In 2010, M. Sakai and the first author showed that the topological complexity of a space $X$ coincides with the fibrewise unpointed L-S category of a pointed fibrewise space $\proj_{1} : X \times X \to X$ with the diagonal map $\Delta : X \to X \times X$ as its section.
In this paper, we describe our algorithm how to determine the fibrewise L-S category or the Topological Complexity of a topological spherical space form. Especially, for $S^3/Q_8$ where $Q_8$ is the quaternion group, we write a python code to realise the algorithm to determine its Topological Complexity.

§ INTORODUCTION

Topological complexity was introduced in <cit.> by Michael Farber as a numerical homotopy invariant.
It attracts many authors including people working on similar homotopy invariant of Lusternik-Schnirelmann category, L-S category, for short.
Recently, many authors started to use a `normalized' or a `reduced' version of it including Farber himself.
In this paper, we use the symbol `$\tc{}$' for the reduced version of it to distinguish from the original denoted by $\TC{}$.

The reduced version of topological complexity is defined as follows:
let $X$ be a path-connected space, $\path{X}=\map{[0,1],X}$ and $\varpi : \path{X} \to M \times M$ the projection given by $\varpi(u) = (u(0),u(1))$.
Using a projection $p_{t} : \path{X} \to X$ defined by $p_{t}(u)=u(t)$, $t \in \{0,1\}$, we may write $\varpi=p_{0} \times p_{1}$.
Topological complexity of $X$, denoted by $\tc{X}$, is the least integer $n$ such that there is an open covering $U_0, \dots, U_n$ of $M \times M$ on each of which $\varpi$ admits a section.
We remark that, if a subset $A \subset X$ is contractible in $X$, there is a section of $\varpi$ on $A$.

The definition reminds us a well-known homotopy invariant: L-S category of a space $X$, denoted by $\cat{X}$, is the least integer $n$ such that there is an open covering $U_0, \dots,U_n$ of $X$ each of which is contractible in $X$.

We can verify the following relation among these invariants, which is due to Farber <cit.>.

\[ \cat{X} \le \tc{X} \le \cat{X \times X} \le 2\cat{X}.\]

More practically for an abelian group $R$, Farber introduced the zero-divisors ideal $I_{\varpi}(X;R)=\ker \Delta^{*} : H^{*}(X \times X;R) \to H^{*}(X;R)$ and the zero-divisors cup-length $\cupl[\varpi]{X;R}$ for a space $X$ <cit.>, and then the TC-weight $\wgt[\varpi]{z;R}$ for $z \in I_{\varpi}(X;R)$ with Mark Grant <cit.>:

\cupl[\varpi]{X;R} \le \Max\{\wgt[\varpi]{z;R} \mid z \in I_{\varpi}(X;R) \smallsetminus \{0\}\} \le \tc{X}.

In this paper, we adopt fibrewise method and skip the precise definition of the above notions.

Since the theory of topological complexity is growing and spreading rapidly, there are many open problems.
Among such problems, we are especially interested in the topological complexity of spherical space forms, and in particular, real projective spaces, aiming to give a natural and computational way to determine topological complexity, since Farber showed that the topological complexity of a real projective space coincides with its immersion dimension.

In this paper, we focus on the work of Kenso Fujii on the $K$-theory of a spherical space forms obtained as the orbit space of the unit sphere of $\quaternion^{t}$, where $t$ is a non-negative integer and $\quaternion$ is the set of all quaternionic numbers, by the diagonal action of the subgroup $Q_8$ of $\mathrm{Sp}(1)$, represented as $Q_{8}=\langle a, b \midvert a^{4} \!= b^4 \!= aba\bar{b} = 1, \,b^2 \!= a^{2} \rangle$, where $\bar{g}$ stands for $g^{-1}$ for any element $g \!\in\! Q_{8}$.

Our results are stated in <ref> and the proofs are in <ref>-<ref>, but the actual calculation is done with computer.
Because of the size of the computer resources required, we gave up to use the usual method for topological complexity but the fibrewise L-S method by Iwase-Sakai <cit.> and Iwase-Sakai-Tsutaya <cit.>.
It results that we don't need to calculate the bar resolution of $G \times G$ but $G$ fibrewise.
This significantly reduces the computer resources required, while the explicit answer to our equation is still too long to print out.
So we include the algorithm and python code in Appendix, instead.
So we include the algorithm of the program in Appendix, instead.
We hope our method can be applied to more general cases.

We work in the convenient category $\category{NG}$, introduced by Shimakawa-Yoshida-Haraguchi <cit.>, and every chain groups are assumed to be $\field_{2}$-modules in this paper, unless otherwise stated.

§ RESULTS

In this section, $R$ is assumed to be an $\field_{2}$-module.
Let us recall the James fibrewise theory.

Fibrewise spaces and maps are defined as follows.

* A fibrewise space is a tuple $(E,p,X)$ consisting of spaces $E$ and $X$ with a map $p : E \to X$ called a projection.
A fibrewise map from a fibrewise space $(F,q,Y)$ to a fibrewise space $(E,p,X)$ is a pair $(\phi,f)$ of maps $\phi : F \to E$ and $f : Y \to X$ satisfying $p \comp \phi = f \comp q$ as $(\phi,f) : (F,q,Y) \to (E,p,X)$.
$(E,p,X)$ is often denoted simply by $E$, and $(\phi,f)$ by $\phi$.

* A fibrewise pointed space is a pair $(E,s)$ of a fibrewise space $E=(E,p,X)$ and a section $s : X \to E$ of $p$, i.e, $p \comp s = \id_X$.

A fibrewise pointed map from a fibrewise pointed space $(F,r)$ to a fibrewise pointed space $(E,s)$ is a fibrewise map $(\phi,f) : (F,r) \to (E,s)$ satisfying $\phi \comp r = s \comp f$.
$(E,s)$ is often denoted simply by $E$, and $(\phi,f)$ by $\phi$.

For instance, we have a Borel construction $EG \times_{G} X$ over $EG/G$ as a fibrewise space for a topological group $G$ and a $G$-space $X$, where we denote by $EG$ some contractible free $G$-space.
Further, for an adjoint action of $G$ on itself, we obtain the Borel construction denoted by $EG\times_{\ad{}}G$ as a fibrewise pointed space over $EG/G$, which is, in fact, a fibrewise group.

James has also introduced a fibrewise version of an ordinary cohomology as a direct summand of the oridinary cohomology of the total space.
From now on, we use a subscript $B$ to indicate that the notion is `fibrewise notion' over some base space, even if the base space is not the same as $B$. Similarly, we use a superscript $B$ to indicate that the notion is `fibrewise pointed'.

For a fibrewise pointed space $(E,p,X,s)$, the base space $X$ is a retract of $E$, and hence $H^{*}(E,s(X);R) \cong \ker\,s^{*} : H^{*}(E;R) \to H^{*}(X;R)$ can be regarded as a direct summad of $H^{*}(E;R)$.

For a fibrewise pointed space $E=(E,p,X,s)$, $H_{B}^{\ast}(E;R)=H^{*}(E,s(X);R)$ is called its fibrewise pointed cohomology.
Then a fibrewise map $\phi : (F,q,Y) \to (E,p,X)$ induces a homomorphism $\phi^{*} : H_{B}^{\ast}(E;R) \subset H^{\ast}(E;R) \to H^{\ast}(F;R)$, while a fibrewise pointed map $\phi : (F,q,Y,r) \to (E,p,X,s)$ induces a homomorphism $\phi^{*} : H_{B}^{\ast}(E;R) \to H_{B}^{\ast}(F;R) \subset H^{\ast}(F;R)$.

For any map $f : Y \to X$, we have a fibrewise pointed space $E_{f}=(Y \times X,\ifdefined\reviseone{\proj_{1}}\else{\proj_{2}}\fi,Y,(\id_{X} {\times} f){\comp}\Delta)$, where $\proj_{t} : X_{1} \times X_{2} \to X_{t}$ denotes the canonical projection to the $t$-th factor, $t \!=\! 1, \,2$, and $\Delta : X \to X \times X$ denotes the diagonal.

Based on the ideas introduced by James <cit.>, Iwase-Sakai <cit.> introduced a fibrewise version of (unpointed) L-S category as follows.

Let $E=(E,p,X,s)$ be a fibrewise pointed space and $\phi=(\phi,f) : F=(F,q,Y) \to E=(E,p,X)$ be a fibrewise map.
Then the fibrewise (unpointed) L-S category of $\phi$, denoted by $\cat[B]{\phi}$, is the least integer $k \ge 0$ such that there is a cover of $F$ by $(k{+}1)$ open subsets $\{U_i\}$ on each of which $\phi|U_{i}$ is fibrewise homotopic to $s \comp f \!\comp q|_{U_i}$. Then we define a fibrewise (unpointed) L-S category of $E$ by $\catB{E}=\cat[B]{\id_{E}}$.

If we stay in the fibrewise pointed category, we obtain James original fibrewise L-S category $\catBB{E}$ of a fiberwise well-pointed space $E$, using fibrewise pointed spaces and maps.

Firstly, for an extensive use of homotopy theory, we alter the definition of L-S category, following George W. Whitehead by replacing an open cover $\{U_{i}\}$ of a space $X$ with a closed cofibration $F_{i} \hookrightarrow X$  covering $X$.
Then we say $\cat{X} \le t$ if the $t{+}1$-fold diagonal $\Delta^{t+1} : X \to \overset{t+1}\product X$ is compressible into the fat wedge $\overset{t+1}\fatvee X$, where $\overset{t+1}\fatvee X$ is defined by induction on $k \!\ge\! 1$ as follows:

\begin{align*}&
(\overset{1}{\product}X,\overset{1}{\fatvee}X) = (X,\ast)%,
\quad\text{and}\quad%\\&
(\overset{k+1}{\product}X,\overset{k+1}{\fatvee}X) = (\overset{k}{\product}X \times X,\overset{k}{\fatvee}X \times X \cup \overset{k}{\product}X \times\ast).
\end{align*}

Secondly, we also alter the definition of fibrewise L-S category by replacing an open cover $\{U_{i}\}$ of $E$ with a closed cofibration $F_{i} \hookrightarrow E$ covering $E$ for a fibrewise pointed space $E=(E,p,X,s)$.
Then we say $\catB{E} \le t$ if the $t{+}1$-fold fibrewise diagonal $\Delta_{B}^{t+1} : E \to \overset{t+1}{\product_{B}} E$ is compressible into the fibrewise fat wedge $\overset{t+1}{\fatvee_{\!B}} E$, where $\overset{t+1}{\fatvee_{\!B}} E$ is defined by induction on $k \!\ge\! 1$ as follows:

\begin{align*}&
(\overset{1}{\product_{B}}E,\overset{1}{\fatvee_{\!B}}E) = (E,s(X))%,
\quad\text{and}\quad%\\&
(\overset{k+1}{\product_{B}}E,\overset{k+1}{\fatvee_{\!B}}E) = (\overset{k}{\product_{B}}E \!\times_{B}\! E, \overset{k}{\fatvee_{\!B}}E \!\times_{B}\! E \cup \overset{k}{\product_{B}}E \!\times_{B}\! s(X)).
\end{align*}

If we consider a monoidal motion planning, we must choose a fibrewise homotopy $h_{i}$ to keep the diagonal part fixed and thus, we obtain a fibrewise pointed map from $E$ to $\overset{t+1}{\fatvee_{\!B}} E \subset \overset{t+1}{\product_{B}} E$ as a fibrewise pointed compression of the fibrewise diagonal $\Delta_{B}^{t+1} : E \to \overset{t+1}{\product_{B}} E$.

For a fibrewise pointed space $(E,p,X,s)$, we have the fibrewise pointed loop space $\LoopB{E}=(\LoopB{E},{p},X,{s})$ as follows:

\begin{align*}
\LoopB{E} &
= \{(b,\ell) \!\in\! X \times \path{E} \mid p \comp \ell \!=\! c(b), \ \ell(0) \!=\! \ell(1) \!=\! s (b) \},
\\&
{p}=\proj_{1}|\LoopB{E} : \LoopB{E} \subset X \times \path{E} \xrightarrow{\proj_{1}} X,\qquad
\\&
{s}(b)=(b,c \comp s(b)),\quad
c(b)=\text{(the constant path at $b$)},
\end{align*}

together with an $A_{\infty}$-structure for $\LoopB{E}$ defined as follows:

* $E_{B}^{t+1}(\LoopB{E})$ as the fibrewise homotopy pull-back in $\TopBB$ of
$X \hookrightarrow \overset{t+1}{\product_{B}}E \hookleftarrow \overset{t+1}{\fatvee_{\!B}}E$.

