s. A major step in the computation is checking
whether the differentials $d_{2k}$ are injective or not, i.e. whether
$d_{2k}(b\cdot z_{2k-1})$ is zero or not for $b$ a generator of
$H^{*}(BN(n);\mathbb{F}_{2})$.

######  Lemma 5.4.

In the spectral sequence $E$, $d_{2k}((c_{[l]}\odot 1_{n-l})\cdot z_{2k-1})$
is non zero for all $1\leq k,l\leq n$.

###### Proof.

From the multiplicative structure of the spectral sequence $E$, we have

$d_{2k}((c_{[l]}\odot 1_{n-l})\cdot z_{2k-1})=(c_{[l]}\odot 1_{n-l})\cdot
d_{2k}(z_{2k-1})=(c_{[l]}\odot 1_{n-l})\cdot f_{n}^{*}(c_{k}).$

Recall from Proposition 5.2 that $f_{n}^{*}(c_{k})=c_{[k]}\odot
1_{n-k}+\cdots$ in $H^{*}(BN(n);\mathbb{F}_{2})$. Note that this also holds in

$\frac{H^{*}(BN(n);\mathbb{F}_{2})}{(f_{n}^{*}(c_{1}),\dots,f_{n}^{*}(c_{k-1}))}.$

Without loss of any generality we assume $k\geq l$. The other case is similar.
Using Hopf ring distributivity,

$(c_{[l]}\odot 1_{n-l})\cdot(c_{[k]}\odot 1_{n-k})=c_{[l]}^{2}\odot
c_{[k-l]}\odot 1_{n-k}+\cdots$

which is always non-zero. We illustrate this using a skyline diagram with
$k=4$, $l=3$, and $n=6$ in Figure 2.

$\cdot$$=$$+$$\cdots$$(c_{[3]}\odot 1_{3})$$(c_{[4]}\odot
1_{2})$$(c_{[3]}^{2}\odot c\odot 1_{2})$ Figure 2. Skyline diagram for the dot
product of $c_{[3]}\odot 1_{3}$ and $c_{[4]}\odot 1_{2}$

Therefore $(c_{[l]}\odot 1_{n-l})\cdot f_{n}^{*}(c_{k})\neq 0$ for all $1\leq
k,l\leq n$. ∎

The previous lemma us that the differentials are always non-zero on classes of
the form $c_{[m]}\odot 1_{n-m}$. Therefore, the only classes that can be
mapped to zero by the differentials $d_{2k}$ are the rank zero decorated
gathered blocks. Here the rank function is as defined in Definition 4.5.
Hence, we only need to consider these classes when we are checking the
injectivity of the differentials. This simplifies our computations in §6.

###  5.2. The real case

A similar machinery can be used to compute
$H^{*}(\overline{\operatorname{Fl}}_{n};\mathbb{F}_{p})$ for all
$n\in\mathbb{N}$ and for all primes $p$.

Recall from [15, Corollary 4D.3] that

$H^{*}(SO(n);\mathbb{F}_{2})\cong\left\\{\begin{array}[]{ll}\Lambda(\\{a_{1},\dots,a_{n-1}\\})&\mbox{if
}p=2\\\ \Lambda(\\{a_{3},\dots,a_{4\lfloor\frac{n}{2}\rfloor-1}\\})&\mbox{if
}p>2,n\mbox{ odd}\\\
\Lambda(\\{a_{3},\dots,a_{4\lfloor\frac{n}{2}\rfloor-5},a^{\prime}_{n-1}\\})&\mbox{if
}p>2,n\mbox{ even}\end{array}\right.,$

where generators are indexed by their degree.

The generators of the mod $p$ cohomology of $SO(n)$ are transgressive in the
Serre spectral sequence associated with the fiber sequence $SO(n)\to ESO(n)\to
BSO(n)$. Moreover, $d_{i}(a_{i-1})=w_{i}$ if $p=2$,
$d_{i}(a_{i-1})=\mathcal{p}_{i}$ if $p>2$, and $d_{n}(a^{\prime}_{n-1})$ is
the universal Euler class $X_{n}\in H^{n}(BSO(n);\mathbb{F}_{p})$ if $p>2$ and
$n$ is even.

