=\mathcal{F}_{k}H_{*}(\Omega^{d}\Sigma^{d}S^{L};\mathbb{F}_{2})/\mathcal{F}_{k-1}H_{*}(\Omega^{d}\Sigma^{d}S^{L};\mathbb{F}_{2}).$

The results presented in [33, Sec. 3 and Sec.4] imply that the isomorphisms
$\alpha_{d,k}$ and $\overline{\alpha}_{d,k}$ are induced by the isomorphism
$(\alpha_{d})_{*}$.

Now, directly from Theorem 4.9 and Theorem 4.13, we get a description of
homology of the unordered configuration space with $\mathbb{F}_{2}$
coefficients, see [26, Sec. 4.4].

######  Corollary 4.14.

Let $d\geq 2$, $k\geq 1$ and $L\geq 1$ be integers. There is an isomorphism of
graded vector space

$H_{*-kL}(\operatorname{F}(\mathbb{R}^{d},k)/\mathfrak{S}_{k};\mathbb{F}_{2})\cong\mathcal{D}_{k}H_{*}(\Omega^{d}\Sigma^{d}S^{L};\mathbb{F}_{2}).$

Alternatively, we can describe the homology of the unordered configurations
space $\operatorname{F}(\mathbb{R}^{d},k)/\mathfrak{S}_{k}$ with
$\mathbb{F}_{2}$ coefficients in the following way.

######  Corollary 4.15.

Let $d\geq 2$, $k\geq 1$, $L\geq 1$ and $0\leq i\leq(d-1)(k-1)$ be integers.
The homology of the unordered configurations space

$H_{i}(\operatorname{F}(\mathbb{R}^{d},k)/\mathfrak{S}_{k};\mathbb{F}_{2})$

is isomorphic to the $\mathbb{F}_{2}$ vector space spanned by all monomials of
degree $i+kL$ of the polynomial ring

$\mathbb{F}_{2}\big{[}\\{u_{L}\\}\cup\\{Q_{i_{1}}Q_{i_{2}}\cdots
Q_{i_{s}}u_{L}\,:\,s\geq 1,\,1\leq i_{1}\leq i_{2}\leq\cdots\leq i_{s}\leq
d-1\\}\big{]}$

whose weights are exactly $k$.

In order to illustrate the previous result we give a few simple examples of
evaluation of the homology of the unordered configuration space.

######  Example 4.16.

From Corollary 4.15 we have that the homology
$H_{*}(\operatorname{F}(\mathbb{R}^{d},k)/\mathfrak{S}_{k};\mathbb{F}_{2})$ of
the unordered configuration space is

  1. (1)

for the case $d=2$ and $k=2$ spanned in

dimension $0$ by the monomial: |  $u_{L}^{2}$,
---|---
dimension $1$ by the monomial: |  $Q_{1}u_{L}$,

(Thus, indeed
$H_{*}(\operatorname{F}(\mathbb{R}^{2},2)/\mathfrak{S}_{2};\mathbb{F}_{2})\cong
H_{*}(\mathbb{R}{\rm P}^{1};\mathbb{F}_{2})$ as expected.)

  2. (2)

for the case $d=2$ and $k=3$ spanned in

dimension $0$ by the monomial: |  $u_{L}^{3}$,
---|---
dimension $1$ by the monomial: |  $u_{L}^{3}\,(Q_{1}u_{L})$,

  3. (3)

for the case $d=2$ and $k=4$ spanned in

dimension $0$ by the monomial: |  $u_{L}^{4}$,
---|---
dimension $1$ by the monomial: |  $u_{L}^{2}\,(Q_{1}u_{L})$,
dimension $2$ by the monomial: |  $(Q_{1}u_{L})\,(Q_{1}u_{L})$,
dimension $3$ by the monomial: |  $Q_{1}Q_{1}u_{L}$,

  4. (4)

for the case $d=3$ and $k=2$ spanned in

dimension $0$ by the monomial: |  $u_{L}^{2}$,
---|---
dimension $1$ by the monomial: |  $Q_{1}u_{L}$,
dimension $1$ by the monomial: |  $Q_{2}u_{L}$,

(Again, we see that
$H_{*}(\operatorname{F}(\mathbb{R}^{3},2)/\mathfrak{S}_{2};\mathbb{F}_{2})\cong
H_{*}(\mathbb{R}{\rm P}^{2};\mathbb{F}_{2})$ as expected.)

  5. (5)

for the case $d=3$ and $k=4$ spanned in

dimension $0$ by the monomial: |  $u_{L}^{4}$,
---|---
dimension $1$ by the monomial: |  $u_{L}^{2}\,(Q_{1}u_{L})$,
dimension $2$ by the monomials: |  $u_{L}^{2}\,(Q_{2}u_{L})$ and $(Q_{1}u_{L})\,(Q_{1}u_{L})$,
dimension $3$ by the monomials: |  $(Q_{1}u_{L})\,(Q_{2}u_{L})$ and $Q_{1}Q_{1}u_{L}$,
dimension $4$ by the monomial: |  $(Q_{2}u_{L})\,(Q_{2}u_{L})$,
dimension $5$ by the monomial: |  $Q_{1}Q_{2}u_{L}$,
dimension $6$ by the monomial: |  $Q_{2}Q_{2}u_{L}$,

Thus, we have that

$H_{i}(\operatorname{F}(\mathbb{R}^{3},4)/\mathfrak{S}_{4};\mathbb{F}_{2})\cong\begin{cases}\mathbb{F}_{2},&i=0,1,4,5,6,\\\
\mathbb{F}_{2}^{\oplus 2},&i=3,4,\\\ 0,&\text{otherwise}.\end{cases}$

With this result we collected all necessary ingredients that we need for the
proof of Theorem 4.8.

####  4.2.2. Proof of the dual Epimorphism Theorem

Let the integers $d\geq 2$ and $m\geq 1$ be fixed. Now we prove that the
following homomorphism, induced by a structural map of the little cubes operad
$\mathcal{C}_{d}$,

$(\mu_{d,2^{m}})_{*}\colon
H_{*}((\mathcal{C}_{d}(2^{m-1})/\mathfrak{S}_{2^{m-1}}\times\mathcal{C}_{d}(2^{m-1})/\mathfrak{S}_{2^{m-1}})\times_{\mathbb{Z}_{2}}\mathcal{C}_{d}(2);\mathbb{F}_{2})\longrightarrow\\\
H_{*}(\mathcal{C}_{d}(2^{m})/\mathfrak{S}_{2^{m}};\mathbb{F}_{2})$

is an epimorphism.

Let $L\geq 1$ be an arbitrary integer, and consider the free
$\mathcal{C}_{d}$-space $\mathcal{C}_{d}(S^{L})$. From Theorem 4.10, using the
the weak homotopy equivalence

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

we get the following isomorphisms

$H_{*}(\mathcal{C}_{d}(S^{L});\mathbb{F}_{2})\cong
H_{*}(\Omega^{d}\Sigma^{d}S^{L};\mathbb{F}_{2})\cong\bigoplus_{k\geq
0}\mathcal{D}_{k}H_{*}(\Omega^{d}\Sigma^{d}S^{L};\mathbb{F}_{2})\cong\\\
\mathbb{F}_{2}\big{[}\\{u_{L}\\}\cup\\{Q_{i_{1}}Q_{i_{2}}\cdots
Q_{i_{s}}u_{L}\,:\,s\geq 1,\,1\leq i_{1}\leq i_{2}\leq\cdots\leq i_{s}\leq
d-1\\}\big{]}.$

Since $\mathcal{C}_{d}(S^{L})$ is a $\mathcal{C}_{d}$-space, as explained in
Lemma 7.6, there exists a map

$\textstyle{\Theta_{2}\colon(\mathcal{C}_{d}(S^{L})\times\mathcal{C}_{d}(S^{L}))\times\mathcal{C}_{d}(2)\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\mathcal{C}_{d}(S^{L}).}$

(In the following we will abuse notation and all maps induced by $\Theta_{2}$
will be denoted in the same way.) On the level of space filtration, for
integers $k_{1}\geq 0$ and $k_{2}\geq 0$, we have that $\Theta_{2}$ induces a
map

$\textstyle{\Theta_{2}\colon(\mathrm{F}_{k_{1}}\mathcal{C}_{d}(S^{L})\times\mathrm{F}_{k_{2}}\mathcal{C}_{d}(S^{L}))\times\mathcal{C}_{d}(2)\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\mathrm{F}_{k_{1}+k_{2}}\mathcal{C}_{d}(S^{L}).}$

Thus, for any integer $k\geq 0$ we have an induced map on quotient spaces

$\textstyle{\Theta_{2}\colon(\mathrm{D}_{k}\mathcal{C}_{d}(S^{L})\times\mathrm{D}_{k}\mathcal{C}_{d}(S^{L}))\times\mathcal{C}_{d}(2)\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\mathrm{D}_{2k}\mathcal{C}_{d}(S^{L}).}$

The symmetric group $\mathfrak{S}_{2}\cong\mathbb{Z}_{2}$ acts naturally on:
the space $\mathcal{C}_{d}(2)$ of pairs of little $d$-cubes by interchanging
the cubes, and on the product
$\mathrm{D}_{k}\mathcal{C}_{d}(S^{L})\times\mathrm{D}_{k}\mathcal{C}_{d}(S^{L})$
by interchanging the factors. If the trivial action on the space
$\mathrm{D}_{2k}\mathcal{C}_{d}(S^{L})$ is assumed, the equivariance property
of the structural map of the little cubes operad implies that the last
$\Theta_{2}$ map is an $\mathfrak{S}_{2}$-equivariant map. Consequently, it
induces maps such that the following diagram commutes

$\textstyle{(\mathcal{C}_{d}(S^{L})\times\mathcal{C}_{d}(S^{L}))\times_{\mathbb{Z}_{2}}\mathcal{C}_{d}(2)\
\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\Theta_{2}}$$\textstyle{\
\mathcal{C}_{d}(S^{L})}$$\textstyle{(\mathrm{F}_{k}\mathcal{C}_{d}(S^{L})\times\mathrm{F}_{k}\mathcal{C}_{d}(S^{L}))\times_{\mathbb{Z}_{2}}\mathcal{C}_{d}(2)\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\
}$$\scriptstyle{\Theta_{2}}$$\textstyle{\
\mathrm{F}_{2k}\mathcal{C}_{d}(S^{L})\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{(\mathrm{D}_{k}\mathcal{C}_{d}(S^{L})\times\mathrm{D}_{k}\mathcal{C}_{d}(S^{L}))\times_{\mathbb{Z}_{2}}\mathcal{C}_{2}(2)\ignorespaces\ignorespaces\ignorespaces\ignorespaces\
}$$\scriptstyle{\Theta_{2}}$$\textstyle{\
\mathrm{D}_{2k}\mathcal{C}_{d}(S^{L}).}$

Here the vertical maps are either inclusions or quotient maps. Passing to
homology we get the following commutative diagram:

$\textstyle{H_{*}((\mathcal{C}_{d}(S^{L})\times\mathcal{C}_{d}(S^{L}))\times_{\mathbb{Z}_{2}}\mathcal{C}_{d}(2);\mathbb{F}_{2})\ignorespaces\ignorespaces\ignorespaces\ignorespaces\
}$$\scriptstyle{(\Theta_{2})_{*}}$$\textstyle{\
H_{*}(\mathcal{C}_{d}(S^{L});\mathbb{F}_{2})}$$\textstyle{H_{*}((\mathrm{F}_{k}\mathcal{C}_{d}(S^{L})\times\mathrm{F}_{k}\mathcal{C}_{d}(S^{L}))\times_{\mathbb{Z}_{2}}\mathcal{C}_{d}(2);\mathbb{F}_{2})\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\
}$$\scriptstyle{(\Theta_{2})_{*}}$$\textstyle{\
H_{*}(\mathrm{F}_{2k}\mathcal{C}_{d}(S^{L});\mathbb{F}_{2})\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{H_{*}((\mathrm{D}_{k}\mathcal{C}_{d}(S^{L})\times\mathrm{D}_{k}\mathcal{C}_{d}(S^{L}))\times_{\mathbb{Z}_{2}}\mathcal{C}_{2}(2);\mathbb{F}_{2})\ignorespaces\ignorespaces\ignorespaces\ignorespaces\
}$$\scriptstyle{(\Theta_{2})_{*}}$$\textstyle{\
H_{*}(\mathrm{D}_{2k}\mathcal{C}_{d}(S^{L});\mathbb{F}_{2}).}$ (58)

In the case when $k=2^{m-1}$ the homomorphism

$(\Theta_{2})_{*}\colon
H_{*}((\mathrm{D}_{k}\mathcal{C}_{d}(S^{L})\times\mathrm{D}_{k}\mathcal{C}_{d}(S^{L}))\times_{\mathbb{Z}_{2}}\mathcal{C}_{2}(2);\mathbb{F}_{2})\longrightarrow\\\
H_{*}(\mathrm{D}_{2k}\mathcal{C}_{d}(S^{L});\mathbb{F}_{2})$ (59)

coincides with the map $(\mu_{d,2^{m}})_{*}$ from (56) — after appropriate
dimension shift by $-2kL$. Therefore in order to complete the proof of Theorem
4.8 it suffices to prove that the homomorphism $(\Theta_{2})_{*}$, from (59),
is surjective.

The map
$H_{*}(\mathrm{F}_{k}\mathcal{C}_{d}(S^{L});\mathbb{F}_{2})\longrightarrow
H_{*}(\mathrm{D}_{k}\mathcal{C}_{d}(S^{L});\mathbb{F}_{2})$ is surjective for
every $k$. Therefore, using Corollary 3.9 and commutativity of the diagram
(58), we should first consider the homomorphism

$(\Theta_{2})_{*}\colon
H_{*}((\mathrm{F}_{k}\mathcal{C}_{d}(S^{L})\times\mathrm{F}_{k}\mathcal{C}_{d}(S^{L}))\times_{\mathbb{Z}_{2}}\mathcal{C}_{2}(2);\mathbb{F}_{2})\longrightarrow\\\
H_{*}(\mathrm{F}_{2k}\mathcal{C}_{d}(S^{L});\mathbb{F}_{2}).$ (60)

For $d\geq 2$, let

$\mathcal{I}_{d}:=\\{(i_{1},\dots,i_{s}):s\geq 1,\,1\leq i_{1}\leq\cdots\leq
i_{s}\leq d-1\\}$

be the set of all admissible iterations. Then, every element
$I\in\mathcal{I}_{d}$ defines the iterated Araki–Kudo–Dyer–Lashof homology
operation

$Q_{I}:=Q_{i_{1}}Q_{i_{2}}\cdots Q_{i_{s}}.$

Hence, as we have seen in Theorem 4.13, the homology
$H_{*}(\mathrm{F}_{k}\mathcal{C}_{d}(S^{L});\mathbb{F}_{2})$ can be identified
with the vector space spanned by all monomials in, variables from the set
$\\{Q_{I}u_{L}:I\in\mathcal{I}_{d}\\}\cup\\{u_{L}\\}$ with weight at most $k$.
Furthermore, from definition of Araki–Kudo–Dyer–Lashof homology operations
given in Section 7.1.6, it follows that for every $Q_{I}u_{L}$,
$I\in\mathcal{I}_{d}$, of weight exactly $2^{m}$, there exists $i\geq 1$ and
an element $Q_{J}u_{L}$ or $u_{L}$, $J\in\mathcal{I}_{d}$, of weight exactly
$2^{m-1}$ such that

$Q_{I}u_{L}=(\Theta_{2})_{*}((Q_{J}u_{L}\otimes
Q_{J}u_{L})\otimes_{\mathbb{Z}_{2}}f_{i}).$ (61)

It is important to notice that this does not mean that the homomorphism
$(\Theta_{2})_{*}$ from (60) is surjective.

Now we specialize to the case $k=2^{m-1}$ and analyze the map (59):

$(\Theta_{2})_{*}\colon
H_{*}((\mathrm{D}_{2^{m-1}}\mathcal{C}_{d}(S^{L})\times\mathrm{D}_{2^{m-1}}\mathcal{C}_{d}(S^{L}))\times_{\mathbb{Z}_{2}}\mathcal{C}_{d}(2);\mathbb{F}_{2})\longrightarrow
H_{*}(\mathrm{D}_{2^{m}}\mathcal{C}_{d}(S^{L});\mathbb{F}_{2})$

and prove that it is surjective. The vector space
$H_{*}(\mathrm{D}_{2^{m}}\mathcal{C}_{d}(S^{L});\mathbb{F}_{2})$ is a quotient
of $H_{*}(\mathrm{F}_{2^{m}}\mathcal{C}_{d}(S^{L});\mathbb{F}_{2})$. It can be
identified with the vector space spanned by all monomials of weight exactly
$2^{m}$ in variables $\\{u_{L}\\}\cup\\{Q_{I}u_{L}:I\in\mathcal{I}_{d}\\}$.
From (61) we that all elements of the form $Q_{I}u_{L}$ of weight $2^{m}$
(variables) are in the image of $(\Theta_{2})_{*}$. It remains to show that
all other monomials are also in $\operatorname{im}((\Theta_{2})_{*})$.

Let $Q_{I_{1}}u_{L}\cdots Q_{I_{t}}u_{L}$ with $t\geq 2$, $I_{1},\dots
I_{t}\in\mathcal{I}_{d}\cup\\{0\\}$ and $Q_{0}u_{L}:=u_{L}$, be a typical
element of the basis of
$H_{*}(\mathrm{D}_{2^{m}}\mathcal{C}_{d}(S^{L});\mathbb{F}_{2})$ which is not
a variable. Then,

$2^{m}=\omega(Q_{I_{1}}u_{L}\cdots
Q_{I_{t}}u_{L})=\omega(Q_{I_{1}}u_{L})+\dots+\omega(Q_{I_{t}}u_{L})=2^{b_{1}}+\cdots+2^{b_{t}},$

where

$\omega(Q_{I_{1}}u_{L})=2^{b_{1}},\ \omega(Q_{I_{2}}u_{L})=2^{b_{2}},\ \dots,\
\omega(Q_{I_{t}}u_{L})=2^{b_{t}}.$

Without loss of generality, we can assume that there exists an integer
$1\leq\ell\leq t-1$ such that

$\omega(Q_{I_{1}}u_{L}\cdots
Q_{I_{\ell}}u_{L})=2^{m-1}\qquad\text{and}\qquad\omega(Q_{I_{\ell+1}}u_{L}\cdots
Q_{I_{t}}u_{L})=2^{m-1}.$

Consequently,

$Q_{I_{1}}u_{L}\cdots Q_{I_{\ell}}u_{L}\ \in\
H_{*}(\mathrm{D}_{2^{m-1}}\mathcal{C}_{d}(S^{L});\mathbb{F}_{2})$

and

$Q_{I_{\ell+1}}u_{L}\cdots Q_{I_{t}}u_{L}\ \in\
H_{*}(\mathrm{D}_{2^{m-1}}\mathcal{C}_{d}(S^{L});\mathbb{F}_{2}).$

The product structure on the homology of $\mathcal{C}_{d}$-space
$\mathcal{C}_{d}(S^{L})$ is introduced by an arbitrary element of
$(\vec{c}_{1},\vec{c}_{2})\in\mathcal{C}_{d}(2)$ via the map

$\Theta_{2}(\vec{c}_{1},\vec{c}_{2})\colon\mathcal{C}_{d}(S^{L})\times\mathcal{C}_{d}(S^{L})\longrightarrow\mathcal{C}_{d}(S^{L}),$

see [76, Lem. 1.9(i)]. The invariance of the action of the little cubes operad
with respected to the actions of related symmetric groups implies that the
product map $\Theta_{2}(\vec{c}_{1},\vec{c}_{2})$ factors thought the induced
map

$\Theta_{2}\colon(\mathcal{C}_{d}(S^{L})\times\mathcal{C}_{d}(S^{L}))\times_{\mathbb{Z}_{2}}\mathcal{C}_{d}(2)\longrightarrow\mathcal{C}_{d}(S^{L}).$

Hence, on the level of homology with $\mathbb{F}_{2}$ coefficients we have
that

$\displaystyle Q_{I_{1}}u_{L}\dots Q_{I_{t}}u_{L}$
$\displaystyle=(Q_{I_{1}}u_{L}\cdots
Q_{I_{\ell}}u_{L})\cdot(Q_{I_{\ell+1}}u_{L}\cdots Q_{I_{t}}u_{L})$
$\displaystyle=(\Theta_{2})_{*}\big{(}(Q_{I_{1}}u_{L}\cdots
Q_{I_{\ell}}u_{L})\otimes(Q_{I_{\ell+1}}u_{L}\cdots
Q_{I_{t}}u_{L})\otimes_{\mathbb{Z}_{2}}h_{0}\big{)},$

where $h_{0}$ is the generator of $H_{0}(\mathbb{R}{\rm
P}^{d-1};\mathcal{M})$, as explained in Section 7.4.5. Thus, we have proved
that all additive generators of
$H_{*}(\mathrm{D}_{2^{m}}\mathcal{C}_{d}(S^{L});\mathbb{F}_{2})$ are in the
image of $(\Theta_{2})_{*}$ implying that the map $(\Theta_{2})_{*}$ in (59)
is surjective. This concludes the proof of Theorem 4.8 and consequently we
have proved Theorem 4.1.

###  4.3. An unexpected corollary

In this section we take a small detour from the objectives of this book and
present an unexpected corollary which is an artefact of the proof of Theorem
4.8. This result shares the spirit with the the classical result of Michael
Atiyah [10] about the complex representation ring of the symmetric group
$R(\mathfrak{S}_{k})\cong K_{\mathfrak{S}_{k}}^{*}(\mathrm{pt})$ and the
modern breakthrough of Chad Giusti, Paolo Salvatore & Dev Sinha [55, Thm. 1.2]
in describing the cohomology of the symmetric groups collected disjointly
together $H^{*}(\coprod_{k\geq 1}\mathrm{B}\mathfrak{S}_{k};\mathbb{F}_{2})$.

####  4.3.1. Motivation

In 1966 Atiyah studied the following direct sum of abelian groups

$R_{*}:=\bigoplus_{k\geq
0}\hom(R(\mathfrak{S}_{k}),\mathbb{Z})\cong\bigoplus_{k\geq
0}\hom(K_{\mathfrak{S}_{k}}^{*}(\mathrm{pt}),\mathbb{Z})$

which was additionally equipped with the structure of a commutative ring via
the product induced by the inclusion maps
$\mathfrak{S}_{r}\times\mathfrak{S}_{s}\longrightarrow\mathfrak{S}_{r+s}$. For
details and original presentation see [10, p. 169]. He showed [10, Cor. 1.3]
that the ring $R_{*}$ is isomorphic to the polynomial ring

$\mathbb{Z}[\sigma^{1},\dots,\sigma^{k},\dots]$

on generators $\sigma^{k}\colon R(\mathfrak{S}_{k})\longrightarrow\mathbb{Z}$,
for $k\geq 1$, given by the dimension of the fixed
$\mathfrak{S}_{k}$-submodule.

