$(N-(d+1)\ell+1)$-dimensional complex inverse.

The result we prove next corrects [14, Thm. 5.1]. In particular, we follow the
outline of the proof of [14, Thm. 5.1] and alternate at a single place.

######  Theorem 5.26.

Let $d\geq 1$ be an integer. There is no complex $k$-regular embedding
$\mathbb{R}^{d}\longrightarrow\mathbb{C}^{N}$ for $N<\frac{1}{2}(M+k)$, where

$M:=\begin{cases}(d-1)(k-\alpha(k)),&d\text{ is a power of }2,\\\
(d-1)(k-\epsilon(k)),&d\text{ is not a power of }2.\end{cases}$

###### Proof.

Consider a complex $k$-regular embedding
$\mathbb{R}^{d}\longrightarrow\mathbb{C}^{N}$. From Lemma 5.25 the complex
vector bundle $\xi_{\mathbb{R}^{d},k}^{\mathcal{C}}$ admits an
$(N-k)$-dimensional complex inverse. Hence, the real vector bundle
$\xi_{\mathbb{R}^{d},k}^{\oplus 2}$ admits a $2(N-k)$-dimensional real
inverse. In particular, the real vector bundle $\xi_{\mathbb{R}^{d},k}$ admits
a $(2N-k)$-dimensional real inverse. Since
$\bar{w}_{M}(\xi_{\mathbb{R}^{d},k})\neq 0$, Theorem 5.13, we have that

$2N-k\geq M\quad\Longleftrightarrow\quad
N\geq\tfrac{1}{2}(M+k).\vspace{-10pt}$

∎

######  Remark 5.27.

Notice that, compared to [14, Thm. 5.1], a correction was needed for the case
when $d$ is not a power of $2$.

Next we correct [14, Thm. 6.1]. Again we proceed in the footsteps of the proof
of [14, Thm. 6.1].

######  Theorem 5.28.

Let $d\geq 1$ and $\ell\geq 1$ be integers. There is no complex $\ell$-skew
embedding $\mathbb{C}^{d}\longrightarrow\mathbb{C}^{N}$ for

$N\leq d+\frac{1}{2}\big{(}M-\ell-2\big{)},$

where

$M:=\begin{cases}2^{\gamma(d)+1}-2d-1,&\ell=2,\\\
(2d-1)(\ell-\alpha(\ell)),&d\text{ is a power of }2,\\\
(2^{\gamma(d)+1}-2d-1)(\ell-\epsilon(\ell))/2,&d\geq 3\text{ is not power of
}2.\end{cases}$

###### Proof.

Consider a complex $\ell$-skew embedding
$\mathbb{C}^{d}\longrightarrow\mathbb{C}^{N}$. According to Lemma 5.25 the
complex vector bundle $(\xi_{\mathbb{C}^{d},\ell}^{\mathbb{C}})^{\oplus(d+1)}$
admits an $(N-(d+1)\ell+1)$-dimensional complex inverse. Hence, the real
vector bundle $\xi_{\mathbb{R}^{2d},\ell}^{\oplus 2(d+1)}$ admits a
$2(N-(d+1)\ell+1)$-dimensional real inverse. Consequently, the real vector
bundle $\xi_{\mathbb{R}^{2d},\ell}^{\oplus(2d+1)}$ admits a
$(2N-2(d+1)\ell+2+\ell)$-dimensional real inverse. From Theorem 5.17 follows
that $\bar{w}_{M}(\xi_{\mathbb{R}^{2d},\ell}^{\oplus(2d+1)})\neq 0$ and
therefore:

$2N-2(d+1)\ell+2+\ell\geq M\quad\Longleftrightarrow\quad
N\geq\frac{1}{2}\big{(}2(d+1)-2-\ell+M\big{)}.$

∎

This completes the corrections of the paper [14].

##  6\. More bounds for highly regular embeddings

In this section we present computations that yield additional bounds for the
existence of highly regular embeddings. For this we present an alternative
approach in the study of Stiefel–Whitney classes of the vector bundle
$\xi_{\mathbb{R}^{d},2^{m}}$. In particular, we utilize a specific
decomposition of the pull-back vector bundle
$\rho_{d,2^{m}}^{*}\xi_{\mathbb{R}^{d},2^{m}}$ of the vector bundle
$\xi_{\mathbb{R}^{d},2^{m}}$ via the map
$\rho_{d,2^{m}}\colon\operatorname{Pe}(\mathbb{R}^{d},2^{m})/\mathcal{S}_{2^{m}}\longrightarrow\operatorname{F}(\mathbb{R}^{d},2^{m})/\mathfrak{S}_{2^{m}}$.

First, we introduce several real $\mathcal{S}_{2^{m}}$-representations and
consider properties of the associated vector bundles.

###  6.1. Examples of $\mathcal{S}_{2^{m}}$-representations and associated
vector bundles

In this text the real vector space $\mathbb{R}^{k}$ is often considered as the
real $k$-dimensional $\mathfrak{S}_{k}$-representation with the action defined
by

$\pi\cdot(a_{1},\ldots,a_{k}):=(a_{\pi^{-1}(1)},\ldots,a_{\pi^{-1}(k)}),$

where $\pi\in\mathfrak{S}_{k}$ and $(a_{1},\ldots,a_{k})\in\mathbb{R}^{k}$.
Furthermore, its $(n-1)$-dimensional vector subspace

$W_{k}:=\\{(a_{1},\ldots,a_{k})\in\mathbb{R}^{k}:a_{1}+\cdots+a_{k}=0\\}$

is an $\mathfrak{S}_{k}$-subrepresentation. For every subgroup $G$ of
$\mathfrak{S}_{k}$ both vector spaces $\mathbb{R}^{k}$ and $W_{k}$ become real
$G$-representations via the corresponding inclusion homomorphism
$G\hookrightarrow\mathfrak{S}_{k}$.

####  6.1.1. Examples of $\mathcal{S}_{2^{m}}$-representations

Let $k=2^{m}$ for some integer $m\geq 1$. Recall that the group
$\mathcal{S}_{2^{m}}$, as a subgroup of $\mathfrak{S}_{2^{m}}$, was introduced
in Definition 2.1 and the corresponding inclusion homomorphism was denotes by
$\iota_{m}\colon\mathcal{S}_{2^{m}}\longrightarrow\mathfrak{S}_{2^{m}}$.
Inductively, we can describe the group $\mathcal{S}_{2^{m}}$ via the exact
sequence of groups

$\textstyle{1\ignorespaces\ignorespaces\ignorespaces\ignorespaces\
}$$\textstyle{\
\mathcal{S}_{2^{m-1}}\times\mathcal{S}_{2^{m-1}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\
}$$\scriptstyle{\vartheta_{m}}$$\textstyle{\
\mathcal{S}_{2^{m}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\
}$$\scriptstyle{\varsigma_{m}}$$\textstyle{\
\mathbb{Z}_{2}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\
}$$\textstyle{\ 1,}$ (82)

where the inclusion map $\vartheta_{m}$ is defined in the expected way.

Now we define inductively a sequence of real
$\mathcal{S}_{2^{m}}$-representations. Let $M_{m}[m]$ be the
$\mathcal{S}_{2^{m}}$-representation obtained as the pull-back of the real
$1$-dimensional $\mathbb{Z}_{2}$-representation $W_{2}$ via the surjection
$\varsigma_{m}$ from the exact sequence (82). This means that $M_{m}[m]=W_{2}$
as a vector space, and that $\pi\cdot v:=\varsigma_{m}(\pi)\cdot v$ for every
$v\in M_{m}[m]$ and every $\pi\in\mathcal{S}_{2^{m}}$.

Assume that for every integer $1\leq i\leq m-1$ we have defined the sequence
of $\mathcal{S}_{2^{i}}$-representations $M_{i}[1],\dots,M_{i}[i]$ with
$\dim(M_{i}[j])=2^{i-j}$, $1\leq j\leq i$. Next, for $i=m$ and $1\leq j\leq
m-1$, we define the $\mathcal{S}_{2^{m}}$-representation $M_{m}[j]$ to be, on
the level of vector spaces, the direct sum $M_{m}[j]:=M_{m-1}[j]\oplus
M_{m-1}[j]$. The action of $\mathcal{S}_{2^{m}}$ on $M_{m}[j]$ is given by:

$\displaystyle(h_{1},h_{2})\cdot(v_{1},v_{2})$ $\displaystyle:=(h_{1}\cdot
v_{1},h_{2}\cdot v_{2}),$
$\displaystyle(h_{1},h_{2},\omega)\cdot(v_{1},v_{2})$
$\displaystyle:=(h_{2}\cdot v_{2},h_{1}\cdot v_{1}),$

where
$(h_{1},h_{2})\in\mathcal{S}_{2^{m-1}}\times\mathcal{S}_{2^{m-1}}\subseteq(\mathcal{S}_{2^{m-1}}\times\mathcal{S}_{2^{m-1}})\rtimes\mathbb{Z}_{2}$,
$\omega$ is the generator of the subgroup
$\mathbb{Z}_{2}\subseteq(\mathcal{S}_{2^{m-1}}\times\mathcal{S}_{2^{m-1}})\rtimes\mathbb{Z}_{2}$,
and $(v_{1},v_{2})\in M_{m-1}[j]\oplus M_{m-1}[j]$. It follows directly that
$\dim(M_{m}[j])=2^{m-j}$ for all $1\leq j\leq m$.

The vector space $W_{2^{m}}$, now considered as an
$\mathcal{S}_{2^{m}}$-representation, we denote by $L_{m}$. The following
decomposition of $\mathcal{S}_{2^{m}}$-representations holds.

######  Lemma 6.1.

For every integer $m\geq 1$ there is an isomorphism of real
$\mathcal{S}_{2^{m}}$-representations

$L_{m}\cong M_{m}[1]\oplus\dots\oplus M_{m}[m].$

####  6.1.2. Associated vector bundles

Let $m\geq 1$ be an integer, and let $d\geq 1$ be an integer, or $d=\infty$.
To every $\mathcal{S}_{2^{m}}$-representation introduced in the previous
section we associate real vector bundles in the following way:

$\textstyle{\lambda_{d,m}:\quad
L_{m}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ }$$\textstyle{\
\operatorname{Pe}(\mathbb{R}^{d},2^{m})\times_{\mathcal{S}_{2^{m}}}L_{m}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\
}$$\textstyle{\ \operatorname{Pe}(\mathbb{R}^{d},2^{m})/\mathcal{S}_{2^{m}},}$
(83)

$\textstyle{\mu_{d,m}[j]:\quad
M_{m}[j]\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ }$$\textstyle{\
\operatorname{Pe}(\mathbb{R}^{d},2^{m})\times_{\mathcal{S}_{2^{m}}}M_{m}[j]\ignorespaces\ignorespaces\ignorespaces\ignorespaces\
}$$\textstyle{\ \operatorname{Pe}(\mathbb{R}^{d},2^{m})/\mathcal{S}_{2^{m}},}$
(84)

where $1\leq j\leq m$. Note that

  * —

$\lambda_{1,d}=\mu_{1,d}[1]$ is the Hopf line bundle over $\mathbb{R}{\rm
P}^{d-1}\cong\operatorname{Pe}(\mathbb{R}^{d},2)/\mathcal{S}_{2}$,

  * —

$\mu_{d,m}[m]$ is the pull-back vector bundle of the line bundle
$\lambda_{d,1}$ via the map

$\big{(}\operatorname{Pe}(\mathbb{R}^{d},2^{m-1})/\mathcal{S}_{2^{m-1}}\times\operatorname{Pe}(\mathbb{R}^{d},2^{m-1}\big{)}/\mathcal{S}_{2^{m-1}})\times_{\mathbb{Z}_{2}}\operatorname{Pe}(\mathbb{R}^{d},2)\longrightarrow\operatorname{Pe}(\mathbb{R}^{d},2)/\mathcal{S}_{2},$

  * —

$\lambda_{d,m}$ and $\mu_{d,m}[j]$ are pull-backs of $\lambda_{\infty,m}$ and
$\mu_{\infty,m}[j]$ along the map

$\kappa_{d,2^{m}}/\mathcal{S}_{2^{m}}\colon\operatorname{Pe}(\mathbb{R}^{d},2^{m})/\mathcal{S}_{2^{m}}\longrightarrow\operatorname{Pe}(\mathbb{R}^{\infty},2^{m})/\mathcal{S}_{2^{m}},$

  * —

$\lambda_{d,m}$ is the pull-back of $\zeta_{\mathbb{R}^{d},2^{m}}$ via the map

$\rho_{d,2^{m}}\colon\operatorname{Pe}(\mathbb{R}^{d},2^{m})/\mathcal{S}_{2^{m}}\longrightarrow\operatorname{F}(\mathbb{R}^{d},2^{m})/\mathfrak{S}_{2^{m}}.$

The vector bundles $\mu_{d,m}[j]$ for different values of $m$, are connected
via the wreath square of bundles operations as follows:

$S^{2}\mu_{\infty,m-1}[j]\cong\mu_{\infty,m}[j]\qquad\text{and}\qquad
S^{2,d}\mu_{d,m-1}[j]\cong\mu_{d,m}[j],$ (85)

for all integers $d\geq 2$, $m\geq 1$, and all $1\leq j\leq m-1$. For details
about wreath square operations see Section 7.3.

The decomposition from Lemma 6.1 implies the following Whitney sum
decomposition on the vector bundle $\lambda_{d,m}$.

######  Lemma 6.2.

For every integer $m\geq 1$, and every integer $d\geq 1$ or $d=\infty$, there
is an isomorphism of real $\mathcal{S}_{2^{m}}$-representations

$\lambda_{d,m}\cong\mu_{d,m}[1]\oplus\dots\oplus\mu_{d,m}[m].$

Using the relation (44) and the decomposition from the previous lemma we get
the following equalities.

######  Lemma 6.3.

For every integer $m\geq 1$ and every integer $1\leq j\leq m$:

$w_{2^{m}-1}(\lambda_{m,\infty})=D_{m,0}\qquad\text{and}\qquad
w_{2^{m-j}}(\mu_{m,\infty}[j])=V_{m,m-j}.$

From the fact that
$w(\zeta_{\mathbb{R}^{d},2^{m}})=w(\xi_{\mathbb{R}^{d},2^{m}})$, Lemma 5.5,
Lemma 5.7, and Theorem 4.1, the injectivity of the homomorphism

$\rho_{d,2^{m}}^{*}\colon
H^{*}(\operatorname{F}(\mathbb{R}^{d},2^{m})/\mathfrak{S}_{2^{m}};\mathbb{F}_{2})\longrightarrow
H^{*}(\operatorname{Pe}(\mathbb{R}^{d},2^{m})/\mathcal{S}_{2^{m}};\mathbb{F}_{2}),$

we deduce the following facts.

######  Lemma 6.4.

Let $m\geq 1$ be an integer.

  1. (1)

If $d=2^{a}$ for an integer $a\geq 1$, then

$\overline{w}_{(d-1)(2^{m}-1)}(\lambda_{d,m})=\overline{w}_{(d-1)(2^{m}-1)}(\rho_{d,2^{m}}^{*}\zeta_{\mathbb{R}^{d},2^{m}})=\\\
\overline{w}_{(d-1)(2^{m}-1)}(\rho_{d,2^{m}}^{*}\xi_{\mathbb{R}^{d},2^{m}})=\rho_{d,2^{m}}^{*}\big{(}\overline{w}_{(d-1)(2^{m}-1)}(\xi_{\mathbb{R}^{d},2^{m}})\big{)}\neq
0.$

  2. (2)

If $d\geq 2$ is an integer, then

$\overline{w}_{(d-1)2^{m-1}}(\lambda_{d,m})=\overline{w}_{(d-1)2^{m-1}}(\rho_{d,2^{m}}^{*}\zeta_{\mathbb{R}^{d},2^{m}})=\\\
\overline{w}_{(d-1)2^{m-1}}(\rho_{d,2^{m}}^{*}\xi_{\mathbb{R}^{d},2^{m}})=\rho_{d,2^{m}}^{*}\big{(}\overline{w}_{(d-1)2^{m-1}}(\xi_{\mathbb{R}^{d},2^{m}})\big{)}\neq
0.$

Furthermore, from the fact that $\mu_{d,m}[m]$ is the pull-back of the line
bundle $\lambda_{d,1}$, and the description of the cohomology of
$\operatorname{Pe}(\mathbb{R}^{d},2^{m})/\mathcal{S}_{2^{m}}$ given in Section
3 we compute $\mu_{d,m}[m]$.

######  Lemma 6.5.

Let $d\geq 1$ and $m\geq 1$ be integers. Then

$w(\mu_{d,m}[m])=1+f,$

where the class $f\in
H^{1}(\operatorname{Pe}(\mathbb{R}^{d},2^{m})/\mathcal{S}_{2^{m}};\mathbb{F}_{2})$
was denoted by $V_{m,1}$ in the proof of Lemma 3.10. In particular $f^{d}=0$.

###  6.2. The Key Lemma and its consequences

Let $m\geq 1$ and $d\geq 1$ be integers. For arbitrary non-negative integers
$r_{1},\dots,r_{m}$ we define the vector bundle
$\psi_{d,m}[r_{1},\dots,r_{m}]$ by:

$\psi_{d,m}[r_{1},\dots,r_{m}]:=\mu_{d,m}[1]^{\oplus
r_{1}}\oplus\cdots\oplus\mu_{d,m}[m]^{\oplus r_{m}}.$

Now we will prove the key technical lemma of this section.

######  Lemma 6.6 (The Key Lemma).

Let $m\geq 1$, $d\geq 1$ and $\ell$ be integers, and let $r_{1},\dots,r_{m}$
be non-negative integers. If the binomial coefficient

${(r_{1}2^{m-2}+\dots+r_{m-1}2^{0})+r_{m}-(d-1+\ell)(2^{m-1}-1)-\ell}\choose{d-1}$
(86)

is odd, then

$w_{(d-1)(2^{m}-1)}(\lambda_{d,m}^{-\ell}\oplus\psi_{d,m}[r_{1},\dots,r_{m}])\neq
0$

as an element of the cohomology group
$H^{(d-1)(2^{m}-1)}(\operatorname{Pe}(\mathbb{R}^{d},2^{m})/\mathcal{S}_{2^{m}};\mathbb{F}_{2})$.
Here $\lambda_{d,m}^{-\ell}$ denotes an inverse of the vector bundle
$\lambda_{d,m}^{\oplus\ell}$.

###### Proof.

We prove the claim of the lemma by induction on $m\geq 1$. In the case $m=1$
the condition (86) reads ${{r_{1}-\ell}\choose{d-1}}\neq 0$ in
$\mathbb{F}_{2}$. Furthermore, $\psi_{d,1}[r_{1}]=\mu_{d,1}[1]^{\oplus r_{1}}$
and $\lambda_{d,1}\cong\mu_{d,1}[1]$. Consequently,
$\lambda_{d,1}^{-\ell}\oplus\psi_{d,1}[r_{1}]\cong\mu_{d,1}[1]^{\oplus(r_{1}-\ell)}$,
and so

$w(\lambda_{d,1}^{-\ell}\oplus\psi_{d,1}[r_{1}])=w(\mu_{d,1}[1]^{\oplus(r_{1}-\ell)})=w(\mu_{d,1}[1])^{r_{1}-\ell}=(1+f)^{r_{1}-\ell}$

in
$H^{*}(\operatorname{Pe}(\mathbb{R}^{d},2)/\mathcal{S}_{2};\mathbb{F}_{2})\cong
H^{*}(\mathbb{R}{\rm P}^{d-1};\mathbb{F}_{2})=\mathbb{F}_{2}[f]/\langle
f\rangle$. Here we use the fact that the vector bundle $\mu_{d,1}[1]$ is the
Hopf line bundle, and so $w(\mu_{d,1}[1])=1+f$. Hence, we have that

$w_{(d-1)(2^{1}-1)}(\lambda_{d,1}^{-\ell}\oplus\psi_{d,1}[r_{1}])=w_{d-1}(\lambda_{d,1}^{-\ell}\oplus\psi_{d,1}[r_{1}])={{r_{1}-\ell}\choose{d-1}}w_{1}^{d-1}\neq
0.$

Let $m\geq 2$, and assume, as an induction hypothesis, that

$w_{(d-1)(2^{m-1}-1)}(\lambda_{d,m-1}^{-\ell}\oplus\psi_{d,m-1}[r_{1},\dots,r_{m-1}])\neq
0$

as an element of
$H^{(d-1)(2^{m-1}-1)}(\operatorname{Pe}(\mathbb{R}^{d},2^{m-1})/\mathcal{S}_{2^{m-1}};\mathbb{F}_{2})$.
Then there exists a vector bundle $\varpi$ of dimension $(d-1)(2^{m-1}-1)$
over $\operatorname{Pe}(\mathbb{R}^{d},2^{m-1})/\mathcal{S}_{2^{m-1}}$ with
the property that for some integers $N_{0}\geq\ell$ and $N_{1}\geq 1$ there is
an isomorphism of vector bundles

$\displaystyle\varpi\oplus\lambda_{d,m-1}^{\oplus\ell}\oplus\tau_{d,m-1}^{\oplus
N_{0}}$
$\displaystyle\cong\psi_{d,m-1}[r_{1},\dots,r_{m-1}]\oplus\tau_{d,m-1}^{\oplus
N_{1}}$ $\displaystyle\cong\mu_{d,m-1}[1]^{\oplus
r_{1}}\oplus\cdots\oplus\mu_{d,m-1}[m-1]^{\oplus
r_{m-1}}\oplus\tau_{d,m-1}^{\oplus N_{1}}.$ (87)

Here $\tau_{d,m-1}$ denotes the trivial line bundle over
$\operatorname{Pe}(\mathbb{R}^{d},2^{m-1})/\mathcal{S}_{2^{m-1}}$.
Consequently,

$w_{(d-1)(2^{m-1}-1)}(\varpi)=w_{(d-1)(2^{m-1}-1)}(\lambda_{d,m-1}^{-\ell}\oplus\psi_{d,m-1}[r_{1},\dots,r_{m-1}])\neq
0$ (88)

does not vanish — by induction hypothesis.