* $P_{\!B}^{t}(\LoopB{E}) = (P_{\!B}^{t}(\LoopB{E}),p_{t+1},X,s_{t+1})$ is the fibrewise homotopy pull-back in $\TopBB$ of $E \xrightarrow{\Delta_{B}^{t+1}} \overset{t+1}{\product_{B}}E \hooklongleftarrow \overset{t+1}{\fatvee_{\!B}}E$, where $p_{t+1}^{-1}(b) = P^{t}(\Loop{p^{-1}(b)})$ for any $b \in X$.

* $e^{E}_{t} : P_{\!B}^{t}(\LoopB{E}) \to E$ is induced from the inclusion $\overset{t+1}{\fatvee_{\!B}}E \hookrightarrow \overset{t+1}{\product_{B}}E$ by the fibrewise diagonal $\Delta_{B}^{t+1} : E \to \overset{t+1}{\product_{B}}E$, and is an extension of $e_{t} : P^{t}\Loop{F} \to F$ for the fibre $F$ of $p$.

* $p_{t+1}^{\LoopB{E}} : E_{B}^{t+1}(\LoopB{E}) \to P_{\!B}^{t}(\LoopB{E})$ is induced from the section $s : X \to E$.

We remark that the section of $\overset{t+1}{\product_{B}}E$ is given by $\Delta_{B}^{t+1}{\comp}s : X \xrightarrow{s} E \xrightarrow{\Delta_{B}^{t+1}} \overset{t+1}{\product_{B}}E$.

When the base point or a section for a fibrewise space is a closed cofibration, Iwase and Sakai showed that a fibrewise L-S category can be characterized in terms of fibrewise $A_{\infty}$-structure.

Let $E$ be a fibrewise pointed space and $t\!\ge\!0$. Then we have

$\catB{E} \le t$ $\iff$ $e_{t}^{E} : P^{t}_{\!B}(\LoopB{E}) \hookrightarrow P^{\infty}_{\!B}(\LoopB{E}) \underset{\simeq}{\xrightarrow{e_{\infty}^{E}}} E$ has a right homotopy inverse.

It enables us to define a stronger homotopy invariant, module weight.

For $u \!\in\! H^{*}_{\!B}(E;R) \!\subset\! H^{*}(E;R)$, we define

\begin{align*}&
\cupl[B]{E;R} = \Max\{ t\!\ge\! 0 \midvert \exists\,\{u_{1},\ldots,u_{t} \!\in\! H_{B}^{*}(E;R)\}\ \ u_{1}{\cdot}\cdots{\cdot}u_{t}\!\neq\!0\},
\\&
\wgt[B]{u;R} = \Max\{t \!\ge\! 0 \midvert \forall\,{\phi : F \!\to\! E} \ \catB{\phi}\!<\!t \Rightarrow \phi^{*}(u)\!=\!0\},
\\&
\wgt[B]{E;R} = \Min\{t \!\ge\! 0 \midvert (e_{t}^{E})^{\ast}  \ \text{is monic}\},
\\&
\Mwgt[B]{E;\Gamma} = \Min\{t \!\ge\! 0 \midvert \Img(e_{t}^{E})^{\ast}  \ \text{is a direct summand of} \ H^{*}(P^{t}_{\!B}(\LoopB{E});\field_{2}) \ \text{as a $\Gamma$-module}\},
\end{align*}

where $\Gamma$ is an $\field_{2}$-subalgebra of $\mathcal{A}_{2}$ the modulo $2$ Steenrod algebra.

Now, let us clarify the relationship between the above invariants and fibrewise L-S category. In <cit.>, $\wgt[B]{u;R}$ is defined with $\wgtBB{u;R}$ which is introduced to give a lower bound for $\catBB{E}$ the original James fibrewise L-S category.
Since we do not know the equality of $\catB{E}$ and $\catBB{E}$ as well as the equality of $\tcm{X}$ and $\tc{X}$ until now, let us state the following.

For a fibrewise pointed space $(E,p,X,s)$, we obtain the following.

$\wgt[B]{u {\cdot} v;R} \ge \wgt[B]{u;R} + \wgt[B]{v;R}$
for $u, v \!\in\! H^{*}(E,X;R) \smallsetminus \{0\}$,

$\wgt[B]{u;R} = \Max\{ t \!\ge\! 0 \midvert (e_{t-1}^{E})^{\ast}(u) \!=\! 0\}$
for $u \!\in\! H^{*}(E,X;R) \smallsetminus \{0\}$,

$\wgt[B]{E;R}=\Max\{ \wgt[B]{u} \midvert u \!\in\! H_{B}^{\ast}(E;R) \smallsetminus \{0\}\}$,

Let us assume that $\wgt[B]{u;R}=m$ and $\wgt[B]{v;R}=n$, and that $\phi : (F,q,Y) \to (E,p,X)$ be a fibrewise map with $\catB{\phi}<m{+}n$.
Then there is a cover of $F$ by $m{+}n$ open subsets $\{U_{i}\}$, at most, each on which $\phi|U_{i}$ is fibrewise homotopic to $s \comp f \!\comp q |U_{i}$.
Let $U = U_{1} \cup \cdots \cup U_{m}$ and $V = U_{m+1} \cdots \cup U_{m+n}$ to satisfy $\catB{\phi|{U}}<m$ and $\catB{\phi|{V}}<n$.
Hence $\phi^{*}(u)|_{U}=(\phi|{U})^{*}(u)=0$ in $H_{B}^{\ast}(U;R)$ and $\phi^{*}(v)|_{V}=(\phi|{V})^{*}(v)=0$ in $H_{B}^{\ast}(V;R)$.
Then we obtain $\phi^{*}(u{\cdot}v)=\phi^{*}(u){\cdot}\phi^{*}(v)=0$ by the definition of cup-products, which implies $\wgtB{u{\cdot}v;R} \ge m+n$.

Let $\wgt[B]{u;R}=m$. We can easily see that the filtration $X=P_{\!B}^{0}(\LoopB{E}) \subset P_{\!B}^{1}(\LoopB{E}) \subset \cdots \subset P_{\!B}^{t}(\LoopB{E})$ gives a fibrewise version of a cone decomposition of a fibrewise space $P_{\!B}^{t}(\LoopB{E})$, $t \!\ge\! 1$. It implies that $\catB{P_{\!B}^{t}(\LoopB{E})} \le t$ and $\catB{e^{E}_{t}} \le t$, and hence we obtain $(e_{m-1}^{E})^{\ast}(u) \!=\! 0$ and $\Max\{ t \!\ge\! 0 \midvert (e_{t-1}^{E})^{\ast}(u) \!=\! 0\} \ge m = \wgt[B]{u;R}$.
Conversely assume that $\Max\{ t \!\ge\! 0 \midvert (e_{t-1}^{E})^{\ast}(u) \!=\! 0\} =m$.
If $(\phi,f) : (F,q,Y) \to (E,p,X)$ satisfies $\catB{\phi}<m$, then there exits an open cover of $F$ by at most $m$ open subsets $U_{i}$ on each of which $\phi|U_{i}$ is fibrewise homotopic to $s \comp f \!\comp q |_{U_{i}}$.
By standard arguments of homotopy theory, we may assume that there exits at most $m$ closed cofibrations $F_{i} \hookrightarrow F$.
Then by extending the homotopy onto $F$ to obtain a fibrewise map $r : F \to \overset{m}{\fatvee_{\!B}}E \subset \overset{m+1}{\product_{B}}E$ a fibrewise compression of the fibrewise diagonal $\Delta^{m}_{B} : E \to \overset{m-1}{\product_{B}}E$, which gives a fibrewise map $\psi : F \to P_{\!B}^{m-1}(\LoopB{E})$ which is a lift of $\phi : F \to E$ on $e^{E}_{m-1} : P_{\!B}^{m-1}(\LoopB{E}) \to E$.
Since $(e^{E}_{m-1})^{*}(u)=0$, we have $\phi^{*}(u)=0$, and it implies (<ref>).

$\wgt[B]{E}=t \!\ge\! 0$ if and only if $(e_{t}^{E})^{\ast}$ is monic and $(e_{t-1}^{E})^{\ast}$ has non-trivial kernel, which is equivalent to that, for any $u \!\neq\! 0$ in $H_{B}^{\ast}(E;R)$, $(e_{t}^{E})^{\ast}(u) \!\neq\! 0$ but there is an element $v \!\in\! H_{B}^{\ast}(E;R)$ such that $(e_{t-1}^{E})^{\ast}(v) \!=\! 0$, in other words, $\wgt[B]{u} \!\leqq\! t$ for all $u \!\in\! H_{B}^{\ast}(E;R)$ but $\wgt[B]{v}=t$.

We have
$\catB{E} \ge \wgt[B]{E;R} \ge \cupl[B]{E;R}$.

Let $\catB{E} = t$. Then there is a map $s : E \to P^{t}_{\!B}(\LoopB{E})$ such that $e_{t}^{E}\comp s = \id_E$, which implies that $s^{\ast} \comp (e_{t}^{E})^{\ast} = (\id_E)^{\ast}$.
Thus, $(e_{t}^{E})^{\ast} : H_{B}^{\ast}(E;R) \to H_{B}^{\ast}(P^{t}_{\!B}(\LoopB{E});R)$ is monic, and hence $\catB{E} \ge \wgt[B]{E;R}$.
The latter part is obtained from $\wgt[B]{u;R} \!\ge\! 1$ by Proposition <ref> (<ref>).

Though it is not necessary in our arguments, <cit.> says the following equalities for the fibrewise well-pointed space $\double{X}=E_{\id_{X}}$, while we skip the details.

* $\tc{X} = \catB{\double{X}}$,
$\cupl[\varpi]{X;R} = \cuplB{\double{X};R}$.

Also the proof of <cit.> claiming $\wgt[\varpi]{u;R} = \wgtBB{u;R}$ for $u \!\in\! H_{B}^{\ast}(\double{X};R)$ works fine to obtain $\wgt[\varpi]{u;R} = \wgtB{u;R}$ for $u \!\in\! H_{B}^{\ast}(\double{X};R)$, while we skip the details, too.

Using the above observations, we obtain the following.

Let $m \ge 2$.
Assume that a group $G$ acts on $S^{m}$ orientation-preservingly and freely, satisfying $\cat{X}=\wgt{X;\field_{2}}=m$ where $X=S^{m}/G$, and thus the weight of the generator $z \in H^{m}(X;\field_{2}) \cong \field_{2}$ is equal to $m$, where $\field_{2}$ denotes the prime field of characteristic $2$.
Let $z \otimes z$ be the generator of $H_{B}^{2m}(\double{X};\field_{2}) = H^{2m}(X \times X,\Delta(X);\field_{2}) \cong \field_{2}$.
Then we have the following.

* The three statements (<ref>), (<ref>) and (<ref>) below are equivalent.

* The three equivalent statements (<ref>), (<ref>) and (<ref>) implies $\catB{\double{X}}=\tc{X}= 2m$.

*  $\Mwgt[B]{\double{X};\field_{2}} \ge 2m$,
$\wgt[B]{\double{X};\field_{2}} \ge 2m$,
$\wgt[B]{z \otimes z;\field_{2}} \ge 2m$.

Since we know that $\wgt[B]{z \otimes z} \le \wgt[B]{X;\field_{2}} \le \Mwgt[B]{X;\field_{2}} \le \cat[B]{X} \le \cat{X \times X} \le \dim{X \times X}$ $=$ $2m$, it is straitforward to obtain (<ref>) $\Rightarrow$ (<ref>) $\Rightarrow$ (<ref>) $\Rightarrow$ $\cat[B]{X} = \tc{X}$ $=$ $2m$.
So we are left to show (<ref>) $\Rightarrow$ (<ref>):
let $\wgt[B]{z \otimes z} < 2m$ and $(e_{2m-1}^{E})^{\ast}(z \otimes z) \!\ne\! 0$.
Let $W=X \times X$ and $W_{0}=W \smallsetminus D^{2m}$ the once punctured submanifold.
Then we have $W \homeo W_{0} \cup_{f} D^{2m}$, where $f$ is an attaching map, and hence we obtain $H^{*}(E;\field_{2}) \cong H^{*}(E_{0};\field_{2}) \oplus H^{*}(D^{2m},S^{2m-1};\field_{2})$, where the latter direct summand is isomorphic to $\field_{2}$ generated by $z \otimes z$.
Since $\catB{E_{0}} \le 2m{-}1$, the inclusion $E_{0} \hookrightarrow E$ has a lift to $e^{E}_{2m-1} : P_{\!B}^{2m-1}(\LoopB{E}) \to E$.
Thus the direct summand $H^{*}(E_{0};\field_{2}) $ mapped to a direct summand of $H^{*}(P_{\!B}^{2m-1}(\LoopB{E});\field_{2})$ by $(e^{E}_{2m-1})^{*}$.
By the hypothesis, the entire module $H^{*}(E;\field_{2})$ mapped to a direct summand of $P_{\!B}^{2m-1}(\LoopB{E})$ by $(e^{E}_{2m-1})^{*}$, and hence we obtain $\Mwgt[B]{\double{X}} < 2m$.