The study of the homology and cohomology of symmetric groups has a rich and
exciting history with many spectacular breakthroughs and crucial applications
in different areas of mathematics. The major contributions go back to the
extraordinary work of Nakaoka [86] [87] [88], Daniel Quillen [90] [89],
Quillen & Boris Venkov [91], Huỳnh Mùi [82], Benjamin Mann [73], Adem, John
Maginnis & Milgram, [1], Adem & Milgram [2], all the way towards the modern
results by Mark Feshbach [51] and Giusti, Salvatore & Sinha [55].

A detailed understanding of the cohomology of a particular symmetric group is
still a challenging problem in general. On the other hand, Giusti, Salvatore &
Sinha [55], considering all symmetric groups together, like Atiyah did, gave a
compact description of the cohomology $H^{*}\big{(}\coprod_{k\geq
1}\mathrm{B}\mathfrak{S}_{k};\mathbb{F}_{2}\big{)}$ in the language of Hopf
rings where the operations are naturally introduced.

In more details, a Hopf ring is a five-tuple $(V,\odot,\cdot,\Delta,S)$ of the
vectors space $V$, two multiplications $\odot$ and $\cdot$, one
comultiplication $\Delta$, and an antipode $S$ such that:

  1. (1)

$(V,\odot,\Delta,S)$ is a Hopf algebra,

  2. (2)

$(V,\cdot,\Delta)$ is a bialgebra, and

  3. (3)

$u\cdot(v\odot w)=\sum_{\Delta u=\sum u^{\prime}\otimes
u^{\prime\prime}}(u^{\prime}\cdot v)\odot(u^{\prime\prime}\cdot w)$ for all
$u,v,w\in V$.

Next, the cohomology $H^{*}\big{(}\coprod_{k\geq
1}\mathrm{B}\mathfrak{S}_{k};\mathbb{F}_{2}\big{)}$ can be equipped with a
structure of a Hopf ring where:

  * —

the first product $\odot$ is the so call transfer product, for a definition
consult [55, Sec. 3],

  * —

the second product $\cdot$ is the cup product, extended to be zero on the
classes coming from different disjoint components,

  * —

the coproduct $\Delta$ on is the dual to the standard Pontryagin product on
the homology of $H_{*}\big{(}\coprod_{k\geq
1}\mathrm{B}\mathfrak{S}_{k};\mathbb{F}_{2}\big{)}$, that is the coproduct
induced by the natural inclusions of the symmetric groups
$\mathfrak{S}_{r}\times\mathfrak{S}_{s}\longrightarrow\mathfrak{S}_{r+s}$, and
finally

  * —

the anipode $S$ is just the identity map.

Now the main result of Giusti, Salvatore & Sinha [55, Thm. 1.2] is as follows.

######  Theorem 4.17.

The cohomology $H^{*}\big{(}\coprod_{k\geq
1}\mathrm{B}\mathfrak{S}_{k};\mathbb{F}_{2}\big{)}$, as a Hopf ring, is
generated by the unit classes on each component and classes $u_{\ell,n}\in
H^{*}(\mathrm{B}\mathfrak{S}_{n2^{\ell}};\mathbb{F}_{2})$. The coproduct is
generated by the formula

$\Delta u_{\ell,n}=\sum_{r+s=n}u_{\ell,r}\otimes u_{\ell,s},$

while the relations between the transfer products on generators are given by

$u_{\ell,r}\odot u_{\ell,s}={r+s\choose r}\,u_{\ell,r+s}.$

The second (cup-)product $\cdot$ between the generators from different
components is zero, and there are no further relations between products of the
generators. In addition, the antipode is the identity.

####  4.3.2. Corollary

Motivated by the work of Atiyah and Giusti, Salvatore & Sinha, for a fixed
integer $d\geq 2$ or $d=\infty$, we describe the $\mathbb{F}_{2}$ homology of
the space of all finite subsets of $\mathbb{R}^{d}$ with additional base
point, that is $H_{*}\big{(}\coprod_{k\geq
0}\operatorname{F}(\mathbb{R}^{d},k)/\mathfrak{S}_{k};\mathbb{F}_{2}\big{)}$.
Here, $\operatorname{F}(\mathbb{R}^{d},0)$ and
$\operatorname{F}(\mathbb{R}^{d},0)/\mathfrak{S}_{0}$ stand for base points.

In addition to the additive structure on the homology, which can be already
read from Corollary 4.15, we can also identify the ring structure with the
respect to a naturally induced multiplication. For a clear definition of the
product structure on this homology we use the little $d$-cubes operad
isomorphic model and set

$\mathcal{T}_{d}:=H_{*}\Big{(}\coprod_{k\geq
0}\mathcal{C}_{d}(k)/\mathfrak{S}_{k};\mathbb{F}_{2}\Big{)}\cong
H_{*}\Big{(}\coprod_{k\geq
0}\operatorname{F}(\mathbb{R}^{d},k)/\mathfrak{S}_{k};\mathbb{F}_{2}\Big{)}.$

Recall that by Definition 7.1 the space $\mathcal{C}_{d}(0)$ is just a point
and therefore coincides with the $\operatorname{F}(\mathbb{R}^{d},0)$. Now,
the product structure on $\mathcal{T}_{d}$, we care about, is induced by the
structural maps of the little $d$-cubes operad

$\mu_{\mathcal{C}_{d}}(\vec{c}_{1},\vec{c}_{2})\colon\mathcal{C}_{d}(r)\times\mathcal{C}_{d}(s)\longrightarrow\mathcal{C}_{d}(r+s),$
(62)

where $(\vec{c}_{1},\vec{c}_{2})\in\mathcal{C}_{d}(2)$ is assumed to be an
arbitrary but fixed pair of little $d$-cubes.

######  Theorem 4.18.

Let $d\geq 2$ be an integer or $d=\infty$. Then $\mathcal{T}_{d}$ is
isomorphic to the polynomial ring

$\mathcal{R}_{d}:=\mathbb{F}_{2}\big{[}\\{u_{0}\\}\cup\\{\mathcal{Q}_{i_{1}}\mathcal{Q}_{i_{2}}\cdots\mathcal{Q}_{i_{s}}u_{0}\,:\,s\geq
1,\,1\leq i_{1}\leq i_{2}\leq\cdots\leq i_{s}\leq d-1\\}\big{]}$

on generators

$u_{0}\in H_{0}(\mathcal{C}_{d}(1)/\mathfrak{S}_{1};\mathbb{F}_{2})$

and

$\mathcal{Q}_{i_{1}}\mathcal{Q}_{i_{2}}\cdots\mathcal{Q}_{i_{s}}u_{0}\in
H_{i}(\mathcal{C}_{d}(2^{s})/\mathfrak{S}_{2^{s}};\mathbb{F}_{2})$

where $s\geq 1$, $1\leq i_{1}\leq i_{2}\leq\cdots\leq i_{s}\leq d-1$, and
$i=i_{1}+2i_{2}+2^{2}i_{3}+\dots+2^{s-1}i_{s}$. The unit of the ring
$\mathcal{R}_{d}$ is the generator of the $0$-homology of the base point, that
is $1\in H_{0}(\mathcal{C}_{d}(0)/\mathfrak{S}_{0};\mathbb{F}_{2})$.

###### Proof.

The ring $\mathcal{T}_{d}$ has two gradings, one with respect to the number of
cubes (points) and the second one with respect to the homology degree:

$\mathcal{T}_{d}=H_{*}\Big{(}\coprod_{k\geq
0}\mathcal{C}_{d}(k)/\mathfrak{S}_{k};\mathbb{F}_{2}\Big{)}\cong\bigoplus_{k\geq
0}H_{*}(\mathcal{C}_{d}(k)/\mathfrak{S}_{k};\mathbb{F}_{2})\cong\\\
\bigoplus_{k\geq 0}\bigoplus_{i\geq
0}H_{i}(\mathcal{C}_{d}(k)/\mathfrak{S}_{k};\mathbb{F}_{2}).$

Indeed the product, induced by the map (62), is the composition homomorphism:

---
$\textstyle{H_{i}(\mathcal{C}_{d}(r)/\mathfrak{S}_{r};\mathbb{F}_{2})\otimes
H_{j}(\mathcal{C}_{d}(s)/\mathfrak{S}_{k};\mathbb{F}_{2})\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\
}$$\scriptstyle{\times}$$\textstyle{\
H_{i+j}(\mathcal{C}_{d}(r)/\mathfrak{S}_{r}\times\mathcal{C}_{d}(s)/\mathfrak{S}_{k};\mathbb{F}_{2})\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{(\mu_{\mathcal{C}_{d}}(\vec{c}_{1},\vec{c}_{2}))_{*}}$$\textstyle{H_{i+j}(\mathcal{C}_{d}(r+s)/\mathfrak{S}_{r+s};\mathbb{F}_{2}).}$

The horizontal map is the homology cross product which is an isomorphism in
our situation; consult for example [21, Thm. VI.1.6].

In order to simplify notation for $d\geq 2$ we set

$\mathcal{I}_{d}:=\\{(i_{1},\dots,i_{s}):s\geq 1,\,1\leq i_{1}\leq\cdots\leq
i_{s}\leq d-1\\}.$

Let $I=(i_{1},\dots,i_{s})\in\mathcal{I}_{d}$, then we set the length of $I$
to be $|I|:=s$. Like in the previous section, we denote the iterated
Araki–Kudo–Dyer–Lashof homology operation $Q_{i_{1}}Q_{i_{2}}\cdots Q_{i_{s}}$
applied on $u_{L}$ by $Q_{I}u_{L}$ where
$I=(i_{1},\dots,i_{s})\in\mathcal{I}_{d}$. Furthermore, we write
$\mathcal{Q}_{I}u_{0}:=\mathcal{Q}_{i_{1}}\mathcal{Q}_{i_{2}}\cdots\mathcal{Q}_{i_{s}}u_{0}$
for a typical generator of the polynomial ring $\mathcal{R}$ which differs
from the generator $u_{0}$. Recall, that $\deg(u_{L})=L$, and

$\deg(Q_{I}u_{l})=\deg(Q_{i_{1}}Q_{i_{2}}\cdots
Q_{i_{s}}u_{L})=i_{1}+2i_{2}+4i_{3}+\cdots+2^{s-1}i_{s}+2^{s}L.$

On the other hand, we have set that

$\deg(\mathcal{Q}_{I}u_{0})=\deg(\mathcal{Q}_{i_{1}}\mathcal{Q}_{i_{2}}\cdots\mathcal{Q}_{i_{s}}u_{0})=i_{1}+2i_{2}+4i_{3}+\cdots+2^{s-1}i_{s},$

with $\deg(u_{0})=0$. In other words, if $L$ could be zero the rings
$\mathcal{T}_{d}$ and $\mathcal{R}_{d}$ would coincide precisely.

From Corollary 4.15 we know that the homology
$H_{i}(\mathcal{C}_{d}(k)/\mathfrak{S}_{k};\mathbb{F}_{2})$, for $0\leq
i\leq(d-1)(k-1)$, is the $\mathbb{F}_{2}$ vector space spanned by all
monomials of degree $i+kL$ of the polynomial ring
$\mathbb{F}_{2}[\\{u_{L}\\}\cup\\{Q_{I}u_{L}:I\in\mathcal{I}_{d}\\}]$ whose
weights are exactly $k$. It is important to keep in mind that the base
elements of this vector space are indexed by, but are not equal to, the
iterated Araki–Kudo–Dyer–Lashof homology operations applied to the class
$u_{L}$. Furthermore, recall that for
$I=(i_{1},\dots,i_{s})\in\mathcal{I}_{d}$ the weight of $Q_{I}u_{L}$ is
$2^{s}$, while the weight of $u_{L}$ is $1$. Then we the ring
$\mathcal{T}_{d}$ can be decomposed into a direct sum of vector spaces as
follows:

$\mathcal{T}_{d}=H_{*}\Big{(}\coprod_{k\geq
0}\mathcal{C}_{d}(k)/\mathfrak{S}_{k};\mathbb{F}_{2}\Big{)}\\\
\cong\bigoplus_{k\geq
0}H_{*}(\mathcal{C}_{d}(k)/\mathfrak{S}_{k};\mathbb{F}_{2})\cong\bigoplus_{k\geq
0}\bigoplus_{i\geq
0}H_{i}(\mathcal{C}_{d}(k)/\mathfrak{S}_{k};\mathbb{F}_{2})\\\
\cong\bigoplus_{k\geq 0}\bigoplus_{i\geq
0}\bigg{\langle}{\small(Q_{I_{1}}u_{L})\cdots(Q_{I_{t}}u_{L})\,u_{L}^{a}:\begin{array}[]{l}\omega((Q_{I_{1}}u_{L})\cdots(Q_{I_{t}}u_{L})\,u_{L}^{a})=k\\\
\deg((Q_{I_{1}}u_{L})\cdots(Q_{I_{t}}u_{L})\,u_{L}^{a})=i+kL\end{array}}\bigg{\rangle}\qquad\quad\
\\\ \cong\bigoplus_{k\geq 0}\bigoplus_{i\geq
0}\bigg{\langle}{\small(Q_{I_{1}}u_{L})\cdots(Q_{I_{t}}u_{L})\,u_{L}^{a}:\begin{array}[]{l}2^{s_{1}}+\cdots+2^{s_{t}}+a=k\\\
\iota(I_{1},\dots,I_{t})+(2^{s_{1}}+\cdots+2^{s_{t}}+a)L=i+kL\end{array}}\bigg{\rangle}\\\
\cong\bigoplus_{k\geq 0}\bigoplus_{i\geq
0}\bigg{\langle}{\small(Q_{I_{1}}u_{L})\cdots(Q_{I_{t}}u_{L})\,u_{L}^{a}:\begin{array}[]{l}2^{s_{1}}+\cdots+2^{s_{t}}+a=k\\\
\iota(I_{1},\dots,I_{t})=i\end{array}}\bigg{\rangle}.$

Here we took
$I_{1}=(i_{11},i_{12},\dots,i_{1s_{1}}),\dots,I_{t}=(i_{t1},i_{t2},\dots,i_{ts_{t}})$
and denoted by $\iota(I_{1},\dots,I_{t})$ the sum
$(i_{11}+\dots+2^{s_{1}-1}i_{1s_{1}})+\dots+(i_{t1}+\dots+2^{s_{t}-1}i_{ts_{t}})$,
that is

$\iota(I_{1},\dots,I_{t}):=(i_{11}+\dots+2^{s_{1}-1}i_{1s_{1}})+\dots+(i_{t1}+\dots+2^{s_{t}-1}i_{ts_{t}}).$

On the other hand, in a similar way as above, the polynomial ring
$\mathcal{R}_{d}$ can be decomposed into a direct sum of vector subspaces as
follows:

$\mathcal{R}_{d}:=\mathbb{F}_{2}[\\{u_{0}\\}\cup\\{\mathcal{Q}_{I}u_{0}:I\in\mathcal{I}_{d}\\}]\cong\\\
\bigoplus_{k\geq 0}\bigoplus_{i\geq
0}\bigg{\langle}{\small(\mathcal{Q}_{I_{1}}u_{0})\cdots(\mathcal{Q}_{I_{t}}u_{0})\,u_{0}^{a}\
:\begin{array}[]{l}2^{s_{1}}+\cdots+2^{s_{t}}+a=k\\\
\iota(I_{1},\dots,I_{t})=i\end{array}}\bigg{\rangle}.$

Thus, the bijective correspondence
$Q_{I}u_{L}\longleftrightarrow\mathcal{Q}_{I}u_{0}$, for
$I\in\mathcal{I}_{d}$, and $u_{L}\longleftrightarrow u_{0}$ induces bijection
between the (monomial) basis of $\mathcal{T}_{d}$ and $\mathcal{R}_{d}$, which
further on induces an isomorphism of $\mathcal{T}_{d}$ and $\mathcal{R}_{d}$
considered as $\mathbb{F}_{2}$ vector spaces.

It remains to show that the induced isomorphism on the underlying vector space
structures of $\mathcal{T}_{d}$ and $\mathcal{R}_{d}$ also respects the
corresponding product structures. This fact follows from the commutativity of
the following diagram:

$\textstyle{H_{i}(\mathcal{C}_{d}(r)/\mathfrak{S}_{r})\otimes
H_{j}(\mathcal{C}_{d}(s)/\mathfrak{S}_{s})\ignorespaces\ignorespaces\ignorespaces\ignorespaces\
}$$\scriptstyle{=}$$\textstyle{\
\ignorespaces\ignorespaces\ignorespaces\ignorespaces\widetilde{H}_{i+rL}(\mathrm{D}_{r}\mathcal{C}_{d}(S^{L}))\otimes\widetilde{H}_{j+sL}(\mathrm{D}_{s}\mathcal{C}_{d}(S^{L}))\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\cong}$$\scriptstyle{\cong}$$\textstyle{H_{i}(\mathcal{C}_{d}(r)/\mathfrak{S}_{r})\otimes
H_{j}(\mathcal{C}_{d}(s)/\mathfrak{S}_{s})\ignorespaces\ignorespaces\ignorespaces\ignorespaces\
}$$\scriptstyle{\cong}$$\textstyle{\
\ignorespaces\ignorespaces\ignorespaces\ignorespaces\widetilde{H}_{i+rL}(\operatorname{Th}(\xi_{d,r}^{\oplus
L}))\otimes\widetilde{H}_{j+sL}(\operatorname{Th}(\xi_{d,s}^{\oplus
L}))\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\cong}$$\scriptstyle{\cong}$$\textstyle{H_{i+j}(\mathcal{C}_{d}(r)/\mathfrak{S}_{r}\times\mathcal{C}_{d}(s)/\mathfrak{S}_{s})\ignorespaces\ignorespaces\ignorespaces\ignorespaces\
}$$\scriptstyle{=}$$\textstyle{\
\ignorespaces\ignorespaces\ignorespaces\ignorespaces\widetilde{H}_{i+j+(r+s)L}(\operatorname{Th}(\xi_{d,r}^{\oplus
L})\wedge\operatorname{Th}(\xi_{d,s}^{\oplus
L}))\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\cong}$$\scriptstyle{\cong}$$\textstyle{H_{i+j}(\mathcal{C}_{d}(r)/\mathfrak{S}_{r}\times\mathcal{C}_{d}(s)/\mathfrak{S}_{s})\ignorespaces\ignorespaces\ignorespaces\ignorespaces\
}$$\scriptstyle{(\mu_{\mathcal{C}_{d}}(\vec{c}_{1},\vec{c}_{2}))_{*}}$$\textstyle{\
\ignorespaces\ignorespaces\ignorespaces\ignorespaces\widetilde{H}_{i+j+(r+s)L}(\operatorname{Th}(\xi_{d,r}^{\oplus
L}\times\xi_{d,s}^{\oplus
L}))\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\cong}$$\scriptstyle{(\mu_{\mathcal{C}_{d}}(\vec{c}_{1},\vec{c}_{2}))_{*}}$$\textstyle{H_{i+j}(\mathcal{C}_{d}(r+s)/\mathfrak{S}_{r+s})\ignorespaces\ignorespaces\ignorespaces\ignorespaces\
}$$\scriptstyle{=}$$\textstyle{\
\ignorespaces\ignorespaces\ignorespaces\ignorespaces\widetilde{H}_{i+j+(r+s)L}(\operatorname{Th}(\xi_{d,r+s}^{\oplus
L}))\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\cong}$$\scriptstyle{\cong}$$\textstyle{H_{i+j}(\mathcal{C}_{d}(r+s)/\mathfrak{S}_{r+s})\
}$$\textstyle{\
\ignorespaces\ignorespaces\ignorespaces\ignorespaces\widetilde{H}_{i+j+(r+s)L}(\mathrm{D}_{r+s}\mathcal{C}_{d}(S^{L}))}$$\scriptstyle{\cong}$
(63)

where the coefficients in all homologies are assumed to be in the field
$\mathbb{F}_{2}$. (Recall, here $\operatorname{Th}(\xi)$ stands for the Thom
space of the vector bundle $\xi$.) Thus, we have proved that $\mathcal{T}_{d}$
and $\mathcal{R}_{d}$ are isomorphic as rings. ∎

######  Remark 4.19.

It is important to observe that alternatively we could see the disjoint union
$\coprod_{k\geq 0}\operatorname{F}(\mathbb{R}^{d},k)/\mathfrak{S}_{k}$ as the
free $\mathcal{C}_{d}$-space $\mathcal{C}_{d}(S^{d})$ generated by the sphere
$S^{0}$. Then the homology $H_{*}(\coprod_{k\geq
0}\operatorname{F}(\mathbb{R}^{d},k)/\mathfrak{S}_{k};\mathbb{F}_{p})$, for
$p$ a prime, was described already in 1976 by Cohen [33, Thm. 3.1] using the
framework of an appropriate class of Hopf algebras.

##  5\. On highly regular embeddings – revised

The study of highly regular embeddings has a long history, which starts in
1957 with the work of Borsuk [20] on the existence $k$-regular embeddings.
Boltjanskiĭ, Ryškov & Šaškin [19] in 1963 gave the first lower bounds for the
existence of $2k$-regular embeddings. In the 1970’s and 1980’s the accumulated
knowledge about the topology of the unordered configuration space allowed for
further progress, which was made by Cohen & Handel [36], Chisholm [29], Handel
[57], and Handel & Segal [59]. In the 1990’s the existence of $k$-regular
embeddings was related to the notion of $k$-neighbourly submanifolds; this was
considered by Vassiliev [100] and Handel [58]. The result of Chisholm [29,
Theorem 2] is the strongest result from that time.

In the first decade of 21st century the notion of $\ell$-skew embeddings was
introduced and studied by Ghomi & Tabachnikov [54]. Furthermore, combining the
notions of $k$-regular embeddings and of $\ell$-skew embeddings, in 2006
Stojenović [96] defined the notion of $k$-regular-$\ell$-skew embeddings and
gave the first bounds for their existence.

The second decade of the new century brought the first non-elementary
constructions of $k$-regular embedding. In 2019, Jarosław Buczyński, Tadeusz
Januszkiewicz, Joachim Jelisiejew & Mateusz Michałek [24], using advanced
methods of algebraic geometry, constructed $k$-regular embeddings of finite
dimensional real and complex vector spaces.

In this section we revise the results of Blagojević, Lück and Ziegler
presented in the paper [15]. The proofs of [15, Thm. 2.1, Thm. 3.1, Thm. 4.1]
relied essentially on a result of Hu’ng [64, (4.7)] that turned out to be
incorrect, see Remark 4.6. Specifically, [15, Lem. 2.15] is not true when $d$
is not a power of $2$.