Now we apply $(d-1)$-partial wreath square $S^{2,d}$ to the isomorphism of
vector bundles (6.2) and get the following isomorphism of vector bundles

$S^{2,d}\varpi\oplus\lambda_{d,m}^{\oplus\ell}\oplus\tau_{d,m}^{\oplus
N_{0}}\oplus\mu_{d,m}[m]^{\oplus N_{0}-\ell}\cong\\\ \mu_{d,m}[1]^{\oplus
r_{1}}\oplus\cdots\oplus\mu_{d,m}[m-1]^{\oplus
r_{m-1}}\oplus\tau_{d,m}^{\oplus N_{1}}\oplus\mu_{d,m}[m]^{\oplus N_{1}},$
(89)

over the base space
$S^{2,d}\operatorname{Pe}(\mathbb{R}^{d},2^{m-1})/\mathcal{S}_{2^{m-1}}\cong\operatorname{Pe}(\mathbb{R}^{d},2^{m})/\mathcal{S}_{2^{m}}$.
Indeed, by direct inspection and using (85) and (106), we get that

$S^{2,d}\tau_{d-1,m}\cong\tau_{d,m}\oplus\mu_{d,m}[m]\qquad\text{and}\qquad
S^{2,d}\mu_{d,m-1}[j]\cong\mu_{d,m}[j]$

for all $1\leq j\leq m-1$. Consequently, from Lemma 6.2 we get that

$S^{2}\lambda_{d,m-1}\cong
S^{2}(\mu_{d,m-1}[1]\oplus\dots\oplus\mu_{d,m-1}[m-1])\cong\mu_{d,m}[1]\oplus\dots\oplus\mu_{d,m}[m-1].$

Collecting all these facts together we compute

$\varpi\oplus\lambda_{d,m-1}^{\oplus\ell}\oplus\tau_{d,m-1}^{\oplus N_{0}}=\\\
\varpi\oplus(\mu_{d,m-1}[1]\oplus\dots\oplus\mu_{d,m-1}[m-1])^{\oplus\ell}\oplus\tau_{d,m-1}^{\oplus
N_{0}}\\\ \lx@xy@svg{\hbox{\raise 0.0pt\hbox{\kern
4.66496pt\hbox{\ignorespaces\ignorespaces\ignorespaces\hbox{\vtop{\kern
0.0pt\offinterlineskip\halign{\entry@#!@&&\entry<EMAIL_ADDRESS>0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern 3.0pt\raise
0.0pt\hbox{$\textstyle{~{}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$}}}}}}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces{\hbox{\kern
0.0pt\raise-3.0pt\hbox{\hbox{\kern 0.0pt\raise
0.0pt\hbox{\lx@xy@stopper}}}}}{\hbox{\lx@xy@droprule}}\ignorespaces\ignorespaces\ignorespaces{\hbox{\kern
0.0pt\raise-15.0pt\hbox{{}\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern
3.0pt\hbox{\hbox{\kern
0.0pt\raise-3.0861pt\hbox{$\scriptstyle{S^{2,d}}$}}}\kern
3.0pt}}}}}}\ignorespaces{\hbox{\kern 0.0pt\raise-27.0pt\hbox{\hbox{\kern
0.0pt\raise
0.0pt\hbox{\lx@xy@tip{1}\lx@xy@tip{-1}}}}}}{\hbox{\lx@xy@droprule}}{\hbox{\lx<EMAIL_ADDRESS>0.0pt\raise 0.0pt\hbox{\hbox{\kern 3.0pt\raise
0.0pt\hbox{$\textstyle{~{}}$}}}}}}}\ignorespaces}}}}\ignorespaces\\\
S^{2,d}\varpi\oplus(\mu_{d,m}[1]\oplus\dots\oplus\mu_{d,m}[m-1])^{\oplus\ell}\oplus\tau_{d,m}^{N_{0}}\oplus\mu_{d,m}[m]^{\oplus
N_{0}}\\\ \cong
S^{2,d}\varpi\oplus\lambda_{d,m}^{\oplus\ell}\oplus\tau_{d,m}^{N_{0}}\oplus\mu_{m}[m]^{\oplus
N_{0}-\ell}.$

On the other hand directly follows that

$\mu_{d,m-1}[1]^{\oplus r_{1}}\oplus\cdots\oplus\mu_{d,m-1}[m-1]^{\oplus
r_{m-1}}\oplus\tau_{d,m-1}^{\oplus N_{1}}\\\ \lx@xy@svg{\hbox{\raise
0.0pt\hbox{\kern
4.66496pt\hbox{\ignorespaces\ignorespaces\ignorespaces\hbox{\vtop{\kern
0.0pt\offinterlineskip\halign{\entry@#!@&&\entry<EMAIL_ADDRESS>0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern 3.0pt\raise
0.0pt\hbox{$\textstyle{~{}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$}}}}}}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces{\hbox{\kern
0.0pt\raise-3.0pt\hbox{\hbox{\kern 0.0pt\raise
0.0pt\hbox{\lx@xy@stopper}}}}}{\hbox{\lx@xy@droprule}}\ignorespaces\ignorespaces\ignorespaces{\hbox{\kern
0.0pt\raise-15.0pt\hbox{{}\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern
3.0pt\hbox{\hbox{\kern
0.0pt\raise-3.0861pt\hbox{$\scriptstyle{S^{2,d}}$}}}\kern
3.0pt}}}}}}\ignorespaces{\hbox{\kern 0.0pt\raise-27.0pt\hbox{\hbox{\kern
0.0pt\raise
0.0pt\hbox{\lx@xy@tip{1}\lx@xy@tip{-1}}}}}}{\hbox{\lx@xy@droprule}}{\hbox{\lx<EMAIL_ADDRESS>0.0pt\raise 0.0pt\hbox{\hbox{\kern 3.0pt\raise
0.0pt\hbox{$\textstyle{~{}}$}}}}}}}\ignorespaces}}}}\ignorespaces\\\
\mu_{d,m}[1]^{\oplus r_{1}}\oplus\cdots\oplus\mu_{d,m}[m-1]^{\oplus
r_{m-1}}\oplus\tau_{d,m}^{\oplus N_{1}}\oplus\mu_{d,m}[m]^{\oplus N_{1}}.$

Thus, the isomorphism (89) holds.

For an arbitrary vector bundle $\eta$ over
$\operatorname{Pe}(\mathbb{R}^{d},2^{m})/\mathcal{S}_{2^{m}}$ the isomorphism
of vector bundles (89) yields the isomorphism

$\eta\oplus
S^{2,d}\varpi\oplus\lambda_{d,m}^{\oplus\ell}\oplus\tau_{d,m}^{\oplus
N_{0}}\oplus\mu_{d,m}[m]^{\oplus N_{0}-\ell}\cong\\\
\eta\oplus\mu_{d,m}[1]^{\oplus r_{1}}\oplus\cdots\oplus\mu_{d,m}[m-1]^{\oplus
r_{m-1}}\oplus\tau_{d,m}^{\oplus N_{1}}\oplus\mu_{d,m}[m]^{\oplus N_{1}}.$
(90)

Take $\eta$ to be stable equivalent to the vector bundle $\mu_{d,m}[m]^{\oplus
r_{m}+N_{0}-\ell-N_{1}}$. Then the vector bundles

$\eta\oplus
S^{2,d}\varpi\qquad\text{and}\qquad\lambda_{d,m}^{-\ell}\oplus\psi_{d,m}[r_{1},\dots,r_{m}]$

are stable equivalent. Indeed, if $\eta$ is stable equivalent to
$\mu_{d,m}[m]^{\oplus r_{m}+N_{0}-\ell-N_{1}}$, then there exist integers
$M_{1},M_{2}\geq 0$, and an isomorphism

$\eta\oplus\tau_{d,m}^{\oplus M_{1}}\cong\mu_{d,m}[m]^{\oplus
r_{m}+N_{0}-\ell-N_{1}}\oplus\tau_{d,m}^{\oplus M_{2}}.$

Now the isomorphism (90) implies that

$\eta\oplus
S^{2,d}\varpi\oplus\lambda_{d,m}^{\oplus\ell}\oplus\tau_{d,m}^{\oplus
N_{0}+M_{1}}\oplus\mu_{d,m}[m]^{\oplus N_{0}-\ell}\cong\\\
(\mu_{d,m}[1]^{\oplus r_{1}}\oplus\cdots\oplus\mu_{d,m}[m-1]^{\oplus
r_{m-1}})\oplus\mu_{d,m}[m]^{\oplus r_{m}}\oplus\\\ \mu_{d,m}[m]^{\oplus
N_{0}-\ell-N_{1}}\oplus\tau_{d,m}^{\oplus
N_{1}+M_{2}}\oplus\mu_{d,m}[m]^{\oplus N_{1}}.$

Hence,

$(\eta\oplus
S^{2,d}\varpi)\oplus\lambda_{d,m}^{\oplus\ell}\oplus\tau_{d,m}^{\oplus
N_{0}+M_{1}}\oplus\mu_{d,m}[m]^{\oplus N_{0}-\ell}\cong\\\
\psi_{d,m}[r_{1},\dots,r_{m}]\oplus\tau_{d,m}^{\oplus
N_{1}+M_{2}}\oplus\mu_{d,m}[m]^{\oplus N_{0}-\ell},$

and consequently the vector bundles $\eta\oplus S^{2,d}\varpi$ and
$\lambda_{d,m}^{-\ell}\oplus\psi_{d,m}[r_{1},\dots,r_{m}]$ are stable
equivalent. In particular,

$w(\lambda_{d,m}^{-\ell}\oplus\psi_{d,m}[r_{1},\dots,r_{m}])=w(\eta\oplus
S^{2,d}\varpi)=w(\mu_{d,m}[m]^{\oplus r_{m}+N_{0}-\ell-N_{1}}\oplus
S^{2,d}\varpi).$

Now we continue computation of the total Stiefel–Whitney class of the vector
bundle $\lambda_{d,m}^{-\ell}\oplus\psi_{d,m}[r_{1},\dots,r_{m}]$ as follows:

$\displaystyle w(\lambda_{d,m}^{-\ell}\oplus\psi_{d,m}[r_{1},\dots,r_{m}])$
$\displaystyle=$ $\displaystyle w(\mu_{d,m}[m]^{\oplus r_{m}+N_{0}-\ell-
N_{1}}\oplus S^{2,d}\varpi)$ $\displaystyle=$ $\displaystyle
w(\mu_{d,m}[m])^{\oplus r_{m}+N_{0}-\ell-N_{1}}\cdot w(S^{2,d}\varpi)$
$\displaystyle\overset{\text{Lem.\,\ref{lem : SW class of mu[m]}}}{=}$
$\displaystyle(1+f)^{r_{m}+N_{0}-\ell-N_{1}}\cdot w(S^{2,d}\varpi)$
$\displaystyle=$ $\displaystyle(1+f)^{r_{m}+N_{0}-\ell-N_{1}}\cdot
s^{2,d}(\varpi)$ $\displaystyle=$
$\displaystyle\Big{(}\sum_{i=0}^{r_{m}+N_{0}-\ell-N_{1}}{{r_{m}+N_{0}-\ell-
N_{1}\choose i}}\,f^{i}\Big{)}\cdot s^{2,d}(\varpi)$
$\displaystyle\overset{f^{d}=0}{=}$
$\displaystyle\Big{(}\sum_{i=0}^{d-1}{{r_{m}+N_{0}-\ell-N_{1}\choose
i}}\,f^{i}\Big{)}\cdot s^{2,d}(\varpi).$

From Corollary 7.17 applied to the $(d-1)(2^{m-1}-1)$-dimensional vector
bundle $\varpi$ we have that

$s^{2,d}(\varpi)=\sum_{0\leq r<s\leq N}T(w_{r}(\varpi)\otimes
w_{s}(\varpi))+\\\ \sum_{0\leq r\leq N}\,\sum_{0\leq
j\leq\min\\{N-r,d-1\\}}{N-r\choose j}P(w_{r}(\varpi))f^{j},$

where $N=(d-1)(2^{m-1}-1)$.

Combining the last two equations we get

$\displaystyle w(\lambda_{d,m}^{-\ell}\oplus\psi_{d,m}[r_{1},\dots,r_{m}])$
$\displaystyle=$ $\displaystyle\Big{(}\sum_{i=0}^{d-1}{{r_{m}+N_{0}-\ell-
N_{1}\choose i}}\,f^{i}\Big{)}\cdot$ $\displaystyle\Big{(}\sum_{0\leq r<s\leq
N}T(w_{r}(\varpi)\otimes w_{s}(\varpi))+$ $\displaystyle\sum_{0\leq r\leq
N}\,\sum_{0\leq j\leq\min\\{N-r,d-1\\}}{N-r\choose
j}P(w_{r}(\varpi))f^{j}\Big{)}$ $\displaystyle\overset{\eqref{eq :
multiplication Q and t-a}}{=}$ $\displaystyle\sum_{0\leq r<s\leq
N}T(w_{r}(\varpi)\otimes w_{s}(\varpi))+$
$\displaystyle\sum_{i=0}^{d-1}\sum_{0\leq r\leq N}\sum_{0\leq
j\leq\min\\{N-r,d-1\\}}$ $\displaystyle{{r_{m}+N_{0}-\ell-N_{1}\choose
i}}{N-r\choose j}P(w_{r}(\varpi))\,f^{i+j}.$

Note that

  * —

$\deg(T(w_{r}(\varpi)\otimes w_{s}(\varpi)))\leq 2N-1=(d-1)(2^{m}-2)-1$, and

  * —

$\deg(P(w_{r}(\varpi))f^{i+j})\leq 2N+d-1=(d-1)(2^{m}-1)$.

Consequently, we have that

$w_{(d-1)(2^{m}-1)}(\lambda_{d,m}^{-\ell}\oplus\psi_{d,m}[r_{1},\dots,r_{m}])={{r_{m}+N_{0}-\ell-
N_{1}\choose d-1}}P(w_{N}(\varpi))\,f^{d-1}.$

Before making final arguments that the Stiefel–Whitney class

$w_{(d-1)(2^{m}-1)}(\lambda_{d,m}^{-\ell}\oplus\psi_{d,m}[r_{1},\dots,r_{m}])$

does not vanish let us review all the assumptions we have:

  * —

${{(r_{1}2^{m-2}+\dots+r_{m-1}2^{0})+r_{m}-(d-1+\ell)(2^{m-1}-1)-\ell}\choose{d-1}}\neq
0\in\mathbb{F}_{2}$ — the assumption (86) from the statement of the lemma, and

  * —

$w_{(d-1)(2^{m-1}-1)}(\varpi)\neq 0$ — the induction hypothesis (88).

Hence, by the induction hypothesis we know that the class
$P(w_{N}(\varpi))\,f^{d-1}$ does not vanish. For the binomial coefficient
${{r_{m}+N_{0}-\ell-N_{1}\choose d-1}}$ note that the isomorphism of vector
bundles (6.2), by evaluating dimensions, implies that

$N_{0}-N_{1}=(r_{1}2^{m-2}+\dots+r_{m-1}2^{0})-(d-1+\ell)(2^{m-1}-1),$

or in other word

$r_{m}+N_{0}-\ell-
N_{1}=(r_{1}2^{m-2}+\dots+r_{m-1}2^{0})+r_{m}-(d-1+\ell)(2^{m-1}-1)-\ell.$

Therefore, by the assumption (86)

${{r_{m}+N_{0}-\ell-N_{1}\choose d-1}}\neq 0\in\mathbb{F}_{2}.$

This completes the proof that

$w_{(d-1)(2^{m}-1)}(\lambda_{d,m}^{-\ell}\oplus\psi_{d,m}[r_{1},\dots,r_{m}])\neq
0.$

∎

######  Remark 6.7.

Observe that the proof of the previous lemma actually yields the following
equivalence:

${{(r_{1}2^{m-2}+\dots+r_{m-1}2^{0})+r_{m}-(d-1+\ell)(2^{m-1}-1)-\ell}\choose{d-1}}\neq
0\in\mathbb{F}_{2}$

if and only if

$w_{(d-1)(2^{m}-1)}(\lambda_{d,m}^{-\ell}\oplus\psi_{d,m}[r_{1},\dots,r_{m}])\neq
0.$

######  Remark 6.8.

The proof of Lemma 6.6 can be simplified as follows. In the step when made a
choice of the vector bundle $\eta$, to be stable equivalent to the vector
bundle $\mu_{m}[m]^{\oplus r_{m}+N_{0}-\ell-N_{1}}$, we could have in addition
asked that $\eta$ is in addition $(d-1)$-dimensional. (Note that $\mu_{m}[m]$
is a pull-back vector bundle of the vector bundle $\lambda_{d,1}$ over
$\mathbb{R}{\rm P}^{d-1}$.) Then the Stiefel–Whitney class

$w_{(d-1)(2^{m}-1)}(\lambda_{d,m}^{-\ell}\oplus\psi_{d,m}[r_{1},\dots,r_{m}])=w_{(d-1)(2^{m}-1)}(\eta\oplus
S^{2,d}\varpi)$

is actually the mod $2$ Euler class $\mathfrak{e}(\eta\oplus S^{2,d}\varpi)$
of the vector bundle $\eta\oplus S^{2,d}\varpi$. Indeed, $\dim(\eta\oplus
S^{2,d}\varpi)=(d-1)(2^{m}-1)$. Thus, using the product formula for Euler
classes [80, Prop. 9.6] we have that

$w_{(d-1)(2^{m}-1)}(\eta\oplus S^{2,d}\varpi)=\mathfrak{e}(\eta\oplus
S^{2,d}\varpi)=\mathfrak{e}(\eta)\cdot\mathfrak{e}(S^{2,d}\varpi).$

Here $\mathfrak{e}(\cdot)$ denotes the mod $2$ Euler class of the respected
vector bundle. Now from Corollary 7.18 and the description of the cohomology
$H^{*}(\operatorname{Pe}(\mathbb{R}^{d},2^{m-1})/\mathcal{S}_{2^{m-1}};\mathbb{F}_{2})$
of the base space we have that

$w_{(d-1)(2^{m}-1)}(\eta\oplus
S^{2,d}\varpi)=\mathfrak{e}(\eta)\cdot\mathfrak{e}(S^{2,d}\varpi)=\\\
{{r_{m}+N_{0}-\ell-N_{1}\choose d-1}}P(w_{N}(\varpi))\,f^{d-1}\neq 0.$

Thus, there was no need to evaluate the total Stiefel–Whitney class of the
vector bundle $\eta\oplus S^{2,d}\varpi$ in full.

After proving Lemma 6.6 we want to discuss how to utilize it. In other words,
for which integer parameters $d\geq 2$, $m\geq 1$, $r_{1},\dots,r_{m}\geq 0$
the assumption (86) is satisfied.

######  Lemma 6.9.

Let $d\geq 2$, $m\geq 1$ and $\ell$ be integers, and let $d=2^{t}+e$ for some
integers $t\geq 1$ and $0\leq e\leq 2^{t}-1$. If

  1. (1)

$\ell=1$, $r_{1}=0$, and $r_{2}=\dots=r_{m}=2e$, or

  2. (2)

$\ell=d+1$, and $r_{1}=\dots=r_{m}=2e$, or

  3. (3)

$\ell=-(d-1+k2^{t+1})$ for some integer $k$, and $r_{1}=\dots=r_{m}=0$,

then

${{(r_{1}2^{m-2}+\dots+r_{m-1}2^{0})+r_{m}-(d-1+\ell)(2^{m-1}-1)-\ell}\choose{d-1}}=1\in\mathbb{F}_{2}.$

###### Proof.

(1) Let $\ell=1$, $r_{1}=0$, and $r_{2}=\dots=r_{m}=2e$. Then

$(r_{1}2^{m-2}+\dots+r_{m-1}2^{0})+r_{m}-(d-1+\ell)(2^{m-1}-1)-\ell=\\\
2e(2^{m-3}+\dots+2^{0})+2e-d(2^{m-1}-1)-1=2^{m-1}(e-d)+2e+d-1=\\\
2^{m-1}(e-d)+d-1=\begin{cases}d-1-2^{m-1+t},&m\geq 2,\\\ -1,&m=1.\end{cases}$

Now, ${-1\choose d-1}=_{\mathbb{F}_{2}}1\in\mathbb{F}_{2}$ and
${d-1-2^{m-1+t}\choose d-1}=_{\mathbb{F}_{2}}{d-1\choose
d-1}=_{\mathbb{F}_{2}}1\in\mathbb{F}_{2}$, because for $m\geq 2$ we have that
$m-1+t\geq t+1$ and $d-1<2^{m+1}$.

(2) Let $\ell=d+1$, and $r_{1}=\dots=r_{m}=2e$. Then

$(r_{1}2^{m-2}+\dots+r_{m-1}2^{0})+r_{m}-(d-1+\ell)(2^{m-1}-1)-\ell=\\\
2e(2^{m-2}+\dots+2^{0})+2e-2d(2^{m-1}-1)-d-1=2^{m}(e-d)+d-1\\\ =d-1-2^{m+t}.$

Now ${d-1-2^{m+t}\choose d-1}=_{\mathbb{F}_{2}}1\in\mathbb{F}_{2}$, because
$d-1<2^{t+1}\leq 2^{t+m}$, and in the ring of formal power series
$\mathbb{F}_{2}[[T]]$ the following equality holds

$(1+T)^{d-1-2^{m+t}}=(1+T)^{d-1}(1+T)^{-2^{m+t}}=\\\
(1+T)^{d-1}(1+T^{2^{m+t}})^{-1}=(1+T)^{d-1}\sum_{j\geq 0}T^{j2^{m+t}}.$

(3) Let $\ell=-(d-1+j2^{t+1})$ for some integer $k\geq 1$, and
$r_{1}=\dots=r_{m}=0$. Then

$(r_{1}2^{m-2}+\dots+r_{m-1}2^{0})+r_{m}-(d-1+\ell)(2^{m-1}-1)-\ell=\\\
-(d-1+\ell)(2^{m-1}-1)-\ell=d-1+k2^{m+t}$

Hence, ${d-1+k2^{m+t}\choose d-1}=_{\mathbb{F}_{2}}1\in\mathbb{F}_{2}$,
because $d-1<2^{m+t}$, and in the ring $\mathbb{F}_{2}[[T]]$ the following
equality holds

$(1+T)^{d-1+k2^{m+t}}=(1+T)^{d-1}(1+T)^{k2^{m+t}}=\\\
(1+T)^{d-1}(1+T^{2^{m+t}})^{k}=(1+T)^{d-1}\sum_{j=0}^{k}{k\choose
j}T^{j2^{m+t}}.$

∎

Now from Lemma 6.6 and Lemma 6.9 we get the following particular results.

######  Corollary 6.10.

Let $d\geq 2$, $m\geq 1$ and $k$ be integers, and let $d=2^{t}+e$ for some
integers $t\geq 1$ and $0\leq e\leq 2^{t}-1$. Then

  1. (1)

$w_{(d-1)(2^{m}-1)}(\lambda_{d,m}^{-1}\oplus(\mu_{d,m}[2]\oplus\cdots\oplus\mu_{d,m}[m])^{\oplus
2e})\neq 0$,

  2. (2)

$w_{(d-1)(2^{m}-1)}(\lambda_{d,m}^{-(d+1)}\oplus(\mu_{d,m}[1]\oplus\cdots\oplus\mu_{d,m}[m])^{\oplus
2e})\neq 0$, and

  3. (3)

$w_{(d-1)(2^{m}-1)}(\lambda_{d,m}^{d-1+k2^{t+1}})\neq 0$.

######  Remark 6.11.

In the case (1) of the previous corollary for $e=0$ we get an alternative
proof of Lemma 5.5. On the other hand, in the case (2) for $e=0$ we have a
particular case of Theorem 5.17 (3).

###  6.3. Additional bounds for the existence of highly regular embeddings

In this section we use consequences of the Key Lemma to derive further bounds
for the existence of highly regular embeddings.

First, we use Corollary 6.10 to get specific results which are relevant for
the study of highly regular embeddings.

######  Corollary 6.12.

Let $d\geq 2$ and $m\geq 1$ be integers, and let $d=2^{t}+e$ for some integers
$t\geq 1$ and $0\leq e\leq 2^{t}-1$. Then exist integers $a$ and $b$ with the
property that

$(d-1)(2^{m}-1)-2(2^{m-1}-1)e\leq a\leq(d-1)(2^{m}-1),$
$(d-1)(2^{m}-1)-2(2^{m}-1)e\leq b\leq(d-1)(2^{m}-1),$

and in addition, the Stiefel–Whitney classes

$w_{a}(\lambda_{d,m}^{-1})\neq 0\qquad\text{and}\qquad
w_{b}(\lambda_{d,m}^{-(d+1)})\neq 0$ (91)

do not vanish.