We remark here that $\wgtB{z \otimes z}<2m$ may not imply $\tc{X}$ $<$ $2m$.
We must know about a higher Hopf invariant to show a result similar to that for L-S category.

From now on, $G$, $M$ and $p$ stands for $Q_8$, $S^3/Q_{8}$ and the canonical projection of the principal $Q_{8}$-bundle $S^{3} \twoheadrightarrow S^3/Q_{8}=M$, respectively.

We show the following in <ref>.

The generator $z \in H^{3}(M;\field_{2}) \cong \field_{2}$ satisfies $\wgt{z}=\cupl{z}=3$ which implies$\cupl{M} = \wgt{M} = \cat{M} = \dim{M} = 3.$

$5 \le \cuplB{\double{M}}\le \catB{\double{M}} = \tc{M} \le 2\cat{M} = 6$.

The following statement is our main result.

$\tc{M} = \catB{\double{M}} = 6$.

In view of Theorem <ref> and Proposition <ref>, it is sufficient to show the following lemma.

$(e_{5}^{\double{M}})^{\ast}(z \otimes z) \!=\! 0$ in $H^{\ast}(P^{5}_{\!B}(\LoopB{\double{M}});\field_{2})$.

§ PROOF OF THEOREM <REF> AND PROPOSITION <REF>

First, we introduce a modified Bar construction of a group $G$ as the realization of a nerve of the category of one object with morphism set $G$, where we only use face relations for the realization so as to obtain $C^{*}(G;\field_{2})=C^{*}(P^{\infty}G;\field_{2})$. Then by Segal <cit.>, it follows that $P^{\infty}G \simeq K(G,1)$.
The cell-structure of the modified Bar construction is as follows:

P^{\infty}G = \bigcup_{t \ge 0} P^{t}G,\quad P^{t}G= \underset{(g_{1},\ldots,g_{t}) \in \reduced{G}^{t}}{\textstyle\bigcup}\,\{g_{1}|g_{2}|\cdots|g_{t}\},

where $\{g_{1}|\cdots|g_{t}\}$ is a $t$-simplex, $(g_{1},\ldots,g_{t}) \in \reduced{G}^{t}$, $\reduced{G}=G \smallsetminus \{e\}$ if $t \!\ge\! 1$, the unique $0$-simplex denoted by $\{\}$ if $t \!=\! 0$.
The boundary of $\{g_{1}|\cdots|g_{t}\}$, $t \!\ge\! 1$ is given by the following formulas:

\begin{align*}
\partial_{i}\{g_{1}|\cdots|g_{t}\} = \begin{cases}
\{g_{2}|\cdots|g_{t}\},&i=0,
\\
\{g_{1}|\cdots|g_{i-1}|g_{i}g_{i+1}|g_{i+2}|\cdots|g_{t}\},&0<i<t, \ g_{i}g_{i+1}\not=e,
\\
\{g_{1}|\cdots|g_{i-1}|g_{i+2}|\cdots|g_{t}\},&0<i<t, \ g_{i}g_{i+1}=e,
\\
\{g_{1}|\cdots|g_{t-1}\},&i=t.
\end{cases}
\end{align*}

\begin{align*}
\partial_{i}\{g_{1}|\cdots|g_{t}\} = \begin{cases}
\{g_{2}|\cdots|g_{t}\},&i=0,
\\
\{g_{1}|\cdots|g_{i-1}|g_{i}g_{i+1}|g_{i+2}|\cdots|g_{t}\},&0<i<t,
\\
\{g_{1}|\cdots|g_{t-1}\},&i=t.
\end{cases}
\end{align*}

The following is well-known (see <cit.>) for $H^{*}(G;\field_{2})=H^{*}(P^{\infty}G;\field_{2})$.

$H^{\ast}(P^{\infty}G;\field_2) = A^{*} \otimes \field_{2}[w]$, $A^{*}=\field_2[x,y]/(x^3,y^3,x^2{+}y^2{+}x{\cdot}y)$, $w \in H^{4}(P^{\infty}G;\field_{2})$ and $x, \,y \in H^{1}(P^{\infty}G;\field_{2})$, where $x\!=\![\alpha]$ and $y\!=\![\beta]$ are given by $\alpha\{a^{m}b^{n}\}=m$ and $\beta\{a^{m}b^{n}\}=n$ on $C_{1}(P^{\infty}G)$, respectively.
In particular, $H^{3}(P^{\infty}G;\field_2) \cong \field_2$ with a generator $z = x^{2}{\cdot}y = x{\cdot}y^{2}$.

Second, we construct a cell complex $BG \simeq K(G,1)$ as a minimal complex realising the group cohomology $H^{*}(G;\field_{2})$ defined by K. Fujii <cit.>, which is actually much smaller than $P^{\infty}G$:

taking quotient from $S^{4t+3}= \{(x_0,\dots ,x_{t}) \in \quaternion^{t+1} \midvert \vert{x_0}\vert^{2} {+} \cdots {+} \vert{x_{t}}\vert^{2} \!=\! 1\}$ by the following action of $G \subset \mathrm{Sp}(1)$, Fujii defined a series of manifolds $N^{t}(2)=S^{4t+3}/G$, $t \!\ge\! 0$:

\[g(x_0,\dots ,x_{t}) = (gx_0,\dots ,gx_{t})\]

for $g \in G \subset \mathrm{Sp}(1)$ and $(x_0,\dots ,x_{t}) \in S^{4t+3} \subset \quaternion^{t+1}$.

Since $\bigcup_{t=0}^{\infty} S^{4t+3}$ is contractible, the canonical projection $p_{\infty} : S^{\infty} = \bigcup_{t=0}^{\infty} S^{4t+3} \to \bigcup_{t=0}^{\infty}N^{t}(2) =:BG$ is a universal principal $G$-bundle.
This tells us that a canonical inclusion $i : M=N^{0}(2) \hookrightarrow BG$ gives a classifying map of the principal $G$-bundle $q : S^{3} \twoheadrightarrow M$, since $q=q_{\infty}|_{S^{3}}$.

We know the following finite cell decomposition of $N^t(2)$, $t \ge 0$.

[Fujii <cit.>]
$S^{4t+3}$ is a $G$-cell complex with the cell decomposition

\left\{ge^{4k}, \,ge^{4k+1}_{1}, \,ge^{4k+1}_{2}, \,ge^{4k+2}_{1}, \,ge^{4k+2}_{2}, \,ge^{4k+3} \mid 0 \!\le\! k \!\le\! t, \ g \in G \right\}

whose boundary formulas in the cellular chain complex with coefficients in $\field_{2}$ is given as follows:

\begin{align*}
\partial &e^{0} = 0, \quad\partial
e^{4k+4} = \sum_{g\in G}ge^{4k+3}\, \ (k \ge 1), \\
\partial &e^{4k+1}_{1} = (a{-}1)e^{4k}, \quad\partial e^{4k+1}_{2} = (b{-}1)e^{4k}, \\
\partial &e^{4k+2}_{1} = (a {+} 1)e^{4k+1}_{1} - (b{+}1)e^{4k+1}_{2}, \quad%
\partial e^{4k+2}_{2} = (ab{+}1)e^{4k+1}_{1} + (a{-}1)e^{4k+1}_{2}, \\
\partial &e^{4k+3} = (a{-}1)e^{4k+2}_{1} - (ab{-}1)e^{4k+2}_{2}.
\end{align*}

The above cell decomposition of $S^{4t+3}$ induces a cell decompostion of $N^{t}(2)$, $t \!\ge\! 0$ as follows.

[Fujii <cit.>]
The manifold $N^{t}(2)$ can be decomposed as the finite cell complex whose cells are given by

\left\{e^{4k}, \,e^{4k+1}_1, \,e^{4k+1}_2, \,e^{4k+2}_1, \,e^{4k+2}_2, \,e^{4k+3} \mid 0\!\le\!k \!\le\!t\right\}

with the following boundary formulas in the cellular chain complex with coefficients in $\field_{2}$ associated with the cell decomposition of $N^{t}(2)$, $t \!\ge\! 0$, above:

\begin{align*}
&\partial e^{0} = 0,\quad \partial e^{4k} = 2^{3}e^{4k-1}_1\, \ (k \ge 1),\\
&\partial e^{4k+1}_1 = 0,\quad \partial e^{4k+1}_2 = 0,\\
&\partial e^{4k+2}_1 = 2e^{4k+1}_1 - 2e^{4k+1}_2, \quad \partial e^{4k+2}_2 = 2e^{4k+1}_1,\\
&\partial e^{4k+3} = 0.
\end{align*}

We remark that the naturality of the above decomposition implies that $N^{k}(2)$ is actually a sub-complex of $N^{t}(2)$ if $0 \le k \le t$, and hence $M$ is a sub-complex of $BG$.

Then, we describe the cohomology groups of $N^{t}(2)$ with coefficients in $\field_{2}$.

The cohomology groups of $BG = \bigcup_{t=0}^{\infty}N^{t}(2)$ are given as follows:

\begin{align*}
H^k(BG;\field_{2}) &=
\begin{cases}
\field_2 \oplus \field_2 & k\equiv 1,2 \mod 4,\\
\field_2 & k\equiv 3,0 \mod 4.
\end{cases}
\end{align*}

Since $BG \simeq P^{\infty}G$, we obtain its multiplicative structure as $H^{\ast}(BG;\integral/(8)) \cong A^{*} \otimes \field_{2}[w]$.

$H^{\ast}(M;\field_2) \cong A^{*}$.
Further, we have $H^{3}(M;\field_2) \cong \field_2$ with $z = x^{2}{\cdot}y = x{\cdot}y^{2}$.

The additive structures are obvious, and so we show the multiplicative structure for $\field_{2}$-coefficient.
Since $\pi_{1}(M) \cong G$, there is a classifying map $i : M \hookrightarrow BG$ inducing the following fibration with fibre $S^{3}$, where the map $S^{3} \twoheadrightarrow M$ is the universal covering of $M$:

\[S^3 \twoheadrightarrow M \overset{i}\hookrightarrow BG.\]

Since the action of $G$ preserves the orientation, the fibration is simple, and hence the $E_{2}$-term of the Serre spectral sequence with coefficients in $\field_{2}$ for the above fibration is

E_{2}^{p,q} \cong H^{p}(BG;\field_{2}) \otimes H^{q}(S^{3};\field_{2}) \cong A^{*} \otimes \field_{2}[w] \otimes \Lambda(s_{3}),

where $s_{3}$ be the generator of $H^{3}(S^{3};\field_{2}) \cong \field_{2}$.
Then $E_{r}^{*,*}$, $r \!\ge\! 2$ has no non-trivial differential other than $d_4$ which is described, by putting $H^{p} = H^{p}(BG;\field_{2})$, as follows.

=by #1
=by #2
by by (,)(6,6)[cc]$#3$

=by 11 by 2
=by 9 by 2

(0,10)	(1,0)
(10,0)	(0,1)

Since $w \in E_{2}^{4,1}$ can not survive in $E_{\infty}$-term, we have $w \in \Img d_{4}$, and hence $d_4(s_3) = w$.
Thus $H^{\ast}(M;\field_2)$ is isomorphic to $E_{5}^{*,*}=A^{*}$ and we have done.

We denote by $\zeta$ and $1$ the cochains dual to $e^{3} \in C_{3}(M)$ and $e^{0} \in C_{0}(M)$, respectively, in $\hom(C^{*}(M);\field_{2})$.
Since $S^{3} = \bigcup\{\,g\sigma \,\vert\, g \!\in\! G\}$ where $\sigma$ runs over all cells of $M$, $p^{*}\zeta(g\sigma)=1$ if and only if $\sigma=e^{3}$ for any $g \in G$ and $p^{*}1(g\sigma)=1$ if and only if $\sigma=\ast$ for any $g \in G$.