###  5.1. $k$-regular embeddings

In this section we correct [15, Sec. 2]. In the following all topological
spaces are Hausdorff spaces and all maps are assumed to be continuous.

######  Definition 5.1.

Let $k\geq 1$ and $N\geq 1$ be integers, and let $X$ be a topological space. A
continuous map $f\colon X\longrightarrow\mathbb{R}^{N}$ is a $k$-regular
embedding if for every $(x_{1},\ldots,x_{k})\in\operatorname{F}(X,k)$ the
vectors $f(x_{1}),\ldots,f(x_{k})$ are linearly independent. In particular,
$0\notin\operatorname{im}(f)$.

For $N\geq 1$ and $1\leq k\leq N$ integers, the Stiefel manifold
$V_{k}(\mathbb{R}^{N})$ of all ordered $k$-frames is a (topological) subspace
of the product $(\mathbb{R}^{N})^{k}$ given by

$V_{k}(\mathbb{R}^{N})=\\{(y_{1},\ldots,y_{k})\in(\mathbb{R}^{N})^{k}:y_{1},\ldots,y_{k}\text{
are linearly independent}\\}.$

The symmetric group $\mathfrak{S}_{k}$ acts freely on the Stiefel manifold by
permuting the vectors in the frame. Now from Definition 5.1 we get a necessary
condition for the existence of a $k$-regular embedding phrased in term of the
existence of an equivariant map.

######  Lemma 5.2.

If there exists a $k$-regular embedding $X\longrightarrow\mathbb{R}^{N}$, then
there exists an $\mathfrak{S}_{k}$-equivariant map

$\operatorname{F}(X,k)\longrightarrow V_{k}(\mathbb{R}^{N}).$ (64)

In order to study the existence of an $\mathfrak{S}_{k}$-equivariant map (64)
we use the criterion of Cohen and Handel [36, Prop. 2.1], see also [15, Lem.
2.11]. Let $\mathbb{R}^{k}$ be endowed with an $\mathfrak{S}_{k}$-action given
by coordinate permutation. Then the subspace
$W_{k}=\\{(a_{1},\ldots,a_{k})\in\mathbb{R}^{k}:a_{1}+\dots+a_{k}=0\\}$ is an
$\mathfrak{S}_{k}$-invariant subspace. Consider the following vector bundles

$\textstyle{\xi_{X,k}\colon}$$\textstyle{\mathbb{R}^{k}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\operatorname{F}(X,k)\times_{\mathfrak{S}_{k}}\mathbb{R}^{k}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\
}$$\textstyle{\
\operatorname{F}(X,k)/\mathfrak{S}_{k},}$$\textstyle{\zeta_{X,k}\colon}$$\textstyle{W_{k}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\operatorname{F}(X,k)\times_{\mathfrak{S}_{k}}W_{k}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\
}$$\textstyle{\
\operatorname{F}(X,k)/\mathfrak{S}_{k},}$$\textstyle{\tau_{X,k}\colon}$$\textstyle{\mathbb{R}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\operatorname{F}(X,k)/\mathfrak{S}_{k}\times\mathbb{R}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\
}$$\textstyle{\ \operatorname{F}(X,k)/\mathfrak{S}_{k},}$ (65)

where $\tau_{X,k}$ is a trivial line bundle. An obvious decomposition holds:

$\xi_{X,k}\cong\zeta_{X,k}\oplus\tau_{X,k}.$ (66)

The bundle $\xi_{d,k}$ introduced in (57) is the pull-back vector bundle of
the vector bundle $\xi_{\mathbb{R}^{d},k}$ along the homotopy equivalence
$\mathcal{C}_{d}(k)/\mathfrak{S}_{k}\longrightarrow\operatorname{F}(\mathbb{R}^{d},k)/\mathfrak{S}_{k}$
which is induced by the evaluation at centers of cubes map
$\operatorname{ev}_{d,n}\colon\mathcal{C}_{d}(n)\longrightarrow\operatorname{F}(\mathbb{R}^{d},n)$
from Lemma 7.9.

######  Lemma 5.3.

An $\mathfrak{S}_{k}$-equivariant map $\operatorname{F}(X,k)\longrightarrow
V_{k}(\mathbb{R}^{N})$ exists if and only if the $k$-dimensional vector bundle
$\xi_{X,k}$ admits an $(N-k)$-dimensional inverse.

As a direct consequence of Lemmas 5.2 and 5.3 we get a criterion for the non-
existence of a $k$-regular embedding in term of the dual Stiefel–Whitney class
of the vector bundle $\xi_{X,k}$. We recall only a particular case of [15,
Lem. 2.12 (2)] that we use here.

######  Lemma 5.4.

Let $d\geq 1$ and $k\geq 1$ be integers. If the dual Stiefel–Whitney class

$\overline{w}_{N-k+1}(\xi_{\mathbb{R}^{d},k})\neq 0$

does not vanish, then there cannot be any $k$-regular embedding
$\mathbb{R}^{d}\longrightarrow\mathbb{R}^{N}$.

The criterion for the non-existence of a $k$-regular embedding
$X\longrightarrow\mathbb{R}^{N}$ given in the previous lemma motivates the
study of Stiefel–Whitney classes of the vector bundle $\xi_{X,k}$. In [15,
Thm. 2.13], based on the work of Hu’ng [64, (4.7)], the following theorem
about the dual Stiefel–Whitney classes of $\xi_{\mathbb{R}^{d},k}$ was proved.

> Theorem 2.13. Let $k,d\geq 1$ be integers. Then the dual Stiefel–Whitney
> class
>
> $\overline{w}_{(d-1)(k-\alpha(k))}(\xi_{\mathbb{R}^{d},k})$
>
> does not vanish.

The previous theorem and the criterion given in Lemma 5.4 implied [15, Thm.
2.1]:

> Theorem 2.1. Let $k,d\geq 1$ be integers. There is no $k$-regular embedding
> $\mathbb{R}^{d}\longrightarrow\mathbb{R}^{N}$ for
>
> $N\leq d(k-\alpha(k))+\alpha(k)-1,$
>
> where $\alpha(k)$ denotes the number of ones in the dyadic presentation of
> $k$.

The proof of the key theorem [15, Thm. 2.13] was done in two steps. First, the
special case when $k$ is a power of $2$ was established [15, Lem. 2.15], and
then the general case for arbitrary integer $k\geq 2$ was derived [15, Lem.
2.17]. In the following we quote both lemmas as in they were written in the
original.

> Lemma 2.15. Let $d\geq 1$ be an integer, and $k=2^{m}$ for some $m\geq 1$.
> Then the dual Stiefel–Whitney class
> $\overline{w}_{(d-1)(k-1)}(\xi_{\mathbb{R}^{d},k})$ does not vanish.
>
> Lemma 2.17. Let $d,k\geq 1$ be integers. Then the dual Stiefel–Whitney class
> $\overline{w}_{(d-1)(k-\alpha(k))}(\xi_{\mathbb{R}^{d},k})$ does not vanish.

The proof of [15, Lem. 2.15] contains a gap at the place where [64, (4.7)] was
applied and is not true for $d\geq 1$ not a power of $2$. More precisely, in
the proof stands:

> The following decomposition of
> $H^{*}(\operatorname{F}(\mathbb{R}^{d},k)/\mathfrak{S}_{k})$ was proved by
> Hu’ng in [20, (4.7), page 279]:
>
> $\quad(5)\quad
> H^{*}(\operatorname{F}(\mathbb{R}^{d},k)/\mathfrak{S}_{k})\cong\alpha_{d,k}^{*}\big{(}\ker(\operatorname{res}^{\mathfrak{S}_{k}}_{E_{m}})\oplus\mathbb{F}_{2}[q_{m,m-1},\ldots,q_{m,1}]\big{)}\oplus\alpha_{d,k}^{*}(\langle
> q_{m,0}\rangle).$
>
> In particular, this implies that
>
>
> $\alpha_{d,k}^{*}\big{(}\ker(\operatorname{res}^{\mathfrak{S}_{k}}_{E_{m}})\oplus\mathbb{F}_{2}[q_{m,m-1},\ldots,q_{m,1}]\big{)}\cap\alpha_{d,k}^{*}(\langle
> q_{m,0}\rangle)=\\{0\\}.$

Consequently, [15, Lem. 2.15], [15, Lem. 2.17], [15, Thm. 2.13] and [15, Thm.
2.1] do not hold. Furthermore, a technical result stated in [15, Cor. 2.16] is
incorrect. In the following we correct these gaps.

For the corrections we will combine different results we established so far.
Many of them can be summarized in the commutative diagram (68). The
cohomologies appearing in the diagram (68) are assumed to be with the field
$\mathbb{F}_{2}$ coefficients.

First, we show that [15, Lem. 2.15], [15, Lem. 2.17] and consequently [15,
Thm. 2.13] hold, as stated, in the case when the dimension $d$ is a power of
$2$. For that we use the following result from [14, Thm. 3.1]: For integers
$d\geq 2$ and $k\geq 2$

$\operatorname{height}(H^{*}(\operatorname{F}(\mathbb{R}^{d},k)/\mathfrak{S}_{k};\mathbb{F}_{2}))\leq\min\\{2^{t}:2^{t}\geq
d\\},$ (67)

where
$\operatorname{height}(\operatorname{F}(\mathbb{R}^{d},k)/\mathfrak{S}_{k};\mathbb{F}_{2}))$
is the height of the algebra
$H^{*}(\operatorname{F}(\mathbb{R}^{d},k)/\mathfrak{S}_{k};\mathbb{F}_{2})$.
This result, obtained as an application of Kahn–Priddy [66, Thm. pp. 103], is
the best possible as explained in [14, Sec. 3]. It is worth mentioning that a
similar upper bound for the hight of the cohomology ring
$H^{*}(\operatorname{F}(\mathbb{R}^{d},k)/\mathfrak{S}_{k};\mathbb{F}_{p})$,
where $p$ is now an odd prime, was given in [14, Thm. 3.2].

Recall, for an algebra $A$ over a field $\mathbb{F}$, the height of an element
$a\in A{\setminus}A^{*}$ is a natural number or infinity
$\operatorname{height}(a):=\min\\{n\in\mathbb{N}:a^{n}=0\\}$. Here $A^{*}$
denotes the group of all invertible elements of the algebra $A$. The height of
the algebra $A$ is defined to be
$\operatorname{height}(A):=\max\\{\operatorname{height}(a):a\in
A{\setminus}A^{*}\\}$.

$\textstyle{\vspace{-10pt}H^{*}(\mathfrak{S}_{2^{m}})\cong
H^{*}(F(\mathbb{R}^{\infty},2^{m})/\mathfrak{S}_{2^{m}})\cong\mathbb{F}_{2}[w_{m,0},\dots,w_{m,m-1}]\oplus\ker(\operatorname{res}^{\mathfrak{S}_{k}}_{\mathcal{E}_{m}})\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\
}$$\scriptstyle{\rho_{\infty,2^{m}}^{*}}$$\textstyle{\
H^{*}(\mathcal{S}_{2^{m}})\cong
H^{*}(\operatorname{Pe}(\mathbb{R}^{\infty},2^{m})/\mathcal{S}_{2^{m}})\cong\mathbb{F}_{2}[V_{m,1},\dots,V_{m,m}]\oplus
I^{*}(\mathbb{R}^{\infty},k)\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{(\kappa_{d,2^{m}}/\mathcal{S}_{2^{m}})^{*}}$$\textstyle{H^{*}(F(\mathbb{R}^{d},2^{m})/\mathfrak{S}_{2^{m}})\ignorespaces\ignorespaces\ignorespaces\ignorespaces\
}$$\scriptstyle{\rho_{d,2^{m}}^{*}}$$\textstyle{\
H^{*}(\operatorname{Pe}(\mathbb{R}^{d},2^{m})/\mathcal{S}_{2^{m}})\cong\mathbb{F}_{2}[V_{m,1},\dots,V_{m,m}]/\langle
V_{m,1}^{d},\dots,V_{m,m}^{d}\rangle\oplus
I^{*}(\mathbb{R}^{d},k)\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$projection$\textstyle{\mathbb{F}_{2}[V_{m,1},\dots,V_{m,m}]/\langle
V_{m,1}^{d},\dots,V_{m,m}^{d}\rangle\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\text{isomorphism
induced by }\chi_{m}}$$\textstyle{\mathbb{F}_{2}[V_{1},\dots,V_{m}]/\langle
V_{1}^{d},\dots,V_{m}^{d}\rangle}$ (68)

We prove the following lemma which can be extracted from work of Chisholm
[29]. For alternative proof see also [42, Prop. 5.2].

######  Lemma 5.5.

Let $d=2^{a}$ and $k=2^{m}$ for some integers $a\geq 1$ and $m\geq 1$. Then
the dual Stiefel–Whitney class

$\overline{w}_{(d-1)(k-1)}(\xi_{\mathbb{R}^{d},k})=\overline{w}_{(2^{a}-1)(2^{m}-1)}(\xi_{\mathbb{R}^{d},2^{m}})$

does not vanish.

###### Proof.

Let us fix $d=2^{a}$ and $k=2^{m}$ for integers $a\geq 1$ and $m\geq 1$. For
simplicity denote by $w:=w(\xi_{\mathbb{R}^{d},2^{m}})$ the total
Stiefel–Whitney class of the vector bundle $\xi_{\mathbb{R}^{d},2^{m}}$ and by
$\overline{w}:=\overline{w}(\xi_{\mathbb{R}^{d},2^{m}})$ the associated total
dual Stiefel–Whitney, that is $w\cdot\overline{w}=1$. From the decomposition
(66) we get that
$w_{i}:=w_{i}(\xi_{\mathbb{R}^{d},2^{m}})=w_{i}(\zeta_{\mathbb{R}^{d},2^{m}})=0$
for all $i\geq 2^{m}$, and so $w=1+w_{1}+\cdots+w_{2^{m}-1}\in
H^{*}(\operatorname{F}(\mathbb{R}^{d},2^{m})/\mathfrak{S}_{2^{m}};\mathbb{F}_{2})$.
From the inequality (67) we have that

$\operatorname{height}(H^{*}(\operatorname{F}(\mathbb{R}^{d},2^{m})/\mathfrak{S}_{2^{m}};\mathbb{F}_{2}))\leq\min\\{2^{t}:2^{t}\geq
d\\}=2^{a}=d.$

Thus, we compute

$w^{d}=w^{2^{a}}=(1+w_{1}+\dots+w_{2^{m}-1})^{2^{a}}=1+w_{1}^{2^{a}}+\dots+w_{2^{m}-1}^{2^{a}}=1,$

and so $w^{d}=w\cdot w^{d-1}=1$. Therefore,

$\displaystyle\overline{w}$
$\displaystyle=w^{d-1}=w^{2^{a}-1}=w^{1+2^{1}+2^{2}+\dots+2^{a-1}}=(1+w_{1}+\dots+w_{2^{m}-1})^{1+2^{1}+2^{2}+\dots+2^{a-1}}$
$\displaystyle=\prod_{i=0}^{a-1}\big{(}1+w_{1}^{2^{i}}+\dots+w_{2^{m}-1}^{2^{i}}\big{)}=w_{2^{m}-1}^{\sum_{i=0}^{a-1}2^{i}}+\big{(}\text{\small
terms of degree}<(d-1)(2^{m}-1)\big{)}$
$\displaystyle=w_{2^{m}-1}^{d-1}+\big{(}\text{\small terms of
degree}<\deg(w_{2^{m}-1}^{d-1})=(d-1)(2^{m}-1)\big{)}.$

In summary,

$\overline{w}=w_{2^{m}-1}^{d-1}+\big{(}\text{\small terms of
degree}<\deg(w_{2^{m}-1}^{d-1})=(d-1)(2^{m}-1)\big{)}\\\ \in
H^{*}(\operatorname{F}(\mathbb{R}^{d},2^{m})/\mathfrak{S}_{2^{m}};\mathbb{F}_{2}).$
(69)

The fact that $w_{2^{m}-1}^{d-1}\neq 0$ follows from general obstruction
computation in [16, Cor. 4.5]. Nevertheless, we present a different argument
that follows from the results of Sections 3 and 4. First we apply the map
$\rho_{d,2^{m}}^{*}$ to the relation (69) and get

$\rho_{d,2^{m}}^{*}(\overline{w})=\rho_{d,2^{m}}^{*}(w_{2^{m}-1})^{d-1}+\big{(}\text{\small
terms of degree}<\deg(w_{2^{m}-1}^{d-1})\big{)}\\\ \in
H^{*}(\operatorname{Pe}(\mathbb{R}^{d},2^{m})/\mathcal{S}_{2^{m}};\mathbb{F}_{2}).$
(70)

Recall that according to Theorem 3.11 we have that

$H^{*}(\operatorname{Pe}(\mathbb{R}^{d},2^{m})/\mathcal{S}_{2^{m}};\mathbb{F}_{2})\cong\mathbb{F}_{2}[V_{m,1},\dots,V_{m,m}]/\langle
V_{m,1}^{d},\ldots,V_{m,m}^{d}\rangle\oplus\mathrm{I}^{*}(\mathbb{R}^{d},2^{m}).$

Furthermore, from the proof of Lemma 3.13, the commutative diagram (68) and
the relation (44) we have that

$\rho_{d,2^{m}}^{*}(w_{2^{m}-1})=V_{m,1}\cdots
V_{m,m}\in\mathbb{F}_{2}[V_{m,1},\dots,V_{m,m}]/\langle
V_{m,1}^{d},\ldots,V_{m,m}^{d}\rangle\\\ \subseteq
H^{*}(\operatorname{Pe}(\mathbb{R}^{d},2^{m})/\mathcal{S}_{2^{m}};\mathbb{F}_{2}).$

Combining these facts we get

$\rho_{d,2^{m}}^{*}(\overline{w})=(V_{m,1}\cdots
V_{m,m})^{d-1}+\big{(}\text{\small terms of
degree}<\deg(w_{2^{m}-1}^{d-1})\big{)}\\\ \in
H^{*}(\operatorname{Pe}(\mathbb{R}^{d},2^{m})/\mathcal{S}_{2^{m}};\mathbb{F}_{2}),$

and furthermore

$\rho_{d,2^{m}}^{*}(\overline{w}_{(d-1)(2^{m}-1)})=\rho_{d,2^{m}}^{*}(w_{2^{m}-1})^{d-1}=(V_{m,1}\cdots
V_{m,m})^{d-1}\neq 0.$

Consequently, $\overline{w}_{(d-1)(2^{m}-1)}\neq 0$, and the proof of the
lemma is concluded. ∎

The extension of the previous lemma (for dimension $d=2^{a}$) to the case of
where $k\geq 1$ is an arbitrary integer (still for dimension $d=2^{a}$) can be
done as in the proof of [15, Lem. 2.17]. For the sake of completeness we give
a detailed proof using the presentation from [15, Proof of Lem. 2.17]. Here
$\alpha(k)$ denotes the number of $1$s in the binary presentation of the
integer $k\geq 1$.

######  Lemma 5.6.

Let $d=2^{a}$ for some integer $a\geq 1$, and let $k\geq 1$ be an integer.
Then the dual Stiefel–Whitney class

$\overline{w}_{(d-1)(k-\alpha(k))}(\xi_{\mathbb{R}^{d},k})=\overline{w}_{(2^{a}-1)(k-\alpha(k))}(\xi_{\mathbb{R}^{d},k})$

does not vanish.

###### Proof.

Let $r:=\alpha(k)$ be the number of $1$s in the binary presentation of the
integer $k\geq 1$, and let $k=2^{b_{1}}+\dots+2^{b_{r}}$ where $0\leq
b_{1}<b_{2}<\dots<b_{r}$. Consider a morphism between vector bundles
$\prod_{i=1}^{r}\xi_{\mathbb{R}^{d},2^{b_{i}}}$ 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^{b_{i}}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\
}$$\scriptstyle{\Theta}$$\textstyle{\
\xi_{\mathbb{R}^{d},k}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\prod_{i=1}^{r}\operatorname{F}(\mathbb{R}^{d},2^{b_{i}})/\mathfrak{S}_{2^{b_{i}}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\
}$$\scriptstyle{\theta}$$\textstyle{\
\operatorname{F}(\mathbb{R}^{d},k)/{\mathfrak{S}_{k}}.}$

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

$(\mathcal{C}_{d}(2^{b_{1}})\times\dots\times\mathcal{C}_{d}(2^{b_{r}}))\times\mathcal{C}_{d}(r)\longrightarrow\mathcal{C}_{d}(2^{b_{1}}+\dots+2^{b_{r}}).$

Alternatively, fix embeddings
$e_{i}\colon\mathbb{R}^{d}\longrightarrow\mathbb{R}^{d}$ for $1\leq i\leq r$
such that their images are pairwise disjoint open $d$-balls. They induces
embeddings
$\operatorname{F}(\mathbb{R}^{d},\ell)\longrightarrow\operatorname{F}(\mathbb{R}^{d},\ell)$
denoted by the same letter $e_{i}$ for all natural numbers $\ell$. Thus, the
map $\theta$ is induced by the map
$\prod_{i=1}^{r}\operatorname{F}(\mathbb{R}^{d},2^{b_{i}})\longrightarrow\operatorname{F}(\mathbb{R}^{d},k)$
defined by

$\big{(}(x_{1,1},\dots,x_{1,2^{b_{1}}}),\dots,(x_{r,1},\dots,x_{r,2^{b_{r}}})\big{)}\longmapsto\\\
e_{1}(x_{1,1},\dots,x_{1,2^{b_{1}}})\times\cdots\times
e_{r}(x_{r,1},\dots,x_{r,2^{b_{r}}}).$

The map $\Theta$ that covers $\theta$ is given by

$\big{(}(x_{1,1},\dots,x_{1,2^{b_{1}}};v_{1}),\dots,(x_{r,1},\dots,x_{r,2^{b_{r}}};v_{r})\big{)}\longmapsto\\\
\big{(}e_{1}(x_{1,1},\dots,x_{1,2^{b_{1}}})\times\cdots\times
e_{r}(x_{r,1},\dots,x_{r,2^{b_{r}}}),v_{1}\times\cdots\times v_{r}\big{)}.$

Consequently, the pullback vector bundle is the product vector bundle
$\theta^{*}\xi_{\mathbb{R}^{d},k}\cong\prod_{i=1}^{r}\xi_{\mathbb{R}^{d},2^{b_{i}}}$.
Now the naturality property of the Stiefel–Whitney classes [80, Ax. 2, p. 37]
implies that in cohomology we get

$\theta^{*}(\overline{w}_{(d-1)(k-r)}(\xi_{\mathbb{R}^{d},k}))=\overline{w}_{(d-1)(k-r)}\Big{(}\prod_{i=1}^{r}\xi_{\mathbb{R}^{d},2^{b_{i}}}\Big{)}.$