By comparison with the Serre spectral sequence of the fibration
$SO(n)\to\overline{\operatorname{Fl}}_{n}(\mathbb{R})\to BSO(n)$ we deduce
that the generators of the cohomology of $SO(n)$ are transgressive and
transgresses to the pullback of the corresponding characteristic class via
$g_{n}$.

The pullbacks of the Stiefel-Whitney and Pontrjagin classes can be computed
from Theorem 4.26 and Proposition 4.31 by restriction from the stable
cohomology as done for Chern classes in Proposition 5.2.

######  Proposition 5.5.

The following statements are true:

  1. (1)

Let $\mathcal{X}_{n}$ denote the set of sequences
$\underline{a}=(a_{0},a_{1},a_{2},\dots)\in\mathcal{X}$ such that $\sum_{i\geq
0}a_{i}2^{i}\leq n$ and $\mathcal{X}_{n,k}$ denote the set of sequences
$\underline{a}\in\mathcal{X}_{n}$ such that $a_{0}+\sum_{i\geq
1}a_{i}(2^{i}-1)=k$. Then, in mod $2$ cohomology,

$g_{n}^{*}(w_{k})=\sum_{\underline{a}\in\mathcal{X}_{n,k}}\operatorname{res}_{n}\left(w_{[a_{0}]}\odot\big{(}\bigodot_{i,a_{i}\neq
0}\gamma_{i,a_{i}}\big{)}\odot 1_{m}\right),$

where $m\in\mathbb{N}$ is such that the addend is in component $n$.

  2. (2)

Let $p$ be an odd prime. Let $\mathcal{Y}_{n}$ denote the set of sequences
$\underline{a}=(a_{1},a_{2},\dots)\in\mathcal{Y}$ such that $\sum_{i\geq
1}a_{i}p^{i}\leq n$ and $\mathcal{Y}_{n,k}$ denote the set of sequences
$\underline{a}\in\mathcal{Y}_{n}$ such that $\sum_{i\geq 1}a_{i}(p^{i}-1)=2k$.
Then, in mod $p$ cohomology,

$g_{n}^{*}(\mathcal{p}_{k})=\sum_{\underline{a}\in\mathcal{Y}_{n,k}}\big{(}\bigodot_{i,a_{i}\not=0}\gamma_{i,a_{i}}\big{)}\odot
1_{m},$

where $m\in\mathbb{N}$ is such that the addend is in component $n$.

On the contrary, the calculation of $g_{n}^{*}(X_{n})$ when $n$ is even
requires an additional analysis.

######  Proposition 5.6.

Let $p>2$ be a prime. Then $g_{n}^{*}(X_{n})=0$ for all $n>0$ even.

###### Proof.

We preliminarly recall that
$H^{*}(B\operatorname{B}_{n}^{+};\mathbb{F}_{p})\cong
H^{*}(B\Sigma_{n};\mathbb{F}_{p})$ by Theorem 3.7. We prove this identity by
induction on $n$. If $n=2$ (base of the induction), then $X_{2}=0$ because
$H^{*}(B\Sigma_{2};\mathbb{F}_{p})$ is trivial. We now assume that $n>2$.
Then, by the formula for the Euler class of direct sums (see for instance
[4]), $\Delta(X_{n})=\sum_{m=0}^{\frac{n}{2}}X_{2m}\otimes X_{n-2m}$. Hence,
by induction hypothesis, $X_{n}$ is primitive. As
$H^{*}(B\Sigma_{n};\mathbb{F}_{p})$ does not contain any non-zero primitive,
this implies $X_{n}=0$. ∎

######  Example 5.7.

$\mathcal{X}_{4,3}=\\{(3,0,0,\dots),(2,1,0,\dots),(0,0,1,0,\dots)\\}$.
Therefore, by Proposition 5.5 we have

$g_{4}^{*}(w_{3})=(w_{[3]}\odot
1_{2})^{0}+(w_{[2]}\odot\gamma_{1,1})^{0}+\gamma_{2,1}^{+}+\gamma_{2,1}^{-}.$