[Theorem <ref>]
Since $z=x^{2}{\cdot}y\not=0$ in $H^{*}(M;\field_{2})$, we have $\wgt{z}=\cupl{z}=3$, and hence$\cupl{M} = \wgt{M} = \cat{M} = \dim{M} = 3.$

Let us denote $X =  x \otimes 1 + 1 \otimes x$ and $Y = y \otimes 1 + 1 \otimes y$, respectively, which are in the zero-divisors-ideal $\ker \Delta^{*} = H_{B}^{*}(\double{M};\field_{2})$.
Thus, we have $X^3{\cdot}Y^2 \neq 0$ and hence we obtain

[Proposition <ref>]
$5 \le \cuplB{\double{M}}\le \catB{\double{M}} = \tc{M} \le 2\cat{M} = 6$.

§ PROOF OF LEMMA <REF>

In general, $P^{t}_{\!B}\LoopB{\double{M}}$,  $t \!\ge\! 0$,  the fibrewise projective $t$-space of the fiberwise loop space of $\double{M}$ has a misterious structure, while the following is known (cf. <cit.>).

There is a fibrewise homotopy equivalence $\tilde{f}_{0} : \LoopB{\double{BG}} \to S^{\infty}\times_{\ad{}}G$, such that

$\tilde{f}_{0} : \LoopB{\double{BG}} \to S^{\infty}\times_{\ad{}}G$ is a fibrewise $A_{\infty}$-map.

$\tilde{f}_{0}$ induces a fibrewise homotopy equivalence $\tilde{f}$ $:$ $P^{t}_{\!B}\LoopB{\double{BG}} \to S^{\infty}\times_{\ad{}}P^{t}G$, where $P^{t}_{\!B}\LoopB{\double{BG}}$ is the fibrewise projective $t$-space and
$P^{t}G$ is the $t$-skeleton of $P^{\infty}G$ with adjoint action of $G$ on $P^{\infty}G$ given by

\[
h\{g_1|g_2|\cdots|g_t\} = \{hg_1\bar{h}|hg_2\bar{h}|\cdots | hg_t\bar{h}\},
\]

where $h \!\in\! G$ and $\{g_1|g_2|\cdots|g_t\}$ is a $t$-cell in $P^{\infty}G$ indexed by $(g_{1},\ldots,g_{t}) \in G^{t}$, $t \!\ge\! 0$.

We know, by a result of Benson <cit.>, there is a fibrewise pointed homotopy equivalence $\tilde{f}_{0} : \LoopB{\double{BG}} \to S^{\infty}\times_{\ad{}}G$ over $BG$.

Since $\tilde{f}_{\!0}$ satisfies that $\tilde{f}_{\!0}(\alpha \cdot \beta) = \tilde{f}_{\!0}(\alpha){\cdot} \tilde{f}_{\!0}(\beta)$ on each fibre and each fibre of $EG\times_{\ad{}}G$ is a discrete set, we obtain that $\tilde{f}_{\!0}$ is a fibrewise $A_{\infty}$-map.

Thus $\tilde{f}_{\!0}$ induces a fibrewise maps $\tilde{f} : P^{t}_{\!B}\LoopB{\double{BG}} \to S^{\infty}\times_{\ad{}}P^{t}G$ and
$\tilde{f}_{\!E} : E^{t}_{\!B}\LoopB{\double{BG}} \to S^{\infty}\times_{\ad{}}E^{t}G$
for all $t \ge 0$ and the following commutative diagram of fibrewise spaces.

\begin{diagram}
\node{E^{t+1}_{\!B}\LoopB{\double{BG}}}
\arrow{s}
\arrow{e}
\node{P^{t}_{\!B}\LoopB{\double{BG}}}
\arrow{s}
\arrow{e}
\node{\double{BG}}
\arrow{s,=}
\\
\node{S^{\infty}\times_{\ad{}}E^{t+1}G}
\arrow{e}
\node{S^{\infty}\times_{\ad{}}P^{t}G}
\arrow{e}
\node{\double{BG}}
\end{diagram}

By employing a similar arguments as in <cit.>.

By restricting the fibrewise structure to a subspace $M \overset{i}\hookrightarrow BG$, we obtain a fibrewise pointed space $E_{i}=\double{BG}|_{M}=\ifdefined\reviseone{(M \times BG,\proj_{1},M,(\id {\times} i)\comp\Delta)}\else{(BG \times M,\proj_{2},M,(i {\times} \id)\comp\Delta)}\fi$ and the following.

There is a fibrewise $A_{\infty}$-map $f_{\!0} : \LoopB{E_{i}} \to S^{3}\times_{\ad{}}G$ over $M=S^{3}/G$, i.e,

(i) $f_{\!0} : \LoopB{E_{i}} \to S^{3}\times_{\ad{}}G$ is a fibrewise homotopy equivalence.

(ii) $f_{\!0}$ induces a fibrewise homotopy equivalence $f : P^{t}_{\!B}\LoopB{E_{i}} \to S^{3}\times_{\ad{}}P^{t}G$, $t \!\ge\! 0$.

\[\xymatrix{
M	\ar@{^{(}->}[d]_{i}
& S^3\times_{\ad{}}P^{t}G
\ar@{^{(}->}[d]_{\iota}
\ar[l]
& P^{t}_{\!B}\LoopB{E_{i}}
\ar@/_15pt/@{->}[ll]
\ar@{^{(}->}[d]
\ar@{->}[l]^{f}
\\
BG & S^{\infty}\times_{\ad{}}P^{t}G
\ar[l]
& P^{t}_{\!B}\LoopB{\double{BG}}
\ar@/^15pt/@{->}[ll]
\ar[l]_{\tilde{f}}

We know that $P^{t}_{\!B}\LoopB{E_{i}} = i^{*}(P^{t}_{\!B}\LoopB{\double{BG}})$ is a fibrewise space over $M$ given by the pull-back of $M \overset{i}\hookrightarrow BG \leftarrow P^{t}_{\!B}\LoopB{\double{BG}}$, and there is a fibrewise homotopy equivalence $f : P^{t}_{\!B}\LoopB{E_{i}} \to S^3\times_{\ad{}}P^{t}G$ since $S^3\times_{\ad{}}P^{t}G$ is the pull-back of $M \overset{i}\hookrightarrow BG \leftarrow S^{\infty}\times_{\ad{}}P^{t}G $ and $\tilde{f} : P^{t}_{\!B}\LoopB{\double{BG}} \to S^{\infty}\times_{\ad{}}P^{t}G$ is a fibrewise homotopy equivalence by Lemma <ref>.

Next, we give the boundary formulas for $S^{3}\times_{\ad{}}P^{t}G$, $t \!\ge\! 0$.
The cell structure of $S^{3}\times_{\ad{}}P^{t}G$ can be described by product cells of $S^{3}$ and $P^{t}G$ as follows.

\[ S^{3}\times_{\ad{}}P^{t}G  =\bigcup_{\sigma \in \{ \text{$G$-cells of $S^{3}$}\}, \,\omega \in \reduced{G}^{t}} \sigma{\times}\{\omega\}/\sim, \]

where `$\sim$' is given by $g\sigma{\times}\{g\omega\reduced{G}\} \sim \sigma{\times}\{\omega\}, \, g \!\in\! G$.
Let us denote by $[\sigma| \{\omega\}]$ the equivalence class of $\sigma{\times}\{\omega\}$.
By Proposition <ref>, we obtain modulo $2$ boundary formulas for $S^{3}\times_{\ad{}}P^{t}G$.

The boundaries of product cells

\[
[e^{4k} | \{\omega\} ],  [e^{4k+1}_{1} | \{\omega\} ],  \,[e^{4k+1}_{2} | \{\omega\} ],  \,[e^{4k+2}_{1} | \{\omega\} ],  \,[e^{4k+2}_{2} | \{\omega\} ], \,[e^{4k+3} | \{\omega\} ],\quad k \ge 0,
\]

are described as a chain in the chain complex with coefficients in $\field_{2}$ as follows.

\begin{align*}&
\partial[e^{0}| \{\omega\} ]=[e^{0}| \{\partial\omega\} ],\qquad
\partial[e^{4k} | \{\omega\} ] = \sum_{g\in G}[e^{4k-1} | \{\overline{g}\omega g\}]+[e^{4k} | \{\partial\omega\} ]\quad (k \ge 1),
\\&
\partial[e^{4k+1}_{1}| \{\omega\} ]=[e^{4k}| \{\omega\} ]+[e^{4k}| \{\bar{a}\omega a\} ]+[e^{4k+1}_{1}| \{\partial\omega\} ], \\
&\partial[e^{4k+1}_{2}| \{\omega\} ]=[e^{4k}| \{\omega\} ]+[e^{4k}| \{\bar{b}\omega b\} ]+[e^{4k+1}_{2}| \{\partial\omega\} ],
\\&
\partial[e^{4k+2}_{1}| \{\omega\} ]=[e^{4k+1}_{1} | \{\omega\} ]+[e^{4k+1}_{2}| \{\omega\} ]+[e^{4k+1}_{1}| \{\bar{a}\omega a\} ]+[e^{4k+1}_{2}| \{\bar{b}\omega b\} ]+[e^{4k+2}_{1}| \{\partial\omega\} ],
\\&
\partial[e^{4k+2}_{2}| \{\omega\} ]=[e^{4k+1}_{1}| \{\omega\} ]+[e^{4k+1}_{2}| \{\omega\} ]+[e^{4k+1}_{1}| \{\overline{ab} \omega ab\} ]+[e^{4k+1}_{2}| \{\bar{a}\omega a\} ]+[e^{4k+2}_{2}| \{\partial\omega\} ],
\\&
\partial[e^{4k+3}| \{\omega\} ]=[e^{4k+2}_{1}| \{\omega\} ]+[e^{4k+2}_{2}| \{\omega\} ]+[e^{4k+2}_{2}| \{\overline{ab}\omega ab\} ]+[e^{4k+2}_{1}| \{\bar{a}\omega a\} ]+[e^{4k+3}| \{\partial\omega\} ],
\end{align*}

where we abbreviate $\displaystyle[\sigma | \{\partial\omega\}] = \overset{t}{\underset{\substack{0 \le i \le t\\\dim{\partial_{i} \omega} = t{-}1}}{\textstyle\sum}} (-1)^{i}[\sigma | \{ \partial_{i} \omega \}]$ for $\sigma$ a $G$-cell of $S^{3}$ and $\omega \in \reduced{G}^{t}$.
where we abbreviate $\displaystyle[\sigma | \{\partial\omega\}] = \overset{t}{\underset{0 \le i \le t}{\textstyle\sum}} (-1)^{i}[\sigma | \{ \partial_{i} \omega \}]$ for $\sigma$ a $G$-cell of $S^{3}$ and $\omega \in \reduced{G}^{t}$.

From Theorem <ref>, there is a fibrewise homotopy equivariance $f : P^{5}_{\!B}\LoopB{E_{i}} \xrightarrow{\simeq} S^{3}\times_{\ad{}}P^{5}G$.