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

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

Now we compute

$\displaystyle\theta^{*}(\overline{w}_{(d-1)(k-r)}(\xi_{\mathbb{R}^{d},k}))$
$\displaystyle=\overline{w}_{(d-1)(k-r)}\Big{(}\prod_{i=1}^{r}\xi_{\mathbb{R}^{d},2^{b_{i}}}\Big{)}$
$\displaystyle=\sum_{s_{1}+\cdots+s_{r}=(d-1)(k-r)}\overline{w}_{s_{1}}(\xi_{\mathbb{R}^{d},2^{b_{1}}})\times\cdots\times\overline{w}_{s_{r}}(\xi_{\mathbb{R}^{d},2^{b_{r}}}).$

From the Künneth formula [21, Thm. VI.3.2] we have that each term of the
previous sum
$\overline{w}_{s_{1}}(\xi_{\mathbb{R}^{d},2^{b_{1}}})\times\cdots\times\overline{w}_{s_{r}}(\xi_{\mathbb{R}^{d},2^{b_{r}}})$
belongs to a different direct summand in the following direct decomposition of
the $(d-1)(k-r)$th cohomology

$H^{(d-1)(k-r)}\Big{(}\prod_{i=1}^{r}\operatorname{F}(\mathbb{R}^{d},2^{b_{i}})/\mathfrak{S}_{2^{b_{i}}};\mathbb{F}_{2}\Big{)}\cong\\\
\bigoplus_{s_{1}+\cdots+s_{r}=(d-1)(k-r)}H^{s_{1}}(\operatorname{F}(\mathbb{R}^{d},2^{b_{1}})/\mathfrak{S}_{2^{b_{1}}};\mathbb{F}_{2})\otimes\cdots\otimes
H^{s_{r}}(\operatorname{F}(\mathbb{R}^{d},2^{b_{r}})/\mathfrak{S}_{2^{b_{r}}};\mathbb{F}_{2}).$

Thus, we have the equivalence

$\overline{w}_{(d-1)(k-r)}\Big{(}\prod_{i=1}^{r}\xi_{\mathbb{R}^{d},2^{b_{i}}}\Big{)}\neq
0\quad\Longleftrightarrow\\\
\overline{w}_{s_{1}}(\xi_{\mathbb{R}^{d},2^{b_{1}}})\times\cdots\times\overline{w}_{s_{r}}(\xi_{\mathbb{R}^{d},2^{b_{r}}})\neq
0\;\text{\small for some}\;s_{1}+\cdots+s_{r}=(d-1)(k-r).$

Now, since $d$ is a power of $2$, Lemma 5.5 implies that
$\overline{w}_{(d-1)(2^{b_{i}}-1)}(\xi_{\mathbb{R}^{d},2^{b_{i}}})\neq 0$ for
all $1\leq i\leq r$, and so

$\overline{w}_{(d-1)(2^{b_{1}}-1)}(\xi_{\mathbb{R}^{d},2^{b_{1}}})\times\cdots\times\overline{w}_{(d-1)(2^{b_{r}}-1)}(\xi_{\mathbb{R}^{d},2^{b_{r}}})\neq
0.$

Hence, $\theta^{*}(\overline{w}_{(d-1)(k-r)}(\xi_{\mathbb{R}^{d},k}))\neq 0$,
and furthermore $\overline{w}_{(d-1)(k-\alpha(k))}(\xi_{\mathbb{R}^{d},k})\neq
0$. ∎

Next we consider the case when $d$, the dimension of the Euclidean space
$\mathbb{R}^{d}$, is not a power of $2$. Like in the previous situation this
is done in two separate steps depending whether $k$ is power of $2$ or not.
Now we give two corrections of [15, Lem. 2.15]. In the first one, next lemma,
we do not take into account the dyadic presentation of the dimensions $d$.

######  Lemma 5.7.

Let $d\geq 3$ be an integer that is not a power of $2$, and let $k=2^{m}$ for
some integer $m\geq 1$. Then the dual Stiefel–Whitney class

$\overline{w}_{(d-1)k/2}(\xi_{\mathbb{R}^{d},k})=\overline{w}_{(d-1)2^{m-1}}(\xi_{\mathbb{R}^{d},2^{m}})$

does not vanish.

###### Proof.

Let us fix an integer $d\geq 3$ which is not a power of $2$. Then there exists
an integer $a\geq 2$ such that $2^{a-1}+1\leq d\leq 2^{a}-1$. Furthermore, let
$k=2^{m}$ where $m\geq 1$ is an integer. Then $d-1$ can be presented as
$d-1=2^{a_{1}}+\dots+2^{a_{q}}$ where $0\leq a_{1}<\dots<a_{q}=a-1$. As
before, for simplicity we again denote by $w=w(\xi_{\mathbb{R}^{d},2^{m}})$
the total Stiefel–Whitney class of the vector bundle
$\xi_{\mathbb{R}^{d},2^{m}}$ and by
$\overline{w}=\overline{w}(\xi_{\mathbb{R}^{d},2^{m}})$ the associated total
dual Stiefel–Whitney. Hence, $w\cdot\overline{w}=1$. Now the inequality (67)
implies that

$\operatorname{height}(H^{*}(\operatorname{F}(\mathbb{R}^{d},2^{m})/\mathfrak{S}_{2^{m}};\mathbb{F}_{2}))\leq\min\\{2^{t}:2^{t}\geq
d\\}=2^{a}.$

Consequently,
$w^{2^{a}}=(1+w_{1}+\dots+w_{2^{m}-1})^{2^{a}}=1+w_{1}^{2^{a}}+\dots+w_{2^{m}-1}^{2^{a}}=1$,
and so

$\displaystyle\overline{w}$ $\displaystyle=w^{2^{a}-1}$
$\displaystyle=(1+w_{1}+\dots+w_{2^{m}-1})^{1+2^{1}+2^{2}+\dots+2^{a-1}}$
$\displaystyle=\prod_{i=0}^{a-1}\big{(}1+w_{1}^{2^{i}}+\dots+w_{2^{m}-1}^{2^{i}}\big{)}.$
(71)

Now we apply the monomorphism

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

from Theorem 4.1 to the equality (5.1) and use the decomposition of the
cohomology

$H^{*}(\operatorname{Pe}(\mathbb{R}^{d},2^{m})/\mathcal{S}_{2^{m}};\mathbb{F}_{2})\cong\mathbb{F}_{2}[V_{m,1},\dots,V_{m,m}]/\langle
V_{m,1}^{d},\ldots,V_{m,m}^{d}\rangle\oplus\mathrm{I}^{*}(\mathbb{R}^{d},2^{m}),$

given in Theorem 3.11, to get that

$\rho_{d,2^{m}}^{*}(\overline{w})=\prod_{i=0}^{a-1}\big{(}1+D_{m,m-1}^{2^{i}}+\dots+D_{m,0}^{2^{i}}\big{)}+R\\\
\ \in\ \mathbb{F}_{2}[V_{m,1},\dots,V_{m,m}]/\langle
V_{m,1}^{d},\ldots,V_{m,m}^{d}\rangle\oplus\mathrm{I}^{*}(\mathbb{R}^{d},2^{m}),$

where

  * –

$D_{m,r}=(\kappa_{d,2^{m}}/\mathcal{S}_{2^{m}})^{*}(D_{m,r})=\rho_{d,2^{m}}^{*}(w_{2^{m}-2^{r}})$,
for $0\leq r\leq m-1$, with the obvious abuse of notation, see (40), and

  * –

$R\in\mathrm{I}^{*}(\mathbb{R}^{d},2^{m})$.

Let us denote by $\pi$ the following composition of the maps from the diagram
(68):

$\textstyle{\mathbb{F}_{2}[V_{m,1},\dots,V_{m,m}]/\langle
V_{m,1}^{d},\dots,V_{m,m}^{d}\rangle\oplus
I^{*}(\mathbb{R}^{d},2^{m})\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$projection$\textstyle{\mathbb{F}_{2}[V_{m,1},\dots,V_{m,m}]/\langle
V_{m,1}^{d},\dots,V_{m,m}^{d}\rangle\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\chi_{m}}$$\textstyle{\mathbb{F}_{2}[V_{1},\dots,V_{m}]/\langle
V_{1}^{d},\dots,V_{m}^{d}\rangle,}$

where the first map is the projection on a direct summand and the second map
is induced by the change of variables
$\chi_{m}\in\operatorname{GL}_{m}(\mathbb{F}_{2})$. Next we apply $\pi$ on
$\rho_{d,2^{m}}^{*}(\overline{w})$ and get:

$\pi(\rho_{d,2^{m}}^{*}(\overline{w}))=\prod_{i=0}^{a-1}\big{(}1+\pi(D_{m,m-1})^{2^{i}}+\dots+\pi(D_{m,0})^{2^{i}}\big{)}.$

Since the change of the variables $\chi_{m}$ transforms
$\mathrm{L}_{m}(\mathbb{F}_{2})$-invariants into
$\mathrm{U}_{m}(\mathbb{F}_{2})$-invariants and Dickson polynomials,
$\operatorname{GL}_{m}(\mathbb{F}_{2})$-invariants, can be presented in terms
of $\mathrm{U}_{m}(\mathbb{F}_{2})$-invariants, as explained in (7.2.1) and
(105), we have

$\displaystyle\pi(\rho_{d,2^{m}}^{*}(\overline{w}))$
$\displaystyle=\prod_{i=0}^{a-1}\big{(}1+\pi(D_{m,m-1})^{2^{i}}+\dots+\pi(D_{m,0})^{2^{i}}\big{)}$
$\displaystyle=\prod_{i=0}^{a-1}\Big{(}1+\big{(}V_{1}^{2^{m-1}}+V_{2}^{2^{m-2}}+\cdots+V_{m}^{2^{0}}\big{)}^{2^{i}}+\dots+$
$\displaystyle\quad\big{(}\sum_{1\leq j_{1}<\cdots<j_{r}\leq m}(V_{1}\cdots
V_{j_{1}-1})^{2^{r}}\,(V_{j_{1}+1}\cdots
V_{j_{2}-1})^{2^{r-1}}\cdots(V_{j_{r}+1}\cdots V_{m})^{2^{0}}\big{)}^{2^{i}}$
$\displaystyle\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad+\dots+$
$\displaystyle\quad\big{(}V_{1}\cdots V_{m}\big{)}^{2^{i}}\Big{)}$
$\displaystyle=\prod_{i=0}^{a-1}\Big{(}1+\big{(}V_{1}^{2^{i+m-1}}+V_{2}^{2^{i+m-2}}+\cdots+V_{m}^{2^{i}}\big{)}+\dots+$
$\displaystyle\quad\big{(}\sum_{1\leq j_{1}<\cdots<j_{r}\leq m}(V_{1}\cdots
V_{j_{1}-1})^{2^{i+r}}\,(V_{j_{1}+1}\cdots
V_{j_{2}-1})^{2^{i+r-1}}\cdots(V_{j_{r}+1}\cdots V_{m})^{2^{i}}\big{)}$
$\displaystyle\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad+\dots+$
$\displaystyle\quad\big{(}V_{1}\cdots V_{m}\big{)}^{2^{i}}\Big{)}.$

Since $d-1=2^{a_{1}}+\dots+2^{a_{q}}$ and $0\leq a_{1}<\dots<a_{q}=a-1$, by
choosing terms $V_{m}^{2^{a_{\ell}+0}}$ from factors in the product indexed by
$i=a_{1},\dots,a_{q}$ we get that

$\pi(\rho_{d,2^{m}}^{*}(\overline{w}_{(d-1)2^{m-1}}))=V_{m}^{d-1}+S\ \in\
\mathbb{F}_{2}[V_{1},\dots,V_{m}]/\langle V_{1}^{d},\dots,V_{m}^{d}\rangle,$

where $S$ denotes a sum of some monomials in $V_{1},\dots,V_{m}$ of degree
$(d-1)2^{m-1}=(d-1)k/2$ which are all different from $V_{m}^{d-1}$.
Consequently, $\pi(\rho_{d,k}^{*}(\overline{w}_{(d-1)k/2}))\neq 0$ and
$\overline{w}_{(d-1)k/2}\neq 0$. ∎

######  Remark 5.8.

In general, without analyzing the dyadic presentation of $d-1$ in more detail,
we cannot give a better result. For example, in the case $k=2^{2}=4$ and $d=3$
we have

$\displaystyle\pi(\rho_{3,4}^{*}(\overline{w}))$
$\displaystyle=(1+\pi(D_{2,1})+\pi(D_{2,0}))(1+\pi(D_{2,1})^{2}+\pi(D_{2,0})^{2})$
$\displaystyle=(1+V_{1}^{2}+V_{2}+V_{1}V_{2})(1+V_{1}^{4}+V_{2}^{2}+V_{1}^{2}V_{2}^{2})$
$\displaystyle=(1+V_{1}^{2}+V_{2}+V_{1}V_{2})(1+V_{2}^{2}+V_{1}^{2}V_{2}^{2})$
$\displaystyle=1+(V_{1}^{2}+V_{2})+V_{1}V_{2}+V_{2}^{2}\in\mathbb{F}_{2}[V_{1},V_{2}]/\langle
V_{1}^{3},V_{2}^{3}\rangle.$

Hence, $\pi(\rho_{3,4}^{*}(\overline{w}_{4}))=V_{2}^{2}\neq 0$ and
$\pi(\rho_{3,4}^{*}(\overline{w}_{i}))=0$ for all $i>k(d-1)/2=4$.

######  Remark 5.9.

On the other hand a better result can be obtained, as the following example
will show. Let $k=2^{2}=4$ and $d=6$. We compute

$\displaystyle\pi(\rho_{6,4}^{*}(\overline{w}))$
$\displaystyle=(1+\pi(D_{2,1})+\pi(D_{2,0}))(1+\pi(D_{2,1})^{2}+\pi(D_{2,0})^{2})$
$\displaystyle\quad\ (1+\pi(D_{2,1})^{4}+\pi(D_{2,0})^{4})$
$\displaystyle=(1+V_{1}^{2}+V_{2}+V_{1}V_{2})(1+V_{1}^{4}+V_{2}^{2}+V_{1}^{2}V_{2}^{2})$
$\displaystyle\quad\ (1+V_{1}^{8}+V_{2}^{4}+V_{1}^{4}V_{2}^{4})$
$\displaystyle=1+(V_{1}^{2}+V_{2})+V_{1}V_{2}+(V_{1}^{4}+V_{2}^{2})+(V_{1}^{4}V_{2}+V_{2}^{3})+$
$\displaystyle\quad\
(V_{1}V_{2}^{3}+V_{1}^{5}V_{2})+(V_{1}^{2}V_{2}^{3}+V_{1}^{4}V_{2}^{2}+V_{2}^{4})+V_{1}^{3}V_{2}^{3}+(V_{1}^{2}V_{2}^{4}+V_{2}^{5})+$
$\displaystyle\quad\ V_{1}V_{2}^{5}\in\mathbb{F}_{2}[V_{1},V_{2}]/\langle
V_{1}^{6},V_{2}^{6}\rangle.$

Thus, $\pi(\rho_{3,4}^{*}(\overline{w}_{10}))=V_{1}^{2}V_{2}^{4}+V_{2}^{5}\neq
0$, as Lemma 5.7 predicts, but actually more is true
$\pi(\rho_{3,4}^{*}(\overline{w}_{11}))=V_{1}V_{2}^{5}\neq 0$. A natural
question arises: Can Lemma 5.7 be improved?

######  Remark 5.10.

The non-vanishing of the dual Stiefel–Whitney class
$\overline{w}_{(d-1)k/2}(\xi_{\mathbb{R}^{d},k})$ in the case when $d\geq 3$
is not a power of $2$ and $k=2^{m}$ which was established in Lemma 5.7 can be
also obtain as follows. Consider the embedding

$\underbrace{\operatorname{F}(\mathbb{R}^{d},2)\times\dots\times\operatorname{F}(\mathbb{R}^{d},2)}_{2^{m-1}}\longrightarrow\operatorname{F}(\mathbb{R}^{d},2^{m})$

induced by the fixed embeddings
$e_{i}\colon\mathbb{R}^{d}\longrightarrow\mathbb{R}^{d}$ for $1\leq i\leq
2^{m-1}$ such that their images are pairwise disjoint open $d$-balls. It
induces a continuous map

$\underbrace{\operatorname{F}(\mathbb{R}^{d},2)/\mathbb{Z}_{2}\times\dots\times\operatorname{F}(\mathbb{R}^{d},2)/\mathbb{Z}_{2}}_{2^{m-1}}\longrightarrow\operatorname{F}(\mathbb{R}^{d},2^{m})/\mathfrak{S}_{2^{m}}.$

Then the pull-back of the vector bundle $\xi_{\mathbb{R}^{d},k}$ is isomorphic
to the product vector bundle $\prod_{i=1}^{2^{m-1}}\xi_{\mathbb{R}^{d},2}$.
Now, the embedding $S^{d-1}\longrightarrow\operatorname{F}(\mathbb{R}^{d},2)$
given by $x\longmapsto(x,-x)$ induces a continuous map $\mathbb{R}{\rm
P}^{d-1}\longrightarrow\operatorname{F}(\mathbb{R}^{d},2)/\mathbb{Z}_{2}$ such
that the pullback bundle of the vector bundle $\xi_{\mathbb{R}^{d},2}$ is
isomorphic to the Whitney sum of the tautological line vector bundle
$\gamma_{1}^{d-1}$ and a trivial line bundle. Thus, the pull-back of the
vector bundle $\xi_{\mathbb{R}^{d},k}$ along the composition

$\underbrace{\mathbb{R}{\rm P}^{d-1}\times\dots\times\mathbb{R}{\rm
P}^{d-1}}_{2^{m-1}}\longrightarrow\underbrace{\operatorname{F}(\mathbb{R}^{d},2)/\mathbb{Z}_{2}\times\dots\times\operatorname{F}(\mathbb{R}^{d},2)/\mathbb{Z}_{2}}_{2^{m-1}}\longrightarrow\operatorname{F}(\mathbb{R}^{d},2^{m})/\mathfrak{S}_{2^{m}}$

is the product vector bundle
$\gamma:=\prod_{i=1}^{2^{m-1}}(\gamma_{1}^{d-1}\oplus\tau_{1}^{d-1})$. Here
$\tau_{1}^{d-1}$ denotes the trivial line vector bundle over the real
projective space $\mathbb{R}{\rm P}^{d-1}$. Now we compute

$\overline{w}(\gamma)=\overline{w}\big{(}\prod_{i=1}^{2^{m-1}}(\gamma_{1}^{d-1}\oplus\tau_{1}^{d-1})\big{)}=\bigtimes_{i=1}^{2^{m-1}}\overline{w}(\gamma_{1}^{d-1}\oplus\tau_{1}^{d-1})=\bigtimes_{i=1}^{2^{m-1}}\overline{w}(\gamma_{1}^{d-1}).$

Here “$\bigtimes$” denotes the cross product in cohomology. Since
$w(\gamma_{1}^{d-1})=1+u$, where $u\in H^{*}(\mathbb{R}{\rm
P}^{d-1};\mathbb{F}_{2})=\mathbb{F}_{2}[u]/\langle u^{d}\rangle$, we have that
$\overline{w}(\gamma_{1}^{d-1})=w(\gamma_{1}^{d-1})^{-1}=1+u+\dots+u^{d-1}$.
Therefore,

$\overline{w}_{(d-1)2^{m-1}}(\gamma)=\overline{w}_{(d-1)k/2}(\gamma)=\underbrace{u\times\dots\times
u}_{2^{m-1}}\neq 0.$

Having this simpler argument in mind it is natural to ask why we chose to
present the more involved proof of the same fact. The reason lies in the
method of the proof of Lemma 5.7 which is used once again for the proof of the
next lemma.

Now, using a similar line of arguments as in the proof of Lemma 5.7 and taking
into account the dyadic presentation of $d-1$ we get the following particular
result.

######  Lemma 5.11.

Let $d\geq 6$ be an even integer such that for some $a\geq 3$ holds
$2^{a-1}+1\leq d\leq 2^{a}-1$, and let $k=2^{m}$ for some integer $m\geq 1$.
Then the dual Stiefel–Whitney class

$\overline{w}_{dk/2-1}(\xi_{\mathbb{R}^{d},k})=\overline{w}_{d2^{m-1}-1}(\xi_{\mathbb{R}^{d},2^{m}})$

does not vanish.

###### Proof.

Let us assume that $d-1=2^{a_{1}}+\dots+2^{a_{q}}$ where
$0=a_{1}<\dots<a_{q}=a-1$. Following the steps of the proof of Lemma 5.7 we
reach again the equality:

$\displaystyle\pi(\rho_{d,2^{m}}^{*}(\overline{w}))$
$\displaystyle=\prod_{i=0}^{a-1}\big{(}1+\pi(D_{m,m-1})^{2^{i}}+\dots+\pi(D_{m,0})^{2^{i}}\big{)}$
(72)
$\displaystyle=\prod_{i=0}^{a-1}\Big{(}1+\big{(}V_{1}^{2^{m-1}}+V_{2}^{2^{m-2}}+\cdots+V_{m}^{2^{0}}\big{)}^{2^{i}}+\dots+$
$\displaystyle\quad\big{(}\sum_{1\leq j_{1}<\cdots<j_{r}\leq m}(V_{1}\cdots
V_{j_{1}-1})^{2^{r}}\,(V_{j_{1}+1}\cdots
V_{j_{2}-1})^{2^{r-1}}\cdots(V_{j_{r}+1}\cdots V_{m})^{2^{0}}\big{)}^{2^{i}}$
$\displaystyle\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad+\dots+$
$\displaystyle\quad\big{(}V_{1}\cdots V_{m}\big{)}^{2^{i}}\Big{)}$
$\displaystyle=\prod_{i=0}^{a-1}\Big{(}1+\big{(}V_{1}^{2^{i+m-1}}+V_{2}^{2^{i+m-2}}+\cdots+V_{m}^{2^{i}}\big{)}+\dots+$
$\displaystyle\quad\big{(}\sum_{1\leq j_{1}<\cdots<j_{r}\leq m}(V_{1}\cdots
V_{j_{1}-1})^{2^{i+r}}\,(V_{j_{1}+1}\cdots
V_{j_{2}-1})^{2^{i+r-1}}\cdots(V_{j_{r}+1}\cdots V_{m})^{2^{i}}\big{)}$
$\displaystyle\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad+\dots+$
$\displaystyle\quad\big{(}V_{1}\cdots V_{m}\big{)}^{2^{i}}\Big{)}.$

Only now we show that

$\pi(\rho_{d,2^{m}}^{*}(\overline{w}_{d2^{m-1}-1}))=\\\ V_{1}\cdots
V_{m-1}V_{m}^{d-1}+S\ \in\ \mathbb{F}_{2}[V_{1},\dots,V_{m}]/\langle
V_{1}^{d},\dots,V_{m}^{d}\rangle,$

where $S$ is a sum of monomials in $V_{1},\dots,V_{m}$ of degree $d2^{m-1}-1$
which are different from $V_{1}\cdots V_{m-1}V_{m}^{d-1}$. Hence,
$\pi(\rho_{d,k}^{*}(\overline{w}_{d2^{m-1}-1}))\neq 0$, and consequently
$\overline{w}_{d2^{m-1}-1}\neq 0$.