###### Proof.

From Corollary 6.10 (i) we have that

$w_{(d-1)(2^{m}-1)}(\lambda_{d,m}^{-1}\oplus(\mu_{d,m}[2]\oplus\cdots\oplus\mu_{d,m}[m])^{\oplus
2e})\neq 0.$ (92)

On the other hand

$w_{(d-1)(2^{m}-1)}(\lambda_{d,m}^{-1}\oplus(\mu_{d,m}[2]\oplus\cdots\oplus\mu_{d,m}[m])^{\oplus
2e})=\\\ \sum_{i=0}^{(d-1)(2^{m}-1)}w_{i}(\lambda_{d,m}^{-1})\cdot
w_{(d-1)(2^{m}-1)-i}((\mu_{d,m}[2]\oplus\cdots\oplus\mu_{d,m}[m])^{\oplus
2e}),$

where

$\dim(\mu_{d,m}[2]\oplus\cdots\oplus\mu_{d,m}[m])^{\oplus
2e}=2e\cdot\dim(\mu_{d,m}[2]\oplus\cdots\oplus\mu_{d,m}[m])=\\\
2e(2^{m-2}+\dots+2^{0})=2e(2^{m-1}-1).$ (93)

Now (92) and (93) imply that at lease one Stiefel–Whitney class
$w_{i}(\lambda_{d,m}^{-1})\neq 0$ does not vanish for

$(d-1)(2^{m}-1)-i\leq
2e(2^{m-1}-1)=2e\cdot\dim(\mu_{d,m}[2]\oplus\cdots\oplus\mu_{d,m}[m])$

and $i\leq(d-1)(2^{m}-1)$. Hence we proved the first claim of (91).

For the second part of (91) recall Corollary 6.10 (ii):

$w_{(d-1)(2^{m}-1)}(\lambda_{d,m}^{-(d+1)}\oplus(\mu_{d,m}[1]\oplus\cdots\oplus\mu_{d,m}[m])^{\oplus
2e})\neq 0.$ (94)

Similarly,

$w_{(d-1)(2^{m}-1)}(\lambda_{d,m}^{-(d+1)}\oplus(\mu_{d,m}[1]\oplus\cdots\oplus\mu_{d,m}[m])^{\oplus
2e})=\\\ \sum_{i=0}^{(d-1)(2^{m}-1)}w_{i}(\lambda_{d,m}^{-(d+1)})\cdot
w_{(d-1)(2^{m}-1)-i}((\mu_{d,m}[1]\oplus\cdots\oplus\mu_{d,m}[m])^{\oplus
2e}),$

where

$\dim(\mu_{d,m}[1]\oplus\cdots\oplus\mu_{d,m}[m])^{\oplus
2e}=2e\cdot\dim(\mu_{d,m}[1]\oplus\cdots\oplus\mu_{d,m}[m])=\\\
2e(2^{m-1}+\dots+2^{0})=2e(2^{m-1}-1).$ (95)

In the same way, now (94) and (95), imply that there exists at lease one
Stiefel–Whitney class $w_{i}(\lambda_{d,m}^{-(d+1)})\neq 0$ which does not
vanish where

$(d-1)(2^{m}-1)-i\leq
2e(2^{m}-1)=2e\cdot\dim(\mu_{d,m}[1]\oplus\cdots\oplus\mu_{d,m}[m])$

and $i\leq(d-1)(2^{m}-1)$. Thus, we proved the second part of (91). ∎

######  Remark 6.13.

Since the vector bundle $\lambda_{d,m}$ is the pull-back of the vector bundle
$\zeta_{\mathbb{R}^{d},2^{m}}$, and
$w(\zeta_{\mathbb{R}^{d},2^{m}})=w(\xi_{\mathbb{R}^{d},2^{m}})$, the previous
corollary implies, under identical assumptions on parameters, that

$\overline{w}_{a}(\xi_{\mathbb{R}^{d},2^{m}})\neq
0\qquad\text{and}\qquad\overline{w}_{b}(\xi_{\mathbb{R}^{d},2^{m}}^{\oplus
d+1})\neq 0.$

Next, we use Corollary 6.12 to get further insights on the dual
Stiefel–Whitney classes $\overline{w}(\xi_{\mathbb{R}^{d},k})$ and
$\overline{w}(\xi_{\mathbb{R}^{d},k}^{\oplus d+1})$, but this time without
restricting to the case when $k$ is a power of $2$.

######  Corollary 6.14.

Let $d\geq 2$ and $k\geq 1$ be integers, and let $d=2^{t}+e$ for some integers
$t\geq 1$ and $0\leq e\leq 2^{t}-1$. Then exist integers $A$ and $B$ with the
property that

$(d-e-1)(k-\alpha(k))+e(\alpha(k)-\epsilon(k))\leq A\leq(d-1)(k-1),$
$(d-2e-1)(k-\alpha(k))\leq B\leq(d-1)(k-1),$

and in addition, the dual Stiefel–Whitney classes

$\overline{w}_{A}(\xi_{\mathbb{R}^{d},k})\neq
0\qquad\text{and}\qquad\overline{w}_{B}(\xi_{\mathbb{R}^{d},k}^{\oplus
d+1})\neq 0$ (96)

do not vanish. Recall that $\epsilon(k)=1$ for $k$ odd, and $\epsilon(k)=0$
for $k$ even.

###### Proof.

Let $r:=\alpha(k)$ be the number of $1$s in the binary presentation of the
integer $k\geq 1$, and let $k=2^{k_{1}}+\dots+2^{k_{r}}$ where $0\leq
k_{1}<k_{2}<\dots<k_{r}$. Like in the proof of Lemma 5.6, consider a morphism
between vector bundles $\prod_{i=1}^{r}\xi_{\mathbb{R}^{d},2^{k_{i}}}$ and
$\xi_{\mathbb{R}^{d},k}$ where the following commutative square is a pullback
diagram:

$\textstyle{\prod_{i=1}^{r}\xi_{\mathbb{R}^{d},2^{k_{i}}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\
}$$\scriptstyle{\Theta}$$\textstyle{\
\xi_{\mathbb{R}^{d},k}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\prod_{i=1}^{r}\operatorname{F}(\mathbb{R}^{d},2^{k_{i}})/\mathfrak{S}_{2^{k_{i}}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\
}$$\scriptstyle{\theta}$$\textstyle{\
\operatorname{F}(\mathbb{R}^{d},k)/{\mathfrak{S}_{k}}.}$

Recall, that the map $\theta$ is induced, up to an equivariant homotopy, from
a restriction of the little cubes operad structural map.

Thus, we have that
$\theta^{*}\xi_{\mathbb{R}^{d},k}\cong\prod_{i=1}^{r}\xi_{\mathbb{R}^{d},2^{k_{i}}}$,
and consequently

$\theta^{*}(\overline{w}(\xi_{\mathbb{R}^{d},k}))=\overline{w}\Big{(}\prod_{i=1}^{r}\xi_{\mathbb{R}^{d},2^{k_{i}}}\Big{)}\qquad\text{and}\qquad\theta^{*}(\overline{w}(\xi_{\mathbb{R}^{d},k}^{\oplus
d+1}))=\overline{w}\Big{(}\prod_{i=1}^{r}\xi_{\mathbb{R}^{d},2^{k_{i}}}^{\oplus
d+1}\Big{)}.$ (97)

The product formula for Stiefel–Whitney classes [80, Pr. 4-A, p. 54] implies
that

$\overline{w}\Big{(}\prod_{i=1}^{r}\xi_{\mathbb{R}^{d},2^{k_{i}}}\Big{)}=\overline{w}(\xi_{\mathbb{R}^{d},2^{k_{1}}})\times\cdots\times\overline{w}(\xi_{\mathbb{R}^{d},2^{k_{r}}}),$

and similarly

$\overline{w}\Big{(}\prod_{i=1}^{r}\xi_{\mathbb{R}^{d},2^{k_{i}}}^{\oplus
d+1}\Big{)}=\overline{w}(\xi_{\mathbb{R}^{d},2^{k_{1}}}^{\oplus
d+1})\times\cdots\times\overline{w}(\xi_{\mathbb{R}^{d},2^{k_{r}}}^{\oplus
d+1}).$

From Corollary 6.12 and for every $0\leq k_{1}<k_{2}<\dots<k_{r}$ there exist
integers $a_{1},\dots,a_{r}$, and integers $b_{1},\dots,b_{r}$, with the
property that

$\begin{cases}a_{1}=0,&k_{1}=0,\\\
a_{1}\geq(d-1)(2^{k_{1}}-1)-2(2^{k_{1}-1}-1)e,&k_{1}\geq 1,\\\
a_{i}\geq(d-1)(2^{k_{i}}-1)-2(2^{k_{i}-1}-1)e,&2\leq i\leq r,\end{cases}$

and

$\begin{cases}b_{1}=(d-1)(2^{k_{1}}-1)-2(2^{k_{1}}-1)e=0,&k_{1}=0,\\\
b_{1}\geq(d-1)(2^{k_{1}}-1)-2(2^{k_{1}}-1)e,&k_{1}\geq 1,\\\
b_{i}\geq(d-1)(2^{k_{i}}-1)-2(2^{k_{i}}-1)e,&2\leq i\leq r,\end{cases}$

and in addition the dual Stiefel– Whitney classes

$\overline{w}_{a_{i}}(\xi_{\mathbb{R}^{d},2^{k_{i}}})\neq
0\qquad\text{and}\qquad\overline{w}_{b_{i}}(\xi_{\mathbb{R}^{d},2^{k_{i}}}^{\oplus
d+1})\neq 0.$

do not vanish.

Let denotes the sums of integers $a_{1},\dots,a_{r}$ and $b_{1},\dots,b_{r}$
by

$A:=\sum_{i=0}^{r}a_{i}\geq(d-e-1)(k-\alpha(k))+e(\alpha(k)-\epsilon(k)),$

and

$B:=\sum_{i=0}^{r}b_{i}\geq(d-2e-1)(k-\alpha(k)).$

Now, from the product formula for Stiefel–Whitney classes and the Künneth
formula we conclude that

$\overline{w}_{A}\Big{(}\prod_{i=1}^{r}\xi_{\mathbb{R}^{d},2^{k_{i}}}\Big{)}\neq
0\qquad\text{and}\qquad\overline{w}_{B}\Big{(}\prod_{i=1}^{r}\xi_{\mathbb{R}^{d},2^{k_{i}}}^{\oplus
d+1}\Big{)}\neq 0.$

(For details of the last argument consult for example the proof of Lemma 5.6.)
Finally, from the pull-backs (97) we conclude that

$\overline{w}_{A}(\xi_{\mathbb{R}^{d},k})\neq
0\qquad\text{and}\qquad\overline{w}_{B}(\xi_{\mathbb{R}^{d},k}^{\oplus
d+1})\neq 0.$

∎

Combining the results of Corollary 6.14 and the approach used in the proof of
Theorem 5.21 we get the following result.

######  Corollary 6.15.

Let $d\geq 2$, $k\geq 1$ and $\ell\geq 1$ be integers, and let $d=2^{t}+e$ for
some integers $t\geq 1$ and $0\leq e\leq 2^{t}-1$. Then exists an integer $C$
with the property that

$(d-e-1)(k-\alpha(k))+e(\alpha(k)-\epsilon(k))+(d-2e-1)(\ell-\alpha(\ell))\leq\\\
C\leq(d-1)(k+\ell-2),$

and in addition, the dual Stiefel–Whitney class

$\overline{w}_{C}(\xi_{\mathbb{R}^{d},k}\times\xi_{\mathbb{R}^{d},\ell}^{\oplus
d+1})\neq 0$ (98)

does not vanish.

###### Proof.

We start as in the proof of Theorem 5.21. To compute the dual Stiefel–Whitney
class of the product vector bundle
$\xi_{\mathbb{R}^{d},k}\times\xi_{\mathbb{R}^{d},\ell}^{\oplus(d+1)}$ we apply
the product formula [80, Problem 4-A, page 54]:

$\overline{w}\big{(}\xi_{\mathbb{R}^{d},k}\times\xi_{\mathbb{R}^{d},\ell}^{\oplus(d+1)}\big{)}=\overline{w}(\xi_{\mathbb{R}^{d},k})\times\overline{w}\big{(}\xi_{\mathbb{R}^{d},\ell}^{\oplus(d+1)}\big{)}.$

Hence, for fixed $r\geq 0$ we have that

$\overline{w}_{r}\big{(}\xi_{\mathbb{R}^{d},k}\times\xi_{\mathbb{R}^{d},\ell}^{\oplus(d+1)}\big{)}=\sum_{i+j=r}\overline{w}_{i}(\xi_{\mathbb{R}^{d},k})\times\overline{w}_{j}\big{(}\xi_{\mathbb{R}^{d},\ell}^{\oplus(d+1)}\big{)}\\\
\ \in\
H^{r}(\operatorname{F}(\mathbb{R}^{d},k)/\mathfrak{S}_{k}\times\operatorname{F}(\mathbb{R}^{d},\ell)/\mathfrak{S}_{\ell};\mathbb{F}_{2}).$

From the Künneth formula we get that each of the terms
$\overline{w}_{i}(\xi_{\mathbb{R}^{d},k})\times\overline{w}_{j}\big{(}\xi_{\mathbb{R}^{d},\ell}^{\oplus(d+1)}\big{)}$
in the previous sum belongs to a different direct summand of the cohomology

$H^{r}(\operatorname{F}(\mathbb{R}^{d},k)/\mathfrak{S}_{k}\times\operatorname{F}(\mathbb{R}^{d},\ell)/\mathfrak{S}_{\ell};\mathbb{F}_{2})\cong\\\
\bigoplus_{i+j=r}H^{i}(\operatorname{F}(\mathbb{R}^{d},k)/\mathfrak{S}_{k};\mathbb{F}_{2})\otimes
H^{j}(\operatorname{F}(\mathbb{R}^{d},\ell)/\mathfrak{S}_{\ell};\mathbb{F}_{2}).$

Therefore, the following equivalence holds:

$\overline{w}_{r}\big{(}\xi_{\mathbb{R}^{d},k}\times\xi_{\mathbb{R}^{d},\ell}^{\oplus(d+1)}\big{)}\neq
0\Longleftrightarrow\overline{w}_{i}(\xi_{\mathbb{R}^{d},k})\times\overline{w}_{j}\big{(}\xi_{\mathbb{R}^{d},\ell}^{\oplus(d+1)}\big{)}\neq
0\;\text{for some}\;i+j=r.$

Now, from Corollary 6.14 we know that there are integers $A$ and $B$ such that

$(d-e-1)(k-\alpha(k))+e(\alpha(k)-\epsilon(k))\leq A\leq(d-1)(k-1),$
$(d-2e-1)(\ell-\alpha(\ell))\leq B\leq(d-1)(\ell-1),$

and

$\overline{w}_{A}(\xi_{\mathbb{R}^{d},k})\neq
0\qquad\text{and}\qquad\overline{w}_{B}(\xi_{\mathbb{R}^{d},\ell}^{\oplus
d+1})\neq 0.$

Consequently, for $C=A+B$ we have that
$\overline{w}_{A}(\xi_{\mathbb{R}^{d},k})\times\overline{w}_{B}\big{(}\xi_{\mathbb{R}^{d},\ell}^{\oplus(d+1)}\big{)}\neq
0$ and so
$\overline{w}_{C}(\xi_{\mathbb{R}^{d},k}\times\xi_{\mathbb{R}^{d},\ell}^{\oplus
d+1})\neq 0$. Obviously, we have that

$(d-e-1)(k-\alpha(k))+e(\alpha(k)-\epsilon(k))+(d-2e-1)(\ell-\alpha(\ell))\leq\\\
C\leq(d-1)(k+\ell-2),$

and we have completed the proof of the corollary. ∎

Finally, using the criteria in Lemma 5.4, Lemma 5.16 and Lemma 5.20 in
combination with Corollary 6.14 and Corollary 6.15 we get the strongest lower
bounds for the existence of highly regular embeddings.

######  Theorem 6.16.

Let $d\geq 2$, $k\geq 1$ and $\ell\geq 1$ be integers, and let $d=2^{t}+e$ for
some integers $t\geq 1$ and $0\leq e\leq 2^{t}-1$. Then

  1. (1)

there is no $k$-regular embedding
$\mathbb{R}^{d}\longrightarrow\mathbb{R}^{N}$ if

$N\leq(d-e-1)(k-\alpha(k))+e(\alpha(k)-\epsilon(k))+k-1,$

  2. (2)

there is no $\ell$-skew embedding
$\mathbb{R}^{d}\longrightarrow\mathbb{R}^{N}$ if

$N\leq(d-2e-1)(\ell-\alpha(\ell))+(d+1)\ell-2,$

  3. (3)

there is no $k$-regular-$\ell$-skew embedding
$\mathbb{R}^{d}\longrightarrow\mathbb{R}^{N}$ if

$N\leq(d-e-1)(k-\alpha(k))+e(\alpha(k)-\epsilon(k))+\\\
(d-2e-1)(\ell-\alpha(\ell))+(d+1)\ell+k-2.$

In order to explain the strength of the previous theorem we demonstrate that
Theorem 5.14, Theorem 5.18 and Theorem 5.22 are consequences of Theorem 6.16.

######  Corollary 6.17.

Let $d\geq 2$ and $k\geq 1$ be integers, and let $d=2^{t}+e$ for some integers
$t\geq 1$ and $0\leq e\leq 2^{t}-1$. Then

Theorem 6.16(1) $\Longrightarrow$ Theorem 5.14.

###### Proof.

We check that Theorem 6.16(1) implies all three cases of Theorem 5.14
independently.

(1)  In Theorem 5.14(1) it is stated that for $d$ being a power of $2$ there
is no $k$-regular embedding $\mathbb{R}^{d}\longrightarrow\mathbb{R}^{N}$ when
$N\leq d(k-\alpha(k))+\alpha(k)-1$. The same results follows from Theorem
6.16(1) by taking $e=0$ and observing that in this case

$(d-e-1)(k-\alpha(k))+e(\alpha(k)-\epsilon(k))+k-1=(d-1)(k-\alpha(k))+k-1=\\\
(d-1)(k-\alpha(k))+k-\alpha(k)+\alpha(k)-1=d(k-\alpha(k))+\alpha(k)-1.$

(2)  Next, in Theorem 5.14(2) it is claimed that for $d$ being not a power of
$2$ there is no $k$-regular embedding
$\mathbb{R}^{d}\longrightarrow\mathbb{R}^{N}$ when
$N\leq\frac{1}{2}(d-1)(k-\epsilon(k))+k-1$. Since $d$ is not a power of $2$ we
have that $e>0$. Set that $k=2k^{\prime}+\epsilon(k)$. Hence,
$\alpha(k)=\alpha(k^{\prime})+\epsilon(k)$. Now, the claim of Theorem 5.14(2)
follows from Theorem 6.16(1) since the following difference is non-negative:

$\big{(}(d-e-1)(k-\alpha(k))+e(\alpha(k)-\epsilon(k))+k-1\big{)}-\big{(}\frac{1}{2}(d-1)(k-\epsilon(k))+k-1\big{)}=\\\
(2^{t}-1-\frac{1}{2}(2^{t}+e-1))k-(2^{t}-1-e)\alpha(k)+\big{(}\frac{1}{2}(2^{t}+e-1)-e\big{)}\epsilon(k)=\\\
\frac{1}{2}(2^{t}-e-1)k-(2^{t}-1-e)\alpha(k)+\frac{1}{2}\big{(}2^{t}-e-1\big{)}\epsilon(k)=\\\
\frac{1}{2}\big{(}2^{t}-e-1\big{)}(k-2\alpha(k)+\epsilon(k))=\big{(}2^{t}-e-1\big{)}(k^{\prime}-\alpha(k^{\prime}))\geq
0.$

(3) Finally, Theorem 5.14(3) says that for $d$ being even and not a power of
$2$ there is no $k$-regular embedding
$\mathbb{R}^{d}\longrightarrow\mathbb{R}^{N}$ when
$N\leq\frac{1}{2}d(k-\epsilon(k))+k-\alpha(k)+\epsilon(k)-1$. We see that this
result is also a consequence of Theorem 6.16(1) by showing that following
difference is non-negative:

$\big{(}(d-e-1)(k-\alpha(k))+e(\alpha(k)-\epsilon(k))+k-1\big{)}-\big{(}\frac{1}{2}d(k-\epsilon(k))+k-\alpha(k)+\epsilon(k)-1\big{)}=\qquad\\\
(2^{t}-1)k-(2^{t}-1)\alpha(k)+e\alpha(k)-e\epsilon(k)-\frac{1}{2}\big{(}2^{t}+e\big{)}k+\frac{1}{2}\big{(}2^{t}+e\big{)}\epsilon(k)+\alpha(k)-\epsilon(k)=\\\
\frac{1}{2}\big{(}2^{t}-e-2\big{)}k-(2^{t}-2-e)\alpha(k)+\frac{1}{2}\big{(}2^{t}-e-2\big{)}\epsilon(k)=\\\
\frac{1}{2}\big{(}2^{t}-e-2\big{)}(k+\epsilon(k)-2\alpha(k))=\big{(}2^{t}-e-2\big{)}(k^{\prime}-\alpha(k^{\prime}))\geq
0.$

Here we have that $e\leq 2^{t}-2$ and $\epsilon(k)=0$, because $e$ is even. As
in the previous computation we set $k=2k^{\prime}+\epsilon(k)=2k^{\prime}$,
and so $\alpha(k)=\alpha(k^{\prime})+\epsilon(k)=\alpha(k^{\prime})$. ∎

######  Corollary 6.18.

Let $d\geq 2$ and $\ell\geq 1$ be integers, and let $d=2^{t}+e$ for some
integers $t\geq 1$ and $0\leq e\leq 2^{t}-1$. Then

Theorem 6.16(2) $\Longrightarrow$ Theorem 5.18.

###### Proof.

We discuss each part of Theorem 5.18 separately.

(1) In Theorem 5.18(1) it is stated that for $d=2$ and $\ell\geq 2$ there is
no $\ell$-skew embedding $\mathbb{R}^{2}\longrightarrow\mathbb{R}^{N}$ when
$N\leq 4\ell-\alpha(\ell)-2$. The same conclusion follows from Theorem 6.16(2)
because in this case $d=2^{t}+e=2\Leftrightarrow t=1,e=0$, and the following
difference vanishes:

$\big{(}(d-2e-1)(\ell-\alpha(\ell))+(d+1)\ell-2\big{)}-\big{(}4\ell-\alpha(\ell)-2\big{)}=\\\
(\ell-\alpha(\ell)+3\ell-2)-(4\ell-\alpha(\ell)-2)=0.$

(2) In Theorem 5.18(2) we have claimed that for $d\geq 1$ and $\ell=2$ there
is no $2$-skew embedding $\mathbb{R}^{d}\longrightarrow\mathbb{R}^{N}$ when
$N\leq 2^{\gamma(d)}+d-1$. Since we have assumed that $d=2^{t}+e$ and $0\leq
e\leq 2^{t}-1$ note that $2^{\gamma(d)}=2^{t+1}$. Again, the case result can
be deduced from Theorem 6.16(2) because the following difference vanishes:

$\big{(}(d-2e-1)(\ell-\alpha(\ell))+(d+1)\ell-2\big{)}-\big{(}2^{\gamma(d)}+d-1\big{)}=\\\
d-2e-1+2d+2-2-2^{\gamma(d)}-d-1=2d-2^{\gamma(d)}-2e=\\\
2^{t+1}+2e-2^{t+1}-2e=0.$

(3) In Theorem 5.18(3) we considered the case when $d=2^{t}+0\geq 2$ is a
power of $2$ and $\ell\geq 2$. We proved that there is no $\ell$-skew
embedding $\mathbb{R}^{d}\longrightarrow\mathbb{R}^{N}$ when $N\leq
2d\ell-(d-1)\alpha(\ell)-2$. To see that Theorem 6.16(2) implies this result
we show that the following difference vanishes:

$\big{(}(d-2e-1)(\ell-\alpha(\ell))+(d+1)\ell-2\big{)}-\big{(}2d\ell-(d-1)\alpha(\ell)-2\big{)}=\\\
(d-1)\ell-(d-1)\alpha(\ell)+(d+1)\ell-2d\ell+(d-1)\alpha(\ell)=0.$

(4) In Theorem 5.18(4) we analysed the case when $d+1\geq 2$ is a power of $2$
and $\ell\geq 2$. We showed that the method based of the computation of dual
Stiefel–Whitney class does not produce any non-trivial result about the
existence of $\ell$-skew embeddings
$\mathbb{R}^{d}\longrightarrow\mathbb{R}^{N}$. In this situation also Theorem
6.16(2) does not give any relevant result.