Similarly, from Proposition 5.5 and the calculations of $\mathcal{X}_{5,k}$ of
Example 5.3 we have

$\displaystyle g_{5}^{*}(w_{2})$ $\displaystyle=(w_{[2]}\odot
1_{3})^{0}+\operatorname{res}(w\odot\gamma_{1,1}\odot 1_{2})=(w_{[2]}\odot
1_{3})^{0}+(\gamma_{1,1}^{2}\odot 1_{3})^{0},$ $\displaystyle
g_{5}^{*}(w_{3})$ $\displaystyle=(w_{[3]}\odot
1_{2})^{0}+(w_{[2]}\odot\gamma_{1,1}\odot
1_{1})^{0}+\operatorname{res}_{n}(w\odot\gamma_{1,2})+(\gamma_{2,1}\odot
1_{1})^{0}$ $\displaystyle=(w_{[3]}\odot
1_{2})^{0}+(w_{[2]}\odot\gamma_{1,1}\odot
1_{1})^{0}+(\gamma_{1,1}^{2}\odot\gamma_{1,1}\odot
1_{1})^{0}+(\gamma_{2,1}\odot 1_{1})^{0},$ $\displaystyle g_{5}^{*}(w_{4})$
$\displaystyle=(w_{[4]}\odot
1_{1})^{0}+(w_{[3]}\odot\gamma_{1,1})^{0}+\operatorname{res}_{n}(w\odot\gamma_{2,1})=(w_{[4]}\odot
1_{1})^{0}+(w_{[3]}\odot\gamma_{1,1})^{0},$ $\displaystyle g_{5}^{*}(w_{5})$
$\displaystyle=(w_{[5]})^{0}.$

Propositions 5.5 and 5.6 and multiplicativity completely determine the
differentials on the Serre spectral sequence associated to
$SO(n)\to\overline{\operatorname{Fl}}_{n}(\mathbb{R})\to
B\operatorname{B}_{n}^{+}$. With our method, we can compute them
systematically, and thus retrieving in a simpler way the results about
$H^{*}(\overline{\operatorname{Fl}}_{n}(\mathbb{R});\mathbb{F}_{2})$ for
$n\leq 5$ proved in [13]. In that article, the authors used a complicated
geometric argument to fully determine the spectral sequences, that we can
completely avoid with our pullback formulas.

##  6\. Unstable cohomology of complete unordered flag varieties

In this section, we present some unstable calculations for the cohomology of
the unordered flag manifolds of low ($n=3,4,5$) orders. We will also present
the mod $p$ cohomology of $\overline{\mathrm{Fl}}_{p}(\mathbb{C})$ for all
prime $p$.

###  6.1. Mod $2$ Cohomology of $\overline{\mathrm{Fl}}_{3}(\mathbb{C})$

Recall from (12) that

$H^{*}(U(3);\mathbb{F}_{2})\cong\Lambda_{\mathbb{F}_{2}}[z_{1},z_{3},z_{5}].$

From Lemma 5.1 and the pullback formula

$\displaystyle d_{2}(z_{1})$ $\displaystyle=c\odot 1_{2}+\gamma_{1,1}^{2}\odot
1_{1},$ $\displaystyle d_{4}(z_{3})$ $\displaystyle=c_{[2]}\odot
1_{1}+c\odot\gamma_{1,1}^{2},$ $\displaystyle d_{6}(z_{5})$
$\displaystyle=c_{[3]}.$

Also, from the Hopf ring structure of $A_{BU(1)}$ in Theorem 2.3,

$H^{*}(BN(3);\mathbb{F}_{2})\cong\frac{\mathbb{F}_{2}[c\odot
1_{2},c_{[2]}\odot 1_{1},c_{[3]},\gamma_{1,1}\odot 1_{1}]}{\big{(}(c\odot
1_{2})\cdot(c_{[2]}\odot 1_{1})\cdot(\gamma_{1,1}\odot
1_{1})+c_{[3]}\cdot(\gamma_{1,1}\odot 1_{1})\big{)}}.$

On the $E_{2}$-page the differential $d_{2}$ is non-zero by Lemma 5.4 and the
following:

$\displaystyle d_{2}((\gamma_{1,1}\odot 1_{1})\cdot z_{1})$
$\displaystyle=(\gamma_{1,1}\odot 1_{1})\cdot(c\odot
1_{2}+\gamma_{1,1}^{2}\odot
1_{1})=c\odot\gamma_{1,1}^{2}+\gamma_{1,1}^{3}\odot 1_{1}.$

This shows that $d_{2}$ is injective on the ideal generated by $z_{1}$. As
before $E_{3}$-page is the same as the $E_{4}$-page as $d_{3}\equiv 0$. Hence,
the $E_{4}$-page is given as follows:

$\displaystyle E_{4}^{*,*}$
$\displaystyle\cong\frac{H^{*}(BN(3);\mathbb{F}_{2})}{\big{(}f_{3}^{*}(c_{1})\big{)}}\otimes\Lambda_{\mathbb{F}_{2}}[z_{3},z_{5}]$
$\displaystyle\cong\frac{\mathbb{F}_{2}[c_{[2]}\odot
1_{1},c_{[3]},\gamma_{1,1}\odot 1_{1}]}{\big{(}(c_{[2]}\odot
1_{1})\cdot(\gamma_{1,1}\odot 1_{1})^{3}+c_{[3]}\cdot(\gamma_{1,1}\odot
1_{1})\big{)}}\otimes\Lambda_{\mathbb{F}_{2}}[z_{3},z_{5}].$

On the $E_{4}$-page the the differential $d_{4}$ is also injective as

$\displaystyle d_{4}((\gamma_{1,1}\odot 1_{3})\cdot z_{3})$
$\displaystyle=(\gamma_{1,1}\odot 1_{1})\cdot(c_{[2]}\odot
1_{1}+c\odot\gamma_{1,1}^{2})=c_{[2]}\gamma_{1,1}\odot
1_{1}+c\odot\gamma_{1,1}^{3}.$

The $E_{6}$-page is therefore given by

$\displaystyle E_{6}^{*,*}$
$\displaystyle\cong\frac{H^{*}(BN(3);\mathbb{F}_{2})}{\big{(}f_{3}^{*}(c_{1}),f_{3}^{*}(c_{2})\big{)}}\otimes\Lambda_{\mathbb{F}_{2}}[z_{5}]$
$\displaystyle\cong\frac{\mathbb{F}_{2}[c_{[3]},\gamma_{1,1}\odot
1_{1}]}{\big{(}(\gamma_{1,1}\odot 1_{1})^{7}+c_{[3]}\cdot(\gamma_{1,1}\odot
1_{1})\big{)}}\otimes\Lambda_{\mathbb{F}_{2}}[z_{5}].$

Note that in $H^{*}(BN(3);\mathbb{F}_{2})/(f_{3}^{*}(c_{1}))$, we have the
relation $c\odot 1_{2}=\gamma_{1,1}^{2}\odot 1_{1}$. Using this identification
we have

$c\odot\gamma_{1,1}^{2}=(c\odot 1_{2})\cdot(\gamma_{1,1}^{2}\odot
1_{1})=(\gamma_{1,1}\odot 1_{1})^{3}$

and hence $c_{[2]}\odot 1_{1}=c\odot\gamma_{1,1}^{2}=(\gamma_{1,1}\odot
1_{1})^{3}$ in
$H^{*}(BN(3);\mathbb{F}_{2})/(f_{3}^{*}(c_{1}),f_{3}^{*}(c_{2}))$. On the
$E_{6}$-page the differential $d_{6}$ is again injective as

$d_{6}((\gamma_{1,1}\odot 1_{1})\cdot z_{5})=c_{[2]}\gamma_{1,1}\odot c.$

Again all higher differentials are zero, and $E_{7}^{*,*}\cong
E_{\infty}^{*,*}$, which gives us the following:

######  Theorem 6.1.

The mod $2$ cohomology ring of $\overline{\mathrm{Fl}}_{3}(\mathbb{C})$ is
given by

$H^{*}(\overline{\mathrm{Fl}}_{3}(\mathbb{C});\mathbb{F}_{2})\cong\frac{H^{*}(BN(3);\mathbb{F}_{2})}{\big{(}f_{3}^{*}(c_{1}),f_{3}^{*}(c_{2}),f_{3}^{*}(c_{3})\big{)}}\cong\frac{\mathbb{F}_{2}[\gamma_{1,1}\odot
1_{1}]}{((\gamma_{1,1}\odot 1_{1})^{7})},$

where $|\gamma_{1,1}\odot 1_{1}|=1$.