Indeed, observe that in every monomial of the $i$th factor of the product (72)
which has the variable $V_{m}$ with positive exponent this exponent is always
the same and equal to $2^{i}$, $0\leq i\leq a-1$. In particular, in the $i$th
factor each monomial with the variable $V_{m}$ is of the form
$p_{i}(V_{1},\dots,V_{m-1})^{2^{i}}V_{m}^{2^{i}}$ where
$p_{i}(V_{1},\dots,V_{m-1})$ is a monomial in variables $V_{1},\dots,V_{m-1}$.
Now, when multiplying out the product (72) the monomial of the form
$p(V_{1},\dots,V_{m-1})V_{m}^{d-1}$ can appear in the final result if and only
if we take non-zero

  * —

monomials $p_{i}(V_{1},\dots,V_{m-1})^{2^{i}}V_{m}^{2^{i}}$ from $i$th factors
where $i\in\\{a_{1},\dots,a_{q}\\}$, and

  * —

monomials $p_{j}^{\prime}(V_{1},\dots,V_{m-1})$ from $j$th factors for
$j\in\\{0,\dots,a-1\\}{\setminus}\\{a_{1},\dots,a_{q}\\}$.

Thus, we have

$p(V_{1},\dots,V_{m-1})V_{m}^{d-1}=\prod_{i\in\\{a_{1},\dots,a_{q}\\}}p_{i}(V_{1},\dots,V_{m-1})^{2^{i}}V_{m}^{2^{i}}\cdot\\\
\prod_{j\in\\{0,\dots,a-1\\}{\setminus}\\{a_{1},\dots,a_{q}\\}}p_{j}^{\prime}(V_{1},\dots,V_{m-1}).$

Observe that, if $p_{i}(V_{1},\dots,V_{m-1})\neq 1$ for some
$i\in\\{a_{1},\dots,a_{q}\\}$, then there exists $1\leq t\leq m-1$ such that
$V_{t}\mid p_{i}(V_{1},\dots,V_{m-1})$. Hence, $V_{t}^{2^{i}}\mid
p_{i}(V_{1},\dots,V_{m-1})^{2^{i}}$.

Now we want to understand in how many ways we can obtain the monomial
$V_{1}\dots V_{m-1}V_{m}^{d-1}$ when we multiply out the product (72). This
means that we need to find all possible $p_{i}$s and $p_{j}^{\prime}$’s such
that

$V_{1}\cdots
V_{m-1}V_{m}^{d-1}=\prod_{i\in\\{a_{1},\dots,a_{q}\\}}p_{i}(V_{1},\dots,V_{m-1})^{2^{i}}V_{m}^{2^{i}}\cdot\\\
\prod_{j\in\\{0,\dots,a-1\\}{\setminus}\\{a_{1},\dots,a_{q}\\}}p_{j}^{\prime}(V_{1},\dots,V_{m-1}).$

From the previous observation and the fact that
$0=a_{1}<a_{2}<\dots<a_{q}=a-1$ we conclude that
$p_{a_{2}}(V_{1},\dots,V_{m-1})=\dots=p_{a_{q}}(V_{1},\dots,V_{m-1})=1$. Thus,
the previous equality becomes

$V_{1}\cdots
V_{m-1}V_{m}^{d-1}=p_{a_{1}}(V_{1},\dots,V_{m-1})^{2^{a_{1}}}V_{m}^{d-1}\cdot\\\
\prod_{j\in\\{0,\dots,a-1\\}{\setminus}\\{a_{1},\dots,a_{q}\\}}p_{j}^{\prime}(V_{1},\dots,V_{m-1}).$

Taking additionally into account that $a_{1}=0$ we have that

$V_{1}\cdots V_{m-1}V_{m}^{d-1}=p_{0}(V_{1},\dots,V_{m-1})V_{m}^{d-1}\cdot\\\
\prod_{j\in\\{1,\dots,a-1\\}{\setminus}\\{a_{2},\dots,a_{q}\\}}p_{j}^{\prime}(V_{1},\dots,V_{m-1}).$

Therefore, the monomial $V_{1}\cdots V_{m-1}V_{m}^{d-1}$ can be obtained only
in the case when we choose

$\displaystyle p_{0}(V_{1},\dots,V_{m-1})$ $\displaystyle=V_{1}\cdots
V_{m-1},$ $\displaystyle p_{i}(V_{1},\dots,V_{m-1})$ $\displaystyle=1,$
$\displaystyle p_{j}^{\prime}(V_{1},\dots,V_{m-1})$ $\displaystyle=1,$

for all $i\in\\{a_{2},\dots,a_{q}\\}$ and all
$j\in\\{1,\dots,a-1\\}{\setminus}\\{a_{2},\dots,a_{q}\\}$.

Indeed, if we assume that
$p_{a_{1}}(V_{1},\dots,V_{m-1})=p_{0}(V_{1},\dots,V_{m-1})\neq 1$, then
obviously we must have $p_{0}(V_{1},\dots,V_{m-1})=V_{1}\cdots V_{m-1}$. This
is possible by taking monomial $V_{1}\dots V_{m-1}V_{m}$ in the $a_{0}$th
factor of the product (72). On the other hand, if we assume that
$p_{a_{1}}(V_{1},\dots,V_{m-1})=p_{0}(V_{1},\dots,V_{m-1})=1$ then we should
have that

$\prod_{j\in\\{1,\dots,a-1\\}{\setminus}\\{a_{2},\dots,a_{q}\\}}p_{j}^{\prime}(V_{1},\dots,V_{m-1})=V_{1}\cdots
V_{m-1}.$

This is not possible since for every $p_{j}^{\prime}(V_{1},\dots,V_{m-1})\neq
1$ in the previous product there is $1\leq t\leq m-1$ such that $V_{t}\mid
p_{j}^{\prime}(V_{1},\dots,V_{m-1})$. Looking closer at the typical monomials
in the product (72) we see that more is true, actually $V_{t}^{2^{j}}\mid
p_{j}^{\prime}(V_{1},\dots,V_{m-1})$ and so

$V_{t}^{2^{j}}\mid\prod_{j\in\\{1,\dots,a-1\\}{\setminus}\\{a_{2},\dots,a_{q}\\}}p_{j}^{\prime}(V_{1},\dots,V_{m-1})=V_{1}\cdots
V_{m-1}.$

Since $j\in\\{1,\dots,a-1\\}{\setminus}\\{a_{2},\dots,a_{q}\\}$ we have that
$2^{j}\geq 2$ and so $V_{t}^{2}\mid V_{1}\cdots V_{m-1}$; contradiction.

Hence, we showed that

$\pi(\rho_{d,2^{m}}^{*}(\overline{w}_{d2^{m-1}-1}))=V_{1}\cdots
V_{m-1}V_{m}^{d-1}+S\ \in\ \mathbb{F}_{2}[V_{1},\dots,V_{m}]/\langle
V_{1}^{d},\dots,V_{m}^{d}\rangle,$

where $S$ is a sum of monomials in $V_{1},\dots,V_{m}$ of degree $d2^{m-1}-1$
which are different from $V_{1}\cdots V_{m-1}V_{m}^{d-1}$. Thus,
$\overline{w}_{d2^{m-1}-1}\neq 0$. ∎

Now we extend the previous two lemmas to the case of an arbitrary integer
$k\geq 1$ and consequently correct [15, Lem. 2.17]. For an integer $k\geq 1$
let $\epsilon(k)$ be the reminder of $k$ modulo $2$, that is $\epsilon(k)=1$
for $k$ odd, and $\epsilon(k)=0$ when $k$ is even.

######  Lemma 5.12.

  1. (1)

Let $d\geq 3$ be an integer which is not a power of $2$, and let $k\geq 1$ be
an integer. Then the dual Stiefel–Whitney class

$\overline{w}_{(d-1)(k-\epsilon(k))/2}(\xi_{\mathbb{R}^{d},k})$

does not vanish.

  2. (2)

Let $d\geq 6$ be an even integer which is not a power of $2$, and let $k\geq
1$ be an integer. Then the dual Stiefel–Whitney class

$\overline{w}_{d(k-\epsilon(k))/2-\alpha(k)+\epsilon(k)}(\xi_{\mathbb{R}^{d},k})$

does not vanish.

###### Proof.

Let $r:=\alpha(k)$ and $k=2^{b_{1}}+\dots+2^{b_{r}}$ where $0\leq
b_{1}<b_{2}<\dots<b_{r}$. As in the proof of Lemma 5.6 we consider a morphism
between vector bundles $\prod_{i=1}^{r}\xi_{\mathbb{R}^{d},2^{b_{i}}}$ 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^{b_{i}}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\
}$$\scriptstyle{\Theta}$$\textstyle{\
\xi_{\mathbb{R}^{d},k}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\prod_{i=1}^{r}\operatorname{F}(\mathbb{R}^{d},2^{b_{i}})/\mathfrak{S}_{2^{b_{i}}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\
}$$\scriptstyle{\theta}$$\textstyle{\
\operatorname{F}(\mathbb{R}^{d},k)/{\mathfrak{S}_{k}}.}$

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

$(\mathcal{C}_{d}(2^{b_{1}})\times\dots\times\mathcal{C}_{d}(2^{b_{r}}))\times\mathcal{C}_{d}(r)\longrightarrow\mathcal{C}_{d}(2^{b_{1}}+\dots+2^{b_{r}}),$

as explained in the proof of Lemma 5.6. The naturality of the Stiefel–Whitney
classes [80, Ax. 2, p. 37] implies that

$\theta^{*}(\overline{w}_{N}(\xi_{\mathbb{R}^{d},k}))=\overline{w}_{N}\Big{(}\prod_{i=1}^{r}\xi_{\mathbb{R}^{d},2^{b_{i}}}\Big{)},$

for any integer $N\geq 0$. Further on, the product formula [80, Pr. 4-A, p.
54] implies that

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

Consequently,

$\displaystyle\theta^{*}(\overline{w}_{N}(\xi_{\mathbb{R}^{d},k}))$
$\displaystyle=\overline{w}_{N}\Big{(}\prod_{i=1}^{r}\xi_{\mathbb{R}^{d},2^{b_{i}}}\Big{)}$
$\displaystyle=\sum_{s_{1}+\cdots+s_{r}=N}\overline{w}_{s_{1}}(\xi_{\mathbb{R}^{d},2^{b_{1}}})\times\cdots\times\overline{w}_{s_{r}}(\xi_{\mathbb{R}^{d},2^{b_{r}}}).$
(73)

According to the Künneth formula [21, Thm. VI.3.2] each term, the cross
product,
$\overline{w}_{s_{1}}(\xi_{\mathbb{R}^{d},2^{b_{1}}})\times\cdots\times\overline{w}_{s_{r}}(\xi_{\mathbb{R}^{d},2^{b_{r}}})$
in the previous sum belongs to a different direct summand of the cohomology

$H^{N}\Big{(}\prod_{i=1}^{r}\operatorname{F}(\mathbb{R}^{d},2^{b_{i}})/\mathfrak{S}_{2^{b_{i}}};\mathbb{F}_{2}\Big{)}\cong\\\
\bigoplus_{s_{1}+\cdots+s_{r}=N}H^{s_{1}}(\operatorname{F}(\mathbb{R}^{d},2^{b_{1}})/\mathfrak{S}_{2^{b_{1}}};\mathbb{F}_{2})\otimes\cdots\otimes
H^{s_{r}}(\operatorname{F}(\mathbb{R}^{d},2^{b_{r}})/\mathfrak{S}_{2^{b_{r}}};\mathbb{F}_{2}).$

Hence the following equivalence holds

$\overline{w}_{N}\Big{(}\prod_{i=1}^{r}\xi_{\mathbb{R}^{d},2^{b_{i}}}\Big{)}\neq
0\Longleftrightarrow\\\
\overline{w}_{s_{1}}(\xi_{\mathbb{R}^{d},2^{b_{1}}})\times\cdots\times\overline{w}_{s_{r}}(\xi_{\mathbb{R}^{d},2^{b_{r}}})\neq
0\;\text{for some}\;s_{1}+\cdots+s_{r}=N.$

To isolate a non-zero summand in (5.1) we use either Lemma 5.7 or Lemma 5.11.

Let $d\geq 3$ be an integer which is not a power of $2$, and let
$N=(d-1)(k-\epsilon(k))/2$. The Lemma 5.7 states that the dual Stiefel–Whitney
class $\overline{w}_{(d-1)2^{b_{i}-1}}(\xi_{\mathbb{R}^{d},2^{b_{i}}})$ does
not vanish when $b_{i}\geq 1$. First consider the case when $k$ is even.
Hence, $\epsilon(k)=0$, and since $k=2^{b_{1}}+\dots+2^{b_{r}}$ we have that
$1\leq b_{1}<b_{2}<\dots<b_{r}$. Now, since
$N=(d-1)2^{b_{1}-1}+\dots+(d-1)2^{b_{r}-1}$ the following summand in (5.1)
does not vanish

$\overline{w}_{(d-1)2^{b_{1}-1}}(\xi_{\mathbb{R}^{d},2^{b_{1}}})\times\cdots\times\overline{w}_{(d-1)2^{b_{r}-1}}(\xi_{\mathbb{R}^{d},2^{b_{r}}})\neq
0.$

In the case when $\epsilon(k)=1$, $k=2^{b_{1}}+\dots+2^{b_{r}}$ and
$0=b_{1}<b_{2}<\dots<b_{r}$, the summand in (5.1) that does not vanish is

$\overline{w}_{0}(\xi_{\mathbb{R}^{d},2^{b_{1}}})\times\overline{w}_{(d-1)2^{b_{2}-1}}(\xi_{\mathbb{R}^{d},2^{b_{2}}})\times\cdots\times\overline{w}_{(d-1)2^{b_{r}-1}}(\xi_{\mathbb{R}^{d},2^{b_{r}}})\neq
0.$

Consequently,
$\theta^{*}(\overline{w}_{(d-1)(k-\epsilon(k))/2}(\xi_{\mathbb{R}^{d},k}))\neq
0$, and so $\overline{w}_{(d-1)(k-\epsilon(k))/2}(\xi_{\mathbb{R}^{d},k})\neq
0$.

Let now $d\geq 6$ be an even integer which is not a power of $2$, and let
$N=d(k-\epsilon(k))/2-\alpha(k)+\epsilon(k)$. The Lemma 5.11 states that the
dual Stiefel–Whitney class
$\overline{w}_{d2^{b_{i}-1}-1}(\xi_{\mathbb{R}^{d},2^{b_{i}}})$ does not
vanish when $b_{i}\geq 1$. Again we first consider the case when $k$ is even,
that is $\epsilon(k)=0$. Since $k=2^{b_{1}}+\dots+2^{b_{r}}$ we have that
$1\leq b_{1}<b_{2}<\dots<b_{r}$. Now, since
$N=(d2^{b_{1}-1}-1)+\dots+(d2^{b_{r}-1}-1)$ the following summand in (5.1)
does not vanish

$\overline{w}_{d2^{b_{1}-1}-1}(\xi_{\mathbb{R}^{d},2^{b_{1}}})\times\cdots\times\overline{w}_{d2^{b_{r}-1}-1}(\xi_{\mathbb{R}^{d},2^{b_{r}}})\neq
0.$

In the case of odd $k$ we have that $\epsilon(k)=1$,
$k=2^{b_{1}}+\dots+2^{b_{r}}$ and $0=b_{1}<b_{2}<\dots<b_{r}$. The summand in
(5.1) that does not vanish is

$\overline{w}_{0}(\xi_{\mathbb{R}^{d},2^{b_{1}}})\times\overline{w}_{d2^{b_{2}-1}-1}(\xi_{\mathbb{R}^{d},2^{b_{2}}})\times\cdots\times\overline{w}_{d2^{b_{r}-1}-1}(\xi_{\mathbb{R}^{d},2^{b_{r}}})\neq
0.$

Consequently,
$\theta^{*}(\overline{w}_{d(k-\epsilon(k))/2-\alpha(k)+\epsilon(k)}(\xi_{\mathbb{R}^{d},k}))\neq
0$, and so the dual Stiefel–Whiney class
$\overline{w}_{d(k-\epsilon(k))/2-\alpha(k)+\epsilon(k)}(\xi_{\mathbb{R}^{d},k})$
does not vanish. ∎

Summarizing the results obtained in Lemma 5.5, Lemma 5.6, Lemma 5.7, Lemma
5.11 and Lemma 5.12 we get the following theorem which is a correction of [15,
Thm. 2.13].

######  Theorem 5.13.

Let $k\geq 1$ and $d\geq 1$ be integers.

  1. (1)

If $d$ is a power of $2$, then the dual Stiefel–Whitney class

$\overline{w}_{(d-1)(k-\alpha(k))}(\xi_{\mathbb{R}^{d},k})$

does not vanish.

  2. (2)

If $d$ is not a power of $2$, then the dual Stiefel–Whitney class

$\overline{w}_{(d-1)(k-\epsilon(k))/2}(\xi_{\mathbb{R}^{d},k})$

does not vanish.

  3. (3)

If $d$ is an even integer which is not a power of $2$, then the dual
Stiefel–Whitney class

$\overline{w}_{d(k-\epsilon(k))/2-\alpha(k)+\epsilon(k)}(\xi_{\mathbb{R}^{d},k})$

does not vanish.

Now, we use Theorem 5.13 and the criterion given in Lemma 5.4 to correct the
result stated in [15, Thm. 2.1]. In this way we completed corrections of the
invalid results in [15, Sec. 2].

######  Theorem 5.14.

Let $k\geq 1$ and $d\geq 1$ be integers. Denote by $\alpha(k)$ the number of
$1$s in the dyadic presentation of $k$, and by $\epsilon(k)$ the remainder of
$k$ modulo $2$.

  1. (1)

If $d$ is a power of $2$, then there is no $k$-regular embedding
$\mathbb{R}^{d}\longrightarrow\mathbb{R}^{N}$ for

$N\leq d(k-\alpha(k))+\alpha(k)-1.$

  2. (2)

If $d$ is not a power of $2$, then there is no $k$-regular embedding
$\mathbb{R}^{d}\longrightarrow\mathbb{R}^{N}$ for

$N\leq\frac{1}{2}(d-1)(k-\epsilon(k))+k-1.$

  3. (3)

If $d$ is an even integer which is not a power of $2$, then is no $k$-regular
embedding $\mathbb{R}^{d}\longrightarrow\mathbb{R}^{N}$ for

$N\leq\frac{1}{2}d(k-\epsilon(k))+k-\alpha(k)+\epsilon(k)-1.$

###  5.2. $\ell$-skew embeddings

In this section we revise [15, Sec. 3] and correct the related results [15,
Thm. 3.1, Thm. 3.7]. In order for our presentation to be complete we recall
basic definitions and necessary facts.

The affine subspaces $L_{1},\ldots,L_{\ell}$ of the Euclidean space
$\mathbb{R}^{N}$ are affinely independent if

$\dim_{\operatorname{aff}}(L_{1}\cup\dots\cup
L_{\ell})=(\dim_{\operatorname{aff}}L_{1}+1)+\cdots+(\dim_{\operatorname{aff}}L_{\ell}+1)-1.$

In particular, any two lines in $\mathbb{R}^{3}$ are skew if and only if they
are affinely independent.

Let $M$ be a real smooth $d$-dimensional manifold. Then $TM$ denotes he
tangent bundle of $M$, and $T_{y}M$ stands for the tangent space of $M$ at the
point $y\in M$. For a smooth map $f\colon M\longrightarrow\mathbb{R}^{N}$ we
denote by $df\colon TM\longrightarrow T\mathbb{R}^{N}$ the differential map
between associated tangent vector bundles induced by $f$. Further on, let
$\iota\colon T\mathbb{R}^{N}\longrightarrow\mathbb{R}^{N}$ denotes the map
that sends a tangent vector $v\in T_{x}\mathbb{R}^{N}$ at the point
$x\in\mathbb{R}^{N}$ to the point $x+v$. Here the standard identification
$T_{x}\mathbb{R}^{N}=\mathbb{R}^{N}$ is assumed.

######  Definition 5.15.

Let $\ell\geq 1$ be an integer, and let $M$ be a real smooth $d$-dimensional
manifold. A smooth embedding $f\colon M\longrightarrow\mathbb{R}^{N}$ is an
$\ell$-skew embedding if for every point
$(y_{1},\ldots,y_{\ell})\in\operatorname{F}(M,\ell)$ the affine subspaces

$(\iota\circ df_{y_{1}})(T_{y_{1}}M),\ldots,(\iota\circ
df_{y_{\ell}})(T_{y_{\ell}}M)$

of $\mathbb{R}^{N}$ are affinely independent.

Now, like in the case of $k$-regular embeddings, a criterion for non-existence
of $\ell$-skew embedding can be derived in terms of Stiefel–Whitney class of
appropriate vector bundle over the configuration space. We recall [15, Lem.
3.6].

######  Lemma 5.16.

Let $d\geq 1$ and $\ell\geq 1$ be integers. If the dual Stiefel–Whitney class

$\overline{w}_{N-(d+1)\ell+2}(\xi_{\mathbb{R}^{d},\ell}^{\oplus(d+1)})$

does not vanish, then there is no $\ell$-skew embedding
$\mathbb{R}^{d}\longrightarrow\mathbb{R}^{N}$.

Motivated by the criterion in Lemma 5.16, based on the work of Hu’ng [64,
(4.7)], a relevant study of Stiefel–Whitney classes of the vector bundle
$\xi_{\mathbb{R}^{d},\ell}^{\oplus(d+1)}$ was given in [15, Thm. 3.7].

> Theorem 3.7. Let $d,\ell\geq 1$ be integers. Then the dual Stiefel–Whitney
> class
>
>
> $\overline{w}_{(2^{\gamma(d)}-d-1)(\ell-\alpha(\ell))}\bigl{(}\xi_{\mathbb{R}^{d},\ell}^{\oplus(d+1)}\bigr{)}$
>
> does not vanish.

Here $\gamma(d)=\lfloor\log_{2}d\rfloor+1$ for $d\geq 1$. The result of the
previous theorem in combination with [15, Lem. 3.6] directly implied the
following result [15, Thm. 3.1].

> Theorem 3.1. Let $\ell,d\geq 2$ be integers. There is no $\ell$-skew
> embedding $\mathbb{R}^{d}\longrightarrow\mathbb{R}^{N}$ for
>
> $N\leq 2^{\gamma(d)}(\ell-\alpha(\ell))+(d+1)\alpha(\ell)-2,$
>
> where $\alpha(\ell)$ denotes the number of ones in the dyadic presentation
> of $l$ and $\gamma(d)=\lfloor\log_{2}d\rfloor+1$.