(5) In Theorem 5.18(5) we studied the case where both $d\geq 5$ and $d+1$ are
not powers of $2$ and $\ell\geq 3$. We showed that there cannot be $\ell$-skew
embedding $\mathbb{R}^{d}\longrightarrow\mathbb{R}^{N}$ for
$N\leq\frac{1}{2}(2^{\gamma(d)}-d-1)(\ell-\epsilon(\ell))+(d+1)\ell-2$. In
order to see that Theorem 6.16(2) implies this result we show that the
following difference is non-negative:

$\big{(}(d-2e-1)(\ell-\alpha(\ell))+(d+1)\ell-2\big{)}-\big{(}\frac{1}{2}(2^{\gamma(d)}-d-1)(\ell-\epsilon(\ell))+(d+1)\ell-2\big{)}=\\\
(2^{t}-e-1)\Big{(}\ell-\alpha(\ell)-\frac{1}{2}\big{(}2^{t+1}-2^{t}-e-1\big{)}(\ell-\epsilon(\ell)\Big{)}=\\\
\frac{1}{2}\big{(}2^{t}-e-1\big{)}(\ell-\alpha(\ell)+\epsilon(\ell))\geq 0.$

(6) In the last case, Theorem 5.18(6), we have that $d\geq 5$ and $d+1$ are
not powers of two, $2^{\gamma(d)}-d-1=2^{a_{1}}z$ where $a_{1}\geq 0$ and
$z\geq 1$ is odd, and $\ell\geq 3$. We proved that in this case there is no
$\ell$-skew embedding $\mathbb{R}^{d}\longrightarrow\mathbb{R}^{N}$ for

$N\leq\frac{1}{2}\big{(}2^{\gamma(d)}-d-1+2^{a_{1}}\big{)}(\ell-\epsilon(\ell))-2^{a_{1}}\alpha(\ell)+(d+1)\ell-2.$

In order to see that Theorem 6.16(2) implies this result we first note that in
this case

$d=2^{t}+e,\quad 1\leq e\leq 2^{t}-2,\quad
2^{a_{1}}z=2^{\gamma(d)}-d-1=2^{t}-e-1.$

Now, as in the previous situations we consider the following difference and
show that it is non-negative:

$\big{(}(d-2e-1)(\ell-\alpha(\ell))+(d+1)\ell-2\big{)}-\\\
\Big{(}\frac{1}{2}\big{(}2^{\gamma(d)}-d-1+2^{a_{1}}\big{)}(\ell-\epsilon(\ell))-2^{a_{1}}\alpha(\ell)+(d+1)\ell-2\Big{)}=$
$(2^{t}-e-1)(\ell-\alpha(\ell))\\\
-\frac{1}{2}\big{(}2^{t}-e-1\big{)}(\ell-\epsilon(\ell))-2^{a_{1}-1}(\ell-\epsilon(\ell))+2^{a_{1}}\alpha(\ell)=$
$2^{a_{1}-1}z(\ell-2\alpha(\ell)+\epsilon(\ell))-2^{a_{1}-1}(\ell-2\alpha(\ell)-\epsilon(\ell))=\\\
2^{a_{1}-1}\big{(}(z-1)(\ell-2\alpha(\ell))+(z+1)\epsilon(\ell)\big{)}\geq 0.$

This concludes the proof of the corollary. ∎

In the same way as previous two corollaries we can show the following.

######  Corollary 6.19.

Let $d\geq 2$, $k\geq 1$ and $\ell\geq 1$ be integers, and let $d=2^{t}+e$ for
some integers $t\geq 1$ and $0\leq e\leq 2^{t}-1$. Then

Theorem 6.16(3) $\Longrightarrow$ Theorem 5.22.

###  6.4. Additional bounds for the existence of complex highly regular
embeddings

In this section we derive additional consequence of Lemma 6.6 which yields
further bounds for the existence of complex highly regular embeddings. In
particular, we will improve bounds given in Theorem 5.26 and Theorem 5.28.

Let us denote by $\lambda_{d,m}^{\mathbb{C}}:=\mathbb{C}\otimes\lambda_{d,m}$
and
$\psi^{\mathbb{C}}_{d,m}[r_{1}^{\mathbb{C}},\dots,r_{m}^{\mathbb{C}}]:=\mathbb{C}\otimes\psi_{d,m}$
complex versions of the real vector bundles $\lambda_{d,m}$ and
$\psi_{d,m}[r_{1},\dots,r_{m}]$. Note that there is an isomorphism of real
vector bundles $\lambda_{d,m}^{\mathbb{C}}\cong\lambda_{d,m}^{\oplus 2}$ and

$\psi^{\mathbb{C}}_{d,m}[r_{1}^{\mathbb{C}},\dots,r_{m}^{\mathbb{C}}]\cong\psi_{d,m}[2r_{1}^{\mathbb{C}},\dots,2r_{m}^{\mathbb{C}}].$

The following lemma is a consequence of Lemma 6.6.

######  Lemma 6.20.

Let $m\geq 1$, $d^{\mathbb{C}}\geq 1$ and ${\ell}^{\mathbb{C}}\geq 1$ be
integers, and let $r_{1}^{\mathbb{C}},\dots,r_{m}^{\mathbb{C}}$ be non-
negative integers. If the binomial coefficient

${(r_{1}^{\mathbb{C}}2^{m-2}+\dots+r_{m-1}^{\mathbb{C}}2^{0})+r_{m}^{\mathbb{C}}-(d^{\mathbb{C}}-1+\ell^{\mathbb{C}})(2^{m-1}-1)-\ell^{\mathbb{C}}}\choose{d^{\mathbb{C}}-1}$
(99)

is odd, then

$w_{2(d^{\mathbb{C}}-1)(2^{m}-1)}((\lambda_{2d^{\mathbb{C}}-1,m}^{\mathbb{C}})^{-\ell^{\mathbb{C}}}\oplus\psi_{2d^{\mathbb{C}}-1,m}^{\mathbb{C}}[r_{1}^{\mathbb{C}},\dots,r_{m}^{\mathbb{C}}])\neq
0,$

as an element of the cohomology group
$H^{2(d^{\mathbb{C}}-1)(2^{m}-1)}(\operatorname{Pe}(\mathbb{R}^{d},2^{m})/\mathcal{S}_{2^{m}};\mathbb{F}_{2})$.
Here $(\lambda_{2d^{\mathbb{C}}-1,m}^{\mathbb{C}})^{-\ell^{\mathbb{C}}}$
denotes an inverse of the vector bundle
$(\lambda_{2d^{\mathbb{C}}-1,m}^{\mathbb{C}})^{\oplus\ell}$.

###### Proof.

There is isomorphism of real vector bundles

$(\lambda_{2d^{\mathbb{C}}-1,m}^{\mathbb{C}})^{-\ell^{\mathbb{C}}}\oplus\psi_{2d^{\mathbb{C}}-1,m}^{\mathbb{C}}[r_{1}^{\mathbb{C}},\dots,r_{m}^{\mathbb{C}}]\cong\lambda_{2d^{\mathbb{C}}-1,m}^{-2\ell^{\mathbb{C}}}\oplus\psi_{2d^{\mathbb{C}}-1,m}[2r_{1}^{\mathbb{C}},\dots,2r_{m}^{\mathbb{C}}].$

Thus, it suffices to prove that

$w_{2(d^{\mathbb{C}}-1)(2^{m}-1)}(\lambda_{2d^{\mathbb{C}}-1,m}^{-2\ell^{\mathbb{C}}}\oplus\psi_{2d^{\mathbb{C}}-1,m}[2r_{1}^{\mathbb{C}},\dots,2r_{m}^{\mathbb{C}}])\neq
0.$

If we set that $d=2d^{\mathbb{C}}-1$, $\ell=2\ell^{\mathbb{C}}$, and
$r_{1}:=2r_{1}^{\mathbb{C}},\dots,r_{1}:=2r_{m}^{\mathbb{C}}$, then by Lemma
6.6 we have that the Stiefel–Whitney class

$w_{2(d^{\mathbb{C}}-1)(2^{m}-1)}(\lambda_{2d^{\mathbb{C}}-1,m}^{-2\ell^{\mathbb{C}}}\oplus\psi_{2d^{\mathbb{C}}-1,m}[2r_{1}^{\mathbb{C}},\dots,2r_{m}^{\mathbb{C}}])=\\\
w_{(d-1)(2^{m}-1)}(\lambda_{d,m}^{-\ell}\oplus\psi_{d,m}[r_{1},\dots,r_{m}])\neq
0$

does not vanish if and only if the binomial coefficient

${{(r_{1}2^{m-2}+\dots+r_{m-1}2^{0})+r_{m}-(d-1+\ell)(2^{m-1}-1)-\ell}\choose{d-1}}=\\\
{{2\big{(}(r_{1}^{\mathbb{C}}2^{m-2}+\dots+r_{m-1}^{\mathbb{C}}2^{0})+r_{m}^{\mathbb{C}}-(d^{\mathbb{C}}-1+\ell^{\mathbb{C}})(2^{m-1}-1)-\ell^{\mathbb{C}}\big{)}}\choose{2(d^{\mathbb{C}}-1)}}$

is odd. Since the binomial coefficient $\binom{2a}{2b}$ is odd if and only if
the binomial coefficient $\binom{a}{b}$ is odd, then the assumption (99)
implies that the binomial coefficient

${{2\big{(}(r_{1}^{\mathbb{C}}2^{m-2}+\dots+r_{m-1}^{\mathbb{C}}2^{0})+r_{m}^{\mathbb{C}}-(d^{\mathbb{C}}-1+\ell^{\mathbb{C}})(2^{m-1}-1)-\ell^{\mathbb{C}}\big{)}}\choose{2(d^{\mathbb{C}}-1)}}$

is odd. Hence,
$w_{2(d^{\mathbb{C}}-1)(2^{m}-1)}((\lambda_{2d^{\mathbb{C}}-1,m}^{\mathbb{C}})^{-\ell^{\mathbb{C}}}\oplus\psi_{2d^{\mathbb{C}}-1,m}^{\mathbb{C}}[r_{1}^{\mathbb{C}},\dots,r_{m}^{\mathbb{C}}])\neq
0.$ ∎

Let us denote by $\mu_{d,m}^{\mathbb{C}}[j]:=\mathbb{C}\otimes\mu_{d,m}[j]$,
for all $1\leq j\leq m$, the complex version of the real vector bundle
$\mu_{d,m}[j]$. There is an isomorphism of real vector bundles
$\mu_{d,m}^{\mathbb{C}}[j]\cong\mu_{d,m}[j]^{\oplus 2}$. Now, like in the case
of Lemma 6.9 and Corollary 6.10, we deduce the following consequence of Lemma
6.20.

######  Corollary 6.21.

Let $d^{\mathbb{C}}\geq 2$ and $m\geq 1$ be integers, and let
$d^{\mathbb{C}}=2^{t}+e$ for some integer $t\geq 1$ and $0\leq e\leq 2^{t}-1$.
If $d=2d^{\mathbb{C}}-1$, then

  1. (1)

$w_{2(d^{\mathbb{C}}-1)(2^{m}-1)}\big{(}(\lambda^{\mathbb{C}}_{d,m})^{-1}\oplus(\mu^{\mathbb{C}}_{d,m}[2]\oplus\cdots\oplus\mu^{\mathbb{C}}_{d,m}[m])^{\oplus
2e}\big{)}\not=0$,

  2. (2)

$w_{2(d^{\mathbb{C}}-1)(2^{m}-1)}((\lambda^{\mathbb{C}}_{d,m})^{-(d^{\mathbb{C}}+1)}\oplus(\mu^{\mathbb{C}}_{d,m}[1]\oplus\cdots\oplus\mu^{\mathbb{C}}_{d,m}[m])^{\oplus
2e})\not=0$.

###### Proof.

The proof is almost identical to the proof of Lemma 6.9. For the sake of
completeness we present the proof in detail.

(1) Let $\ell^{\mathbb{C}}=1$, $r_{1}^{\mathbb{C}}=0$, and
$r_{2}^{\mathbb{C}}=\dots=r_{m}^{\mathbb{C}}=2e$. Then

$(r_{1}^{\mathbb{C}}2^{m-2}+\dots+r_{m-1}^{\mathbb{C}}2^{0})+r_{m}^{\mathbb{C}}-(d^{\mathbb{C}}-1+\ell^{\mathbb{C}})(2^{m-1}-1)-\ell^{\mathbb{C}}=\\\
2e(2^{m-3}+\dots+2^{0})+2e-d^{\mathbb{C}}(2^{m-1}-1)-1=2^{m-1}(e-d^{\mathbb{C}})+2e+d^{\mathbb{C}}-1=\\\
2^{m-1}(e-d^{\mathbb{C}})+d^{\mathbb{C}}-1=\\\
\begin{cases}d^{\mathbb{C}}-1-2^{m-1+t},&m\geq 2,\\\ -1,&m=1.\end{cases}$

Since ${{-1}\choose{d^{\mathbb{C}}-1}}=_{\mathbb{F}_{2}}1\in\mathbb{F}_{2}$
and ${d^{\mathbb{C}}-1-2^{m-1+t}\choose
d^{\mathbb{C}}-1}=_{\mathbb{F}_{2}}{d^{\mathbb{C}}-1\choose
d^{\mathbb{C}}-1}=_{\mathbb{F}_{2}}1\in\mathbb{F}_{2}$, the assumption of
Lemma 6.20 is satisfied for the chosen parameters. Consequently,

$w_{2(d^{\mathbb{C}}-1)(2^{m}-1)}\big{(}(\lambda^{\mathbb{C}}_{d,m})^{-1}\oplus(\mu^{\mathbb{C}}_{d,m}[2]\oplus\cdots\oplus\mu^{\mathbb{C}}_{d,m}[m])^{\oplus
2e}\big{)}\not=0.$

(2) Let $\ell^{\mathbb{C}}=d+1$, $r_{1}^{\mathbb{C}}=0$, and
$r_{1}^{\mathbb{C}}=\dots=r_{m}^{\mathbb{C}}=2e$. Hence

$(r_{1}^{\mathbb{C}}2^{m-2}+\dots+r_{m-1}^{\mathbb{C}}2^{0})+r_{m}^{\mathbb{C}}-(d^{\mathbb{C}}-1+\ell^{\mathbb{C}})(2^{m-1}-1)-\ell^{\mathbb{C}}=\\\
2e(2^{m-2}+\dots+2^{0})+2e-2d^{\mathbb{C}}(2^{m-1}-1)-d^{\mathbb{C}}-1=\\\
2^{m}(e-d^{\mathbb{C}})+d^{\mathbb{C}}-1=d^{\mathbb{C}}-1-2^{m+t}.$

Thus, ${d^{\mathbb{C}}-1-2^{m+t}\choose
d^{\mathbb{C}}-1}=_{\mathbb{F}_{2}}1\in\mathbb{F}_{2}$, because
$d^{\mathbb{C}}-1<2^{t+1}\leq 2^{t+m}$. Consequently, the assumption of Lemma
6.20 is satisfied for the chosen parameters, and so

$w_{2(d^{\mathbb{C}}-1)(2^{m}-1)}((\lambda^{\mathbb{C}}_{d,m})^{-(d^{\mathbb{C}}+1)}\oplus(\mu^{\mathbb{C}}_{d,m}[1]\oplus\cdots\oplus\mu^{\mathbb{C}}_{d,m}[m])^{\oplus
2e})\not=0.$

∎

Just as in the proof of Corollary 6.14 we can derive the following estimate
for non-vanishing of the relevant dual Stiefel–Whitney classes.

######  Corollary 6.22.

Let $d^{\mathbb{C}}\geq 2$ and $m\geq 1$ be integers, and let
$d^{\mathbb{C}}=2^{t}+e$ for some integer $t\geq 1$ and $0\leq e\leq 2^{t}-1$.
Set $d=2d^{\mathbb{C}}-1$. Then there exist integers $a$ and $b$ with the
property that

$(d^{\mathbb{C}}-1)(2^{m}-1)-2(2^{m-1}-1)e\leq
a\leq(d^{\mathbb{C}}-1)(2^{m}-1),$
$(d^{\mathbb{C}}-1)(2^{m}-1)-2(2^{m}-1)e\leq
b\leq(d^{\mathbb{C}}-1)(2^{m}-1),$

and in addition, the dual Stiefel–Whitney classes

$\overline{w}_{2a}(\lambda^{\mathbb{C}}_{d,m})\not=0\qquad\text{and}\qquad\overline{w}_{2b}((\lambda^{\mathbb{C}}_{d,m})^{d^{\mathbb{C}}+1})\not=0,$

do not vanish.

Now, using the criteria for the existence of complex $k$-regular embeddings
and complex $\ell$-skew embeddings given in Lemma 5.25, we can directly derive
the following estimates — a complex analogue of Theorem 6.16.

######  Theorem 6.23.

Let $d^{\mathbb{C}}\geq 2$, $m\geq 1$, $k\geq 1$ and $\ell\geq 1$ be integers,
and let $d^{\mathbb{C}}=2^{t}+e$ for some integer $t\geq 1$ and $0\leq e\leq
2^{t}-1$. Set $d=2d^{\mathbb{C}}-1$. Then

  1. (i)

there is no complex $k$-regular embedding
$\mathbb{R}^{d}\longrightarrow\mathbb{C}^{N}$ if

$N\leq(d^{\mathbb{C}}-1-e)(k-\alpha(k))+e(\alpha_{2}(k)-\epsilon(k))+k-1,$

  2. (ii)

there is no complex $\ell$-skew embedding
$\mathbb{C}^{d^{\mathbb{C}}}\longrightarrow\mathbb{C}^{N}$ if

$N\leq(d^{\mathbb{C}}-1-2e)(\ell-\alpha(\ell))+(d^{\mathbb{C}}+1)\ell-2.$

In a similar way as in Corollary 6.17, Corollary 6.18 and Corollary 6.19 it
can be verified that Theorem 6.23 implies Theorem 5.26 and Theorem 5.28.

##  7\. Appendix

###  7.1. Operads

In this section we recall basic notions from the theory of operads that we
use. We follow the framework developed by May in [76], with some slight
modifications in the notation.

Let $\operatorname{\mathrm{T}op}$ be the category of compactly generated weak
Hausdorff spaces with continuous maps as morphisms, and let
$\operatorname{\mathrm{T}op_{pt}}$ denote the category of compactly generated
weak Hausdorff spaces with non-degenerate base points where the morphisms are
base point preserving continuous maps. Furthermore, assume that all the
products as well as function spaces are endowed with compactly generated
topology.

####  7.1.1. Definition and basic example

The notion of an operad and its first formal definition appeared in work of
May in 1970’s. For more details consult the original publication [76, Sec. 1].

######  Definition 7.1.

An operad $\mathcal{O}$ is given by a family of topological spaces in
$\operatorname{\mathrm{T}op}$

$\mathcal{O}:=\\{\mathcal{O}(n):n\geq
0\\}\qquad\text{where}\qquad\mathcal{O}(0):=\\{\mathrm{pt}\\}$

together with a family of continuous maps

$\mu\colon(\mathcal{O}(n_{1})\times\cdots\times\mathcal{O}(n_{k}))\times\mathcal{O}(k)\longrightarrow\mathcal{O}(n)$

where $k,n_{1},\ldots,n_{k}\geq 0$ and $n=n_{1}+\cdots+n_{k}$, such that the
following axioms are satisfied:

  1. (1)

For every $a\in\mathcal{O}(k)$,
$b_{1}\in\mathcal{O}(n_{1}),\ldots,b_{k}\in\mathcal{O}(n_{k})$,
$c_{1}\in\mathcal{O}(m_{1}),\ldots,c_{n}\in\mathcal{O}(m_{n})$

$\mu(c_{1},\ldots,c_{n};\mu(b_{1},\ldots,b_{k};a))=\mu(d_{1},\ldots,d_{k};a)$

where for $1\leq i\leq k$

$d_{i}:=\begin{cases}\mu(c_{n_{1}+\cdots+n_{i-1}+1},\ldots,c_{n_{1}+\cdots+n_{i}};b_{i}),&n_{i}\neq
0\\\ \mathrm{pt},&n_{i}=0.\end{cases}$

Here $n=n_{1}+\cdots+n_{k}$.

  2. (2)

There exists an element $\mathbf{1}\in\mathcal{O}(1)$ with the property that
for every $a\in\mathcal{O}(n)$ and every $b\in\mathcal{O}(k)$

$\mu(a;\mathbf{1})=a,\qquad\qquad\qquad\mu(\mathbf{1},\ldots,\mathbf{1};b)=b.$

  3. (3)

Every space $\mathcal{O}(n)$ is endowed with a right action of the symmetric
group $\mathfrak{S}_{n}$ that fulfills

$\displaystyle\mu(b_{1},\ldots,b_{k};a\cdot\pi)$ $\displaystyle=$
$\displaystyle\mu(b_{\pi^{-1}(1)},\ldots,b_{\pi^{-1}(k)};a)\cdot\pi_{n_{1},\ldots,n_{k}},$
$\displaystyle\mu(b_{1}\cdot\pi_{1},\ldots,b_{k}\cdot\pi_{k};a)$
$\displaystyle=$
$\displaystyle\mu(b_{1},\ldots,b_{k};a)\cdot(\pi_{1},\ldots,\pi_{k}),$

where the permutation

     * —

$\pi_{n_{1},\ldots,n_{k}}\in\mathfrak{S}_{n}$ is given by permuting $k$ blocks

$(1,\ldots,n_{1})(n_{1}+1,\ldots,n_{1}+n_{2})\cdots(n_{1}+\cdots
n_{k-1}+1,\ldots,n_{1}+\cdots n_{k})$

as the permutation $\pi\in\mathfrak{S}_{k}$ permutes $(1,\ldots,k)$, and

     * —

$(\pi_{1},\ldots,\pi_{k})\in\mathfrak{S}_{n}$ denotes the image of
$(\pi_{1},\ldots,\pi_{k})\in\mathfrak{S}_{n_{1}}\times\cdots\times\mathfrak{S}_{n_{k}}$
via the inclusion
$\mathfrak{S}_{n_{1}}\times\cdots\times\mathfrak{S}_{n_{k}}\hookrightarrow\mathfrak{S}_{n}$.

If in addition each action of the symmetric group $\mathfrak{S}_{n}$ on
$\mathcal{O}(n)$ is free the operad $\mathcal{O}$ is called a
$\mathfrak{S}$-free operad. The map $\mu_{\mathcal{O}}:=\mu$ is called the
structural map of the operad $\mathcal{O}$.

The defining requirement on the action of the symmetric group
$\mathfrak{S}_{n}$ on $\mathcal{O}(n)$ for $n\geq 0$, Definition 7.1(3),
directly implies the following property of the structural map.