######  Corollary 6.2.

The Poincaré series of the mod $2$ cohomology ring of
$\overline{\mathrm{Fl}}_{3}(\mathbb{C})$ is

$\Pi_{\overline{\mathrm{Fl}}_{3}(\mathbb{C})}(t)=1+t+t^{2}+t^{3}+t^{4}+t^{5}+t^{6}.$

Note that this agrees with the previous results obtained in Theorem 5.3 and
Corollary 5.4 of [13] using a different spectral sequence.

###  6.2. Mod $2$ Cohomology of $\overline{\mathrm{Fl}}_{4}(\mathbb{C})$

From (12),
$H^{*}(U(4);\mathbb{F}_{2})\cong\Lambda_{\mathbb{F}_{2}}[z_{1},z_{3},z_{5},z_{7}]$.
Also, from Lemma 5.1 and the pullback formula the differentials in the
spectral sequence $E$ associated with
$U(4)\rightarrow\overline{\mathrm{Fl}}_{4}(\mathbb{C})\rightarrow BN(4)$ are
given by

$\displaystyle d_{2}(z_{1})$ $\displaystyle=c\odot 1_{3}+\gamma_{1,1}^{2}\odot
1_{2},$ $\displaystyle d_{4}(z_{3})$ $\displaystyle=c_{[2]}\odot
1_{2}+c\odot\gamma_{1,1}^{2}\odot 1_{1}+\gamma_{1,2}^{2},$ $\displaystyle
d_{6}(z_{5})$ $\displaystyle=c_{[3]}\odot
1_{1}+c_{[2]}\odot\gamma_{1,1}^{2}+\gamma_{2,1}^{2},$ $\displaystyle
d_{8}(z_{7})$ $\displaystyle=c_{[4]}.$

From the Hopf ring structure of $A_{BU(1)}$, we have that
$H^{*}(BN(4);\mathbb{F}_{2})$ is generated by

$\\{c\odot 1_{3},c_{[2]}\odot 1_{3},c_{[3]}\odot
1_{1},c_{[4]},c_{[2]}\odot\gamma_{1,1},\gamma_{1,1}\odot
1_{2},\gamma_{1,2},\gamma_{2,1}\\},$

with all the relations in Theorem 2.3 and Hopf ring distributivity. By Lemma
5.4, the differential $d_{2}$ non-zero on $\\{c\odot 1_{3},c_{[2]}\odot
1_{2},c_{[3]}\odot 1_{1},c_{[4]}\\}$. On the rank zero decorated gathered
blocks $d_{2}$ is given by

$\displaystyle d_{2}((\gamma_{1,1}\odot 1_{2})\cdot z_{1})$
$\displaystyle=(\gamma_{1,1}\odot 1_{2})\cdot(c\odot
1_{3}+\gamma_{1,1}^{2}\odot 1_{2})=c\odot\gamma_{1,1}\odot
1_{1}+\gamma_{1,1}^{3}\odot 1_{2},$ $\displaystyle
d_{2}((c_{[2]}\odot\gamma_{1,1})\cdot z_{1})$
$\displaystyle=(c_{[2]}\odot\gamma_{1,1})\cdot(c\odot
1_{3}+\gamma_{1,1}^{2}\odot 1_{2})=c^{2}\odot c\odot\gamma_{1,1}+\cdots,$
$\displaystyle d_{2}(\gamma_{1,2}\cdot z_{1})$
$\displaystyle=\gamma_{1,2}\cdot(c\odot 1_{3}+\gamma_{1,1}^{2}\odot
1_{2})=\gamma_{1,1}^{3}\odot\gamma_{1,1},$ $\displaystyle
d_{2}(\gamma_{2,1}\cdot z_{1})$ $\displaystyle=\gamma_{2,1}\cdot(c\odot
1_{3}+\gamma_{1,1}^{2}\odot 1_{2})=0.$