The proof of [15, Thm. 3.7] was based on some of the results presented in [15,
Sec. 2]. In particular, in [15, Sec. 3.3.3] an incorrect result was used [15,
Cor. 2.16]. Consequently, [15, Thm. 3.1] does not stand. In the following we
give correct versions, first of [15, Thm. 3.7], and then of [15, Thm. 3.1].

######  Theorem 5.17.

Let $d\geq 1$ and $\ell\geq 2$ be integers.

  1. (1)

If $d=2$, and if $\ell\geq 2$ is an integer, then

$\overline{w}_{\ell-\alpha(\ell)}\bigl{(}\xi_{\mathbb{R}^{d},\ell}^{\oplus(d+1)}\bigr{)}=\overline{w}_{\ell-\alpha(\ell)}\bigl{(}\xi_{\mathbb{R}^{2},\ell}^{\oplus
3}\bigr{)}\neq 0.$

  2. (2)

If $d\geq 1$ is an integer, and if $\ell=2$, then

$\overline{w}_{2^{\gamma(d)}-d-1}\bigl{(}\xi_{\mathbb{R}^{d},\ell}^{\oplus(d+1)}\bigr{)}=\overline{w}_{2^{\gamma(d)}-d-1}\bigl{(}\xi_{\mathbb{R}^{d},2}^{\oplus(d+1)}\bigr{)}\neq
0.$

  3. (3)

If $d\geq 2$ is a power of $2$, and if $\ell\geq 2$ is an integer, then

$\overline{w}_{(d-1)(\ell-\alpha(\ell))}\bigl{(}\xi_{\mathbb{R}^{d},\ell}^{\oplus(d+1)}\bigr{)}\neq
0.$

  4. (4)

If $d+1\geq 2$ is a power of $2$, and if $\ell\geq 2$ is an integer, then

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

  5. (5)

If $d\geq 5$ is an integer which is not a power of $2$, and in addition $d+1$
is not a power of $2$, and if $\ell\geq 3$ is an integer, then

$\overline{w}_{(2^{\gamma(d)}-d-1)(\ell-\epsilon(\ell))/2}\bigl{(}\xi_{\mathbb{R}^{d},\ell}^{\oplus(d+1)}\bigr{)}\neq
0.$

  6. (6)

If $d\geq 5$ is an integer which is not a power of $2$,
$2^{\gamma(d)}-d-1=2^{a_{1}}z$ where $a_{1}\geq 0$ is an integer and $z\geq 1$
is an odd integer, and $d+1$ is not a power of $2$, and if $\ell\geq 3$ is an
integer, then

$\overline{w}_{(2^{\gamma(d)}-d-1+2^{a_{1}})(\ell-\epsilon(\ell))/2-2^{a_{1}}\alpha(\ell)}\bigl{(}\xi_{\mathbb{R}^{d},\ell}^{\oplus(d+1)}\bigr{)}\neq
0.$

###### Proof.

We prove the theorem by discussing all cases separately.

(1) Let $d=2$, and let $\ell\geq 2$ be an integer. Then from [38, Thm. 1] we
get that the bundle $\xi_{\mathbb{R}^{2},\ell}^{\oplus 2}$ is a trivial
bundle. Consequently, $\overline{w}\bigl{(}\xi_{\mathbb{R}^{2},\ell}^{\oplus
3}\bigr{)}=\overline{w}\bigl{(}\xi_{\mathbb{R}^{2},\ell}\bigr{)}$. Since $d=2$
is power of $2$ we can use Lemma 5.6 to get that

$\overline{w}_{\ell-\alpha(\ell)}\bigl{(}\xi_{\mathbb{R}^{d},\ell}^{\oplus(d+1)}\bigr{)}=\overline{w}_{\ell-\alpha(\ell)}\bigl{(}\xi_{\mathbb{R}^{2},\ell}^{\oplus
3}\bigr{)}=\overline{w}_{\ell-\alpha(\ell)}\bigl{(}\xi_{\mathbb{R}^{2},\ell}\bigr{)}\neq
0.$

(2) Let $d\geq 2$ be an integer and let $\ell=2$. In this case the base space
of the vector bundle $\xi_{\mathbb{R}^{2},\ell}$ is the unordered
configuration space $\operatorname{F}(\mathbb{R}^{d},2)/\mathfrak{S}_{2}$. The
$\mathfrak{S}_{2}$-equivariant map $\operatorname{ecy}_{d,2}\colon
S^{d-1}\longrightarrow\operatorname{F}(\mathbb{R}^{d},2)$ given by
$x\longmapsto(x,-x)$ is an $\mathfrak{S}_{2}$-equivariant homotopy
equivalence. Consequently, $\operatorname{ecy}_{d,2}$ induces homotopy
equivalence $\rho_{d,2}\colon
S^{d-1}/\mathfrak{S}_{2}\longrightarrow\operatorname{F}(\mathbb{R}^{d},2)/\mathfrak{S}_{2}$.
Recall that $S^{d-1}/\mathfrak{S}_{2}\cong\mathbb{R}{\rm P}^{d-1}$. It can be
directly checked that the vector bundle $\xi_{\mathbb{R}^{d},2}$ over
$\operatorname{F}(\mathbb{R}^{d},2)/\mathfrak{S}_{2}$ pulls back to the vector
bundle isomorphic to the Whitney sum of the tautological line bundle and
trivial line bundle over the projective space $\mathbb{R}{\rm P}^{d-1}$.
Hence, if we denote the cohomology of the projective space by

$H^{*}(\mathbb{R}{\rm P}^{d-1},\mathbb{F}_{2})\cong
H^{*}(\operatorname{F}(\mathbb{R}^{d},2)/\mathfrak{S}_{2},\mathbb{F}_{2})=\mathbb{F}_{2}[w_{1}]/\langle
w_{1}^{d}\rangle,$

where $\deg(w_{1})=1$, we have that

$w(\xi_{\mathbb{R}^{d},2})=1+w_{1}.$

Now, the total Stiefel–Whitney class of the vector bundle
$\xi_{\mathbb{R}^{d},\ell}^{\oplus(d+1)}$ is

$w\bigl{(}\xi_{\mathbb{R}^{d},\ell}^{\oplus(d+1)}\bigr{)}=w\bigl{(}\xi_{\mathbb{R}^{d},2}^{\oplus(d+1)}\bigr{)}=(1+w_{1})^{d+1}.$

Now notice that $2^{\gamma(d)}$ is the minimal power of $2$ that is greater
that $d$. Therefore,

$\displaystyle
w\bigl{(}\xi_{\mathbb{R}^{d},2}^{\oplus(d+1)}\bigr{)}(1+w_{1})^{2^{\gamma(d)}-d-1}$
$\displaystyle=(1+w_{1})^{d+1}(1+w_{1})^{2^{\gamma(d)}-d-1}$
$\displaystyle=(1+w_{1})^{2^{\gamma(d)}}$ $\displaystyle=1.$

Thus, we have that

$\overline{w}\bigl{(}\xi_{\mathbb{R}^{d},\ell}^{\oplus(d+1)}\bigr{)}=\overline{w}\bigl{(}\xi_{\mathbb{R}^{d},2}^{\oplus(d+1)}\bigr{)}=(1+w_{1})^{2^{\gamma(d)}-d-1}=1+\dots+w_{1}^{2^{\gamma(d)}-d-1}.$

Sine $2^{\gamma(d)}-d-1<d$ we have that $w_{1}^{2^{\gamma(d)}-d-1}\neq 0$, and
so
$\overline{w}_{2^{\gamma(d)}-d-1}\bigl{(}\xi_{\mathbb{R}^{d},2}^{\oplus(d+1)}\bigr{)}\neq
0$.

(3) Let $d=2^{a}$ for $a\geq 1$ an integer, and let $\ell\geq 2$ be an
integer. From the decomposition of vector bundles (66) we have that

$w(\xi_{\mathbb{R}^{d},\ell})=w(\zeta_{\mathbb{R}^{d},\ell})=1+w_{1}+\dots+w_{\ell-1},$

where
$w_{i}:=w_{i}(\xi_{\mathbb{R}^{d},\ell})=w_{i}(\zeta_{\mathbb{R}^{d},\ell})$.
Consequently,

$\displaystyle w\bigl{(}\xi_{\mathbb{R}^{d},\ell}^{\oplus(d+1)}\bigr{)}$
$\displaystyle=(1+w_{1}+\dots+w_{\ell-1})^{d+1}$
$\displaystyle=(1+w_{1}+\dots+w_{\ell-1})^{2^{a}}(1+w_{1}+\dots+w_{\ell-1})$
$\displaystyle=(1+w_{1}^{2^{a}}+\dots+w_{\ell-1}^{2^{a}})(1+w_{1}+\dots+w_{\ell-1})$
$\displaystyle=1+w_{1}+\dots+w_{\ell-1}$
$\displaystyle=w(\xi_{\mathbb{R}^{d},\ell}).$

Here we used that from (67) we know that

$\operatorname{height}((H^{*}(\operatorname{F}(\mathbb{R}^{d},\ell)/\mathfrak{S}_{\ell};\mathbb{F}_{2}))\leq\min\\{2^{t}:2^{t}\geq
d\\}=2^{a}=d.$

Thus, with the additional help of Lemma 5.6, we get that

$\overline{w}_{(d-1)(\ell-\alpha(\ell))}\bigl{(}\xi_{\mathbb{R}^{d},\ell}^{\oplus(d+1)}\bigr{)}=\overline{w}_{(d-1)(\ell-\alpha(\ell))}(\xi_{\mathbb{R}^{d},\ell})\neq
0.$

(4) Let $d=2^{a}-1$ for $a\geq 1$ an integer, and let $\ell\geq 2$ be an
integer. In the footsteps of the proof of the previous case we calculate:

$\displaystyle w\bigl{(}\xi_{\mathbb{R}^{d},\ell}^{\oplus(d+1)}\bigr{)}$
$\displaystyle=(1+w_{1}+\dots+w_{\ell-1})^{d+1}$
$\displaystyle=(1+w_{1}+\dots+w_{\ell-1})^{2^{a}}$
$\displaystyle=1+w_{1}^{2^{a}}+\dots+w_{\ell-1}^{2^{a}}$ $\displaystyle=1.$

In this case we used the fact that

$\operatorname{height}(H^{*}(\operatorname{F}(\mathbb{R}^{d},\ell)/\mathfrak{S}_{\ell};\mathbb{F}_{2}))\leq\min\\{2^{t}:2^{t}\geq
d\\}=2^{a}=d+1.$

Consequently,
$\overline{w}\bigl{(}\xi_{\mathbb{R}^{d},\ell}^{\oplus(d+1)}\bigr{)}=1$.

(5) This case is analyzed in two separate steps depending whether $\ell$ is a
power of $2$ or not.

(5A) Let $d\geq 5$ be an integer such that $2^{a-1}+1\leq d\leq 2^{a}-2$ where
$a\geq 3$ is an integer. We first consider the case when $\ell=2^{m}$ for
$m\geq 2$ an integer. The decomposition of vector bundles (66) implies that

$w(\xi_{\mathbb{R}^{d},2^{m}})=w(\zeta_{\mathbb{R}^{d},2^{m}})=1+w_{1}+\dots+w_{2^{m}-1},$

and consequently,

$w\bigl{(}\xi_{\mathbb{R}^{d},2^{m}}^{\oplus(d+1)}\bigr{)}=(1+w_{1}+\dots+w_{2^{m}-1})^{d+1}.$

From (67) we have that

$\operatorname{height}(H^{*}(\operatorname{F}(\mathbb{R}^{d},2^{m})/\mathfrak{S}_{2^{m}};\mathbb{F}_{2}))\leq\min\\{2^{t}:2^{t}\geq
d\\}=2^{a}.$

Therefore,

$\displaystyle
w\bigl{(}\xi_{\mathbb{R}^{d},2^{m}}^{\oplus(d+1)}\bigr{)}(1+w_{1}+\dots+w_{2^{m}-1})^{2^{a}-d-1}$
$\displaystyle=(1+w_{1}+\dots+w_{2^{m}-1})^{2^{a}}$
$\displaystyle=1+w_{1}^{2^{a}}+\dots+w_{2^{m}-1}^{2^{a}}$ $\displaystyle=1,$

and consequently,

$\overline{w}\bigl{(}\xi_{\mathbb{R}^{d},2^{m}}^{\oplus(d+1)}\bigr{)}=(1+w_{1}+\dots+w_{2^{m}-1})^{2^{a}-d-1}.$

Now, let $2^{a}-d-1=2^{a_{1}}+\dots+2^{a_{q}}$ where $0\leq
a_{1}<\dots<a_{q}\leq a-2$. Then

$\displaystyle\overline{w}\bigl{(}\xi_{\mathbb{R}^{d},2^{m}}^{\oplus(d+1)}\bigr{)}$
$\displaystyle=(1+w_{1}+\dots+w_{2^{m}-1})^{2^{a_{1}}+\dots+2^{a_{q}}}$
$\displaystyle=\prod_{i=1}^{q}\big{(}1+w_{1}^{2^{a_{i}}}+\dots+w_{2^{m}-1}^{2^{a_{i}}}\big{)}.$
(74)

Following the calculation in the proof of Lemma 5.7 we apply the monomorphism

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

from Theorem 4.1 to the equality (5.2), and use the decomposition of the
cohomology

$H^{*}(\operatorname{Pe}(\mathbb{R}^{d},2^{m})/\mathcal{S}_{2^{m}};\mathbb{F}_{2})\cong\mathbb{F}_{2}[V_{m,1},\dots,V_{m,m}]/\langle
V_{m,1}^{d},\ldots,V_{m,m}^{d}\rangle\oplus\mathrm{I}^{*}(\mathbb{R}^{d},2^{m}),$

given in Theorem 3.11, to get that

$\displaystyle\rho_{d,2^{m}}^{*}(\overline{w}\bigl{(}\xi_{\mathbb{R}^{d},2^{m}}^{\oplus(d+1)}\bigr{)})$
$\displaystyle=\prod_{i=1}^{q}\big{(}1+D_{m,m-1}^{2^{a_{i}}}+\dots+D_{m,0}^{2^{a_{i}}}\big{)}+R$
(75) $\displaystyle\ \in\
H^{*}(\operatorname{Pe}(\mathbb{R}^{d},2^{m})/\mathcal{S}_{2^{m}};\mathbb{F}_{2}),$

where

  * —

$D_{m,r}=(\kappa_{d,2^{m}}/\mathcal{S}_{2^{m}})^{*}(D_{m,r})=\rho_{d,2^{m}}^{*}(w_{2^{m}-2^{r}})$,
for $0\leq r\leq m-1$, with the obvious abuse of notation, see (40), and

  * —

$R\in\mathrm{I}^{*}(\mathbb{R}^{d},2^{m})$.

Let $\pi$ be the following composition of the maps from the diagram (68):

$\textstyle{\mathbb{F}_{2}[V_{m,1},\dots,V_{m,m}]/\langle
V_{m,1}^{d},\dots,V_{m,m}^{d}\rangle\oplus
I^{*}(\mathbb{R}^{d},2^{m})\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$projection$\textstyle{\mathbb{F}_{2}[V_{m,1},\dots,V_{m,m}]/\langle
V_{m,1}^{d},\dots,V_{m,m}^{d}\rangle\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\chi_{m}}$$\textstyle{\mathbb{F}_{2}[V_{1},\dots,V_{m}]/\langle
V_{1}^{d},\dots,V_{m}^{d}\rangle.}$

The first map is the projection on a direct summand and the second map is
induced by the change of variables
$\chi_{m}\in\operatorname{GL}_{m}(\mathbb{F}_{2})$. We apply $\pi$ to (75) and
have

$\pi\big{(}\rho_{d,2^{m}}^{*}\big{(}\overline{w}\bigl{(}\xi_{\mathbb{R}^{d},2^{m}}^{\oplus(d+1)}\bigr{)}\big{)}\big{)}=\prod_{i=1}^{q}\big{(}1+\pi(D_{m,m-1})^{2^{i}}+\dots+\pi(D_{m,0})^{2^{i}}\big{)}.$

The change of the variables $\chi_{m}$ transforms
$\mathrm{L}_{m}(\mathbb{F}_{2})$-invariants into
$\mathrm{U}_{m}(\mathbb{F}_{2})$-invariants and Dickson polynomials,
$\operatorname{GL}_{m}(\mathbb{F}_{2})$-invariants, can be presented in terms
of $\mathrm{U}_{m}(\mathbb{F}_{2})$-invariants, as explained in (7.2.1) and
(105). Hence,
$p:=\pi\big{(}\rho_{d,2^{m}}^{*}\big{(}\overline{w}\bigl{(}\xi_{\mathbb{R}^{d},2^{m}}^{\oplus(d+1)}\bigr{)}\big{)}\big{)}$
can be computed further as follows:

$\displaystyle\quad p$
$\displaystyle=\prod_{i=1}^{q}\big{(}1+\pi(D_{m,m-1})^{2^{a_{i}}}+\dots+\pi(D_{m,0})^{2^{a_{i}}}\big{)}$
$\displaystyle=\prod_{i=1}^{q}\Big{(}1+\big{(}V_{1}^{2^{m-1}}+V_{2}^{2^{m-2}}+\cdots+V_{m}^{2^{0}}\big{)}^{2^{a_{i}}}+\dots+$
$\displaystyle\quad\big{(}\sum_{1\leq j_{1}<\cdots<j_{r}\leq m}(V_{1}\cdots
V_{j_{1}-1})^{2^{r}}\,(V_{j_{1}+1}\cdots
V_{j_{2}-1})^{2^{r-1}}\cdots(V_{j_{r}+1}\cdots
V_{m})^{2^{0}}\big{)}^{2^{a_{i}}}$
$\displaystyle\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad+\dots+$
$\displaystyle\quad\big{(}V_{1}\cdots V_{m}\big{)}^{2^{a_{i}}}\Big{)}$
$\displaystyle=\prod_{i=1}^{q}\Big{(}1+\big{(}V_{1}^{2^{a_{i}+m-1}}+V_{2}^{2^{a_{i}+m-2}}+\cdots+V_{m}^{2^{a_{i}}}\big{)}+\dots+$
$\displaystyle\quad\big{(}\sum_{1\leq j_{1}<\cdots<j_{r}\leq m}(V_{1}\cdots
V_{j_{1}-1})^{2^{a_{i}+r}}\,(V_{j_{1}+1}\cdots
V_{j_{2}-1})^{2^{a_{i}+r-1}}\cdots(V_{j_{r}+1}\cdots
V_{m})^{2^{a_{i}}}\big{)}$
$\displaystyle\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad+\dots+$
$\displaystyle\quad\big{(}V_{1}\cdots V_{m}\big{)}^{2^{a_{i}}}\Big{)}.$

Since $2^{a}-d-1=2^{a_{1}}+\dots+2^{a_{q}}\leq d-1$ by choosing terms
$V_{m}^{2^{a_{i}+0}}$ from each factors in the product indexed by
$i=1,\dots,q$ we get that

$p=V_{m}^{2^{a}-d-1}+S\ \in\ \mathbb{F}_{2}[V_{1},\dots,V_{m}]/\langle
V_{1}^{d},\dots,V_{m}^{d}\rangle,$

where $S$ is a sum of some monomials in $V_{1},\dots,V_{m}$ of degree
$(2^{a}-d-1)2^{m-1}=(2^{a}-d-1)\ell/2$ which are all different from
$V_{m}^{d-1}$. Hence, $p\neq 0$ and consequently
$\overline{w}_{(2^{a}-d-1)\ell/2}\bigl{(}\xi_{\mathbb{R}^{d},2^{m}}^{\oplus(d+1)}\bigr{)}=\overline{w}_{(2^{a}-d-1)(\ell-\epsilon(\ell))/2}\bigl{(}\xi_{\mathbb{R}^{d},2^{m}}^{\oplus(d+1)}\bigr{)}\neq
0$.

(5B) Let $d\geq 5$ be an integer that is not a power of $2$, and furthermore
$d+1$ is not a power of $2$. Now we consider the case when $\ell\geq 3$ is not
a power of $2$. Set $r:=\alpha(\ell)\geq 2$ and
$\ell=2^{b_{1}}+\dots+2^{b_{r}}$ where $0\leq b_{1}<b_{2}<\dots<b_{r}$. As in
the proofs of Lemma 5.6 and Lemma 5.12 we consider a morphism between vector
bundles $\prod_{i=1}^{r}\xi_{\mathbb{R}^{d},2^{b_{i}}}$ and
$\xi_{\mathbb{R}^{d},\ell}$ where the following commutative square is a
pullback diagram:

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

That is
$\theta^{*}\xi_{\mathbb{R}^{d},\ell}\cong\prod_{i=1}^{r}\xi_{\mathbb{R}^{d},2^{b_{i}}}$.
The naturality of the Stiefel–Whitney classes [80, Ax. 2, p. 37] gives the
equality

$\theta^{*}(\overline{w}_{(2^{\gamma(d)}-d-1)(\ell-\epsilon(\ell))/2}(\xi_{\mathbb{R}^{d},\ell}^{\oplus(d+1)}))=\overline{w}_{(2^{\gamma(d)}-d-1)(\ell-\epsilon(\ell))/2}\Big{(}\prod_{i=1}^{r}\xi_{\mathbb{R}^{d},2^{b_{i}}}^{\oplus(d+1)}\Big{)}.$

The product formula [80, Pr. 4-A, p. 54] implies that

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

Thus,

$\displaystyle\theta^{*}(\overline{w}_{(2^{\gamma(d)}-d-1)(\ell-\epsilon(\ell))/2}(\xi_{\mathbb{R}^{d},\ell}^{\oplus(d+1)}))$
(76)
$\displaystyle\qquad\qquad=\overline{w}_{(2^{\gamma(d)}-d-1)(\ell-\epsilon(\ell))/2}\Big{(}\prod_{i=1}^{r}\xi_{\mathbb{R}^{d},2^{b_{i}}}^{\oplus(d+1)}\Big{)}$
$\displaystyle\qquad\qquad=\sum_{s_{1}+\cdots+s_{r}=(2^{\gamma(d)}-d-1)(\ell-\epsilon(\ell))/2}\overline{w}_{s_{1}}(\xi_{\mathbb{R}^{d},2^{b_{1}}}^{\oplus(d+1)})\times\cdots\times\overline{w}_{s_{r}}(\xi_{\mathbb{R}^{d},2^{b_{r}}}^{\oplus(d+1)}).$

The Künneth formula [21, Thm. VI.3.2] implies that each term

$\overline{w}_{s_{1}}(\xi_{\mathbb{R}^{d},2^{b_{1}}}^{\oplus(d+1)})\times\cdots\times\overline{w}_{s_{r}}(\xi_{\mathbb{R}^{d},2^{b_{r}}}^{\oplus(d+1)})$

in the previous sum belongs to a different direct summand of the cohomology

$H^{(2^{\gamma(d)}-d-1)(\ell-\epsilon(\ell))/2}\Big{(}\prod_{i=1}^{r}\operatorname{F}(\mathbb{R}^{d},2^{b_{i}})/\mathfrak{S}_{2^{b_{i}}};\mathbb{F}_{2}\Big{)}\cong\\\
\bigoplus_{s_{1}+\cdots+s_{r}=(2^{\gamma(d)}-d-1)(k-\epsilon(k))/2}H^{s_{1}}(\operatorname{F}(\mathbb{R}^{d},2^{b_{1}})/\mathfrak{S}_{2^{b_{1}}};\mathbb{F}_{2})\otimes\cdots\otimes
H^{s_{r}}(\operatorname{F}(\mathbb{R}^{d},2^{b_{r}})/\mathfrak{S}_{2^{b_{r}}};\mathbb{F}_{2}).$

Therefore, the following equivalence holds

$\overline{w}_{(2^{\gamma(d)}-d-1)(\ell-\epsilon(\ell))/2}\Big{(}\prod_{i=1}^{r}\xi_{\mathbb{R}^{d},2^{b_{i}}}^{\oplus(d+1)}\Big{)}\neq
0\quad\Longleftrightarrow\\\
\overline{w}_{s_{1}}(\xi_{\mathbb{R}^{d},2^{b_{1}}}^{\oplus(d+1)})\times\cdots\times\overline{w}_{s_{r}}(\xi_{\mathbb{R}^{d},2^{b_{r}}}^{\oplus(d+1)})\neq
0\;\\\ \text{for
some}\;s_{1}+\cdots+s_{r}=(2^{\gamma(d)}-d-1)(\ell-\epsilon(\ell))/2.$

To isolate a non-zero summand in (76) we use the previous case of this theorem
which states that
$\overline{w}_{(2^{\gamma(d)}-d-1)2^{b_{i}-1}}(\xi_{\mathbb{R}^{d},2^{b_{i}}}^{\oplus(d+1)})\neq
0$ when $b_{i}\geq 1$.