######  Lemma 7.2.

Let $k\geq 1$ be an integer, and let $n_{1},\ldots,n_{k}$ be integers such
that

$n_{1}=\cdots=n_{i_{1}}<n_{i_{1}+1}=\cdots=n_{i_{2}}<\cdots<n_{i_{r-1}+1}=\cdots=n_{i_{r}}$

where $1\leq i_{1}<i_{2}<\cdots<i_{r-1}<i_{r}=k$. Then

  1. (1)

The product
$(\mathcal{O}(n_{1})\times\cdots\times\mathcal{O}(n_{k}))\times\mathcal{O}(k)$
is endowed with a

$\mathcal{S}_{n_{1},\ldots,n_{k};k}=\\\
(\mathfrak{S}_{n_{i_{1}}}^{i_{1}}\rtimes\mathfrak{S}_{i_{1}})\times(\mathfrak{S}_{n_{i_{2}}}^{i_{2}-i_{1}+1}\rtimes\mathfrak{S}_{i_{2}-i_{1}+1})\times\cdots\times(\mathfrak{S}_{n_{i_{r}}}^{i_{r}-i_{r-1}+1}\rtimes\mathfrak{S}_{i_{r}-i_{r-1}+1})$

right action.

  2. (2)

The structural map

$\mu\colon(\mathcal{O}(n_{1})\times\cdots\times\mathcal{O}(n_{k}))\times\mathcal{O}(k)\longrightarrow\mathcal{O}(n),$

with $n:=n_{1}+\cdots+n_{k}$, is an equivariant map with respect to the
natural inclusion map of the groups
$\mathcal{S}_{n_{1},\ldots,n_{k};k}\hookrightarrow\mathfrak{S}_{n}$.

The first and the central example of an operad is the endomorphism operad
associated with any topological space in $\operatorname{\mathrm{T}op_{pt}}$.
The importance of the endomorphism operad will be become apparent after the
definition of an action of an operad on a topological space.

######  Example 7.3.

Let $X$ be a topological space in $\operatorname{\mathrm{T}op_{pt}}$. The
endomorphism operad $\mathcal{E}nd_{X}$ associated to $X$ is defined as
follows:

  * —

$\mathcal{E}nd_{X}(n):=\operatorname{\mathrm{M}or}_{\operatorname{\mathrm{T}op_{pt}}}(X^{n},X)$
where $n\geq 0$ and $X^{0}:=\mathrm{pt}$,

  * —

$\mu(g_{1},\ldots,g_{k};f):=f\circ(g_{1}\times\cdots\times g_{k})$ for
$f\in\mathcal{E}nd_{X}(k)$,
$g_{1}\in\mathcal{E}nd_{X}(n_{1}),\ldots,g_{k}\in\mathcal{E}nd_{X}(n_{k})$,
and

  * —

$(f\cdot\pi)(x_{1},\ldots,x_{k}):=f(x_{\pi^{-1}(1)},\ldots,x_{\pi^{-1}(k)})$
where $f\in\mathcal{E}nd_{X}(k)$, $\pi\in\mathfrak{S}_{k}$, and
$(x_{1},\ldots,x_{k})\in X^{k}$.

The endomorphism operad is not a $\mathfrak{S}$-free operad.

In order to have a category of operads we introduce a notion of a morphism
between operads.

######  Definition 7.4.

Let $\mathcal{O}$ and $\mathcal{D}$ be operads. A morphism of operads
$\Phi\colon\mathcal{O}\longrightarrow\mathcal{D}$ is a family of
$\mathfrak{S}_{n}$-equivariant maps
$\Phi_{n}\colon\mathcal{O}(n)\longrightarrow\mathcal{D}(n)$ such that the
following diagram commute:

$\textstyle{(\mathcal{O}(n_{1})\times\cdots\times\mathcal{O}(n_{k}))\times\mathcal{O}(k)\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\
}$$\scriptstyle{\mu_{\mathcal{O}}}$$\scriptstyle{(\Phi_{n_{1}}\times\cdots\times\Phi_{n_{k}})\times\Phi_{k}}$$\textstyle{\
\mathcal{O}(n)\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\Phi_{n}}$$\textstyle{(\mathcal{D}(n_{1})\times\cdots\times\mathcal{D}(n_{k}))\times\mathcal{D}(k)\ignorespaces\ignorespaces\ignorespaces\ignorespaces\
}$$\scriptstyle{\mu_{\mathcal{D}}}$$\textstyle{\ \mathcal{D}(n)}$

for every collection of integers $k,n_{1},\ldots,n_{k}\geq 0$ where
$n:=n_{1}+\cdots+n_{k}$.

The category of operads $\operatorname{\mathrm{O}p}$ consists of operads as
objects and morphisms of operads as morphisms.

####  7.1.2. $\mathcal{O}$-space

An action of an operad on a topological with a base point is defined as
follows.

######  Definition 7.5.

Let $X$ be a space in $\operatorname{\mathrm{T}op_{pt}}$ and let $\mathcal{O}$
be an operad. An action of the operad $\mathcal{O}$ on the space $X$ is a
morphism of operads $\Theta\colon\mathcal{O}\longrightarrow\mathcal{E}nd_{X}$.
The pairs $(X,\Theta)$ is called an $\mathcal{O}$-space.

A morphism of $\mathcal{O}$-spaces $(X,\Theta)$ and
$(X^{\prime},\Theta^{\prime})$ is a continuous based map $f\colon
X\longrightarrow X^{\prime}$ in $\operatorname{\mathrm{T}op_{pt}}$ such that
for every $n\geq 0$ and every $a\in\mathcal{O}(n)$ the following diagram
commutes

$\textstyle{X^{n}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\
}$$\scriptstyle{\Theta_{n}(a)}$$\scriptstyle{f^{n}}$$\textstyle{\
X\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{f}$$\textstyle{Y^{n}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\
}$$\scriptstyle{\Theta_{n}^{\prime}(a)}$$\textstyle{\ Y.}$

An equivalent definition of the action of an operad on a topological space is
formulated in the following elementary lemma.

######  Lemma 7.6.

Let $X$ be a space in $\operatorname{\mathrm{T}op_{pt}}$, and let
$\mathcal{O}$ be an operad. An action
$\Theta\colon\mathcal{O}\longrightarrow\mathcal{E}nd_{X}$ of the operad
$\mathcal{O}$ on $X$ determines and is determined by the family of continuous
maps

$\Theta_{n}\colon X^{n}\times\mathcal{O}(n)\longrightarrow X,$

where $\Theta_{0}\colon\mathrm{pt}\longrightarrow X$ is the inclusion of the
base point, that satisfy the following properties:

  1. (1)

For every $k,n_{1},\ldots,n_{k}\geq 0$ and $n:=n_{1}+\cdots+n_{k}$ the
following diagram commutes

$\textstyle{X^{n}\times(\mathcal{O}(n_{1})\times\cdots\times\mathcal{O}(n_{k}))\times\mathcal{O}(k)\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\
}$$\scriptstyle{\operatorname{id}\times\mu}$$\scriptstyle{s}$$\textstyle{\
X^{n}\times\mathcal{O}(n)\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\Theta_{n}}$$\textstyle{X}$$\textstyle{\big{(}X^{n_{1}}\times\mathcal{O}(n_{1})\big{)}\times\cdots\times\big{(}X^{n_{k}}\times\mathcal{O}(n_{k})\big{)}\times\mathcal{O}(k)\ignorespaces\ignorespaces\ignorespaces\ignorespaces\
}$$\scriptstyle{\Theta_{n_{1}}\times\cdots\times\Theta_{n_{k}}\times\operatorname{id}}$$\textstyle{\
X^{k}\times\mathcal{O}(k)\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\Theta_{k}}$

where

$\qquad s\colon
X^{n}\times(\mathcal{O}(n_{1})\times\cdots\times\mathcal{O}(n_{k}))\times\mathcal{O}(k)\longrightarrow\\\
\big{(}X^{n_{1}}\times\mathcal{O}(n_{1})\big{)}\times\cdots\times\big{(}X^{n_{k}}\times\mathcal{O}(n_{k})\big{)}\times\mathcal{O}(k)$

is the obvious shuffle homeomorphism.

  2. (2)

For every $x\in X$

$\Theta_{1}(x;\mathbf{1})=x.$

  3. (3)

For every $a\in\mathcal{O}(n)$, $\pi\in\mathfrak{S}_{n}$, and every
$(x_{1},\ldots,x_{n})\in X^{n}$

$\Theta_{n}(x_{1},\ldots,x_{n};a\cdot\pi)=\Theta_{n}(x_{\pi^{-1}(1)},\ldots,x_{\pi^{-1}(n)};a).$

Furthermore, if $f\colon(X,\Theta)\longrightarrow(X^{\prime},\Theta^{\prime})$
is a morphism of $\mathcal{O}$-spaces, then for every $n\geq 0$ the following
diagram commutes

$\textstyle{X^{n}\times\mathcal{O}(n)\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\
}$$\scriptstyle{\Theta_{n}}$$\scriptstyle{f^{n}\times\operatorname{id}}$$\textstyle{\
X\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{f}$$\textstyle{(X^{\prime})^{n}\times\mathcal{O}(n)\ignorespaces\ignorespaces\ignorespaces\ignorespaces\
}$$\scriptstyle{\Theta^{\prime}_{n}}$$\textstyle{\ X^{\prime}.}$

####  7.1.3. Little cubes operad

In this section we give an example of an operad whose structural map is
studied in the central part of this work.

Let $I:=[0,1]\subseteq\mathbb{R}$ be the unit interval, and then
$I^{d}\subseteq\mathbb{R}^{d}$ is the associated $d$-cube. Denote by
$p:=(\tfrac{1}{2},\ldots,\tfrac{1}{2})\in I^{d}$ the centre of the $d$-cube. A
little $d$-cube is simply an embedding of a cube into $I^{d}$ in such a way
that then corresponding edges are parallel, see illustration in Figure 11.

######  Definition 7.7.

Let $d\geq 1$ be an integer. A little $d$-cube is an orientation preserving
affine embedding $\vec{c}\colon I^{d}\longrightarrow I^{d}$ of a $d$-cube that
can be presented as the product map $\vec{c}=c_{1}\times\cdots\times c_{d}$
where each $c_{i}\colon I\longrightarrow I$, $1\leq i\leq d$, is an
orientation preserving affine embedding.

Figure 11. Illustration of a $2$-cube and an element of the little $2$-cube
operad.

Now, the little cubes operad is defined as follows.

######  Definition 7.8.

Let $d\geq 1$ be an integer. The little $d$-cubes operad $\mathcal{C}_{d}$,
for $n\geq 1$, is defined by the family of topological spaces of all
$n$-tuples of little $d$-cubes whose interiors are pairwise disjoint, that is

$\mathcal{C}_{d}(n):=\\{\alpha:=(\vec{c}_{1},\ldots,\vec{c}_{n}):\vec{c}_{i}(\operatorname{int}I^{d})\cap\vec{c}_{j}(\operatorname{int}I^{d})=\varnothing\text{
for }i\neq j\\}.$

The space $\mathcal{C}_{d}(0)$ is the point, containing the unique “embedding”
of the empty set into the cubde $I^{d}$. The $n$-tuple
$\alpha=(\vec{c}_{1},\ldots,\vec{c}_{n})$ of little $d$-cubes can be seen also
as a continuous map $\alpha\colon\coprod_{j=1}^{n}I^{d}\longrightarrow I^{d}$,
where “$\coprod$” denotes the disjoint union. The space $\mathcal{C}_{d}(0)$
is assumed to be a point interpreted as the unique embedding of the empty set
into $I^{d}$. The space $\mathcal{C}_{d}(n)$ is equipped with the subspace
topology induced from the space of all continuous maps
$\coprod_{j=1}^{n}I^{d}\longrightarrow I^{d}$.

The remaining necessary ingredients for the definition the little $d$-cube
operad are given as follows.

  1. (1)

The structural map of the little $d$-cube operad

$\mu\colon(\mathcal{C}_{d}(n_{1})\times\cdots\times\mathcal{C}_{d}(n_{k}))\times\mathcal{C}_{d}(k)\longrightarrow\mathcal{C}_{d}(n),$

$n=n_{1}+\cdots+n_{k}$, is defined as a composition

$\alpha\circ(\beta_{1}\uplus\cdots\uplus\beta_{k})\colon\Big{(}\coprod_{j=1}^{n_{1}}I^{d}\Big{)}\amalg\cdots\amalg\Big{(}\coprod_{j=1}^{n_{k}}I^{d}\Big{)}\longrightarrow\Big{(}\coprod_{j=1}^{k}I^{d}\Big{)}\longrightarrow
I^{d}$

where $\alpha\in\mathcal{C}(k)$ and
$\beta_{1}\in\mathcal{C}_{d}(n_{1}),\ldots,\beta_{k}\in\mathcal{C}_{d}(n_{k})$.

  2. (2)

The element $\mathbf{1}\in\mathcal{C}_{d}(1)$ is the identity map
$\operatorname{id}\colon I^{d}\longrightarrow I^{d}$.

  3. (3)

The right action of the symmetric group $\mathfrak{S}_{n}$ on the space
$\mathcal{C}_{d}(n)$ is given by

$(\vec{c}_{1},\ldots,\vec{c}_{n})\cdot\pi:=(\vec{c}_{\pi(1)},\ldots,\vec{c}_{\pi(n)}).$

The little $d$-cubes operad is an $\mathfrak{S}$-free operad since for each
$n$ the symmetric group $\mathfrak{S}_{n}$ acts freely on
$\mathcal{C}_{d}(n)$.

There exists an $\mathfrak{S}_{n}$-equivariant map of little $d$-cube operad
space $\mathcal{C}_{d}(n)$ into the configuration space
$\operatorname{F}(\mathbb{R}^{d},n)$ given by evaluating each cube at the
centre $p=(\tfrac{1}{2},\ldots,\tfrac{1}{2})\in I^{d}$:

$\operatorname{ev}_{d,n}\colon\mathcal{C}_{d}(n)\longrightarrow\operatorname{F}(\mathbb{R}^{d},n),\qquad(\vec{c}_{1},\ldots,\vec{c}_{n})\longmapsto(\vec{c}_{1}(p),\ldots,\vec{c}_{n}(p)).$
(100)

This map is an $\mathfrak{S}_{n}$-equivariant homotopy equivalence, see [76,
Thm. 4.8].

######  Lemma 7.9.

For integers $d\geq 1$ and $n\geq 1$ the evaluation at centers of cubes map
$\operatorname{ev}_{d,n}\colon\mathcal{C}_{d}(n)\longrightarrow\operatorname{F}(\mathbb{R}^{d},n)$
is an $\mathfrak{S}_{n}$-equivariant homotopy equivalence of the spaces
$\mathcal{C}_{d}(n)$ and $\operatorname{F}(\mathbb{R}^{d},n)$.

####  7.1.4. $\mathcal{C}_{d}$-spaces, an example

In this section we give an example of a family of $\mathcal{C}_{d}$-spaces.
More precisely, for a pointed topological space $X$ we define an action of the
little cubes operad $\mathcal{C}_{d}$ on its $d$-fold loop space
$\Omega^{d}X$. For more details consult for example [76, Thm. 5.1].

In order to define a $\mathcal{C}_{d}$-action on the $d$-fold loop space
$\Omega^{d}X$ we use Lemma 7.6 and give a family of functions

$\Theta_{d,n}\colon(\Omega^{d}X)^{n}\times\mathcal{C}_{d}(n)\longrightarrow\Omega^{d}X.$

For $\omega_{1},\ldots,\omega_{n}\in\Omega^{d}X$ and
$(\vec{c}_{1},\ldots,\vec{c}_{n})\in\mathcal{C}_{d}(n)$ the loop

$\Theta_{d,n}(\omega_{1},\ldots,\omega_{n};\vec{c}_{1},\ldots,\vec{c}_{n})\colon
I^{d}\longrightarrow X$

is defined, when $x\in I^{d}$, by

$\Theta_{d,n}(\omega_{1},\ldots,\omega_{n};\vec{c}_{1},\ldots,\vec{c}_{n})(x):=\begin{cases}\omega_{i}(\vec{c}_{i}^{-1}(x)),&\text{if
}x\in\vec{c}_{i}(I^{d})\text{ for some }0\leq i\leq n,\\\
\mathrm{pt},&\text{otherwise},\end{cases}$

where $\mathrm{pt}\in X$ is the base point. In the case $d=1$ the elements
$\omega_{1},\ldots,\omega_{n}$ are pointed loops and the the map
$\Theta_{1,n}(\omega_{1},\ldots,\omega_{n};\vec{c}_{1},\ldots,\vec{c}_{n})$ is
just a concatenation of loops $\omega_{1},\ldots,\omega_{n}$ specified by the
collection of, pairwise interior disjoint, intervals
$(\vec{c}_{1},\ldots,\vec{c}_{n})\in\mathcal{C}_{1}(n)$. For an illustration
see Figure 12. By direct inspection it can be verified that just defined
family of functions satisfies all the necessary conditions of Lemma 7.6. Thus,
for a space $X$ in $\operatorname{\mathrm{T}op_{pt}}$ the $d$-fold loop space
$\Omega^{d}X$ is a $\mathcal{C}_{d}$-space.

Figure 12. An illustration of the loop
$\Theta_{1,3}(\omega_{1},\omega_{2},\omega_{1};\vec{c}_{1},\vec{c}_{2},\vec{c}_{3})$.

####  7.1.5. $\mathcal{C}_{d}$-spaces, a free $\mathcal{C}_{d}$-space over $X$

Let $X$ be a pointed space with the base point $\mathrm{pt}\in X$, and let
$d\geq 1$ be an integer. Then we define the $\mathcal{C}_{d}$-space associated
to $X$ to be the quotient space

$\mathcal{C}_{d}(X):=\Big{(}\coprod_{m\geq
0}\mathcal{C}_{d}(m)\times_{\mathfrak{S}_{m}}X^{m}\Big{)}/_{\approx}$

where for $(\vec{c}_{1},\ldots,\vec{c}_{m})\in\mathcal{C}_{d}(m)$ and
$(x_{1},\ldots,x_{m-1},\mathrm{pt})\in X^{m}$ we define equivalence relation
generated by

$((\vec{c}_{1},\ldots,\vec{c}_{m-1},\vec{c}_{m}),(x_{1},\ldots,x_{m-1},\mathrm{pt}))\approx((\vec{c}_{1},\ldots,\vec{c}_{m-1}),(x_{1},\ldots,x_{m-1})).$

The $\mathcal{C}_{d}$-action on $\mathcal{C}_{d}(X)$ is induced from the
structural maps of the little cubes operad $\mathcal{C}_{d}$. For more details
on the definition of $\mathcal{C}_{d}(X)$, as a monad associated to the operad
$\mathcal{C}_{d}$, see [76, Constr. 2.4].

The $\mathcal{C}_{d}$-space $\mathcal{C}_{d}(X)$, associated to the pointed
space $X$, can be called the free $\mathcal{C}_{d}$-space generated by $X$,
because for every $\mathcal{C}_{d}$-space $Y$ there exists a bijective
correspondence

$(\text{morphisms of }\mathcal{C}_{d}\text{-space from
}\mathcal{C}_{d}(X)\text{ to }Y)\longleftrightarrow\\\ (\text{morphisms of
topological space from }X\text{ to }Y).$ (101)

Consult [76, Prop 2.8 and Lem. 2.9].

Let $X\longrightarrow\Omega^{d}\Sigma^{d}X$ be the map associated to the
identity map $\operatorname{id}\colon\Sigma^{d}X\longrightarrow\Sigma^{d}X$
along the adjunction relation

$[A,\Omega^{d}B]_{\mathrm{pt}}\longleftrightarrow[\Sigma^{d}A,B]_{\mathrm{pt}}.$

Here $[A,B]_{\mathrm{pt}}$ denotes the set of all homotopy classes of pointed
maps $A\longrightarrow B$ between the pointed spaces $A$ and $B$. Next, let
$\alpha_{d}\colon\mathcal{C}_{d}(X)\longrightarrow\Omega^{d}\Sigma^{d}X$
denotes the morphism of $\mathcal{C}_{d}$-space associated to the map
$X\longrightarrow\Omega^{d}\Sigma^{d}X$ with respect to the correspondence
(101). Now the Approximation theorem [76, Thm. 2.7 and Thm. 6.1] of May states
the following.

######  Theorem 7.10.

Let $d\geq 1$ be an integer, or let $d=\infty$. If $X$ is a path-connected
space in $\operatorname{\mathrm{T}op_{pt}}$, then

$\alpha_{d}\colon\mathcal{C}_{d}(X)\longrightarrow\Omega^{d}\Sigma^{d}X$

is a weak homotopy equivalence.

####  7.1.6. Araki–Kudo–Dyer–Lashof homology operations

Following analogy with Dyer & Lashof [45, Def. 2.2] we review basic properties
of the Araki–Kudo–Dyer–Lashof homology operations as defined by Cohen [33,
Def. 5.6]. In the case $p=2$ the homology operations were first introduced by
Araki & Kudo [70].

Let $d\geq 1$ be an integer, or let $d=\infty$. Let $Y$ be a compactly
generated weak Hausdorff space with non-degenerate base point endowed with an
action of the little cubes operad $\mathcal{C}_{d}$. Then there exists a
sequence of maps

$Q_{i}\colon H_{j}(Y;\mathbb{F}_{2})\longrightarrow
H_{i+2j}(Y;\mathbb{F}_{2}),\qquad\qquad 0\leq i\leq d-1,$

called Araki–Kudo–Dyer–Lashof homology operations. Properties of these
operations are listed in the next proposition, see also [45, Thm. 2.2, Cor. 1]
[33, Sec. 1]. In the following $\lambda_{d-1}\colon
H_{i}(Y;\mathbb{F}_{2})\otimes H_{j}(Y;\mathbb{F}_{2})\longrightarrow
H_{i+j+d-1}(Y;\mathbb{F}_{2})$ denotes the Browder operation, consult [22]
[33, Thm. 1.2].

######  Proposition 7.11.

Let $Y$ and $Z$ be $\mathcal{C}_{d}$-spaces. The following properties hold:

  1. (1)

$Q_{0}(y)=y^{2}$ for every $x\in H_{*}(Y;\mathbb{F}_{2})$.

  2. (2)

$Q_{i}$ is a homomorphism for every $0\leq i\leq d-2$.

  3. (3)

$Q_{d-1}$ is not homomorphism in general and

$Q_{d-1}(y_{1}+y_{2})=Q_{d-1}(y_{1})+Q_{d-1}(y_{2})+\lambda_{d-1}(y_{1},y_{2})$

for $y_{1},y_{2}\in H_{*}(Y;\mathbb{F}_{2})$ with $\deg(y_{1})=\deg(y_{2})$.

  4. (4)

If $Y$ is connected and $1\in H_{0}(Y;\mathbb{F}_{2})$, then $Q_{i}(1)=0$ for
$0<i\leq d-1$.

  5. (5)

The operations $Q_{i}$ are natural with respect to the morphisms of
$\mathcal{C}_{d}$-spaces.

  6. (6)

If $y_{r}\in H_{r}(Y;\mathbb{F}_{2})$, $Z_{s}\in H_{s}(Z;\mathbb{F}_{2})$ and
$0\leq i\leq d-1$, then in $H_{*}(Y\times Z;\mathbb{F}_{2})$ holds

$Q_{i}(y_{r}\otimes z_{s})=\sum_{j=0}^{i}Q_{j}(y_{r})\otimes
Q_{i-j}(z_{s})+\varepsilon_{n}(y_{r},z_{s}),$

where $\varepsilon_{n}(y_{r}\otimes z_{s})$ is the “error term.” For example,
in the case when $Y=Z=\Omega^{d}\Sigma^{d}S^{L}$, $d\geq 2$, and $L\geq 1$,
the “error term” vanishes.