On the other hand, we have a fibrewise map $\lambda=P^{5}_{\!B}\LoopB{\id_{M} \times i} : P^{5}_{\!B}\LoopB{\double{M}} \to P^{5}_{\!B}\LoopB{E_{i}}$ over $M$ in Diag. 2, which is commutative and $P^{5}_{\!B}\LoopB{E_{i}}$ is the pull-back of $M \overset{i}\hookrightarrow BG \leftarrow P^{5}_{\!B}\LoopB{\double{BG}}$.

\[\xymatrix{
M \ar@{=}[d]& P^{5}_{\!B}\LoopB{\double{M}}\ar[d]^{\lambda}\ar[l] \\
M & P^{5}_{\!B}\LoopB{E_{i}} \ar[l]

Now, let us denote by $C_{\ast}(X)$ the celluar chain complex with coefficients in $\field_{2}$ and by $C^{\ast}(X,R)=\hom(C_{\ast}(X),R)$ the cellular cochain complex with coefficients in $\field_{2}$-module $R$.
We define three cochains $c \in C^6(S^{3}\times_{\ad{}}P^{6}G;\field_{2}) = C^6(S^{3}\times_{\ad{}}P^{\infty}G;\field_{2})$, $c' \in C^5(S^{3}\times_{\ad{}}P^{5}G;\field_{2}) = C^5(S^{3}\times_{\ad{}}P^{\infty}G;\field_{2})$ and
$v \in C^1(S^{3}\times_{\ad{}}P^{1}G;\field_{2}) = C^1(S^{3}\times_{\ad{}}P^{\infty}G;\field_{2})$ by the following equations on a generator $[\sigma| \{ h_1 | \cdots | h_{t}\}]$, $t \ge 0$ in $C_{*}(S^{3}\times_{\ad{}}P^{\infty}G)$:

\begin{align*}&
c[\sigma| \{ h_1 | \cdots | h_{t}\}] = p^{*}\zeta(\sigma){\cdot}\alpha^{2}\beta\{h_1 | \cdots | h_{t}\},
\\&
c'[\sigma| \{ h_1 | \cdots | h_{t}\}] = p^{*}\zeta(\sigma){\cdot}\alpha^{2}\{h_1 | \cdots | h_{t}\},
\\&
v[\sigma| \{ h_1 | \cdots | h_{t}\}] = p^{*}1(\sigma){\cdot}\beta\{h_1 | \cdots | h_{t}\}.
\end{align*}

Then we show that $c$, $c'$ and $v$ are well-defined:
since $a^mb^na^kb^l$ $=$ $a^{m+(-1)^nk}b^{n+l}$, for any $g=a^mb^n$ and $h=a^kb^l$, we obtain

$=$ $a^{m+(-1)^{n}k+(-1)^{l}m}b^{l}$,

and hence $\alpha(gh\bar{g})=k=\alpha(h)$ and $\beta(gh\bar{g})=l=\beta(h)$ in $\field_{2}$.
Hence they satisfy

\begin{align*}&
c[g\sigma| {g}\{h_1 | \cdots | h_{t}\}]
= p^{*}\zeta(g\sigma){\cdot}\alpha^2\beta\{gh_1\bar{g} | \cdots | gh_{t}\bar{g}\}
= p^{*}\zeta(\sigma){\cdot}\alpha^2\beta\{h_1 | \cdots | h_{t}\}
= c[\sigma| \{h_1 | \cdots | h_{t}\}],
\\&
c'\{g\sigma| {g}\{h_1 | \cdots | h_{t}\}\}
= p^{*}\zeta(g\sigma){\cdot}\alpha^{2}\{gh_1\bar{g} | \cdots | gh_{t}\bar{g}\}
= p^{*}\zeta(\sigma){\cdot}\alpha^{2}\{h_1 | \cdots | h_{t}\}
= c'[\sigma| \{h_1 | \cdots | h_{t}\}],
\\&
v[g\sigma| {g}\{h_1 | \cdots | h_{t}\}]
= p^{*}1(g\sigma){\cdot}\beta\{gh_1\bar{g} | \cdots | gh_{t}\bar{g}\}
= p^{*}1(\sigma){\cdot}\beta\{h_1 | \cdots | h_{t}\}
= v[\sigma| \{h_1 | \cdots | h_{t}\}].
\end{align*}

Thus $c$, $c'$ and $v$ are well-defined. Since $z$, $\alpha$ and $\beta$ are cocycles, we have the following.

$c$, $c'$ and $v$ are cocycles satisfying $c=v \cupp c'$ in $C^{*}(S^{3}\times_{\ad{}}P^{\infty}G;\field_{2})$.

$(e''_{\infty})^{\ast}(z \otimes z)= [c]$ in $H^{6}(S^{3}\times_{\ad{}}P^{\infty}G;\field_{2})$.

Firstly, since the fibre bundle $\hat{p} : S^{3}\times_{\ad{}}P^{\infty}G \to M$ is fibrewise homotopy equivalent to the product bundle $\proj_{1} : M \times P^{\infty}G \to M$, their Serre spectral sequences are naturally isomorphic.
Secondly, $(e''_{\infty})^{\ast}(z \otimes z)$ is non-zero and in the image of $j^{*} : H^{6}(S^{3} \times_{\ad{}} P^{\infty}G,\hat{p}^{-1}(M^{(2)});\field_{2}) \to H^{6}(S^{3} \times_{\ad{}} P^{\infty}G;\field_{2})$, since it is trivial in $H^{6}(\hat{p}^{-1}(M^{(2)});\field_{2})$.
Thirdly, we have $H^{6}(S^{3} \times_{\ad{}} P^{\infty}G,\hat{p}^{-1}(M^{(2)});\field_{2}) = E^{3,3}_{1}(S^{3} \times_{\ad{}} P^{\infty}G) \cong E^{3,3}_{1}(M \times P^{\infty}G) = \hom(C_{3}(M),H^{3}(P^{\infty}G;\field_{2})) \cong \field_{2}\phi$, where the generator $\phi$ is the homomorphism taking $\phi(e^{3})$ to the non-trivial element in $H^{3}(P^{\infty}G;\field_{2}) \cong \field_{2}z$, $z=[\alpha^{2}\beta]$.
Finally, $\phi$ is the corresponding homomorphism to $[c] \in H^{6}(S^{3} \times_{\ad{}} P^{\infty}G;\field_{2})$ (cf, <cit.>) by definition.
Thus, we obtain $(e''_{\infty})^{\ast}(z \otimes z)=j^{*}\phi=[c]$.

We consider the linear equation on $u \in C^{5}(S^3\times_{\ad{}}P^5G;\field_{2}) = C^{5}(S^3\times_{\ad{}}P^{\infty}G;\field_{2})$ as follows.

\begin{align}%\label{cd1}
\delta_{5} u = c\quad \text{in} \quad C^{6}(S^3\times_{\ad{}}P^5G;\field_{2}).\tag{Eq. \!1}\label{eq:Equation 1}
\end{align}

\[\xymatrix{
P^{5}_{\!B}\LoopB{\double{M}} \ar@/^15pt/@{->}[rr]^-{e^{\double{M}}_{5}} \ar@{^{(}->}[r]  \ar@{>}[d]_{\lambda} & P^{\infty}_{\!B}\LoopB{\double{M}} \ar[r] & \double{M} \ar[d]^{\id_{M} \times i}
\\
P^{5}_{\!B}\LoopB{E_{i}}\ar[d]_-{f} \ar@/^15pt/@{->}[rr]^-{e'_5} \ar@{^{(}->}[r] & \ar[r]_-{\simeq} P^{\infty}_{\!B}\LoopB{E_{i}} \ar[d]^{f} & E_{i} \ar@{=}[d]
\\
S^{3}\times_{\ad{}}P^{5}G \ar@{^{(}->}[r]&
S^{3}\times_{\ad{}}P^{\infty}G \ar[r]_-{e''_{\infty}} &

Since (<ref>) exceeds the acceptable size for our manipulation, we are forced to consider a similar but smaller linear equation on $u' \in C^4(S^3\times_{\ad{}}P^4G;\field_{2}) = C^4(S^3\times_{\ad{}}P^{\infty}G;\field_{2})$ as follows.

\begin{align}%\label{cd2}
\delta u' = c' \quad \text{in} \quad C^{5}(S^3\times_{\ad{}}P^4G;\field_{2}).
\tag{Eq. \!2}\label{eq:Equation 2}
\end{align}

Here, we may consider the equation (<ref>) in $C^{5}(S^3\times_{\ad{}}P^5G;\field_{2})$ as $\delta u' = c'$ with indeterminacy on generators $[{*}|\{h_{1}|h_{2}|h_{3}|h_{4}|h_{5}\}]$ in $C_{5}(S^3\times_{\ad{}}P^5G) = C_{5}(S^3\times_{\ad{}}P^{\infty}G)$.

The following describes the relationship between (<ref>) and (<ref>).

If $u'$ is a solution for (<ref>), then $u = v \cupp u'$ gives a solution for (<ref>).

Let $u' \in C^4(S^3\times_{\ad{}}P^4G;\field_{2}) = C^4(S^3\times_{\ad{}}P^{\infty}G;\field_{2})$ be a solution of (<ref>).
Then, $v \cupp \delta u'$ coincides with $v \cupp c'$ in $C^6(S^3\times_{\ad{}}P^5G;\field_{2})$:
for any generator $[\sigma|\{h_{1}|\cdots|h_{k}\}]$ in $C_{6}(S^3\times_{\ad{}}P^5G)$,

\begin{align*}
(v \cupp \delta u')[\sigma|\{h_{1}|\cdots|h_{k}\}]
&= v[{\ast}|\{h_{1}\}] \cdot \delta u'[\sigma|\{h_{2}|\cdots|h_{k}\}]
\\&
= v[{\ast}|\{h_{1}\}] \cdot c'[\sigma|\{h_{2}|\cdots|h_{k}\}]
= (v \cupp c')[\sigma|\{h_{1}|\cdots|h_{k}\}]
\end{align*}

without indeterminacy, since $3 \!\le\! k \!\le\! 5$.
Then it implies $v \cupp \delta u' = v \cupp c'$ in $C^6(S^3\times_{\ad{}}P^5G;\field_{2})$, and we obtain $\delta u = \delta(v \cupp u') = v \cupp \delta u' = v \cupp c' = c$ in $C^6(S^3\times_{\ad{}}P^5G;\field_{2})$, since $\delta v = 0$.

To show the existence of the solution $u'$ of (<ref>), we use the matrix representations of homomorphisms $\kappa^{*}c'$, $u'$ and $\delta$ which are respectively denoted by a $1 \times d_{5}$ matrix $T_{\!c'}$, a $1 \times d_{4}$ matrix $T_{\!u'}$ and a $d_{5} \times d_{4}$ matrix $T_{\!\delta}$, where we denote $d_{i}=\dim[\field_{2}]{C^{i}(S^3\times_{\ad{}}P^4G;\field_{2})}$,
namely, $d_{4} = 5256$ and $d_{5} = 9280$, while $\dim[\field_{2}]{C^{5}(S^3\times_{\ad{}}P^5G;\field_{2})} = 42048$.
namely, $d_{4} = 3192$ and $d_{5} = 5537$, while $\dim[\field_{2}]{C^{5}(S^3\times_{\ad{}}P^5G;\field_{2})} = 22344$.

Then (<ref>) is realized using matrices above as the following linear equation on $\mathbb{x}={}^tT_{\!u'}$:

\[\tag{Eq. \!M}\label{eq:Equation matrix}
{}^tT_{\!\delta} \mathbb{x} = {}^tT_{\!c'}.
\]

We remark that the existence of the solution of (<ref>) implies the existence of the solution of (<ref>), and hence the existence of the solution of (<ref>) by Proposition <ref>.

On the other hand, we know the following statements from the theory of linear algebra.

There is a solution of (<ref>), provided that the following equation holds:

\[\tag{Eq. \!R}\label{eq:Equation rank}
\mbox{rank } {}^tT_{\!\delta} = \mbox{rank }\left[ {}^tT_{\!\delta} | {}^tT_{\!c'}\right],
\]

where $\left[ {}^tT_{\!\delta} | {}^tT_{\!c'}\right]$ denotes the augmented coefficients matrix of the equation (<ref>).

The size of the augmented coefficients matrix of (<ref>) exceeds the size for manipulation, we are forced to use a computer program to show (<ref>).
Fortunately, our python program stops saying that (<ref>) is correct.

Thus there is a solution $u'$ of (<ref>), and we obtain $\delta u' = \kappa^{*}c'$.
Then by Proposition <ref>, $u=v \cupp u'$ gives a solution of (<ref>), and we obtain $[\kappa^{*}c] = [\delta u] = 0$.

[Lemma <ref>]
$(e_{5}^{\double{M}})^{\ast}(z \otimes z) \!=\! 0$ in $H^{\ast}(P^{5}_{B}(\LoopB{\double{M}});\field_{2})$.

Since ${i}^{*} : H^{*}({BG};\field_{2}) \to H^{*}({M};\field_{2})$ is surjective by the proof of Proposition <ref>, we know that $z \otimes z = (i \times \id)^{*}(z \otimes z)$.
Then using a solution $u$ of (<ref>), we obtain

\begin{align*}
(e_{5}^{\double{M}})^{\ast}(z \otimes z)
&= (e_{5}^{\double{M}})^{\ast}(\id_{M} \times i)^{\ast}(z \otimes z)
\\&= (f\!\comp\lambda)^{\ast}\kappa^{\ast}(e''_{\infty})^{\ast}(z \otimes z)
= (f\!\comp\lambda)^{\ast}\kappa^{\ast}[c]
\ifdefined\reviseone\else
\ \textem{ = (f\!\comp\lambda)^{\ast}[\delta (u) ] }
\fi
= 0,
\end{align*}

where $\kappa : S^3\times_{\ad{}}P^{5}G \hookrightarrow S^3\times_{\ad{}}P^{\infty}G$ denotes the canonical inclusion.