As $d_{2}(\gamma_{2,1}\cdot z_{1})=0$, we have that $d_{2}$ is not injective
on the ideal generated by $z_{1}$ and $\mathrm{ker}(d_{2})$ is the ideal
generated by $(\gamma_{2,1}\cdot z_{1})$. So, the $E_{3}\equiv E_{4}$-page is
given by

$\displaystyle
E_{4}^{*,*}\cong\Big{(}\frac{H^{*}(BN(4);\mathbb{F}_{2})}{\big{(}f_{4}^{*}(c_{1})\big{)}}\oplus(\langle\gamma_{2,1}\cdot
z_{1}\rangle)\Big{)}\otimes\Lambda_{\mathbb{F}_{2}}[z_{3},z_{5},z_{7}],$

where $\langle\gamma_{2,1}\cdot z_{1}\rangle$ denotes the ideal generated by
$\gamma_{2,1}\cdot z_{1}$. As $c\odot 1_{3}=\gamma_{1,1}^{2}\odot 1_{2}$ on
$E_{4}^{*,*}$, we can rewrite the differential $d_{4}$ as
$d_{4}(z_{3})=c_{[2]}\odot 1_{2}+\gamma_{1,1}^{4}\odot
1_{1}+\gamma_{1,2}^{2}$. The injectivity of $d_{4}$, $d_{6}$, and $d_{8}$ on
the ideals generated by $z_{3}$, $z_{5}$, and $z_{7}$ respectively, can be
verified by directly computing these differentials on the rank zero gathered
blocks multiplied by $z_{3}$, $z_{5}$, and $z_{7}$ respectively.

On the first row of the spectral sequence, $\gamma_{1,2}^{2}\gamma_{2,1}\cdot
z_{1}$ is killed by $d_{4}(\gamma_{2,1}\cdot z_{1}z_{3})$,
$\gamma_{2,1}^{3}\cdot z_{1}$ is killed by $d_{6}(\gamma_{2,1}\cdot
z_{1}z_{5})$, and $c_{[4]}\gamma_{2,1}\cdot z_{1}$ is killed by
$d_{8}(\gamma_{2,1}\cdot z_{1}z_{7})$ since

$\displaystyle d_{4}(\gamma_{2,1}\cdot z_{3})$
$\displaystyle=\gamma_{2,1}\cdot(c_{[2]}\odot 1_{2}+\gamma_{1,1}^{4}\odot
1_{1}+\gamma_{1,2}^{2})=\gamma_{1,2}^{2}\gamma_{2,1},$ $\displaystyle
d_{6}(\gamma_{2,1}\cdot z_{5})$ $\displaystyle=\gamma_{2,1}\cdot(c_{[3]}\odot
1_{1}+c_{[2]}\odot\gamma_{1,1}^{2}+\gamma_{2,1}^{2})=\gamma_{2,1}^{3},$
$\displaystyle d_{8}(\gamma_{2,1}\cdot z_{7})$
$\displaystyle=\gamma_{2,1}\cdot c_{[4]}=c_{[4]}\gamma_{2,1}.$

All other higher differentials are zero and we have

$\displaystyle E_{\infty}^{*,*}$
$\displaystyle\cong\frac{H^{*}(BN(4);\mathbb{F}_{2})}{\big{(}f_{4}^{*}(c_{1}),f_{4}^{*}(c_{2}),f_{4}^{*}(c_{3}),f_{4}^{*}(c_{4})\big{)}}\oplus\mathbb{F}_{2}\\{\gamma_{2,1}\cdot
z_{1},\gamma_{1,2}\gamma_{2,1}\cdot z_{1},\gamma_{2,1}^{2}\cdot
z_{1},\gamma_{1,2}\gamma_{2,1}^{2}\cdot z_{1}\\}.$

We record the result of our above computation as the following Theorem.

######  Theorem 6.3.