  7. (7)

If $y_{r}\in H_{r}(Y;\mathbb{F}_{2})$, $y_{s}^{\prime}\in
H_{s}(Y;\mathbb{F}_{2})$ and $0\leq i\leq d-1$, then in
$H_{*}(Y;\mathbb{F}_{2})$ holds

$Q_{i}(y_{r}\cdot y_{s}^{\prime})=\sum_{j=0}^{i}Q_{j}(y_{r})\cdot
Q_{i-j}(y_{s}^{\prime})+\varepsilon_{n}^{\prime}(y_{r},y_{s}^{\prime}),$

where “$\cdot$” denotes the Pontryagin product and
$\varepsilon_{n}^{\prime}(y_{r},y_{s}^{\prime})$ stands for the error term.
For example, in the case when $Y=\Omega^{d}\Sigma^{d}S^{L}$, $d\geq 2$, and
$L\geq 1$, the error term vanishes.

Assume that $Y$ is still a $\mathcal{C}_{d}$-space. Then the operations

$Q_{i}\colon H_{j}(Y;\mathbb{F}_{2})\longrightarrow
H_{i+2j}(Y;\mathbb{F}_{2})$

are defined in [33, Def. 5.6] as follows. The $\mathcal{C}_{d}$-action on $Y$
yields a map

$\Theta_{2}\colon\mathcal{C}_{d}(2)\times(Y\times Y)\longrightarrow Y$

that behaves naturally with respect to the acton of
$\mathfrak{S}_{2}\cong\mathbb{Z}_{2}$. Consequently, it induces the quotient
map (denoted in the same way)

$\Theta_{2}\colon\mathcal{C}_{d}(2)\times_{\mathbb{Z}_{2}}(Y\times
Y)\longrightarrow Y$

where the action on the product $Y\times Y$ is given by interchanging factors.
The space $\mathcal{C}_{d}(2)$ is $\mathbb{Z}_{2}$ equivariantly homotopic to
the sphere $S^{d-1}$ equipped with the antipodal actions. Hence, the
cohomology and homology of $\mathcal{C}_{d}(2)\times_{\mathbb{Z}_{2}}(Y\times
Y)$, with $\mathbb{F}_{2}$-coefficients, is completely described in Section
3.3. Using the notation from Theorem 3.7 we define

$Q_{i}(y):=(\Theta_{2})_{*}\big{(}(y\otimes
y)\otimes_{\mathbb{Z}_{2}}f_{i}\big{)}$

for $y\in H_{*}(Y;\mathbb{F}_{2})$ where $0\leq i\leq d-1$.

###  7.2. The Dickson algebra

In this section we present all the facts about the Dickson algebra that are
relevant for the calculations in this paper. For this we rely on the sources
[2, Sec. III.2], [27], and [101], where some of the results we present
appeared already in the original paper of Dickson [43].

####  7.2.1. Rings of invariants

Let $m\geq 1$ be an integer, let $V$ be an $m$-dimensional vector space over
the field $\mathbb{F}_{2}$, and let $\mathrm{Sym}(V)$ denotes the symmetric
algebra of $V$ over $\mathbb{F}_{2}$. Then for a choice of a basis
$(x_{1},\ldots,x_{m})$ of $V$ there is an isomorphism
$\mathrm{Sym}(V)\cong\mathbb{F}_{2}[x_{1},\ldots,x_{m}]$. The algebra
$\mathrm{Sym}(V)$ is graded by setting $\deg(x_{1})=\cdots=\deg(x_{m})=1$. The
degree of a monomial is defined in the usual way by
$\deg(x_{1}^{\alpha_{1}}\cdots
x_{m}^{\alpha_{m}}):=\alpha_{1}+\cdots+\alpha_{m}$. The set of all homogeneous
polynomials of degree $n$ is denoted by $\mathrm{Sym}^{n}(V)$ where
$\mathrm{Sym}^{0}(V)\cong\mathbb{F}_{2}$. Consequently,
$\mathrm{Sym}(V)\cong\bigoplus_{n\geq 0}\mathrm{Sym}^{n}(V)$.

The general linear group
$\operatorname{GL}(V)\cong\operatorname{GL}_{m}(\mathbb{F}_{2})$ on the vector
space $V$ acts from the left on $\mathrm{Sym}(V)$ and preserves the introduced
grading. If $G$ is a subgroup of $\operatorname{GL}_{m}(\mathbb{F}_{2})$, then
$\mathrm{Sym}(V)^{G}$ denotes the ring of $G$-invariants. In this section we
will discuss the invariants only of the full general linear group
$\operatorname{GL}_{m}(\mathbb{F}_{2})$ and the subgroup
$\mathrm{L}_{m}(\mathbb{F}_{2})\subseteq\operatorname{GL}_{m}(\mathbb{F}_{2})$
of all lower triangular matrices with $1$’s on the main diagonal. It is a
known fact that $\mathrm{L}_{m}(\mathbb{F}_{2})$ is a Sylow $2$-subgroup of
$\operatorname{GL}_{m}(\mathbb{F}_{2})$. In fact, the order of the group
$\mathrm{L}_{m}(\mathbb{F}_{2})$ is $2^{\frac{m(m-1)}{2}}$ while its index in
$\operatorname{GL}_{m}(\mathbb{F}_{2})$ is $(2m-1)!!=1\cdot 3\cdot
5\cdots(2m-1)$.

Let $(x_{1},\ldots,x_{m})$ be a fixed basis of the vector space $V$, and let
us introduce a complete flag of subspaces in $V$ by

$\\{0\\}=V_{0}\subseteq V_{1}\subseteq\cdots\subseteq V_{m-1}\subseteq
V_{m}=V$

where $V_{i}:=\operatorname{span}\\{x_{m},\ldots,x_{m-i+1}\\}$, for $1\leq
i\leq m$. Thus the complete flag we consider is

$\\{0\\}\subseteq\operatorname{span}\\{x_{m}\\}\subseteq\operatorname{span}\\{x_{m},x_{m-1}\\}\subseteq\\\
\cdots\subseteq\operatorname{span}\\{x_{m},x_{m-1},\dots,x_{2}\\}\subseteq\operatorname{span}\\{x_{m},x_{m-1},\dots,x_{2},x_{1}\\}.$

The rings of invariants of the polynomial ring
$S(V)\cong\mathbb{F}_{2}[x_{1},\ldots,x_{m}]$ with respect to the groups
$\mathrm{L}_{m}(\mathbb{F}_{2})$ and $\operatorname{GL}_{m}(\mathbb{F}_{2})$
are denoted as follows:

$\mathfrak{H}_{m}:=\mathrm{Sym}(V)^{\mathrm{L}_{m}(\mathbb{F}_{2})}\cong\mathbb{F}_{2}[x_{1},\ldots,x_{m}]^{\mathrm{L}_{m}(\mathbb{F}_{2})},$

and

$\mathfrak{D}_{m}:=\mathrm{Sym}(V)^{\operatorname{GL}_{m}(\mathbb{F}_{2})}\cong\mathbb{F}_{2}[x_{1},\ldots,x_{m}]^{\operatorname{GL}_{m}(\mathbb{F}_{2})}.$

Assuming the previously introduced notation a result of Mùi [82, Thm. 6.4]
gives the following described the ring of invariants $\mathfrak{H}_{m}$, see
also [27, Thm. 3.1].

######  Theorem 7.12.

Let $h_{i}:=\prod_{v\in V_{i-1}}(x_{m-i+1}+v)$ for $1\leq i\leq m$, a
polynomial of degree $2^{i-1}$ in
$\mathrm{Sym}(V)\cong\mathbb{F}_{2}[x_{1},\ldots,x_{m}]$. Then

$\mathfrak{H}_{m}=\mathbb{F}_{2}[h_{1},\ldots,h_{m}].$

For example, if $m=3$ then

$h_{1}=x_{3},\qquad h_{2}=x_{2}(x_{2}+x_{3}),\qquad
h_{3}=x_{1}(x_{1}+x_{2})(x_{1}+x_{3})(x_{1}+x_{2}+x_{3}).$

For the ring of invariants $\mathfrak{D}_{m}$ we rely on a result of Dickson
[43] and have the following presentation, see also [101, Thm. 1.2].

######  Theorem 7.13.

Let $d_{m,0},\ldots,d_{m,m-1},d_{m,m}$ be the polynomials in
$\mathbb{F}_{2}[x_{1},\ldots,x_{m}]$ defined as the coefficients of the
polynomial

$f_{m}(T):=\prod_{v\in V}(T+v)=\sum_{i=0}^{m}d_{m,i}\,T^{2^{i}}$

in $\mathbb{F}_{2}[x_{1},\ldots,x_{m}][T]$. Then $\deg(d_{m,i})=2^{m}-2^{i}$
for $0\leq i\leq m$, $d_{m,m}=1$, and

$\mathfrak{D}_{m}=\mathbb{F}_{2}[d_{m,0},\ldots,d_{m,m-1}].$

The polynomials $d_{m,0},\ldots,d_{m,m-1}$ are called the Dickson invariants
of $\mathrm{Sym}(V)$ or of the polynomial ring
$\mathbb{F}_{2}[x_{1},\ldots,x_{m}]$. For example, if $m=2$ then

$f_{2}(T)=T(T+x_{1})(T+x_{2})(T+x_{1}+x_{2})=x_{1}x_{2}(x_{1}+x_{2})T+(x_{1}x_{2}+x_{1}^{2}+x_{2}^{2})T^{2}.$

Hence,

$d_{2,0}=x_{1}x_{2}(x_{1}+x_{2})\qquad\text{and}\qquad
d_{2,1}=x_{1}x_{2}+x_{1}^{2}+x_{2}^{2}.$

On the other hand $h_{1}=x_{2}$ and $h_{2}=x_{1}(x_{1}+x_{2})$, and
consequently

$d_{2,0}=h_{1}h_{2}\qquad\text{and}\qquad d_{2,1}=1\cdot
h_{2}+\chi_{2}h_{1}^{2},$

where $\chi_{2}\in\operatorname{GL}_{2}(F_{2})$ is the variable substitution
given by the matrix $\begin{pmatrix}0&1\\\ 1&0\end{pmatrix}$.

The formula connecting generators of the rings of invariants
$\mathfrak{H}_{m}$ and $\mathfrak{D}_{m}$ is as follows, see for more details
[101, Prop. 1.3 (b)].

######  Proposition 7.14.

For $0\leq i\leq m-1$ and setting $d_{m-1,-1}=0$ we have that

$d_{m,i}=(\chi_{m}d_{m-1,i})\,h_{m}+(\chi_{m}d_{m-1,i-1})^{2},$ (102)

where $\chi_{m}\in\operatorname{GL}_{m}(\mathbb{F}_{2})$ is the change of
variables given by the matrix

$\begin{pmatrix}0&0&\cdots&0&1\\\ 0&0&\cdots&1&0\\\ &&\cdots&&\\\
1&0&\cdots&0&0\end{pmatrix}.$

If we set $k_{j}:=\chi_{m}h_{j}$ for $1\leq j\leq m$ then the relation (102)
becomes

$d_{m,i}=d_{m-1,i}k_{m}+d_{m-1,i-1}^{2},$ (103)

where $0\leq i\leq m-1$. It is not hard to see that $k_{1},\ldots,k_{m}$ are
generators of the ring of invariants with respect to the subgroup
$\mathrm{U}_{m}(\mathbb{F}_{2})$ of $\operatorname{GL}_{m}(\mathbb{F}_{2})$
that consists of all upper triangular matrices with $1$’s on the main
diagonal. More precisely,

$\mathrm{Sym}(V)^{\mathrm{U}_{m}(\mathbb{F}_{2})}\cong\mathbb{F}_{2}[x_{1},\ldots,x_{m}]^{\mathrm{U}_{m}(\mathbb{F}_{2})}=\mathbb{F}_{2}[k_{1},\ldots,k_{m}].$

From the relation (103) we can derive a presentation of each Dickson invariant
$d_{m,i}$ in term of $k_{1},\ldots,k_{m}$. Indeed, let $0\leq r\leq m-1$ be an
integer, and let $J=(j_{1},\ldots,j_{r})\in[m]^{r}$ with $1\leq
j_{1}<\cdots<j_{r}\leq m$. We denote particular monomials in
$k_{1},\ldots,k_{n}$ as follows:

$k[J]:=(k_{1}\cdots k_{j_{1}-1})^{2^{r}}\,(k_{j_{1}+1}\cdots
k_{j_{2}-1})^{2^{r-1}}\cdots(k_{j_{r}+1}\cdots k_{m})^{2^{0}}.$

Then as in [27, Thm. 4.3] we have that

$\displaystyle d_{m,r}$ $\displaystyle=$
$\displaystyle\sum_{J\in[m]^{r}\,:\,j_{1}<\cdots<j_{r}}k[J]$ $\displaystyle=$
$\displaystyle\sum_{J\in[m]^{r}\,:\,j_{1}<\cdots<j_{r}}(k_{1}\cdots
k_{j_{1}-1})^{2^{r}}\,(k_{j_{1}+1}\cdots
k_{j_{2}-1})^{2^{r-1}}\cdots(k_{j_{r}+1}\cdots k_{m})^{2^{0}}.$

In particular,

$d_{m,0}=k_{1}\cdots k_{m}\qquad\text{and}\qquad
d_{m,m-1}=k_{1}^{2^{m-1}}+k_{2}^{2^{m-2}}+\cdots+k_{m}^{2^{0}}.$ (105)

####  7.2.2. The Dickson invariants as characteristic classes

Let $m\geq 1$ be an integer. Consider the sequence of group embeddings

$\textstyle{\mathcal{E}_{m}:=\mathbb{Z}_{2}^{\oplus
m}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{(\mathrm{reg})}$$\textstyle{\mathfrak{S}_{2^{m}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\iota_{2^{m}}}$$\textstyle{\mathrm{O}(2^{m}),}$

where

  * $\bullet$

$(\mathrm{reg})\colon\mathbb{Z}_{2}^{\oplus
m}\longrightarrow\mathfrak{S}_{2^{m}}$ is the regular embedding that is given
by the left translation action of $\mathbb{Z}_{2}^{\oplus m}$ on itself,
consult [2, Ex. III.2.7], and

  * $\bullet$

$\iota_{2^{m}}\colon\mathfrak{S}_{2^{m}}\longrightarrow\mathrm{O}(2^{m})$ is
the embedding given by the permutation representation.

This sequence of embeddings induces a sequence of maps of the corresponding
classifying spaces

$\textstyle{\mathrm{B}\mathcal{E}_{m}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\mathrm{B}(\mathrm{reg})}$$\textstyle{\mathrm{B}\mathfrak{S}_{2^{m}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\mathrm{B}(\iota_{2^{m}})}$$\textstyle{\mathrm{B}\mathrm{O}(2^{m}).}$

Let $\gamma_{2^{m}}$ denotes the tautological vector bundle over
$\mathrm{B}\mathrm{O}(2^{m})$, and let us denote the pullbacks as follows

$\xi_{2^{m}}:=\mathrm{B}(\iota_{2^{m}})^{*}\gamma_{2^{m}}\qquad\text{and}\qquad\nu_{2^{m}}:=(\mathrm{B}(\iota_{2^{m}})\circ\mathrm{B}(\mathrm{reg}))^{*}\gamma_{2^{m}}.$

Then the bundles $\gamma_{2^{m}}$, $\xi_{2^{m}}$ and $\nu_{2^{m}}$ induce the
following commutative diagram of vector bundle morphisms:

$\textstyle{\mathrm{E}\mathcal{E}_{m}\times_{\mathcal{E}_{m}}\mathbb{R}^{2^{m}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\
}$$\scriptstyle{\nu_{2^{m}}}$$\textstyle{\
\mathrm{E}\mathfrak{S}_{2^{m}}\times_{\mathfrak{S}_{2^{m}}}\mathbb{R}^{2^{m}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\
}$$\scriptstyle{\xi_{2^{m}}}$$\textstyle{\
\mathrm{E}\mathrm{O}(2^{m})\times_{\mathrm{O}(2^{m})}\mathbb{R}^{2^{m}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\gamma_{2^{m}}}$$\textstyle{\mathrm{B}\mathcal{E}_{m}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\
}$$\scriptstyle{\mathrm{B}(\mathrm{reg})}$$\textstyle{\
\mathrm{B}\mathfrak{S}_{2^{m}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\
}$$\scriptstyle{\mathrm{B}(\iota_{2^{m}})}$$\textstyle{\
\mathrm{B}\mathrm{O}(2^{m}).}$

Recall, that the cohomology ring of the elementary abelian group
$\mathcal{E}_{m}$ with coefficients in the field $\mathbb{F}_{2}$ is a
polynomial ring on $m$ generators in degree $1$, that is
$H^{*}(\mathcal{E}_{m};\mathbb{F}_{2})\cong\mathbb{F}_{2}[y_{1},\ldots,y_{m}]$
with $\deg(y_{1})=\cdots=\deg(y_{m})=1$.

The relationship of the introduced bundles and the Dickson invariants is given
by the following theorem, see [72, Lem. 3.26].

######  Theorem 7.15.

The total Stiefel–Whitney class of the vector bundle $\nu_{2^{m}}$ is

$w(\nu_{2^{m}})=1+d_{m,m-1}+d_{m,m-2}+\cdots+d_{m,0},$

where $d_{m,m-1},\ldots,d_{m,0}$ are the Dickson invariants of
$F_{2}[y_{1},\ldots,y_{m}]\cong H^{*}(\mathcal{E}_{m};\mathbb{F}_{2})$. This
means that

$w_{i}(\nu_{2^{m}})=\begin{cases}d_{m,j},&i=2^{m}-2^{j},\ 0\leq j\leq m-1,\\\
1,&i=0,\\\ 0,&\text{\rm otherwise}.\end{cases}$

###  7.3. The Stiefel–Whitney classes of the wreath square of a vector bundle

In this section we introduce notions called the wreath square of a vector
bundle [11, Sec. 3] and the $(d-1)$-partial wreath square of a vector bundle.
Furthermore, we collect all necessary facts we use in Section 6. Our
presentation partially follows [11, Sec. 3] and is given in the generality
necessary for our computations.

####  7.3.1. The wreath square and the $(d-1)$-partial wreath square of a
vector bundle

Let $X$ be a CW-complex which a priori is not finite. The product $X\times X$
has a natural action of the group $\mathbb{Z}_{2}$ given by interchanging the
copies, that is $\omega\cdot(x_{1},x_{2}):=(x_{2},x_{1})$, where $\omega$
generates $\mathbb{Z}_{2}$ and $(x_{1},x_{2})\in X\times X$. The product
spaces $(X\times X)\times\mathrm{E}\mathbb{Z}_{2}$ and $(X\times X)\times
S^{d-1}$ with the diagonal $\mathbb{Z}_{2}$-actions are free
$\mathbb{Z}_{2}$-spaces. The action on the sphere $S^{d-1}$ is assumed to be
antipodal. Thus, the projection maps

$p_{1}\colon(X\times X)\times\mathrm{E}\mathbb{Z}_{2}\longrightarrow X\times
X\qquad\text{and}\qquad p_{1}\colon(X\times X)\times S^{d-1}\longrightarrow
X\times X$

given by $(x_{1},x_{1},e)\longmapsto(x_{1},x_{2})$ are $\mathbb{Z}_{2}$-map.
Here the model for $\mathrm{E}\mathbb{Z}_{2}$ is assumed to be the infinite
sphere $S^{\infty}:=\operatorname{colim}_{d\to\infty}S^{d-1}$ equipped with
the antipodal action inherited from the action on $S^{d-1}$.

Now, we define the functors

$S^{2}\colon\operatorname{\mathrm{T}op_{cw}}\longrightarrow\operatorname{\mathrm{T}op_{cw}}\qquad\text{and}\qquad
S^{2,d}\colon\operatorname{\mathrm{T}op_{cw}}\longrightarrow\operatorname{\mathrm{T}op_{cw}}$

that are on the objects given by

$S^{2}X:=((X\times X)\times\mathrm{E}\mathbb{Z}_{2})/\mathbb{Z}_{2}=(X\times
X)\times_{\mathbb{Z}_{2}}\mathrm{E}\mathbb{Z}_{2},$

and

$S^{2,d}X:=((X\times X)\times S^{d-1})/\mathbb{Z}_{2}=(X\times
X)\times_{\mathbb{Z}_{2}}S^{d-1},$

where $X$ is a CW-complex and $d\geq 1$ integer. For a morphism $h\colon
X\longrightarrow Y$ of CW-complexes, a continuous map, we set

$S^{2}h:=(h\times h)\times_{\mathbb{Z}_{2}}\operatorname{id}\colon(X\times
X)\times_{\mathbb{Z}_{2}}\mathrm{E}\mathbb{Z}_{2}\longrightarrow(Y\times
Y)\times_{\mathbb{Z}_{2}}\mathrm{E}\mathbb{Z}_{2},$

and

$S^{2,d}h:=(h\times h)\times_{\mathbb{Z}_{2}}\operatorname{id}\colon(X\times
X)\times_{\mathbb{Z}_{2}}S^{d-1}\longrightarrow(Y\times
Y)\times_{\mathbb{Z}_{2}}S^{d-1},$

to be the maps induced by the product maps

$(h\times h)\times\operatorname{id}\colon(X\times
X)\times\mathrm{E}\mathbb{Z}_{2}\longrightarrow(Y\times
Y)\times\mathrm{E}\mathbb{Z}_{2}$

and

$(h\times h)\times\operatorname{id}\colon(X\times X)\times
S^{d-1}\longrightarrow(Y\times Y)\times S^{d-1}$

by passing to the $\mathbb{Z}_{2}$-orbits.

Consider now a real $n$-dimensional vector bundle
$\xi:=(E(\xi)\overset{p_{\xi}}{\longrightarrow}B(\xi))$ over the CW-complex
$B(\xi)$ whose fiber is $F(\xi)$. Here we denote by $E(\xi)$ the total space
of $\xi$, and by $p_{\xi}$ the corresponding projection map. The pull-back
vector bundle $p_{1}^{*}(\xi\times\xi)$ of the product vector bundle
$\xi\times\xi$ along the $p_{1}$:

$\textstyle{E(p_{1}^{*}(\xi\times\xi))\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\
}$$\textstyle{\ E(\xi\times\xi)\cong E(\xi)\times
E(\xi)\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{(B(\xi)\times
B(\xi))\times\mathrm{E}\mathbb{Z}_{2}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\
}$$\scriptstyle{p_{1}}$$\textstyle{\ B(\xi\times\xi)\cong B(\xi)\times
B(\xi),}$

is equipped with a free $\mathbb{Z}_{2}$-action. Moreover, the projection map
of the pull-back bundle
$E(p_{1}^{*}(\xi\times\xi))\longrightarrow(B(\xi)\times
B(\xi))\times\mathrm{E}\mathbb{Z}_{2}$ is a $\mathbb{Z}_{2}$-map. Hence, after
passing to $\mathbb{Z}_{2}$-orbits we get the $2n$-dimensional vector bundle
$S^{2}\xi$ over $S^{2}B(\xi)$:

$E(p_{1}^{*}(\xi\times\xi))/\mathbb{Z}_{2}\longrightarrow(B(\xi)\times
B(\xi))\times_{\mathbb{Z}_{2}}\mathrm{E}\mathbb{Z}_{2}.$

The bundle $S^{2}\xi$ is called the wreath square of the vector bundle $\xi$.