It implies $(e_{5}^{\double{M}})^{\ast}(z \otimes z) \!=\! 0$ in $H^{\ast}(P^{5}_{B}(\LoopB{\double{M}});\field_{2})$, and we are done.

§ APPENDIX A

First, the following is the algorithm to check the equation (<ref>).

$G$ : Cayley table of $Q_8$, $L$ : the set of cells of $S^3/Q_8$, the modulo $2$ boundary formulas of product cells and $\delta u'$ : the cocycle in $C^{5}(S^3\times_{\ad{}}P^4G;\field_{2})$
The status of the equation (<ref>)
$Cells4$ : All $4$-cells in $S^3\times_{\ad{}}P^4G$

$Cells5$ : All $5$-cells in $S^3\times_{\ad{}}P^4G$

$delta$ : the matrix representing $\delta$ of type $(\#Cells5,\#Cells4)$

$Delta$ : the augmeted matrix of type $(\#Cells5,\#Cells4 \!+\! 1)$ ($1$ for $c'$)

We determine $Delta$ using the following algorithm.

$Delta = []$

$s \in Cells5$ /* Comments below are in the case when $s = [e^{2}_{1}|\{a|a\} ]$ */
$(f_s | c_s) = $ the tuple with column '$column$' /* $(-,\dots,-)$ */

$(\delta u')(s) = 1$$(f_s | c_s)_{answer} = 1$ /* $(-,\dots,-,1)$ */$(f_s | c_s)_{answer} = 0$ /* $(-,\dots,-,0)$ */
$t \in C_3$
$t \in \partial s$/* $\partial [e^{2}_{1}|\{a|a\} ] = \{[e^{1}_{2}|\{a|a\}], [e^{1}_{2}|\{a^3|a^3\}], [e^{2}_{1}|\{a^2\} ] \}$ */
$(f_s | c_s)_t = 1$

$(f_s | c_s)_t = 0$
/*$(f_{[e^{2}_{1}|\{a|a\} ]} | c_{[e^{2}_{1}|\{a|a\} ]}) = (0,0,1,\dots, 0)$; in which $1$ is appearing at the place corresponding to $[e^{1}_{2}|\{a|a\}], [e^{1}_{2}|\{a^3|a^3\}]$ or $ [e^{2}_{1}|\{a^2\} ]$*/

/* the tuple $(f_s | c_s)$ consists of 0 or 1 */

add $(f_s | c_s)$ to $Delta$
This gives $Delta = [(f_{s_1} | c_{s_1}),\dots,(f_{s_n} | c_{s_n})]_{s_i \in Cells5}$.
We transform $Delta$ into the reduced row echelon form using Gaussian elimination.

The algorithm to check the equation (<ref>)

The following is the source code written in Python 3 to realize Algorithm 1.

[basicstyle=, frame=single, breaklines=true]
import numpy as np
from itertools import count, product

# z in H^Element(P^HeadΩM)
Manifold = 3    # = dimM
Element = 2     # = degz =< dimM
Head = 4        # >= dimM

# Special Cocycles
def u(a) : # x
return a
def v(a): # y
return a // 4
def z(i): # x^2
return u(i[0]) * u(i[1])

# Cell decomposition of M
SpaceForm = [e0,e11,e12,e21,e22,e3] = ["e0","e11","e12","e21","e22","e3"]

# Group Structure
Group = [e,x,xx,xxx,y,xy,xxy,xxxy] = [0,1,2,3,4,5,6,7]
Mul = [
Inv = [e,xxx,xx,x,xxy,xxxy,y,xy]

# Adjoint Action
def adj(a,b):
return Mul[Inv[a]][Mul[b][a]]
def adl(a,lst):
return [adj(a,i) for i in lst]

# Boundaries of the fibre - the resolution of the group
def difi(i,lst):
n = len(lst)
if i == 0:
return lst[1:]
elif i == n:
return lst[:n-1]
l = lst[:]
l[i-1:i+1] = [Mul[lst[i-1]][lst[i]]]
return l
def partial(s,lst):
ans1 = [(s,difi(i,lst)) for i in range(len(lst)+1)]
ans2 = []
for i in ans1:
if i in ans2:
return ans2

# Boundaries of the Fibrewise resolution
def boundary(s,w):
if s == e3:
return [(e21,w),(e21,adl(x,w)),(e22,w),(e22,adl(Mul[x][y],w))] + partial(s,w)
elif s == e21:
return [(e11,w),(e11,adl(x,w)),(e12,w),(e12,adl(y,w))] + partial(s,w)
elif s == e22:
return [(e11,w),(e11,adl(Mul[x][y],w)),(e12,w),(e12,adl(x,w))] + partial(s,w)
elif s == e11:
return [(e0,w),(e0,adl(x,w))] + partial(s,w)
elif s == e12:
return [(e0,w),(e0,adl(y,w))] + partial(s,w)
elif s == e0:
return partial(s,w)
return 'miss!'

# Cells of the Reduced Projective Space up to dimension 'degree'
NonDegenerates = [7, 3, 6, 2, 5, 1, 4] # 823
lst = [[[]],[],[],[],[],[],[],[],[]]
for a in range(Head):
for i,j in product(NonDegenerates,lst[a]):
if j == []:
w = [i]
w = [i] + j

# Counting Cells of each dimension.
Degree = Element + Manifold	# upper degree
degree = Degree - 1		# lower degree
LowerCells = [] # degree cells
for i in lst[degree]:
for i in lst[degree-1]:
for i in lst[degree-2]:
for i in lst[degree-3]:
UpperCells = [] # Degree cells
for i in lst[Degree]:
for i in lst[Degree-1]:
for i in lst[Degree-2]:
for i in lst[Degree-3]:

TheNumberofLowerCells = len(LowerCells)
TheNumberofUpperCells = len(UpperCells)

LabeltoNum = LowerCells[i]:i for i in range(TheNumberofLowerCells)

print('The Number of ' + str(degree) + '-cells is ' + str(TheNumberofLowerCells) + '.')
print('The Number of ' + str(Degree) + '-cells is ' + str(TheNumberofUpperCells) + '.')

# Initializing Augmented Coefficients Matrix
TheNumberofLowerCellsPlusAns = TheNumberofLowerCells + 1
TheNumberofUpperCellsPlusAns = TheNumberofUpperCells + 1

print("The size of the coefficients matrix delta is 0x1.".format(TheNumberofUpperCells,TheNumberofLowerCells))

# Constructing Delta the set of coodinates whose entries are 1.
num = 0
Delta = [] # the set of coordinates whose entries are 1.
for i in lst[Degree]: # (e0,i) in UpperCells
c = []
J = boundary(e0,i)
for k in boundary(e0,i):
if k in c:
for j in c:
j = str(j)
if j in LowerCells:
num += 1
for i in lst[Degree-1]: # (e11,i) and (e12,i) in UpperCells
c = []
for k in boundary(e11,i):
if k in c:
for j in c:
j = str(j)
if j in LowerCells:
num += 1
c = []
for k in boundary(e12,i):
if k in c:
for j in c:
j = str(j)
if j in LowerCells:
num += 1
for i in lst[Degree-2]: # (e21,i) and (e22,i) in UpperCells
c = []
for k in boundary(e21,i):
if k in c:
for j in c:
j = str(j)
if j in LowerCells:
num += 1
c = []
for k in boundary(e22,i):
if k in c:
for j in c:
j = str(j)
if j in LowerCells:
num += 1
for i in lst[Degree-3]: # (e3,i) in UpperCells
ans = z(i)
if ans == 1:
c = []
for k in boundary(e3,i):
if k in c:
for j in c:
j = str(j)
if j in LowerCells:
num += 1

AugCMat = np.zeros((TheNumberofUpperCells,TheNumberofLowerCellsPlusAns),dtype = bool)  # Augmented Coefficients Matrix
for i in Delta:
AugCMat[i[0]][i[1]] = 1
CMat = np.delete(AugCMat,TheNumberofLowerCells,1)

print("The size of the augmented coefficients matrix Delta is 0x1.".format(TheNumberofUpperCells,TheNumberofLowerCellsPlusAns))

# Transform AugCMat into the reduced row echelon form.
EAugCMat = AugCMat  # row echelon form of AugCMat
count = TheNumberofUpperCells
for j in range(0,TheNumberofLowerCells):
num = -1
for i in range(0,TheNumberofUpperCells):
if num == -1: # no i st M[i][j] = 1
if i < count:
if EAugCMat[i][j] == 1:
num = i
else: # For i > num, add EAugCMat[num] to EAugCMat[i]
if EAugCMat[i][j] == 1:
EAugCMat[i] = EAugCMat[i] ^ EAugCMat[num]
if num > -1: # move num culumn to the bottom
EAugCMat = np.append(EAugCMat,[EAugCMat[num]],axis = 0)
EAugCMat = np.delete(EAugCMat,num,0)
count -= 1 # one more principal line
RankCMat = TheNumberofUpperCells - count
# row echelon form of CMat
ECMat = np.delete(EAugCMat,TheNumberofLowerCells,1)
for i in range(0,count):
ECMat = np.delete(ECMat, 0, 0)
print("The rank of the matrix delta is 0.".format(RankCMat))
num = -1
for i in range(0,TheNumberofUpperCells):
if num == -1: # no i st M[i][TheNumberofLowerCells] = 1
if i < count:
if EAugCMat[i][TheNumberofLowerCells] == 1:
num = i
else: # For i > num, add EAugCMat[num] to EAugCMat[i]
if EAugCMat[i][TheNumberofLowerCells] == 1:
EAugCMat[i] = EAugCMat[i] ^ EAugCMat[num]
if num > -1: # move num culumn to the bottom
EAugCMat = np.append(EAugCMat,[EAugCMat[num]],axis = 0)
EAugCMat = np.delete(EAugCMat,num,0)
count -= 1 # one more principal line
RankAugCMat = TheNumberofUpperCells - count
# row echelon form of AugCMat
for i in range(0,count):
EAugCMat = np.delete(EAugCMat, 0, 0)
print("The rank of the matrix Delta is 0.".format(RankAugCMat))

# Display a special solution
if RankCMat == RankAugCMat:
for i in range(RankCMat):
b = EAugCMat[i][TheNumberofLowerCells]
if b == 1:
for j in range(0,TheNumberofLowerCells):
if ECMat[i][j] == 1:
LengthofSolution = len(Sol)
if LengthofSolution > 0:
print("The length of one particular solution is 0.".format(LengthofSolution))
print("The particular solution is 0".format(LowerCells[Sol[0]]), end=”)
for i in range(1,len(Sol)):
print(" + 0".format(LowerCells[Sol[i]]), end=”)
# Verify the solution
for i in range(TheNumberofUpperCells):
num = AugCMat[i][TheNumberofLowerCells]
for j in Sol:
num = num ^ CMat[i][j]
if num != 0:
if i == TheNumberofUpperCells-1:

The above python program produces the following outputs:

We use python program which produces the following outputs:

[basicstyle=, frame=single, breaklines=true]
The Number of 4-cells is 5256.
The Number of 5-cells is 9280.
The size of the coefficients matrix delta is 9280x5256.
The size of the augmented coefficients matrix Delta is 9280x5257.
The rank of the matrix delta is 3600.
The rank of the matrix Delta is 3600.

[basicstyle=, frame=single, breaklines=true]
The Number of 4-cells is 3192.
The Number of 5-cells is 5537.
The size of the coefficients matrix delta is 5537x3192.
The size of the augmented coefficients matrix Delta is 5537x3193.
The rank of the matrix delta is 2214.
The rank of the matrix Delta is 2214.
The length of one particular solution is 823.