We discuss two separate cases. Let us assume that $\ell$ be even, or in other
words $\epsilon(\ell)=0$. Since $\ell=2^{b_{1}}+\dots+2^{b_{r}}$ it follows
that $1\leq b_{1}<b_{2}<\dots<b_{r}$. Hence, the following summand in (76)
does not vanish

$\overline{w}_{(2^{\gamma(d)}-d-1)2^{b_{1}-1}}(\xi_{\mathbb{R}^{d},2^{b_{1}}}^{\oplus(d+1)})\times\cdots\times\overline{w}_{(2^{\gamma(d)}-d-1)2^{b_{r}-1}}(\xi_{\mathbb{R}^{d},2^{b_{r}}}^{\oplus(d+1)})\neq
0.$

When $\epsilon(\ell)=1$ we have that $0=b_{1}<b_{2}<\dots<b_{r}$, and the
summand in (76) which does not vanish is

$\overline{w}_{0}(\xi_{\mathbb{R}^{d},2^{b_{1}}}^{\oplus(d+1)})\times\overline{w}_{(2^{\gamma(d)}-d-1)2^{b_{2}-1}}(\xi_{\mathbb{R}^{d},2^{b_{2}}}^{\oplus(d+1)})\times\cdots\times\overline{w}_{(2^{\gamma(d)}-d-1)2^{b_{r}-1}}(\xi_{\mathbb{R}^{d},2^{b_{r}}}^{\oplus(d+1)})\neq
0.$

In summary,
$\theta^{*}(\overline{w}_{(2^{\gamma(d)}-d-1)(\ell-\epsilon(\ell))/2}(\xi_{\mathbb{R}^{d},\ell}^{\oplus(d+1)}))\neq
0$ and therefore

$\overline{w}_{(2^{\gamma(d)}-d-1)(\ell-\epsilon(\ell))/2}(\xi_{\mathbb{R}^{d},\ell}^{\oplus(d+1)})\neq
0.$

(6) In this case we follow the footsteps of the proof of the previous claim.
For completeness reasons we discuss all steps of the proof. Again we
distinguish case when $\ell$ is a power of $2$ from the case when $\ell$ is
not a power of $2$

(6A) Let $d\geq 5$ be an integer such that $2^{a-1}+1\leq d\leq 2^{a}-2$ where
$\gamma(d)=a\geq 3$ is an integer. Take $\ell=2^{m}$ for $m\geq 2$ an integer.

Using the decomposition of vector bundles (66) we get that

$w\bigl{(}\xi_{\mathbb{R}^{d},2^{m}}^{\oplus(d+1)}\bigr{)}=w\bigl{(}\zeta_{\mathbb{R}^{d},2^{m}}^{\oplus(d+1)}\bigr{)}=(1+w_{1}+\dots+w_{2^{m}-1})^{d+1}.$

From (67) the height of the algebra
$H^{*}(\operatorname{F}(\mathbb{R}^{d},2^{m})/\mathfrak{S}_{2^{m}};\mathbb{F}_{2})$
is known:

$\operatorname{height}(H^{*}(\operatorname{F}(\mathbb{R}^{d},2^{m})/\mathfrak{S}_{2^{m}};\mathbb{F}_{2}))\leq\min\\{2^{t}:2^{t}\geq
d\\}=2^{a}.$

Therefore,

$\overline{w}\bigl{(}\xi_{\mathbb{R}^{d},2^{m}}^{\oplus(d+1)}\bigr{)}=(1+w_{1}+\dots+w_{2^{m}-1})^{2^{a}-d-1}.$

Let $2^{a}-d-1=2^{a_{1}}+\dots+2^{a_{q}}$ for $0\leq a_{1}<\dots<a_{q}\leq
a-2$. Then

$\displaystyle\overline{w}\bigl{(}\xi_{\mathbb{R}^{d},2^{m}}^{\oplus(d+1)}\bigr{)}$
$\displaystyle=(1+w_{1}+\dots+w_{2^{m}-1})^{2^{a_{1}}+\dots+2^{a_{q}}}$
$\displaystyle=\prod_{i=1}^{q}\big{(}1+w_{1}^{2^{a_{i}}}+\dots+w_{2^{m}-1}^{2^{a_{i}}}\big{)}.$
(77)

We apply the monomorphism

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

from Theorem 4.1 to the equality (5.2). Using the decomposition of the
cohomology

$H^{*}(\operatorname{Pe}(\mathbb{R}^{d},2^{m})/\mathcal{S}_{2^{m}};\mathbb{F}_{2})\cong\mathbb{F}_{2}[V_{m,1},\dots,V_{m,m}]/\langle
V_{m,1}^{d},\ldots,V_{m,m}^{d}\rangle\oplus\mathrm{I}^{*}(\mathbb{R}^{d},2^{m}),$

given in Theorem 3.11, we have

$\displaystyle\rho_{d,2^{m}}^{*}(\overline{w}\bigl{(}\xi_{\mathbb{R}^{d},2^{m}}^{\oplus(d+1)}\bigr{)})$
$\displaystyle=\prod_{i=1}^{q}\big{(}1+D_{m,m-1}^{2^{a_{i}}}+\dots+D_{m,0}^{2^{a_{i}}}\big{)}+R$
(78)

where

  * —

$D_{m,r}=(\kappa_{d,2^{m}}/\mathcal{S}_{2^{m}})^{*}(D_{m,r})=\rho_{d,2^{m}}^{*}(w_{2^{m}-2^{r}})$,
for $0\leq r\leq m-1$, with the obvious abuse of notation, see (40), and

  * —

$R\in\mathrm{I}^{*}(\mathbb{R}^{d},2^{m})$.

Furthermore, let $\pi$ denote the following composition of the maps from the
diagram (68):

$\textstyle{\mathbb{F}_{2}[V_{m,1},\dots,V_{m,m}]/\langle
V_{m,1}^{d},\dots,V_{m,m}^{d}\rangle\oplus
I^{*}(\mathbb{R}^{d},2^{m})\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$projection$\textstyle{\mathbb{F}_{2}[V_{m,1},\dots,V_{m,m}]/\langle
V_{m,1}^{d},\dots,V_{m,m}^{d}\rangle\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\chi_{m}}$$\textstyle{\mathbb{F}_{2}[V_{1},\dots,V_{m}]/\langle
V_{1}^{d},\dots,V_{m}^{d}\rangle,}$

The first map is the projection on a direct summand and the second map is
induced by the change of variables
$\chi_{m}\in\operatorname{GL}_{m}(\mathbb{F}_{2})$. Applying $\pi$ to (78) we
get

$\pi\big{(}\rho_{d,2^{m}}^{*}\big{(}\overline{w}\bigl{(}\xi_{\mathbb{R}^{d},2^{m}}^{\oplus(d+1)}\bigr{)}\big{)}\big{)}=\prod_{i=1}^{q}\big{(}1+\pi(D_{m,m-1})^{2^{i}}+\dots+\pi(D_{m,0})^{2^{i}}\big{)}.$

Recall that the change of the variables $\chi_{m}$ transforms
$\mathrm{L}_{m}(\mathbb{F}_{2})$-invariants into
$\mathrm{U}_{m}(\mathbb{F}_{2})$-invariants and Dickson polynomials,
$\operatorname{GL}_{m}(\mathbb{F}_{2})$-invariants, can be presented in terms
of $\mathrm{U}_{m}(\mathbb{F}_{2})$-invariants, as explained in (7.2.1) and
(105). Now, the element
$p:=\pi\big{(}\rho_{d,2^{m}}^{*}\big{(}\overline{w}\bigl{(}\xi_{\mathbb{R}^{d},2^{m}}^{\oplus(d+1)}\bigr{)}\big{)}\big{)}$
can be expressed as follows:

$\displaystyle\quad p$
$\displaystyle=\prod_{i=1}^{q}\big{(}1+\pi(D_{m,m-1})^{2^{a_{i}}}+\dots+\pi(D_{m,0})^{2^{a_{i}}}\big{)}$
$\displaystyle=\prod_{i=1}^{q}\Big{(}1+\big{(}V_{1}^{2^{m-1}}+V_{2}^{2^{m-2}}+\cdots+V_{m}^{2^{0}}\big{)}^{2^{a_{i}}}+\dots+$
$\displaystyle\quad\big{(}\sum_{1\leq j_{1}<\cdots<j_{r}\leq m}(V_{1}\cdots
V_{j_{1}-1})^{2^{r}}\,(V_{j_{1}+1}\cdots
V_{j_{2}-1})^{2^{r-1}}\cdots(V_{j_{r}+1}\cdots
V_{m})^{2^{0}}\big{)}^{2^{a_{i}}}$
$\displaystyle\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad+\dots+$
$\displaystyle\quad\big{(}V_{1}\cdots V_{m}\big{)}^{2^{a_{i}}}\Big{)}$
$\displaystyle=\prod_{i=1}^{q}\Big{(}1+\big{(}V_{1}^{2^{a_{i}+m-1}}+V_{2}^{2^{a_{i}+m-2}}+\cdots+V_{m}^{2^{a_{i}}}\big{)}+\dots+$
(79) $\displaystyle\quad\big{(}\sum_{1\leq j_{1}<\cdots<j_{r}\leq
m}(V_{1}\cdots V_{j_{1}-1})^{2^{a_{i}+r}}\,(V_{j_{1}+1}\cdots
V_{j_{2}-1})^{2^{a_{i}+r-1}}\cdots(V_{j_{r}+1}\cdots
V_{m})^{2^{a_{i}}}\big{)}$
$\displaystyle\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad+\dots+$
$\displaystyle\quad\big{(}V_{1}\cdots V_{m}\big{)}^{2^{a_{i}}}\Big{)}.$

Now we want to show that

$\pi\big{(}\rho_{d,2^{m}}^{*}\big{(}\overline{w}_{(2^{\gamma(d)}-d-1+2^{a_{1}})2^{m-1}-2^{a_{1}}}\bigl{(}\xi_{\mathbb{R}^{d},2^{m}}^{\oplus(d+1)}\bigr{)}\big{)}\big{)}=\\\
(V_{1}\cdots V_{m-1})^{2^{a_{1}}}V_{m}^{2^{\gamma(d)}-d-1}+S\ \in\
\mathbb{F}_{2}[V_{1},\dots,V_{m}]/\langle V_{1}^{d},\dots,V_{m}^{d}\rangle.$

Here $S$ is a sum of monomials in $V_{1},\dots,V_{m}$ of degree
$(2^{\gamma(d)}-d-1+2^{a_{1}})2^{m-1}-2^{a_{1}}$ which are different from the
monomial $(V_{1}\cdots V_{m-1})^{2^{a_{1}}}V_{m}^{2^{\gamma(d)}-d-1}+S$.
Hence,
$\pi\big{(}\rho_{d,2^{m}}^{*}\big{(}\overline{w}_{(2^{\gamma(d)}-d-1+2^{a_{1}})2^{m-1}-2^{a_{1}}}\bigl{(}\xi_{\mathbb{R}^{d},2^{m}}^{\oplus(d+1)}\bigr{)}\big{)}\big{)}\neq
0$, and consequently

$\overline{w}_{(2^{\gamma(d)}-d-1+2^{a_{1}})2^{m-1}-2^{a_{1}}}\bigl{(}\xi_{\mathbb{R}^{d},2^{m}}^{\oplus(d+1)}\bigr{)}\neq
0.$

Indeed, observe that in every monomial of the $i$th factor of the product
(5.2) which has the variable $V_{m}$, with a positive exponent, the variable
$V_{m}$ has always the same exponent equal to $2^{a_{i}}$, $1\leq i\leq q$. In
particular, in the $i$th factor each monomial with the variable $V_{m}$ is of
the form $p_{i}(V_{1},\dots,V_{m-1})^{2^{a_{i}}}V_{m}^{2^{a_{i}}}$ where
$p_{i}(V_{1},\dots,V_{m-1})$ is a monomial in variables $V_{1},\dots,V_{m-1}$.
Now, when multiplying out the product (5.2), since
$2^{\gamma(d)}-d-1=2^{a_{1}}+\dots+2^{a_{q}}$, the monomial of the form
$p(V_{1},\dots,V_{m-1})V_{m}^{2^{\gamma(d)}-d-1}$ can appear in the final
result if and only if we take from each factor a non-zero monomial of the form
$p_{i}(V_{1},\dots,V_{m-1})^{2^{a_{i}}}V_{m}^{2^{a_{i}}}$. Thus, we have

$p(V_{1},\dots,V_{m-1})V_{m}^{2^{\gamma(d)}-d-1}=\prod_{i=1}^{q}p_{i}(V_{1},\dots,V_{m-1})^{2^{a_{i}}}V_{m}^{2^{a_{i}}}$

Observe that, if $p_{i}(V_{1},\dots,V_{m-1})\neq 1$ for some $1\leq i\leq q$,
then there exists $1\leq t\leq q$ such that $V_{t}\mid
p_{i}(V_{1},\dots,V_{m-1})$. Hence, $V_{t}^{2^{a_{i}}}\mid
p_{i}(V_{1},\dots,V_{m-1})^{2^{a_{i}}}$.

Now we want to count in how many ways we can obtain the monomial

$(V_{1}\cdots V_{m-1})^{2^{a_{1}}}V_{m}^{2^{\gamma(d)}-d-1}$

when we multiply out the product (5.2). This means that we need to find all
possible $p_{i}$s and $p_{j}^{\prime}$’s such that

$(V_{1}\cdots
V_{m-1})^{2^{a_{1}}}V_{m}^{2^{\gamma(d)}-d-1}=\prod_{i=1}^{q}p_{i}(V_{1},\dots,V_{m-1})^{2^{a_{i}}}V_{m}^{2^{a_{i}}}$

From the previous observation and the fact that $0\leq
a_{1}<a_{2}<\dots<a_{q}\leq a-2$ we conclude that
$p_{a_{2}}(V_{1},\dots,V_{m-1})=\dots=p_{a_{q}}(V_{1},\dots,V_{m-1})=1$. Thus,
the previous equality becomes

$(V_{1}\cdots
V_{m-1})^{2^{a_{1}}}V_{m}^{2^{\gamma(d)}-d-1}=p_{1}(V_{1},\dots,V_{m-1})^{2^{a_{i}}}V_{m}^{2^{\gamma(d)}-d-1}.$

Therefore, the monomial $(V_{1}\cdots
V_{m-1})^{2^{a_{1}}}V_{m}^{2^{\gamma(d)}-d-1}$ can be obtained for

$p_{a_{1}}(V_{1},\dots,V_{m-1})=V_{1}\cdots V_{m-1},$

and

$p_{a_{2}}(V_{1},\dots,V_{m-1})=\dots=p_{a_{q}}(V_{1},\dots,V_{m-1})=1.$

Hence, we completed the proof of the non-vanishing of the dual class:

$\overline{w}_{(2^{\gamma(d)}-d-1+2^{a_{1}})2^{m-1}-2^{a_{1}}}\bigl{(}\xi_{\mathbb{R}^{d},2^{m}}^{\oplus(d+1)}\bigr{)}\neq
0.$

(6B) Let $d\geq 5$ be an integer that is not a power of $2$, and furthermore
$d+1$ is not a power of $2$. Consider the case when $\ell\geq 3$ is not a
power of $2$. Set $r:=\alpha(\ell)\geq 2$ and $\ell=2^{b_{1}}+\dots+2^{b_{r}}$
where $0\leq b_{1}<b_{2}<\dots<b_{r}$. As many times before we consider a
morphism between vector bundles
$\prod_{i=1}^{r}\xi_{\mathbb{R}^{d},2^{b_{i}}}$ and
$\xi_{\mathbb{R}^{d},\ell}$ where the following commutative square is a
pullback diagram:

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

In particular,
$\theta^{*}\xi_{\mathbb{R}^{d},\ell}\cong\prod_{i=1}^{r}\xi_{\mathbb{R}^{d},2^{b_{i}}}$.
The naturality of the Stiefel–Whitney classes [80, Ax. 2, p. 37] gives the
equality

$\theta^{*}(\overline{w}(\xi_{\mathbb{R}^{d},\ell}^{\oplus(d+1)}))=\overline{w}\Big{(}\prod_{i=1}^{r}\xi_{\mathbb{R}^{d},2^{b_{i}}}^{\oplus(d+1)}\Big{)},$

while the product formula [80, Pr. 4-A, p. 54] implies that

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

Thus, for every integer $N\geq 0$

$\displaystyle\theta^{*}(\overline{w}_{N}(\xi_{\mathbb{R}^{d},\ell}^{\oplus(d+1)}))$
$\displaystyle=\overline{w}_{N}\Big{(}\prod_{i=1}^{r}\xi_{\mathbb{R}^{d},2^{b_{i}}}^{\oplus(d+1)}\Big{)}$
$\displaystyle=\sum_{s_{1}+\cdots+s_{r}=N}\overline{w}_{s_{1}}(\xi_{\mathbb{R}^{d},2^{b_{1}}}^{\oplus(d+1)})\times\cdots\times\overline{w}_{s_{r}}(\xi_{\mathbb{R}^{d},2^{b_{r}}}^{\oplus(d+1)}).$
(80)

The Künneth formula [21, Thm. VI.3.2] implies that each term

$\overline{w}_{s_{1}}(\xi_{\mathbb{R}^{d},2^{b_{1}}}^{\oplus(d+1)})\times\cdots\times\overline{w}_{s_{r}}(\xi_{\mathbb{R}^{d},2^{b_{r}}}^{\oplus(d+1)})$

in the previous sum belongs to a different direct summand of the cohomology

$H^{N}\big{(}\prod_{i=1}^{r}\operatorname{F}(\mathbb{R}^{d},2^{b_{i}})/\mathfrak{S}_{2^{b_{i}}};\mathbb{F}_{2}\big{)}\cong\\\
\bigoplus_{s_{1}+\cdots+s_{r}=N}H^{s_{1}}(\operatorname{F}(\mathbb{R}^{d},2^{b_{1}})/\mathfrak{S}_{2^{b_{1}}};\mathbb{F}_{2})\otimes\cdots\otimes
H^{s_{r}}(\operatorname{F}(\mathbb{R}^{d},2^{b_{r}})/\mathfrak{S}_{2^{b_{r}}};\mathbb{F}_{2}).$

Therefore, the following equivalence holds

$\overline{w}_{N}\Big{(}\prod_{i=1}^{r}\xi_{\mathbb{R}^{d},2^{b_{i}}}^{\oplus(d+1)}\Big{)}\neq
0\Longleftrightarrow\overline{w}_{s_{1}}(\xi_{\mathbb{R}^{d},2^{b_{1}}}^{\oplus(d+1)})\times\cdots\times\overline{w}_{s_{r}}(\xi_{\mathbb{R}^{d},2^{b_{r}}}^{\oplus(d+1)})\neq
0\;\\\ \text{for some}\;s_{1}+\cdots+s_{r}=N.$

To isolate a non-zero summand in (5.2) for

$N=(2^{\gamma(d)}-d-1+2^{a_{1}})(\ell-\epsilon(\ell))/2-2^{a_{1}}\alpha(\ell)$

we use the previous case of this theorem which states that for $b_{i}\geq 1$:

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

We discuss two separate cases. Let $\ell$ be even, or $\epsilon(\ell)=0$.
Since $\ell=2^{b_{1}}+\dots+2^{b_{r}}$ it follows that $1\leq
b_{1}<b_{2}<\dots<b_{r}$. Thus, the following summand in (5.2) does not vanish

$\overline{w}_{(2^{\gamma(d)}-d-1+2^{a_{1}})2^{b_{1}-1}-2^{a_{1}}}(\xi_{\mathbb{R}^{d},2^{b_{1}}}^{\oplus(d+1)})\times\cdots\times\overline{w}_{(2^{\gamma(d)}-d-1+2^{a_{1}})2^{b_{r}-1}-2^{a_{1}}}(\xi_{\mathbb{R}^{d},2^{b_{r}}}^{\oplus(d+1)})\neq
0.$

When $\epsilon(\ell)=1$ we have that $0=b_{1}<b_{2}<\dots<b_{r}$, and the
summand in (5.2) which does not vanish is

$\overline{w}_{0}(\xi_{\mathbb{R}^{d},2^{b_{1}}}^{\oplus(d+1)})\times\overline{w}_{(2^{\gamma(d)}-d-1+2^{a_{1}})2^{b_{2}-1}-2^{a_{1}}}(\xi_{\mathbb{R}^{d},2^{b_{1}}}^{\oplus(d+1)})\times\cdots\times\\\
\overline{w}_{(2^{\gamma(d)}-d-1+2^{a_{1}})2^{b_{r}-1}-2^{a_{1}}}(\xi_{\mathbb{R}^{d},2^{b_{r}}}^{\oplus(d+1)})\neq
0.$

In summary,

$\overline{w}_{(2^{\gamma(d)}-d-1+2^{a_{1}})(\ell-\epsilon(\ell))/2-2^{a_{1}}\alpha(\ell)}\bigl{(}\xi_{\mathbb{R}^{d},\ell}^{\oplus(d+1)}\bigr{)}\neq
0.$

∎

Now, we use Theorem 5.17 and the criterion from Lemma 5.16 to correct the
result stated in [15, Thm. 3.1]. In this way we completed corrections of the
invalid claims in [15, Sec. 3].