Next we consider the following pull-back diagram induced by the
$\mathbb{Z}_{2}$-inclusion $i\colon S^{d-1}\longrightarrow S^{\infty}$:

$\textstyle{E(((\operatorname{id}\times\operatorname{id})\times_{\mathbb{Z}_{2}}i)^{*}S^{2}\xi)\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\
}$$\textstyle{\
E(p_{1}^{*}(\xi\times\xi))\ignorespaces\ignorespaces\ignorespaces\ignorespaces/\mathbb{Z}_{2}=E(S^{2}\xi)}$$\textstyle{(B(\xi)\times
B(\xi))\times_{\mathbb{Z}_{2}}S^{d-1}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\
}$$\scriptstyle{(\operatorname{id}\times\operatorname{id})\times_{\mathbb{Z}_{2}}i}$$\textstyle{\
(B(\xi)\times
B(\xi))\times_{\mathbb{Z}_{2}}\mathrm{E}\mathbb{Z}_{2}=B(S^{2}\xi).}$

The pull-back vector bundle
$((\operatorname{id}\times\operatorname{id})\times_{\mathbb{Z}_{2}}i)^{*}S^{2}\xi$
is called the $(d-1)$-partial wreath square of the vector bundle $\xi$ and is
denoted by $S^{2,d}\xi$. In other words,

$S^{2,d}\xi:=((\operatorname{id}\times\operatorname{id})\times_{\mathbb{Z}_{2}}i)^{*}S^{2}\xi,$

with the base space $S^{2,d}B(\xi)$.

The wreath square, as well as the $(d-1)$-partial wreath square, of a vector
bundle is natural with respect to morphisms of vector bundles. Indeed, a
morphism between vector bundles $\xi\longrightarrow\eta$ induces morphisms
between associated wreath squares $S^{2}\xi\longrightarrow S^{2}\eta$ and
$(d-1)$-partial wreath squares $S^{2,d}\xi\longrightarrow S^{2,d}\eta$. These
morphisms satisfy all expected properties with respect to the composition of
morphisms. Furthermore, the wreath square and the $(d-1)$-partial wreath
square of a vector bundle behave naturally with respect to the Whitney sum of
vector bundles. This means that for arbitrary vector bundles $\xi$ and $\eta$
there are isomorphisms of vector bundles

$S^{2}(\xi\oplus\eta)\cong S^{2}(\xi)\oplus S^{2}(\eta)\qquad\text{and}\qquad
S^{2,d}(\xi\oplus\eta)\cong S^{2,d}(\xi)\oplus S^{2,d}(\eta).$ (106)

####  7.3.2. Cohomology of $B(S^{2}\xi)=S^{2}B(\xi)$

In this section, based on the material presented in Section 3.3, we describe
the cohomology of a typical base space of the wreath square of a vector
bundle.

Let $X$ be the base space of the vector bundle $\xi$, that is $X=B(\xi)$.
Assume that $X$ in addition is a CW-complex. Then the base space of the vector
bundle $S^{2}\xi$ is $S^{2}X$, and also the total space of the fiber bundle

$\textstyle{X\times X\ignorespaces\ignorespaces\ignorespaces\ignorespaces\
}$$\textstyle{\ (X\times
X)\times_{\mathbb{Z}_{2}}\mathrm{E}\mathbb{Z}_{2}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\
}$$\textstyle{\ \mathrm{B}\mathbb{Z}_{2}.}$ (107)

Note that similarly the base space of the vector bundle $S^{2,d}\xi$ is the
total space of the fiber bundle

$\textstyle{X\times X\ignorespaces\ignorespaces\ignorespaces\ignorespaces\
}$$\textstyle{\ (X\times
X)\times_{\mathbb{Z}_{2}}S^{d-1}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\
}$$\textstyle{\ \mathbb{R}{\rm P}^{d-1}.}$ (108)

The Serre spectral sequence associated to the fiber bundle (107) has
$E_{2}$-term given by

$E_{2}^{i,j}(X)=H^{i}(\mathrm{B}\mathbb{Z}_{2};\mathcal{H}^{j}(X\times
X;\mathbb{F}_{2}))\cong H^{i}(\mathbb{Z}_{2};H^{j}(X\times
X;\mathbb{F}_{2})).$ (109)

As discussed in Section 3.3 this spectral sequence collapses at the
$E_{2}$-term, that means $E_{2}^{i,j}(X)\cong E_{\infty}^{i,j}(X)$ for all
$i,j\in\mathbb{Z}$. For more details consult Theorem 3.4 or [2, Thm. IV.1.7].

In the description of the total Stiefel–Whitney classes of the wreath square
of a vector bundle the following maps turn out to be very useful. At first,
consider the map (not a homomorphism)

$P\colon H^{j}(X;\mathbb{F}_{2})\longrightarrow H^{2j}(X\times
X;\mathbb{F}_{2})^{\mathbb{Z}_{2}}\cong E^{0,2j}_{2}\cong E^{0,2j}_{\infty}$

given by

$P(u):=u\otimes u,$

for $u\in H^{j}(X;\mathbb{F}_{2})$ and $j\geq 0$ an integer. By a direct
inspection we see that the map $P$ is not an additive map, but it is a
multiplicative map. The second map we consider is

$T\colon H^{j}(X\times X;\mathbb{F}_{2})\longrightarrow H^{j}(X\times
X;\mathbb{F}_{2})^{\mathbb{Z}_{2}}$

defined by

$T(u\otimes u^{\prime}):=u\otimes u^{\prime}+u^{\prime}\otimes u,$

where $u\otimes u^{\prime}\in H^{j}(X\times X;\mathbb{F}_{2})$ and $j\geq 0$
is an integer. The map $T$ is an additive map, but not a multiplicative map.

With the help of just introduced maps $P$ and $T$, based on Lemma 3.2 and
Theorem 3.4, we can describe the $E_{\infty}$-term of the Serre spectral
sequence (68) as follows:

  * —

$E^{*,0}_{2}\cong E^{*,0}_{\infty}\cong
H^{*}(\mathbb{Z}_{2};\mathbb{F}_{2})\cong\mathbb{F}_{2}[f]$, $\deg(f)=1$,

  * —

$E^{0,*}_{2}\cong E^{0,*}_{\infty}\cong H^{*}(X\times
X;\mathbb{F}_{2})^{\mathbb{Z}_{2}}$,

  * —

$E^{0,j}_{2}\cong E^{0,j}_{\infty}\cong P(H^{j/2}(X;\mathbb{F}_{2}))\oplus
T(H^{j}(X\times X;\mathbb{F}_{2}))$ for $j\geq 2$ even,

  * —

$E^{i,j}_{2}\cong E^{i,j}_{\infty}\cong P(H^{j/2}(X;\mathbb{F}_{2}))\otimes
H^{*}(\mathbb{Z}_{2};\mathbb{F}_{2})$ for $j\geq 2$ even, and $i\geq 1$.

Furthermore, we have that the generator $f$ of
$H^{*}(\mathbb{Z}_{2};\mathbb{F}_{2})$ annihilates the image of the map $T$,
that is

$T(H^{*}(X\times X;\mathbb{F}_{2}))\cdot f=0.$ (110)

Note that the Serre spectral sequence associated to the fiber bundle (108) can
be describe in a similar way — as discussed in Section 3.3.

####  7.3.3. The total Stiefel–Whitney class of the wreath square of a vector
bundle

Let $\xi$ be a real $n$-dimensional vector bundle over the CW-complex
$B(\xi)$. To the vector bundle $\xi$ we associate characteristic classes

$s^{2}(\xi):=w(S^{2}\xi)\in H^{*}(B(S^{2}\xi);\mathbb{F}_{2}),$

and

$s^{2,d}(\xi):=w(S^{2,d}\xi)\in H^{*}(B(S^{2,d}\xi);\mathbb{F}_{2}).$

These are the total Stiefel–Whitney classes of the real $2n$-dimensional
vector bundles $S^{2}\xi$ and $S^{2,d}\xi$. The assignments

$\xi\longmapsto s^{2}(\xi)\qquad\text{and}\qquad\xi\longmapsto s^{2,d}(\xi)$

are natural with respect to the continuous maps. This means that for a
continuous map $h\colon K\longrightarrow B(\xi)$ from a CW-complex $K$ into
the base space $B(\xi)$ of $\xi$ the following equalities hold

$(S^{2}h)^{*}(s^{2}(\xi))=s^{2}(h^{*}\xi)\qquad\text{and}\qquad(S^{2,d}h)^{*}(s^{2,d}(\xi))=s^{2,d}(h^{*}\xi)$

Here $h^{*}\xi$ denotes the pull-back vector bundle of $\xi$ along the map
$h$.

The characteristic class $s^{2}(\xi)$ was calculated, in term of
Stiefel–Whitney classes of $\xi$, in [11, Thm. 3.4] and is given in the
following theorem.

######  Theorem 7.16.

Let $\xi$ be a real $n$-dimensional vector bundle over a $CW$-complex. Then

$s^{2}(\xi)=w(S^{2}\xi)=\sum_{0\leq r<s\leq n}T(w_{r}(\xi)\otimes
w_{s}(\xi))+\sum_{0\leq r\leq n}P(w_{r}(\xi))\cdot(1+f)^{n-r}$

In particular, $\xi$ is $1$ dimensional, then

$s^{2}(\xi)=w(S^{2}\xi)=T(w_{0}(\xi)\otimes
w_{1}(\xi))+P(w_{0}(\xi))\cdot(1+f)+P(w_{1}(\xi))=\\\ 1+\big{(}f+1\otimes
w_{1}(\xi)+w_{1}(\xi)\otimes 1\big{)}+w_{1}(\xi)\otimes w_{1}(\xi).$

From the computations in Section 3.3 and the fact that by definition
$S^{2,d}\xi=((\operatorname{id}\times\operatorname{id})\times_{\mathbb{Z}_{2}}i)^{*}S^{2}\xi$
we get the following consequence of Theorem 7.16.

######  Corollary 7.17.

Let $\xi$ be a real $n$-dimensional vector bundle over a $CW$-complex, and let
$d\geq 2$ be an integer. Then

$s^{2,d}(\xi)=w(S^{2,d}\xi)=\\\ \sum_{0\leq r<s\leq n}T(w_{r}(\xi)\otimes
w_{s}(\xi))+\sum_{0\leq r\leq n}\,\sum_{0\leq
j\leq\min\\{n-r,d-1\\}}{n-r\choose j}P(w_{r}(\xi))f^{j}.$

In particular, Theorem 7.16 and Corollary 7.17 give formulas for the
evaluation of mod $2$ Euler classes (top Stiefel–Whitney class) of the vector
bundles $S^{2}\xi$ and $S^{2,d}\xi$ in term of the mod $2$ Euler class of the
vector bundle $\xi$. In the following we present these formulas and show them
with a direct proof which does not rely on the previous claims.

######  Corollary 7.18.

Let $\xi$ be a real $n$-dimensional vector bundle over a $CW$-complex. Then

$\mathfrak{e}(S^{2}\xi)=P(\mathfrak{e}(\xi))\qquad\text{or}\qquad
w_{2n}(S^{2}\xi)=P(w_{n}(\xi)),$

and

$\mathfrak{e}(S^{2,d}\xi)=P(\mathfrak{e}(\xi))\qquad\text{or}\qquad
w_{2n}(S^{2,d}\xi)=P(w_{n}(\xi)).$

In particular, if $\mathfrak{e}(\xi)\neq 0$, then $\mathfrak{e}(S^{2}\xi)\neq
0$ and $\mathfrak{e}(S^{2,d}\xi)\neq 0$.

Here $\mathfrak{e}(S^{2}\xi)$, $\mathfrak{e}(S^{2,d}\xi)$ and
$\mathfrak{e}(\xi)$ denote mod $2$ Euler classes of the vector bundles
$S^{2}\xi$, $S^{2,d}\xi$ and $\xi$, respectively. Note the abuse of notation:
the map $P$ is not the same in the formulas for $\mathfrak{e}(S^{2}\xi)$ and
$\mathfrak{e}(S^{2,d}\xi)$, since it operates on different spectral sequences.

###### Proof.

Let $u\in H^{n}(D(\xi),S(\xi);\mathbb{F}_{2})$ be the Thom class of the
$n$-dimensional vector bundle $\xi$. Here $D(\xi)$ and $S(\xi)$ denote
respectively the disk and sphere bundles associate to the vector bundle $\xi$.
The mod $2$ Euler class of the vector bundle $\xi$, by definition, equals to
$\mathfrak{e}(\xi)=i_{\xi}^{*}(u)$ where
$i_{\xi}\colon(E(\xi),\varnothing)\longrightarrow(D(\xi),S(\xi))$ is the zero
section.

Consider the following commutative diagram

$\textstyle{H^{n}(D(\xi),S(\xi))\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\
}$$\scriptstyle{i_{\xi}^{*}}$$\scriptstyle{P}$$\textstyle{\
H^{n}(E(\xi))\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{P}$$\textstyle{H^{2n}(((D(\xi),S(\xi))\times(D(\xi),S(\xi)))\times_{\mathbb{Z}_{2}}\mathrm{E}\mathbb{Z}_{2})\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\
}$$\scriptstyle{(Pi_{\xi})^{*}}$$\textstyle{\ H^{2n}((E(\xi)\times
E(\xi))\times_{\mathbb{Z}_{2}}\mathrm{E}\mathbb{Z}_{2})\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{H^{2n}(((D(\xi),S(\xi))\times(D(\xi),S(\xi)))\times_{\mathbb{Z}_{2}}S^{d-1})\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\
}$$\scriptstyle{(Pi_{\xi})^{*}}$$\textstyle{\ H^{2n}((E(\xi)\times
E(\xi))\times_{\mathbb{Z}_{2}}S^{d-1})\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{H^{2n}(D(S^{2}\xi),S(\xi))\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\
}$$\scriptstyle{i_{S^{2}\xi}^{*}}$$\textstyle{\ H^{2n}((E(\xi)\times
E(\xi))\times_{\mathbb{Z}_{2}}\mathrm{E}\mathbb{Z}_{2})\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{H^{2n}(D(S^{2,d}\xi),S(\xi))\ignorespaces\ignorespaces\ignorespaces\ignorespaces\
}$$\scriptstyle{i_{S^{2,d}\xi}^{*}}$$\textstyle{\ H^{2n}((E(\xi)\times
E(\xi))\times_{\mathbb{Z}_{2}}S^{d-1})}$

where the coefficients are assumed to be in the field $\mathbb{F}_{2}$. Note
that the curly left arrows are isomorphism while the right ones are
equalities.

We claim that $P(u)$, appropriately interpreted, is the Thom class of the
vector bundle $S^{2}\xi$, respectively $S^{2,d}\xi$. To check this it is
enough to show that it restricts to the generator in each fibre. Thus we can
reduce to the case in which $E(\xi)$ and $\mathrm{B}\mathbb{Z}_{2}$,
respectively $\mathbb{R}{\rm P}^{d-1}$, are just points. Then it is an
elementary fact that the square map $P$:

$\big{(}H^{n}(D(\mathbb{R}^{n}),S(\mathbb{R}^{n});\mathbb{F}_{2})\cong\mathbb{F}_{2}\big{)}\longrightarrow\big{(}H^{2n}(D(\mathbb{R}^{2n}),S(\mathbb{R}^{2n});\mathbb{F}_{2})\cong\mathbb{F}_{2}\big{)}$

maps the generator to the generator.

The assertion now follows from the commutativity of the correspond diagram:

$\mathfrak{e}(S^{2}\xi)=i_{S^{2}\xi}^{*}(P(u))=P(i_{\xi}^{*}(u))=P(\mathfrak{e}(\xi)),$

and similarly

$\mathfrak{e}(S^{2,d}\xi)=i_{S^{2,d}\xi}^{*}(P(u))=P(i_{\xi}^{*}(u))=P(\mathfrak{e}(\xi)).$

∎

###  7.4. Miscellaneous calculations

In order give a smoother presentation of the main computational components in
the main body of the paper in this section we present details of some
auxiliary computations.

####  7.4.1. Detecting group cohomology

In this section we review some basic facts from classical work of Quillen [90]
in generality we need in this paper. For more details consult for example [2,
Sec. IV.4 and VI.1].

Let $G$ be a finite group, and let $p$ be a prime. The family of subgroups
$\\{H_{i}:i\in I\\}$ of the group $G$ detects the cohomology of $G$ modulo
$\mathbb{F}_{p}$, or is a detecting family of subgroups, if the homomorphism

$H^{*}(G;\mathbb{F}_{p})\longrightarrow\prod_{i\in
I}H^{*}(H_{i};\mathbb{F}_{p})$

induced by the restrictions $\operatorname{res}^{G}_{H_{i}}$ is a
monomorphism. If $G^{(p)}$ is Sylow $p$-subgroup of $G$, then the restriction
$\operatorname{res}^{G}_{G^{(p)}}$ is a monomorphism, and consequently
$G^{(p)}$ detects the cohomology of $G$ modulo $\mathbb{F}_{p}$.

First we recall an auxiliary lemma that is particularly useful for us, see
[72, Lem. 3.22].

######  Lemma 7.19.

Let $G$ be a finite group, and let $p$ be a prime. Then $G\wr\mathbb{Z}_{p}$
is detected by $G\times\mathbb{Z}_{p}$ and $\underbrace{G\times\cdots\times
G}_{p\text{ times}}$.

Next, let $p$ be a prime, and let $n=p^{m}$ for an integer $m\geq 1$. Consider
symmetric group $\mathfrak{S}_{n}=\mathfrak{S}_{p^{m}}$ as a group of
permutation of the set $\mathbb{Z}_{p}^{\oplus m}$. Then the elementary
abelian group $\mathcal{E}_{m}$ of all translations (seen as permutations) of
the vector space $\mathbb{Z}_{p}^{\oplus m}$, the so called regular embedded
subgroup [2, Ex. III.2.7], is isomorphic to the elementary abelian group
$\mathbb{Z}_{p}^{\oplus m}$. Let

$\mathcal{S}_{p^{m}}=E_{m,1}\wr\cdots\wr
E_{m,m}\cong\underbrace{\mathbb{Z}_{p}\wr\cdots\wr\mathbb{Z}_{p}}_{m\text{
times}}$

denotes the Sylow $p$-subgroup of $\mathfrak{S}_{p^{m}}$ containing
$\mathcal{E}_{m}$. Here $E_{m,i}\cong\mathbb{Z}_{p}$ is the subgroup of
$\mathcal{E}_{m}$ generated by the $i$th basis element
$(0,\ldots,0,1,0,\ldots,0)$ of the vector space $\mathbb{Z}_{p}^{\oplus m}$.

Now we proceed with the following detection property of the cohomology of the
symmetric group $\mathfrak{S}_{p^{m}}$ and its Sylow $p$-subgroup
$\mathcal{S}_{p^{m}}$, consult [2, Cor. VI.1.4].

######  Theorem 7.20.

Let $p$ be a prime, and let $n=p^{m}$ for an integer $m\geq 1$.

  1. (1)

The cohomology $H^{*}(\mathfrak{S}_{p^{m}};\mathbb{F}_{p})$ of the symmetric
group $\mathfrak{S}_{p^{m}}$ is detected modulo $\mathbb{F}_{p}$ by the
elementary abelian subgroup $\mathcal{E}_{m}$ and the product subgroup

$\underbrace{\mathfrak{S}_{p^{m-1}}\times\cdots\times\mathfrak{S}_{p^{m-1}}}_{p\text{
times}}.$

  2. (2)

The cohomology $H^{*}(\mathcal{S}_{p^{m}};\mathbb{F}_{p})$ of the Sylow
$p$-subgroup $\mathcal{S}_{p^{m}}$ of the symmetric group
$\mathfrak{S}_{p^{m}}$ is detected modulo $\mathbb{F}_{p}$ by the elementary
abelian subgroup $\mathcal{E}_{m}$ and the product subgroup

$\underbrace{\mathcal{S}_{p^{m-1}}\times\cdots\times\mathcal{S}_{p^{m-1}}}_{p\text{
times}}.$

Finally we state the classical result of Quillen about detection with the
family of elementary abelian subgroups, see [2, Thm. VI.1.2].

######  Theorem 7.21.

Let $n\geq 1$ be an integer, and let $p$ be a prime. The cohomology
$H^{*}(\mathfrak{S}_{n};\mathbb{F}_{p})$ of the symmetric group
$\mathfrak{S}_{n}$ with $\mathbb{F}_{p}$ coefficients is detected by the
family of its elementary abelian $p$-subgroups.

####  7.4.2. The image of a restriction homomorphism

Let $G$ be a finite group and let $H$ a subgroup of $G$. In this section we
want to give a description of the image of the restriction homomorphism

$\operatorname{im}\big{(}\operatorname{res}^{G}_{H}\colon
H^{*}(G;M)\longrightarrow H^{*}(H;M)\big{)}$

where $M$ is a trivial $G$-module.

Let $G$ be a (finite) group. Any contractible free $G$-CW complex equipped
with the right $G$ cellular action is called a model for an $\mathrm{E}G$
space. The Milnor model is given by
$\mathrm{E}G=\operatorname{colim}_{n\in\mathbb{N}}G^{*n}$ where $G$ stands for
a $0$-dimensional free $G$-simplicial complex whose vertices are indexed by
the elements of the group $G$ and the action on $G$ is given by the right
translation, and $G^{*n}$ is an $n$-fold join of the $0$-dimensional
simplicial complex with induced diagonal (right) action. A typical point in
$\mathrm{E}G$ can be presented as follows

$\sum_{i\geq
1}\lambda_{i}g_{i}\equiv(\lambda_{1}g_{1},\lambda_{2}g_{2},\lambda_{3}g_{3},\ldots),$

where $g_{i}\in G$ and $\lambda_{i}\geq 0$ for all $i\geq 1$, the set
$I:=\\{\lambda_{i}\neq 0:i\geq 1\\}$ is finite, and $\sum_{i\in
I}\lambda_{i}=1$. The quotient space $\mathrm{B}G=\mathrm{E}G/G$ is called a
classifying space a the group $G$. For a trivial $G$-module $M$ the group
cohomology $H^{*}(G;M)$ can be defined as a singular cohomology
$H^{*}(\mathrm{B}G;M)$.

Let $H$ and $G$ be (finite) groups, and let $f\colon H\longrightarrow G$ be a
homomorphism. Then $f$ induces the following $G$-equivariant map
$\mathrm{E}(f)\colon\mathrm{E}H\longrightarrow\mathrm{E}G$ by

$\sum_{i\geq
1}\lambda_{i}h_{i}\equiv(\lambda_{1}h_{1},\lambda_{2}h_{2},\lambda_{3}h_{3},\ldots)\longmapsto\\\
\sum_{i\geq
1}\lambda_{i}g(h_{i})\equiv(\lambda_{1}f(h_{1}),\lambda_{2}f(h_{2}),\lambda_{3}f(h_{3}),\ldots)$

where the $H$-action on $G$ is induced by the homomorphism $f$. Since the map
$\mathrm{E}(f)$ is $H$-equivariant map it induces a map between quotient
spaces $\mathrm{B}(f)\colon\mathrm{B}H\longrightarrow\mathrm{B}G$. In
particular, if $H$ is a subgroup of $G$, $i\colon H\longrightarrow G$ the
inclusion map, then the induced homomorphism in cohomology $\mathrm{B}(i)^{*}$
by definition is the restriction homomorphism $\operatorname{res}^{G}_{H}$,
that is $\mathrm{B}(i)^{*}=\operatorname{res}^{G}_{H}$.