######  Theorem 5.18.

Let $\ell\geq 1$ and $d\geq 2$ be integers.

  1. (1)

If $d=2$, and if $\ell\geq 2$ is an integer, then there is no $\ell$-skew
embedding $\mathbb{R}^{2}\longrightarrow\mathbb{R}^{N}$ for

$N\leq 4\ell-\alpha(\ell)-2.$

  2. (2)

If $d\geq 1$ is an integer, and if $\ell=2$, then there is no $2$-skew
embedding $\mathbb{R}^{d}\longrightarrow\mathbb{R}^{N}$ for

$N\leq 2^{\gamma(d)}+d-1.$

  3. (3)

If $d\geq 2$ is a power of $2$, and if $\ell\geq 2$ is an integer, then there
is no $\ell$-skew embedding $\mathbb{R}^{d}\longrightarrow\mathbb{R}^{N}$ for

$N\leq 2d\ell-(d-1)\alpha(\ell)-2.$

  4. (4)

If $d+1\geq 2$ is a power of $2$, and if $\ell\geq 2$ is an integer, then this
method does not produce any non-trivial result about the existence of
$\ell$-skew embeddings $\mathbb{R}^{d}\longrightarrow\mathbb{R}^{N}$.

  5. (5)

If $d\geq 5$ is an integer which is not a power of $2$, and in addition $d+1$
is not a power $2$, and if $\ell\geq 3$ is an integer, then there is no
$\ell$-skew embedding $\mathbb{R}^{d}\longrightarrow\mathbb{R}^{N}$ for

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

  6. (6)

If $d\geq 5$ is an integer which is not a power of $2$,
$2^{\gamma(d)}-d-1=2^{a_{1}}z$ where $a_{1}\geq 0$ is an integer and $z\geq 1$
is an odd integer, and $d+1$ is not a power of $2$, and if $\ell\geq 3$ is an
integer, then there is no $\ell$-skew embedding
$\mathbb{R}^{d}\longrightarrow\mathbb{R}^{N}$ for

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

###  5.3. $k$-regular-$\ell$-skew embeddings

In this section we revise [15, Sec. 4] and correct the related results [15,
Thm. 4.1, Thm. 4.8]. First we recall some basic notions on
$k$-regular-$\ell$-skew embeddings.

######  Definition 5.19.

Let $k\geq 1$ and $\ell\geq 1$ be an integer, and let $M$ be a real smooth
$d$-dimensional manifold. A smooth embedding $f\colon
M\longrightarrow\mathbb{R}^{N}$ is $k$-regular-$\ell$-skew embedding if for
every $(x_{1},\ldots,x_{k},y_{1},\ldots,y_{\ell})$ in
$\operatorname{F}(M,k+\ell)$ the affine subspaces

$\\{f(x_{1})\\},\ldots,\\{f(x_{k})\\},(\iota\circ
df_{y_{1}})(T_{y_{1}}M),\ldots,(\iota\circ df_{y_{\ell}})(T_{y_{\ell}}M)$

of $\mathbb{R}^{N}$ are affinely independent.

Now, like in the case of $k$-regular embeddings and $\ell$-skew embeddings a
criterion for non-existence of $k$-regular-$\ell$-skew embedding can be
derived in terms of Stiefel–Whitney class of appropriate vector bundle over
the relevant configuration space. We state a consequence of [15, Lem. 4.6] and
[15, Lem. 4.7] used for the proof of [15, Thm. 4.1].

######  Lemma 5.20.

Let $d\geq 1$, $k\geq 1$ and $\ell\geq 1$ be integers. If the dual
Stiefel–Whitney class

$\overline{w}_{N-(d+1)\ell-k+2}(\xi_{\mathbb{R}^{d},k}\times\xi_{\mathbb{R}^{d},\ell}^{\oplus(d+1)})$

does not vanish, then there is no $k$-regular-$\ell$-skew embedding
$\mathbb{R}^{d}\longrightarrow\mathbb{R}^{N}$.

Like in the case of $k$-regular embeddings and $\ell$-skew embeddings, now
based on invalid results in [15, Thm. 2.13] and [15, Thm. 3.7] the following
theorem was proved [15, Thm. 4.8]:

> Theorem 4.8. Let $\ell,k,d\geq 2$ be integers. The dual Stiefel–Whitney
> class
>
>
> $\overline{w}_{(d-1)(k-\alpha(k))+(2^{\gamma(d)}-d-1)(\ell-\alpha(\ell))}(\xi_{\mathbb{R}^{d},k}\times\xi_{\mathbb{R}^{d},\ell}^{\oplus(d+1)})$
>
> does not vanish.

The result of the previous theorem in combination with [15, Lem. 4.6] and [15,
Lem. 4.4] directly implied the following result [15, Thm. 4.1].

> Theorem 4.1. et $\ell,d\geq 2$ be integers. There is no
> $k$-regular-$\ell$-skew embedding
> $\mathbb{R}^{d}\longrightarrow\mathbb{R}^{N}$ for
>
>
> $N\leq(d-1)(k-\alpha(k))+(2^{\gamma(d)}-d-1)(\ell-\alpha(\ell))+(d+1)\ell+k-2,$
>
> where $\alpha(c)$ denotes the number of ones in the dyadic presentation of
> $c$, and $\gamma(d):=\lfloor\log_{2}d\rfloor+1$.

The proof of [15, Thm. 4.8] was based on incorrect results [15, Cor. 2.13] and
[15, Cor. 3.7]. Since we corrected these results we can now give correct
versions, first of [15, Thm. 4.8], and then of [15, Thm. 4.1].

######  Theorem 5.21.

Let $d\geq 1$, $k\geq 1$, and $\ell\geq 1$ be integers.

  1. (1)

If $d=2$, and if $k\geq 1$ and $\ell\geq 1$ are integers, then

$\overline{w}_{k-\alpha(k)+\ell-\alpha(\ell)}\big{(}\xi_{\mathbb{R}^{d},k}\times\xi_{\mathbb{R}^{d},\ell}^{\oplus(d+1)}\big{)}\neq
0.$

  2. (2)

If $d\geq 2$ is a power of $2$, $\ell=2$, and if $k\geq 1$ is an integer, then

$\overline{w}_{(d-1)(k-\alpha(k)+1)}\big{(}\xi_{\mathbb{R}^{d},k}\times\xi_{\mathbb{R}^{d},\ell}^{\oplus(d+1)}\big{)}\neq
0.$

  3. (3)

If $d\geq 3$ is not a power of $2$, $\ell=2$, and if $k\geq 1$ is an integer,
then

$\overline{w}_{(d-1)(k-\epsilon(k))/2+2^{\gamma(d)}-d-1}\big{(}\xi_{\mathbb{R}^{d},k}\times\xi_{\mathbb{R}^{d},\ell}^{\oplus(d+1)}\big{)}\neq
0.$

  4. (4)

If $d\geq 2$ is a power of $2$, and if $k\geq 1$ and $\ell\geq 1$ are
integers, then

$\overline{w}_{(d-1)(k-\alpha(k)+\ell-\alpha(\ell))}\big{(}\xi_{\mathbb{R}^{d},k}\times\xi_{\mathbb{R}^{d},\ell}^{\oplus(d+1)}\big{)}\neq
0.$

  5. (5)

If $d+1\geq 2$ is a power of $2$, and if $k\geq 1$ and $\ell\geq 1$ are
integers, then

$\overline{w}_{(d-1)(k-\epsilon(k))/2}\big{(}\xi_{\mathbb{R}^{d},k}\times\xi_{\mathbb{R}^{d},\ell}^{\oplus(d+1)}\big{)}\neq
0.$

  6. (6)

If $d\geq 5$ is an integer which is not a power of $2$, and in addition $d+1$
is a not power of $2$, and if $k\geq 1$ and $\ell\geq 1$ are integers, then

$\overline{w}_{(d-1)(k-\epsilon(k))/2+(2^{\gamma(d)}-d-1)(\ell-\epsilon(\ell))/2}\big{(}\xi_{\mathbb{R}^{d},k}\times\xi_{\mathbb{R}^{d},\ell}^{\oplus(d+1)}\big{)}\neq
0.$

  7. (7)

If $d\geq 6$ be an even integer which is not a power of $2$,
$2^{\gamma(d)}-d-1=2^{a_{1}}z$ where $a_{1}\geq 0$ is an integer and $z\geq 1$
is an odd integer, and if $k\geq 1$ and $\ell\geq 3$ are integers, then

$\overline{w}_{d(k-\epsilon(k))/2-\alpha(k)+\epsilon(k)+(2^{\gamma(d)}-d-1+2^{a_{1}})(\ell-\epsilon(\ell))/2-2^{a_{1}}\alpha(\ell)}\big{(}\xi_{\mathbb{R}^{d},k}\times\xi_{\mathbb{R}^{d},\ell}^{\oplus(d+1)}\big{)}\neq
0.$

###### Proof.

In order 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 use the product formula [80, Problem 4-A, page 54]:

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

where “$\times$” on the right hand side denotes the cross product in
cohomology. In particular, for a fixed integer $r\geq 0$ we have that

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

The Künneth formula [21, Thm. VI.3.2] implies 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, using Theorem 5.13 and Theorem 5.17 to prove all cases of the theorem.

(1) Let $d=2$, and let $k\geq 1$ and $\ell\geq 1$ be integers. Then

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

and consequently
$\overline{w}_{k-\alpha(k)+\ell-\alpha(\ell)}\big{(}\xi_{\mathbb{R}^{d},k}\times\xi_{\mathbb{R}^{d},\ell}^{\oplus(d+1)}\big{)}\neq
0$.

(2) Let $d\geq 2$ be a power of $2$, $\ell=2$, and let $k\geq 1$ be an
integer. Then

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

and consequently
$\overline{w}_{(d-1)(k-\alpha(k)+1)}\big{(}\xi_{\mathbb{R}^{d},k}\times\xi_{\mathbb{R}^{d},\ell}^{\oplus(d+1)}\big{)}\neq
0$.

(3) Let $d\geq 2$ be not a power of $2$, $\ell=2$, and let $k\geq 1$ be an
integer. Then

$\overline{w}_{(d-1)(k-\epsilon(k))/2}(\xi_{\mathbb{R}^{d},k})\neq
0\qquad\text{and}\qquad\overline{w}_{2^{\gamma(d)}-d-1}\big{(}\xi_{\mathbb{R}^{d},\ell}^{\oplus(d+1)}\big{)}\neq
0,$

and consequently
$\overline{w}_{(d-1)(k-\epsilon(k))/2+2^{\gamma(d)}-d-1}\big{(}\xi_{\mathbb{R}^{d},k}\times\xi_{\mathbb{R}^{d},\ell}^{\oplus(d+1)}\big{)}\neq
0$.

(4) Let $d\geq 2$ be not a power of $2$, and let $k\geq 1$ and $\ell\geq 1$ be
integers. Then

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

and consequently
$\overline{w}_{(d-1)(k-\alpha(k)+\ell-\alpha(\ell))}\big{(}\xi_{\mathbb{R}^{d},k}\times\xi_{\mathbb{R}^{d},\ell}^{\oplus(d+1)}\big{)}\neq
0$.

(5) Let $d+1\geq 2$ be a power of $2$, and let $k\geq 1$ and $\ell\geq 1$ be
integers. Then

$\overline{w}_{(d-1)(k-\epsilon(k))/2}(\xi_{\mathbb{R}^{d},k})\neq
0\qquad\text{and}\qquad\overline{w}_{0}\big{(}\xi_{\mathbb{R}^{d},\ell}^{\oplus(d+1)}\big{)}\neq
0,$

and consequently
$\overline{w}_{(d-1)(k-\epsilon(k))/2}\big{(}\xi_{\mathbb{R}^{d},k}\times\xi_{\mathbb{R}^{d},\ell}^{\oplus(d+1)}\big{)}\neq
0$

(6) Let $d\geq 5$ be an integer which is not a power of $2$, and in addition
$d+1$ is a not power of $2$, and let $k\geq 1$ and $\ell\geq 1$ be integers.
Then

$\overline{w}_{(d-1)(k-\epsilon(k))/2}(\xi_{\mathbb{R}^{d},k})\neq
0\qquad\text{and}\qquad\overline{w}_{(2^{\gamma(d)}-d-1)(\ell-\epsilon(\ell))/2}\big{(}\xi_{\mathbb{R}^{d},\ell}^{\oplus(d+1)}\big{)}\neq
0,$

and consequently
$\overline{w}_{(d-1)(k-\epsilon(k))/2+(2^{\gamma(d)}-d-1)(\ell-\epsilon(\ell))/2}\big{(}\xi_{\mathbb{R}^{d},k}\times\xi_{\mathbb{R}^{d},\ell}^{\oplus(d+1)}\big{)}\neq
0$.

(7) Let $d\geq 6$ be an even integer which is not a power of $2$,
$2^{\gamma(d)}-d-1=2^{a_{1}}z$ where $a_{1}\geq 0$ is an integer and $z\geq 1$
is an odd integer, and let $k\geq 1$ and $\ell\geq 3$ be integers. Then

$\overline{w}_{d(k-\epsilon(k))/2-\alpha(k)+\epsilon(k)}(\xi_{\mathbb{R}^{d},k})\neq
0\qquad\text{and}\qquad\overline{w}_{d(k-\epsilon(k))/2-\alpha(k)+\epsilon(k)}(\xi_{\mathbb{R}^{d},k})\neq
0,$

and consequently

$\overline{w}_{d(k-\epsilon(k))/2-\alpha(k)+\epsilon(k)+(2^{\gamma(d)}-d-1+2^{a_{1}})(\ell-\epsilon(\ell))/2-2^{a_{1}}\alpha(\ell)}\big{(}\xi_{\mathbb{R}^{d},k}\times\xi_{\mathbb{R}^{d},\ell}^{\oplus(d+1)}\big{)}\neq
0.$

∎

Like in the previous situation, we use Theorem 5.21 and the criterion from
Lemma 5.20 to correct the result stated in [15, Thm. 4.1]. In this way we
completed corrections of the invalid results in [15, Sec. 4].

######  Theorem 5.22.

Let $d\geq 1$, $k\geq 1$, and $\ell\geq 1$ be integers.

  1. (1)

If $d=2$, and if $k\geq 1$ and $\ell\geq 1$ are integers, then there is no
$k$-regular-$\ell$-skew embedding
$\mathbb{R}^{2}\longrightarrow\mathbb{R}^{N}$ for

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

  2. (2)

If $d\geq 2$ is a power of $2$, $\ell=2$, and if $k\geq 1$ is an integer, then
there is no $k$-regular-$2$-skew embedding
$\mathbb{R}^{d}\longrightarrow\mathbb{R}^{N}$ for

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

  3. (3)

If $d\geq 2$ is not a power of $2$, $\ell=2$, and if $k\geq 1$ is an integer,
then there is no $k$-regular-$2$-skew embedding
$\mathbb{R}^{d}\longrightarrow\mathbb{R}^{N}$ for

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

  4. (4)

If $d\geq 2$ is not a power of $2$, and if $k\geq 1$ and $\ell\geq 1$ are
integers, then there is no $k$-regular-$\ell$-skew embedding
$\mathbb{R}^{2}\longrightarrow\mathbb{R}^{N}$ for

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

  5. (5)

If $d+1\geq 2$ is a power of $2$, and if $k\geq 1$ and $\ell\geq 1$ are
integers, then there is no $k$-regular-$\ell$-skew embedding
$\mathbb{R}^{d}\longrightarrow\mathbb{R}^{N}$ for

$N\leq(d+1)\ell+k-2+\frac{1}{2}(d-1)(k-\epsilon(k)).$

  6. (6)

If $d\geq 5$ is an integer which is not a power of $2$, and in addition $d+1$
is a not power of $2$, and if $k\geq 1$ and $\ell\geq 1$ are integers, then
there is no $k$-regular-$\ell$-skew embedding
$\mathbb{R}^{d}\longrightarrow\mathbb{R}^{N}$ for

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

  7. (7)

If $d\geq 6$ be an even integer which is not a power of $2$,
$2^{\gamma(d)}-d-1=2^{a_{1}}z$ where $a_{1}\geq 0$ is an integer and $z\geq 1$
is an odd integer, and if $k\geq 1$ and $\ell\geq 3$ are integers, then there
is no $k$-regular-$\ell$-skew embedding
$\mathbb{R}^{d}\longrightarrow\mathbb{R}^{N}$ for

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

This concludes all corrections of the paper [15].

###  5.4. Complex highly regular embeddings

A gap in the decomposition [64, (4.7)] created incorrectness that we have
already discussed in the study of real highly regular embeddings [15], which
in turn implied two results on complex highly regular embeddings [14, Thm.
5.1, Thm. 6.1] that now also need to be corrected. For completeness we recall
some basic notions about complex highly regular embeddings.

First we introduce a notion of complex $k$-regular embedding in a similar way
as in the case of real $k$-regular embedding, see Section 5.1.

######  Definition 5.23.

Let $k\geq 1$ be an integer, and let $X$ be a topological space. A continuous
map $f\colon X\longrightarrow\mathbb{C}^{N}$ is a complex $k$-regular
embedding if for every $(x_{1},\ldots,x_{k})\in\operatorname{F}(X,k)$ the
vectors $f(x_{1}),\ldots,f(x_{n})$ of the complex vector space
$\mathbb{C}^{N}$ are linearly independent.

Next we introduce a notion of complex $\ell$-regular embedding. For a real
analogue consult Section 5.2. A collection of complex affine subspaces
$\\{L_{1},\ldots,L_{\ell}\\}$ of the complex vector space $\mathbb{C}^{N}$ is
affinely independent if the following equality holds

$\dim_{\operatorname{aff}}^{\mathbb{C}}\operatorname{span}(L_{1}\cup\dots\cup
L_{\ell})=(\dim_{\operatorname{aff}}^{\mathbb{C}}L_{1}+1)+\cdots+(\dim_{\operatorname{aff}}^{\mathbb{C}}L_{\ell}+1)-1.$

Let $M$ be a complex $d$-dimensional manifold. The associated complex tangent
bundle of $M$ is denote by $TM$. For a point $y\in M$ we denote by $T_{y}M$
the corresponding tangent space to $M$. If $f\colon
M\longrightarrow\mathbb{C}^{N}$ is a smooth complex map, then $df\colon
TM\longrightarrow T\mathbb{C}^{N}$ denotes the complex differential map
between tangent complex vector bundles induced by $f$. Furthermore, let
$\iota\colon T\mathbb{C}^{N}\longrightarrow\mathbb{C}^{N}$ be the map which
sends a tangent vector $v\in T_{x}\mathbb{C}^{N}$ at a point
$x\in\mathbb{C}^{N}$ to the sum $x+v$, where the standard identification
$T_{x}\mathbb{C}^{N}=\mathbb{C}^{N}$ is assumed.

######  Definition 5.24.

Let $\ell\geq 1$ be an integer, and let $M$ be a smooth complex
$d$-dimensional manifold. A smooth complex embedding $f\colon
M\longrightarrow\mathbb{C}^{N}$ is an complex $\ell$-skew embedding if for
every $(y_{1},\ldots,y_{\ell})\in\operatorname{F}(M,\ell)$ the collection of
complex affine subspaces $\\{(\iota\circ
df_{y_{1}})(T_{y_{1}}M),\ldots,(\iota\circ df_{y_{\ell}})(T_{y_{\ell}}M)\\}$
of $\mathbb{C}^{N}$ is affinely independent.

In the following we will work with complex vector bundles that are analogues
of real vector bundles introduced in (65), see also [14, Sec. 4]. Hence,
consider the complex vector bundles:

$\textstyle{\xi_{X,k}^{\mathbb{C}}\colon}$$\textstyle{\mathbb{C}^{k}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\operatorname{F}(X,k)\times_{\mathfrak{S}_{k}}\mathbb{C}^{k}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\operatorname{F}(X,k)/\mathfrak{S}_{k},}$$\textstyle{\zeta_{X,k}^{\mathbb{C}}\colon}$$\textstyle{W_{k}^{C}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\operatorname{F}(X,k)\times_{\mathfrak{S}_{k}}W_{k}^{\mathbb{C}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\operatorname{F}(X,k)/\mathfrak{S}_{k},}$$\textstyle{\tau_{X,k}^{\mathbb{C}}\colon}$$\textstyle{\mathbb{C}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\operatorname{F}(X,k)/\mathfrak{S}_{k}\times\mathbb{C}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\operatorname{F}(X,k)/\mathfrak{S}_{k},}$
(81)

where
$W_{k}^{\mathbb{C}}=\\{(b_{1},\dots,b_{k})\in\mathbb{C}^{k}:b_{1}+\dots+b_{k}=0\\}$
is an $\mathfrak{S}_{k}$-invariant subspace of $\mathbb{C}^{k}$. It is obvious
that on the level of underlying real vector bundles the following bundle
isomorphisms hold:

$\xi_{X,k}^{\mathbb{C}}\cong\xi_{X,k}^{\oplus
2},\qquad\zeta_{X,k}^{\mathbb{C}}\cong\zeta_{X,k}^{\oplus
2},\qquad\tau_{X,k}^{\mathbb{C}}\cong\tau_{X,k}^{\oplus 2}.$

Like in the real case a criterion for an existence of complex $k$-regular
embeddings and of complex $\ell$-skew embeddings can be phrased in terms of
the vector bundle $\xi_{X,k}^{\mathbb{C}}$. Let us recall the relevant special
cases of [14, Lem. 5.7] and [14, Lem. 6.6].

######  Lemma 5.25.

Let $k\geq 1$, $\ell\geq 1$, $d\geq 1$ and $N\geq 1$ be integers.

  1. (1)

If there exists a complex $k$-regular embedding
$\mathbb{R}^{d}\longrightarrow\mathbb{C}^{N}$, then the complex vector bundle
$\xi_{\mathbb{R}^{d},k}^{\mathbb{C}}$ admits an $(N-k)$-dimensional complex
inverse.

  2. (2)

If there exists a complex $\ell$-skew embedding
$\mathbb{C}^{d}\longrightarrow\mathbb{C}^{N}$, then the complex vector bundle
$(\xi_{\mathbb{C}^{d},\ell}^{\mathbb{C}})^{\oplus(d+1)}$ admits an