We prove an auxiliary lemma following [44, Prop. I.6.14] and [2, Thm. II.1.9].

######  Lemma 7.22.

Let $G$ be a finite group.

  1. (i)

Any two continuous $G$-equivariant maps
$f,g\colon\mathrm{E}G\longrightarrow\mathrm{E}G$ are $G$-homotopic.

  2. (ii)

Let $a\in G$, and let $k_{a}\colon G\longrightarrow G$ be the conjugation
homomorphism $k_{a}(g):=aga^{-1}$. Then the induced map
$\mathrm{B}(k)\colon\mathrm{B}G\longrightarrow\mathrm{B}G$ is homotopic to the
identity $\operatorname{id}\colon\mathrm{B}G\longrightarrow\mathrm{B}G$.

###### Proof.

(i) This is a presentation of the proof of [44, Prop. I.6.14]. For
$e\in\mathrm{E}G$ let us denote images $f(e)$ and $g(e)$ as follows

$f(e)=(\lambda_{1}(e)f_{1}(e),\lambda_{2}(e)f_{2}(e),\ldots)\quad\text{and}\quad
g(e)=(\mu_{1}(e)g_{1}(e),\mu_{2}(e)g_{2}(e),\ldots).$

We define two additional maps
$\bar{f},\bar{g}\colon\mathrm{E}G\longrightarrow\mathrm{E}G$ by

$\bar{f}(e)=(\lambda_{1}(e)f_{1}(e),0,\lambda_{2}(e)f_{2}(e),0,\ldots)$

and

$\bar{g}(e)=(0,\mu_{1}(e)g_{1}(e),0,\mu_{2}(e)g_{2}(e),0,\ldots).$

The pairs of maps $f$, $\bar{f}$ and $g$, $\bar{g}$ are $G$-homotopic, that is
$f\simeq_{G}\bar{f}$ and $g\simeq_{G}\bar{g}$. In order to construct a
$G$-homotopy, for example, between $f$ and $\bar{f}$ we proceed as follows.
Let $j\geq 1$ be an integer and let the $G$-homotopy
$H_{j}\colon\mathrm{E}G\times I\longrightarrow\mathrm{E}G$ be defined by

$H_{j}(e,t):=\big{(}\lambda_{1}(e)f_{1}(e),\ldots,\lambda_{j}(e)f_{j}(e),\\\
t\lambda_{j+1}(e)f_{j+1}(e),(1-t)\lambda_{j+1}(e)f_{j+1}(e),\\\
t\lambda_{j+2}(e)f_{j+2}(e),(1-t)\lambda_{j+2}(e)f_{j+2}(e),\ldots\big{)}.$

Starting with $\bar{f}(e)=H_{1}(e,0)$ and applying consecutively
$H_{1},H_{2},\ldots$ we reach $f(e)$. Hence we have define a $G$-homotopy
between $\bar{f}$ and $f$. Since in the presentation of every point
$e=(\lambda_{1}g_{1},\lambda_{2}g_{2},\ldots)\in\mathrm{E}G$ there are only
finitely many non-zero $\lambda$’s the definition we just gave is correct and
the map is moreover continuous. Therefore, $f\simeq_{G}\bar{f}$ and
$g\simeq_{G}\bar{g}$.

It suffices to prove that $\bar{f}\simeq_{G}\bar{g}$. For this we give the
$G$-homotopy $H\colon\mathrm{E}G\times I\longrightarrow\mathrm{E}G$ by

$H(e,t):=\big{(}(1-t)\lambda_{1}(e)f_{1}(e),\,t\mu_{1}(e)g_{1}(e),\,(1-t)\lambda_{2}(e)f_{2}(e),\,t\mu_{2}(e)g_{2}(e),\ldots\big{)}.$

This concludes the proof of part (i).

(ii) The homomorphisms $\operatorname{id}\colon G\longrightarrow G$ and
$k_{a}\colon G\longrightarrow G$ induce $G$-equivariant maps

$\mathrm{E}(\operatorname{id})=\operatorname{id}\colon\mathrm{E}G\longrightarrow\mathrm{E}G\qquad\text{and}\qquad\mathrm{E}(k_{a})\colon\mathrm{E}G\longrightarrow\mathrm{E}G.$

From the part (i) of this lemma we have that
$\mathrm{E}(\operatorname{id})=\operatorname{id}$ and $\mathrm{E}(k_{a})$ are
$G$-homotopic. Consequently, $\mathrm{B}(\operatorname{id})=\operatorname{id}$
and $\mathrm{B}(k_{a})$ are homotopic. ∎

Now we are ready to give a description of the image a restriction that we use
in the central part of the paper. Consult [2, Lem. II.3.1].

######  Lemma 7.23.

Let $G$ be a finite group, $H$ its subgroup, $N_{G}(H)$ the normalizer of $H$
in $G$, $W_{G}(H):=N_{G}(H)/H$ the corresponding Weyl group, and $M$ a trivial
$G$-module. There is an action of $W_{G}(H)$ on $H^{*}(H;M)$ such that

$\operatorname{im}\big{(}\operatorname{res}^{G}_{H}\colon
H^{*}(G;M)\longmapsto H^{*}(H;M)\big{)}\subseteq H^{*}(H;M)^{W_{G}(H)}.$

###### Proof.

First we introduce an action of the normalizer $N_{G}(H)$ on $\mathrm{B}H$.
Let $a\in N_{G}(H)$, and let $k_{a}\colon H\longrightarrow H$ be a
homomorphism defined by $k_{a}(h):=aha^{-1}$ for $h\in H$. For $a,b\in
N_{G}(H)$ the following relation obviously holds $k_{ab}=k_{a}\circ k_{b}$.
Applying the functor $\mathrm{B}$ on the maps $k_{a}$, for every $a\in
N_{G}(H)$, we get an action of $N_{G}(H)$ on $\mathrm{B}H$. This action
naturally extends to an action on the cohomology

$N_{G}(H)\times H^{*}(H;M)\longrightarrow H^{*}(H;M).$

In the case when $a\in H\subseteq N_{G}(H)$ from Lemma 7.22 we get that the
homomorphism $\mathrm{B}(k_{a})^{*}\colon H^{*}(H;M)\longrightarrow
H^{*}(H;M)$ is the identity. Consequently the action of $N_{G}(H)$ factors
through an action of the Weyl group $W_{G}(H)$, that gives us a commutative
diagram

$\textstyle{N_{G}(H)\times
H^{*}(H;M)\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\
}$$\textstyle{\ H^{*}(H;M)}$$\textstyle{W_{G}(H)\times
H^{*}(H;M).\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$

The group $N_{G}(H)$ acts on $\mathrm{B}G$ in the identical way. Now for $a\in
N_{G}(H)$ we set $k_{a}\colon G\longrightarrow G$ to be again defined by
$k_{a}(g):=aga^{-1}$ for $g\in G$. Applying the functor $\mathrm{B}$ we now
get an action of $N_{G}(H)$ on $\mathrm{B}G$ and consequently on $H^{*}(G;M)$.
In this case Lemma 7.22 implies that each map $\mathrm{B}(k_{a})$ is homotopic
to the identity. Hence, the induced action of $N_{G}(H)$ on $H^{*}(G;M)$ is a
trivial action.

Since the action of $N_{G}(H)$ on $\mathrm{B}G$ is an extension of the action
of $N_{G}(H)$ on $\mathrm{B}H$ we have the following commutative diagram

$\textstyle{\mathrm{B}H\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\
}$$\scriptstyle{\mathrm{B}(i)}$$\scriptstyle{\mathrm{B}(k_{a})}$$\textstyle{\
\mathrm{B}G\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\mathrm{B}(k_{a})}$$\textstyle{\mathrm{B}H\ignorespaces\ignorespaces\ignorespaces\ignorespaces\
}$$\scriptstyle{\mathrm{B}(i)}$$\textstyle{\ \mathrm{B}G}$ (111)

for every $a\in N_{G}(H)$ where $i\colon H\longrightarrow G$ is the inclusion.
Applying the cohomology functor on the commutative diagram (111) we get that

$\mathrm{B}(i)^{*}\circ\mathrm{B}(k_{a})^{*}=\mathrm{B}(k_{a})^{*}\circ\mathrm{B}(i)^{*}.$

Since $\mathrm{B}(i)^{*}=\operatorname{res}^{G}_{H}$, and
$\mathrm{B}(k_{a})^{*}\colon H^{*}(G;M)\longrightarrow H^{*}(G;M)$ is the
identity for every $a\in N_{G}(H)$, we get that
$\operatorname{res}^{G}_{H}=\mathrm{B}(k_{a})^{*}\circ\operatorname{res}^{G}_{H}$,
and consequently

$\operatorname{im}\big{(}\operatorname{res}^{G}_{H}\colon
H^{*}(G;M)\longmapsto H^{*}(H;M)\big{)}\subseteq
H^{*}(H;M)^{N_{G}(H)}=H^{*}(H;M)^{W_{G}(H)}.$

∎

####  7.4.3. Weyl groups of an elementary abelian group

In this section we identify Weyl groups
$W_{\mathcal{S}_{2^{m}}}(\mathcal{E}_{m})$ and
$W_{\mathfrak{S}_{2^{m}}}(\mathcal{E}_{m})$ for every $m\geq 1$. First,
following Mùi [82, Proof of Lem. II.5.1] we prove the following fact.

######  Lemma 7.24.

Let $m\geq 0$ be an integer. Then

$W_{\mathcal{S}_{2^{m}}}(\mathcal{E}_{m})\cong\mathrm{L}_{m}(\mathbb{F}_{2}).$

###### Proof.

Let $\kappa\colon
N_{\mathcal{S}_{2^{m}}}(\mathcal{E}_{m})\longrightarrow\mathrm{Aut}(\mathcal{E}_{m})$
be the homomorphism defined by $\kappa(a):=k_{a}$, where as before
$k_{a}\colon\mathcal{E}_{m}\longrightarrow\mathcal{E}_{m}$ is the conjugation
automorphism $k_{a}(e)=aea^{-1}$, $e\in\mathcal{E}_{m}$. The kernel of the
homomorphism $\kappa$ is the centralizer of $\mathcal{E}_{m}$ in
$\mathcal{S}_{2^{m}}$, which is
$\ker(\kappa)=C_{\mathcal{S}_{2^{m}}}(\mathcal{E}_{m})$. Furthermore, in our
situation, $C_{\mathcal{S}_{2^{m}}}(\mathcal{E}_{m})=\mathcal{E}_{m}$. Thus,
there is an exact sequence of groups

$\textstyle{1\ignorespaces\ignorespaces\ignorespaces\ignorespaces\
}$$\textstyle{\
\mathcal{E}_{m}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\
}$$\textstyle{\mathbb{N}_{\mathcal{S}_{2^{m}}}(\mathcal{E}_{m})\ignorespaces\ignorespaces\ignorespaces\ignorespaces\
}$$\textstyle{\
\operatorname{im}(\kappa)\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\
1.}$ (112)

In other words $N_{\mathcal{S}_{2^{m}}}(\mathcal{E}_{m})$ is a semi-direct
product $\mathcal{E}_{m}\rtimes\operatorname{im}(\kappa)$. Consequently,

$W_{\mathcal{S}_{2^{m}}}(\mathcal{E}_{m})=N_{\mathcal{S}_{2^{m}}}(\mathcal{E}_{m})/\mathcal{E}_{m}\cong\operatorname{im}(\kappa).$

Since
$\mathrm{Aut}(\mathcal{E}_{m})\cong\operatorname{GL}_{m}(\mathbb{F}_{2})$ we
can say that
$\operatorname{im}(\kappa)\subseteq\operatorname{GL}_{m}(\mathbb{F}_{2})$ and
therefore the Weyl group $W_{\mathcal{S}_{2^{m}}}(\mathcal{E}_{m})$ can be
seen as a subgroup of $\operatorname{GL}_{m}(\mathbb{F}_{2})$. Thus it remains
to identify $\operatorname{im}(\kappa)$, and this will be done in two steps.

First we prove that $\mathrm{L}_{m}(\mathbb{F}_{2})\subseteq
N_{\mathcal{S}_{2^{m}}}(\mathcal{E}_{m})$ using the induction on $m\geq 1$.
Notice that obviously $\mathrm{L}_{m}(\mathbb{F}_{2})\subseteq
N_{\mathfrak{S}_{2^{m}}}(\mathcal{E}_{m})$. For $m=1$ the group
$\mathrm{L}_{1}(\mathbb{F}_{2})$ is trivial group while
$\mathcal{E}_{1}=\mathcal{S}_{2}=N_{\mathcal{S}_{2}}(\mathcal{E}_{1})\cong\mathbb{Z}_{2}$.
Let us assume that for $m\geq 2$ the following inclusions
$\mathrm{L}_{m-1}(\mathbb{F}_{2})\subseteq
N_{\mathcal{S}_{2^{m-1}}}(\mathcal{E}_{m-1})\subseteq\mathcal{S}_{2^{m-1}}$
holds. Consider the embedding
$\mathrm{L}_{m-1}(\mathbb{F}_{2})\longrightarrow\mathrm{L}_{m}(\mathbb{F}_{2})$
of group given by

$A\longmapsto\left(\begin{array}[]{cc}1&0\\\ 0&A\\\ \end{array}\right)$

for $A\in\mathrm{L}_{m-1}(\mathbb{F}_{2})$. Note the “difference” between
zeroes in the upper matrix. Since for every $i\in\mathbb{F}_{2}$ and
$e\in\mathbb{F}_{2}^{m-1}$

$\left(\begin{array}[]{cc}1&0\\\ 0&A\\\
\end{array}\right)\left(\begin{array}[]{c}i\\\
e\end{array}\right)=\left(\begin{array}[]{c}i\\\ Ae\end{array}\right)$

we have that $\mathrm{L}_{m-1}(\mathbb{F}_{2})$ is a subgroup of
$\delta(\mathcal{S}_{2^{m-1}})\subseteq\mathcal{S}_{2^{m-1}}\times\mathcal{S}_{2^{m-1}}$
where $\delta$ denotes the diagonal embedding. An arbitrary element of the
group $\mathrm{L}_{m}(\mathbb{F}_{2})$ can be presented in the form

$\left(\begin{array}[]{cc}1&0\\\ a&A\\\ \end{array}\right)$

where $A\in\mathrm{L}_{m-1}(\mathbb{F}_{2})$ and $a\in\mathbb{F}_{2}^{m-1}$.
Then for $i\in\mathbb{F}_{2}$ and $e\in\mathbb{F}_{2}^{m-1}$ we have

$\left(\begin{array}[]{cc}1&0\\\ a&A\\\
\end{array}\right)\left(\begin{array}[]{c}i\\\
e\end{array}\right)=\left(\begin{array}[]{c}i\\\ Ae+ai\end{array}\right).$

Consequently,
$\mathrm{L}_{m}(\mathbb{F}_{2})\subseteq(\mathrm{L}_{m-1}(\mathbb{F}_{2})\cdot\mathcal{E}_{m-1})\times(\mathrm{L}_{m-1}(\mathbb{F}_{2})\cdot\mathcal{E}_{m-1})$.
From the induction hypothesis we have more

$\mathrm{L}_{m}(\mathbb{F}_{2})\subseteq(\mathrm{L}_{m-1}(\mathbb{F}_{2})\cdot\mathcal{E}_{m-1})\times(\mathrm{L}_{m-1}(\mathbb{F}_{2})\cdot\mathcal{E}_{m-1})\subseteq\mathcal{S}_{2^{m-1}}\times\mathcal{S}_{2^{m-1}}\subseteq\mathcal{S}_{2^{m}}.$

As we have seen $\mathrm{L}_{m}(\mathbb{F}_{2})\subseteq
N_{\mathfrak{S}_{2^{m}}}(\mathcal{E}_{m})$ and thus

$\mathrm{L}_{m}(\mathbb{F}_{2})\subseteq
N_{\mathcal{S}_{2^{m}}}(\mathcal{E}_{m}),$

which concludes the induction.

Now we continue identification of $\operatorname{im}(\kappa)$. Since
$\mathrm{L}_{m}(\mathbb{F}_{2})\cap\mathcal{E}_{m}=\\{1\\}$ we have that the
exact sequence (112) gives us an embedding of $\mathrm{L}_{m}(\mathbb{F}_{2})$
into $\operatorname{im}(\kappa)$. Hence we have the following inclusions of
$2$-groups (with the obvious abuse of notation)

$\mathrm{L}_{m}(\mathbb{F}_{2})\subseteq\operatorname{im}(\kappa)\subseteq\operatorname{GL}_{m}(\mathbb{F}_{2}).$

The image $\operatorname{im}(\kappa)$ is a $2$-group as an image of the
$2$-group $N_{\mathcal{S}_{2^{m}}}(\mathcal{E}_{m})$. Because
$\mathrm{L}_{m}(\mathbb{F}_{2})$ is a Sylow $2$-subgroup of
$\operatorname{GL}_{m}(\mathbb{F}_{2})$ we have that
$\mathrm{L}_{m}(\mathbb{F}_{2})=\operatorname{im}(\kappa)$ and consequently

$W_{\mathcal{S}_{2^{m}}}(\mathcal{E}_{m})\cong\operatorname{im}(\kappa)\cong\mathrm{L}_{m}(\mathbb{F}_{2}).$

∎

Next, adapting the proof of the previous lemma and following [2, Ex. III.2.7]
we determine the Weyl group of $\mathcal{E}_{m}$ now inside the symmetric
group $\mathfrak{S}_{2^{m}}$.

######  Lemma 7.25.

Let $m\geq 0$ be an integer. Then

$W_{\mathfrak{S}_{2^{m}}}(\mathcal{E}_{m})\cong\operatorname{GL}_{m}(\mathbb{F}_{2}).$

###### Proof.

Let $\kappa\colon
N_{\mathfrak{S}_{2^{m}}}(\mathcal{E}_{m})\longrightarrow\mathrm{Aut}(\mathcal{E}_{m})$
be the homomorphism defined by $\kappa(a):=k_{a}$, where as before
$k_{a}\colon\mathcal{E}_{m}\longrightarrow\mathcal{E}_{m}$ is the conjugation
automorphism $k_{a}(e)=aea^{-1}$, $e\in\mathcal{E}_{m}$. As in the proof of
the previous lemma
$\ker(\kappa)=C_{\mathfrak{S}_{2^{m}}}(\mathcal{E}_{m})=\mathcal{E}_{m}$.
Hence, we get an exact sequence of groups

$\textstyle{1\ignorespaces\ignorespaces\ignorespaces\ignorespaces\
}$$\textstyle{\
\mathcal{E}_{m}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\
}$$\textstyle{\
N_{\mathfrak{S}_{2^{m}}}(\mathcal{E}_{m})\ignorespaces\ignorespaces\ignorespaces\ignorespaces\
}$$\textstyle{\
\operatorname{im}(\kappa)\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{1.}$

Consequently, $N_{\mathfrak{S}_{2^{m}}}(\mathcal{E}_{m})$ is a semi-direct
product $\mathcal{E}_{m}\rtimes\operatorname{im}(\kappa)$ implying that

$W_{\mathfrak{S}_{2^{m}}}(\mathcal{E}_{m})=N_{\mathfrak{S}_{2^{m}}}(\mathcal{E}_{m})/\mathcal{E}_{m}\cong\operatorname{im}(\kappa).$

Since
$\mathrm{Aut}(\mathcal{E}_{m})\cong\operatorname{GL}_{m}(\mathbb{F}_{2})$, in
order to complete the proof of the lemma, it suffices to prove that
$\operatorname{im}(\kappa)=\mathrm{Aut}(\mathcal{E}_{m})$. For that fix
$\alpha\in\mathrm{Aut}(\mathcal{E}_{m})$, and denote by
$\bar{\alpha}\in\mathfrak{S}_{2^{m}}$ the permutation given with
$e\longmapsto\alpha(e)$ for $e\in\mathcal{E}_{m}$. Here we use the fact that
$\mathfrak{S}_{2^{m}}$ is group of permutations of the set $\mathcal{E}_{m}$.
Then $\bar{\alpha}\in N_{\mathfrak{S}_{2^{m}}}(\mathcal{E}_{m})$ and
$\kappa(\bar{\alpha})=\alpha$. Indeed, for $e,h\in\mathcal{E}_{m}$ holds:

$\bar{\alpha}e\bar{\alpha}^{-1}(h)=(\bar{\alpha}e)(\alpha^{-1}(h))=\bar{\alpha}(e+\alpha^{-1}(h))=\alpha(e+\alpha^{-1}(h))=\alpha(e)+h.$

consequently we have an equality of permutations
$\bar{\alpha}e\bar{\alpha}^{-1}=\alpha(e)\in\mathcal{E}_{m}$ that implies
$\bar{\alpha}\in N_{\mathfrak{S}_{2^{m}}}(\mathcal{E}_{m})$ and
$\kappa(\bar{\alpha})=\alpha$. This concludes the proof of surjectivity of
$\kappa$ and of the lemma. ∎

####  7.4.4. Cohomology of the real projective space with local coefficients

For $d\geq 2$ we compute the cohomology of the projective space
$H^{*}(\mathbb{R}{\rm P}^{d-1};\mathcal{M})$ where the local coefficient
system $\mathcal{M}$ is additively $\mathbb{F}_{2}\oplus\mathbb{F}_{2}$, and
the action of $\pi_{1}(\mathbb{R}{\rm P}^{d-1})=\langle t\rangle$ on
$\mathcal{M}$ is given by $t\cdot(a_{1},a_{2})=(a_{2},a_{1})$ where
$(a_{1},a_{2})\in\mathbb{F}_{2}\oplus\mathbb{F}_{2}$. We consider two separate
cases: $d=2$ when $\pi_{1}(\mathbb{R}{\rm P}^{d-1})\cong\mathbb{Z}$, and
$d\geq 3$ when $\pi_{1}(\mathbb{R}{\rm P}^{d-1})\cong\mathbb{Z}_{2}$. First we
recall the definition of the cohomology with local coefficients, consult for
example [60, Sec. 3.H].

Let $X$ be a path-connected CW-complex, $\widetilde{X}$ its universal cover,
and let $\pi:=\pi_{1}(X)$ be its fundamental group. Denote by $\mathcal{L}$ a
local coefficient system on $X$, that is a $\mathbb{Z}[\pi]$-module. Assume
that we are given a structure of $\mathbb{Z}[\pi]$-CW-complex on
$\widetilde{X}$ with associated cellular chain complex

$\textstyle{\cdots\ \ignorespaces\ignorespaces\ignorespaces\ignorespaces\
}$$\textstyle{\
C_{n+1}(\widetilde{X})\ignorespaces\ignorespaces\ignorespaces\ignorespaces\
}$$\scriptstyle{d_{n+1}}$$\textstyle{\
C_{n}(\widetilde{X})\ignorespaces\ignorespaces\ignorespaces\ignorespaces\
}$$\scriptstyle{d_{n}}$$\textstyle{\
C_{n-1}(\widetilde{X})\ignorespaces\ignorespaces\ignorespaces\ignorespaces\
}$$\scriptstyle{d_{n-1}}$$\textstyle{\
\cdots}$$\textstyle{\cdots\ignorespaces\ignorespaces\ignorespaces\ignorespaces\
}$$\scriptstyle{d_{2}}$$\textstyle{\
C_{1}(\widetilde{X})\ignorespaces\ignorespaces\ignorespaces\ignorespaces\
}$$\scriptstyle{d_{1}}$$\textstyle{\