The program outputs a list of cells in the form of .
In fact, in our program, $[0,1,2,3,4,5,6,7]=[e,a,a^{2},a^{3},b,ab,a^{2}b,a^{3}b]$.
To compress the list, we transform them into the form of $[e0|7|7|7|6]$ which means the cell $[e^0|\{a^3b|a^3b|a^3b|a^2b\}]$.
Then one particular solution is given by $u'_{*} = [e0|{7|7|7|6}] + [e0|{7|7|6|6}] + [e0|{7|7|6|2}] + [e0|{7|7|2|5}] + [e0|{7|7|2|1}] + [e0|{7|7|5|3}] + [e0|{7|7|5|6}] + [e0|{7|7|5|5}] + [e0|{7|7|5|1}] + [e0|{7|7|1|3}] + [e0|{7|7|1|2}] + [e0|{7|3|6|5}] + [e0|{7|3|6|1}] + [e0|{7|3|5|3}] + [e0|{7|3|1|5}] + [e0|{7|6|7|5}] + [e0|{7|6|7|1}] + [e0|{7|6|3|3}] + [e0|{7|6|3|2}] + [e0|{7|6|3|5}] + [e0|{7|6|3|1}] + [e0|{7|6|6|2}] + [e0|{7|6|6|1}] + [e0|{7|6|2|7}] + [e0|{7|6|2|3}] + [e0|{7|6|2|6}] + [e0|{7|6|2|2}] + [e0|{7|6|2|1}] + [e0|{7|6|2|4}] + [e0|{7|6|5|3}] + [e0|{7|6|5|4}] + [e0|{7|6|1|7}] + [e0|{7|6|1|3}] + [e0|{7|6|1|2}] + [e0|{7|6|1|4}] + [e0|{7|2|7|6}] + [e0|{7|2|3|7}] + [e0|{7|2|3|2}] + [e0|{7|2|3|4}] + [e0|{7|2|6|6}] + [e0|{7|2|6|2}] + [e0|{7|2|2|6}] + [e0|{7|2|2|2}] + [e0|{7|2|2|5}] + [e0|{7|2|2|4}] + [e0|{7|2|5|2}] + [e0|{7|2|5|5}] + [e0|{7|2|5|1}] + [e0|{7|2|1|7}] + [e0|{7|2|1|3}] + [e0|{7|2|1|6}] + [e0|{7|2|1|4}] + [e0|{7|2|4|2}] + [e0|{7|5|7|6}] + [e0|{7|5|3|7}] + [e0|{7|5|3|2}] + [e0|{7|5|3|4}] + [e0|{7|5|6|3}] + [e0|{7|5|6|6}] + [e0|{7|5|6|2}] + [e0|{7|5|6|5}] + [e0|{7|5|2|2}] + [e0|{7|5|2|4}] + [e0|{7|5|5|3}] + [e0|{7|5|5|6}] + [e0|{7|5|5|2}] + [e0|{7|5|5|1}] + [e0|{7|5|1|3}] + [e0|{7|5|1|4}] + [e0|{7|5|4|2}] + [e0|{7|1|7|3}] + [e0|{7|1|7|2}] + [e0|{7|1|7|5}] + [e0|{7|1|3|3}] + [e0|{7|1|3|5}] + [e0|{7|1|6|3}] + [e0|{7|1|2|3}] + [e0|{7|1|2|6}] + [e0|{7|1|2|2}] + [e0|{7|1|2|4}] + [e0|{7|1|5|2}] + [e0|{7|1|5|5}] + [e0|{7|1|5|4}] + [e0|{7|1|1|3}] + [e0|{7|1|1|4}] + [e0|{7|1|4|3}] + [e0|{7|1|4|2}] + [e0|{7|1|4|5}] + [e0|{7|4|7|1}] + [e0|{7|4|3|7}] + [e0|{7|4|3|6}] + [e0|{7|4|6|7}] + [e0|{7|4|6|3}] + [e0|{7|4|6|6}] + [e0|{7|4|2|2}] + [e0|{7|4|2|4}] + [e0|{7|4|5|2}] + [e0|{7|4|5|4}] + [e0|{7|4|1|2}] + [e0|{7|4|4|2}] + [e0|{3|7|7|3}] + [e0|{3|7|7|2}] + [e0|{3|7|7|5}] + [e0|{3|7|7|1}] + [e0|{3|7|3|7}] + [e0|{3|7|3|3}] + [e0|{3|7|3|6}] + [e0|{3|7|3|5}] + [e0|{3|7|3|4}] + [e0|{3|7|6|5}] + [e0|{3|7|6|1}] + [e0|{3|7|2|3}] + [e0|{3|7|2|5}] + [e0|{3|7|5|7}] + [e0|{3|7|5|2}] + [e0|{3|7|5|5}] + [e0|{3|7|1|6}] + [e0|{3|7|1|4}] + [e0|{3|7|4|5}] + [e0|{3|7|4|1}] + [e0|{3|3|6|7}] + [e0|{3|3|2|1}] + [e0|{3|3|5|7}] + [e0|{3|3|5|3}] + [e0|{3|3|5|5}] + [e0|{3|3|1|6}] + [e0|{3|3|1|1}] + [e0|{3|6|7|3}] + [e0|{3|6|7|6}] + [e0|{3|6|7|2}] + [e0|{3|6|7|5}] + [e0|{3|6|3|7}] + [e0|{3|6|3|3}] + [e0|{3|6|3|6}] + [e0|{3|6|3|5}] + [e0|{3|6|3|4}] + [e0|{3|6|6|3}] + [e0|{3|6|6|6}] + [e0|{3|6|6|1}] + [e0|{3|6|6|4}] + [e0|{3|6|2|7}] + [e0|{3|6|2|3}] + [e0|{3|6|2|5}] + [e0|{3|6|2|1}] + [e0|{3|6|5|3}] + [e0|{3|6|5|5}] + [e0|{3|6|5|1}] + [e0|{3|6|1|7}] + [e0|{3|6|1|2}] + [e0|{3|6|1|5}] + [e0|{3|6|1|4}] + [e0|{3|6|4|7}] + [e0|{3|6|4|5}] + [e0|{3|2|7|1}] + [e0|{3|2|3|7}] + [e0|{3|2|3|6}] + [e0|{3|2|6|6}] + [e0|{3|2|6|2}] + [e0|{3|2|6|5}] + [e0|{3|2|6|4}] + [e0|{3|2|2|2}] + [e0|{3|2|2|5}] + [e0|{3|2|2|1}] + [e0|{3|2|2|4}] + [e0|{3|2|5|7}] + [e0|{3|2|5|3}] + [e0|{3|2|5|6}] + [e0|{3|2|5|2}] + [e0|{3|2|5|5}] + [e0|{3|2|5|1}] + [e0|{3|2|1|2}] + [e0|{3|2|1|1}] + [e0|{3|2|4|2}] + [e0|{3|5|7|7}] + [e0|{3|5|7|6}] + [e0|{3|5|7|1}] + [e0|{3|5|3|3}] + [e0|{3|5|3|6}] + [e0|{3|5|3|5}] + [e0|{3|5|3|4}] + [e0|{3|5|6|6}] + [e0|{3|5|6|5}] + [e0|{3|5|6|1}] + [e0|{3|5|2|7}] + [e0|{3|5|2|2}] + [e0|{3|5|2|5}] + [e0|{3|5|2|4}] + [e0|{3|5|5|5}] + [e0|{3|5|5|4}] + [e0|{3|5|1|5}] + [e0|{3|5|4|2}] + [e0|{3|5|4|5}] + [e0|{3|5|4|1}] + [e0|{3|1|7|1}] + [e0|{3|1|3|7}] + [e0|{3|1|3|6}] + [e0|{3|1|3|5}] + [e0|{3|1|6|3}] + [e0|{3|1|6|6}] + [e0|{3|1|6|5}] + [e0|{3|1|2|6}] + [e0|{3|1|2|2}] + [e0|{3|1|2|4}] + [e0|{3|1|5|7}] + [e0|{3|1|5|6}] + [e0|{3|1|5|4}] + [e0|{3|1|1|2}] + [e0|{3|1|4|2}] + [e0|{3|4|6|5}] + [e0|{3|4|6|1}] + [e0|{3|4|5|3}] + [e0|{3|4|1|5}] + [e0|{6|7|7|3}] + [e0|{6|7|3|3}] + [e0|{6|7|3|6}] + [e0|{6|7|3|2}] + [e0|{6|7|5|3}] + [e0|{6|7|5|2}] + [e0|{6|7|5|4}] + [e0|{6|7|1|5}] + [e0|{6|7|4|3}] + [e0|{6|7|4|5}] + [e0|{6|3|7|2}] + [e0|{6|3|7|5}] + [e0|{6|3|7|1}] + [e0|{6|3|3|2}] + [e0|{6|3|3|5}] + [e0|{6|3|3|1}] + [e0|{6|3|6|3}] + [e0|{6|3|6|5}] + [e0|{6|3|6|1}] + 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[e12|{7|7|1}] + [e12|{7|3|7}] + [e11|{7|3|3}] + [e11|{7|3|4}] + [e11|{7|6|7}] + [e12|{7|6|7}] + [e12|{7|6|3}] + [e12|{7|6|6}] + [e11|{7|6|2}] + [e12|{7|6|5}] + [e12|{7|6|1}] + [e11|{7|6|4}] + [e11|{7|2|3}] + [e11|{7|2|6}] + [e11|{7|2|2}] + [e12|{7|2|2}] + [e11|{7|2|5}] + [e11|{7|2|1}] + [e12|{7|2|4}] + [e11|{7|5|3}] + [e12|{7|5|3}] + [e12|{7|5|6}] + [e11|{7|5|2}] + [e11|{7|1|7}] + [e12|{7|1|2}] + [e12|{7|1|5}] + [e12|{7|1|4}] + [e11|{7|4|3}] + [e11|{7|4|2}] + [e12|{7|4|2}] + [e11|{7|4|1}] + [e11|{7|4|4}] + [e11|{3|7|7}] + [e11|{3|7|3}] + [e11|{3|7|6}] + [e11|{3|7|2}] + [e12|{3|7|2}] + [e12|{3|7|1}] + [e11|{3|7|4}] + [e12|{3|7|4}] + [e11|{3|3|7}] + [e12|{3|3|7}] + [e11|{3|3|2}] + [e12|{3|3|2}] + [e11|{3|3|1}] + [e12|{3|6|7}] + [e12|{3|6|3}] + [e12|{3|6|6}] + [e12|{3|6|5}] + [e12|{3|6|1}] + [e11|{3|2|7}] + [e11|{3|2|6}] + [e12|{3|2|2}] + [e11|{3|2|5}] + [e11|{3|2|1}] + [e11|{3|2|4}] + [e12|{3|2|4}] + [e11|{3|5|7}] + [e12|{3|5|3}] + [e11|{3|5|1}] + [e11|{3|1|7}] + [e11|{3|1|6}] + [e12|{3|1|6}] + [e11|{3|1|2}] + [e11|{3|1|5}] + [e12|{3|1|5}] + [e11|{3|4|7}] + [e11|{3|4|3}] + [e11|{3|4|6}] + [e11|{3|4|2}] + [e12|{3|4|2}] + [e11|{3|4|1}] + [e12|{6|7|7}] + [e11|{6|7|3}] + [e12|{6|7|2}] + [e12|{6|7|5}] + [e11|{6|3|7}] + [e12|{6|3|6}] + [e11|{6|3|5}] + [e11|{6|3|4}] + [e12|{6|3|4}] + [e12|{6|6|7}] + [e12|{6|6|3}] + [e11|{6|6|2}] + [e11|{6|6|5}] + [e11|{6|2|7}] + [e12|{6|2|7}] + [e11|{6|2|3}] + [e11|{6|2|2}] + [e11|{6|2|5}] + [e12|{6|2|1}] + [e11|{6|5|7}] + [e12|{6|5|7}] + [e12|{6|5|3}] + [e11|{6|5|2}] + [e12|{6|5|2}] + [e11|{6|5|5}] + [e12|{6|5|1}] + [e11|{6|1|7}] + [e12|{6|1|7}] + [e12|{6|1|6}] + [e11|{6|1|2}] + [e11|{6|1|5}] + [e12|{6|1|5}] + [e11|{6|1|1}] + [e12|{6|1|1}] + [e11|{6|1|4}] + [e12|{6|1|4}] + [e12|{6|4|7}] + [e12|{6|4|3}] + [e11|{6|4|5}] + [e11|{6|4|1}] + [e12|{2|7|7}] + [e11|{2|7|3}] + [e11|{2|7|1}] + [e12|{2|7|1}] + [e11|{2|7|4}] + [e12|{2|3|7}] + [e11|{2|3|3}] + [e12|{2|3|3}] + [e11|{2|3|6}] + [e11|{2|3|5}] + [e11|{2|6|7}] + [e11|{2|6|3}] + [e11|{2|6|6}] + [e12|{2|6|5}] + [e11|{2|6|1}] + [e12|{2|6|1}] + [e11|{2|2|7}] + [e12|{2|2|7}] + [e12|{2|2|3}] + [e11|{2|2|6}] + [e11|{2|5|7}] + [e11|{2|1|3}] + [e11|{5|7|7}] + [e11|{5|7|3}] + [e11|{5|3|7}] + [e11|{5|3|2}] + [e11|{5|3|4}] + [e11|{5|6|7}] + [e11|{5|6|3}] + [e11|{5|6|2}] + [e11|{5|2|7}] + [e11|{5|2|3}] + [e11|{5|2|6}] + [e11|{5|5|3}] + [e11|{1|7|7}] + [e11|{1|7|6}] + [e11|{1|7|2}] + [e11|{1|3|7}] + [e11|{1|3|3}] + [e11|{1|6|2}] + [e11|{1|2|7}] + [e11|{1|2|3}] + [e11|{1|2|6}] + [e11|{1|1|7}] + [e12|{4|7|1}] + [e12|{4|3|7}] + [e12|{4|6|7}] + [e12|{4|6|3}] + [e21|{7|7}] + [e21|{7|6}] + [e21|{7|2}] + [e21|{7|1}] + [e22|{7|1}] + [e22|{7|4}] + [e22|{3|3}] + [e21|{3|6}] + [e22|{3|2}] + [e21|{3|5}] + [e22|{3|5}] + [e22|{3|4}] + [e22|{6|7}] + [e21|{6|3}] + [e22|{6|3}] + [e21|{6|5}] + [e21|{2|7}] + [e22|{2|3}] + [e21|{2|2}] + [e22|{2|2}] + [e3|{7}] + [e3|{3}]
The outputs of the program indicates a solution $u'$ of (<ref>) which is given as follows:

\begin{align*}
u'_{*} = &
[e^0|\{a^3b|a^3b|a^3b|a^{2}b\}]^{*} + [e^0|\{a^3b|a^3b|a^2b|a^2b\}]^{*} + [e^0|\{a^3b|a^3b|a^{2}b|a^2\}]^{*}
\\&
+ [e^0|\{a^3b|a^3b|a^2|ab\}]^{*} + \cdots + [e^3|\{a^3b\}]^{*} + [e^{3}|\{a^{3}\}]^{*},
\end{align*}

where $u'_{*}$ denotes the dual of $u'$ in $C_{4}(S^3\times_{\ad{}}P^{4}G;\field_{2})$.

Thus (<ref>) has one solution at least, and it completes the proof of Lemma <ref>.

If we consider (<ref>) in $C^{5}(S^3\times_{\ad{}}P^{5}G;\field_{2}) = C^{5}(S^3\times_{\ad{}}P^{\infty}G;\field_{2})$ instead of $C^{5}(S^3\times_{\ad{}}P^{4}G;\field_{2})$, we have no solution as the python program produces the following outputs.
[basicstyle=, frame=single, breaklines=true]
The Number of 4-cells is 3192.
The Number of 5-cells is 22344.
The size of the coefficients matrix delta is 22344 * 3192.
The size of the augmented coefficients matrix Delta is 22344 * 3193.
The rank of the matrix delta is 2789.
The rank of the matrix Delta is 2790.
It should imply that $z \otimes x^{2}$ is non-zero in $H^{*}(S^3\times_{\ad{}}P^{5}G;\field_{2})$ and $\wgtB{z \otimes x^{2};\field_{2}}=5$.

§ ACKNOWLEDGEMENTS

The authors are very grateful to Takeshi Nanri for his kind advice to improve python program.
The computation was carried out using the computer resource offered under the category of Trial Use Projects by Research Institute for Information Technology (R.075emI.75ex$2$.1emT), Kyushu University.
In fact, we obtained a direct solution for ${}^{t}T_{\!\delta} \,\mathbb{x} = {}^{t}T_{\!c}$, while it is too big to describe here.
[basicstyle=, frame=single, breaklines=true]
The Number of 5-cells is 22344.
The Number of 6-cells is 38759.
The size of the coefficients matrix delta is 38759x22344.
The size of the augmented coefficients matrix Delta is 38759x22345.
The rank of the matrix delta is 15724.
The rank of the matrix Delta is 15724.
The length of one particular solution is 5546.
This research was supported by Grant-in-Aid for Scientific Research (S) #17H06128 and Exploratory Research #18K18713 from Japan Society for the Promotion of Science.

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author=Milgram, R. James,
title=Cohomology of finite groups,
series=Grundlehren der Mathematischen Wissenschaften [Fundamental
Principles of Mathematical Sciences],
publisher=Springer-Verlag, Berlin,

author=Benson, D. J.,
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title=On the $K$-ring of $S^{4n+3}/H_{m}$,
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author=Iwase, Norio,
author=Sakai, Michihiro,
title=Topological complexity is a fibrewise L-S category,
journal=Topology Appl.,

author=Iwase, Norio,
author=Sakai, Michihiro,
author=Tsutaya, Mitsunobu,
title=A short proof for ${\rm tc}(K)=4$,
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author=James, I. M.,
title=Introduction to fibrewise homotopy theory,
booktitle=Handbook of algebraic topology,
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title=Categories and cohomology theories,

author=Shimakawa, Kazuhisa,
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title=Homology and cohomology via enriched bifunctors,
journal=Kyushu J. Math.,

author=Whitehead, George W.,
title=Elements of homotopy theory,
series=Graduate Texts in Mathematics,
publisher=Springer-Verlag, New York-Berlin,


# Coverage Enhancement for Vehicles

Stefan Runeson  Ericsson  Ali Zaidi  Ericsson  Ana Cantarero  BMW Group

###### Abstract

The Third Generation Partnership Project (3GPP) has standardized Coverage
Enhancement (CE) for Internet-of-Things (IoT) to connect devices in
challenging radio conditions with cellular networks. CE is based on the
principle of prolonged transmission time that exploits the fact that many IoT
applications have relaxed requirements on data rate and latency, and the
coverage can be significantly boosted by repeating transmissions for such
applications. However, CE consumes a lot of radio resources and should be
implemented carefully for different applications. This paper presents an end-
to-end concept for realizing dynamic use of CE for connected vehicles in a
resource efficient way. The proposed framework has the potential of improving
coverage by around 10dB for low data rate connected vehicle applications,
based on the 3GPP Long Term Evolution (LTE) standard.

##  I Introduction

The automotive industry is going through a fast-paced evolution, with the
connected vehicle as one of its focus. A connected vehicle refers to one which
can connect over wireless networks to other devices and its surroundings. This
may go from vehicle-to-vehicle (V2V) and vehicle-to-infrastructure (V2I)
communications, which are designed to increase situation awareness and road
safety, to vehicle-to-network (V2N) communication, which supports a wider area
over a cellular network. This paper is focused on V2N services which can be
benefited from CE and how could this feature be enabled on a vehicular modem.

Cellular V2N communication enables a wide range of services in the vehicle,
including Cooperative Intelligent Transport Systems (C-ITS), Advanced Driver
Assistance Systems (ADAS), dynamic high-definition maps, vehicle-centric
Original Equipment Manufacturer (OEM) telematics, fleet management, logistics
services and infotainment. Many of these services require high data rates and
low latency for their reliability to be guaranteed. In this case, CE is not an
option for optimization, due to the high amount of required radio resources.
There are others however, whose data rate requirements are low and on which
coverage and network connectivity should always be guaranteed. Here, CE could
play a major role for optimization.

These low data rate services include, among others, basic vehicle status
updates to the OEM backend servers, remote vehicle lock or unlock and sharing
of vehicle diagnostics, which are of major importance for fleet management and
OEM logistics. Specially with the evolution of mobility concepts, such as car
sharing, the fleet manager must have knowledge on the location and status of
each vehicle on its fleet. The vehicles on the other side, should be able to
connect to the backend servers when needed, even in areas of poor coverage.
Areas such as underground garages or rural zones are of special interest,
since their network coverage may be worse in comparison with urban zones or
garages above ground.

##  II Concept description

###  II-A 3GPP standardization of Coverage Enhancement

Third Generation Partnership Project (3GPP) has standardized Coverage
Enhancement (CE) techniques for Internet-of-Things (IoT) devices in Release 13
and further evolved it in Release 14 [1]. Coverage Enhancement is primarily
achieved by repetition or retransmission techniques for applications that have
relaxed requirements on data rate and latency [1].

3GPP has standardized two CE modes: i) CE mode A, supporting up to 32 subframe
repetitions with estimated coverage gain of around 10 dB, ii) CE mode B,
supporting up to 2048 subframe repetitions. The repetitions are costly in
terms of spectrum use, which is a scarce resource, and therefore this paper
focuses on CE mode A and low data rate connected vehicle applications.

There is a wide range of LTE UE categories supporting different capabilities
and peak data rates [2]. CE mode A is mandatory for LTE Category-M1 UE, which
is a very low complexity UE suitable for very low cost modems targeting low
data rate connections (below 1Mbps peak rates). The connected vehicles
typically employ higher category LTE UEs (capable of supporting tens or
hundreds of Mbps) due to the need for delivering high data rates. CE mode A is
optional for high category LTE UEs and is beneficial only when the device is
out of normal LTE coverage and is supporting one or multiple low data rate
applications.

###  II-B Usage of Coverage Enhancement

One way to take advantage of the benefits of CE in a vehicle would be to
install an additional modem of LTE Cat-M1. This modem would then only be used
for services with low requirements on data rates and latency. However, the
cost would be increased with dual modems and the administrative effort would
rise with dual Subscriber Identity Modules (SIMs).

The preferred concept is rather to use a single modem of high LTE UE category
and allow the vehicle to dynamically enable the use of CE mode A in this
modem.

When the vehicle is outside normal coverage and only use services with low
requirements on data rates and latency, it can enable the use of CE mode A in
the modem of high LTE UE category.

One example where CE may be used is when the vehicle is parked outside normal
coverage and only uses “remote services”111Transmission of a small information
packet to the vehicle in order to trigger some action, such as starting the
heating or flashing the headlights. and “real time monitoring
services”222Periodic transmissions from electrical vehicles to a backend
server in order to track the current status of the battery., see Fig. 1.

Figure 1: Example of usage of Coverage Enhancement

Another example where CE may be used is an emergency situation outside normal
coverage, where the vehicle may shut down other services and only send
emergency messages.

###  II-C Enable and disable the use of CE mode A in the vehicle’s modem

Different UEs have different capabilities. The capabilities could be what
frequency bands the UE supports, what radio features it supports, to name a
few. The UE informs the mobile network of its capabilities when the UE
attaches to the mobile network. The mobile network takes the individual UE’s
capabilities into account when it interacts with the UE.

During the attach procedure, the base station (eNB) in the mobile network uses
the Radio Resource Control (RRC) message “UE Capability Enquiry” to retrieve
the UE capabilities from the UE. The UE capabilities are stored in the serving
base station (eNB) but also in the Mobility Management Entity (MME), in order
to keep the UE capabilities in the mobile network when the UE moves between
different base stations.

Support for Coverage Enhancement is one such UE capability. The UE may
indicate its support for CE mode A in its UE capabilities by setting the
parameter “ce-ModeA-r13” to “supported” [3]. In order to update the mobile
network with new UE capabilities, the UE must start a new attach procedure,
see Fig. 2. Radio Resource Control (RRC) is the control plane protocol between
UE and eNB. Non-Access Stratum (NAS) is the control plane protocol between UE
and MME. S1 Application Protocol (S1-AP) is the control plane protocol between
eNB and MME.

Figure 2: 3GPP procedures to enable support of CE mode A

###  II-D Restriction of use of Enhanced Coverage

The usage of Enhanced Coverage may require use of extensive radio resources
from the mobile network. 3GPP has standardized a restriction that enables the
mobile network operator to prevent specific subscribers from using Enhanced
Coverage [4].

The UE indicates its capability of support for restriction of use of Enhanced
Coverage in the attach procedure to the MME. MME receives “Enhanced Coverage
Restricted” parameter from the Home Subscriber Server (HSS). This parameter is
retained as part of subscription data in the HSS and indicates whether the
enhanced coverage functionality is restricted (i.e. disallowed) for the UE or
not.

For a UE that indicates support for restriction of use of Enhanced Coverage,
MME sends a “Enhanced Coverage Restricted” parameter to the UE in the “Attach
Accept” message. The UE assumes Enhanced Coverage is allowed unless explicitly
restricted by the mobile network.

If the MME has sent the “Enhanced Coverage Restricted” parameter to the UE,
the MME provides the “Enhanced Coverage Restricted” parameter to the eNB via
S1 Application Protocol (S1-AP), see Fig. 3. Diameter is the name of an
authentication, authorization, and accounting protocol for computer networks.
It is used between MME and HSS.

Figure 3: 3GPP procedures to restrict usage of Enhanced Coverage

Since the “Enhanced Coverage Restricted” parameter is kept as part of
subscription data in the HSS, the mobile network operator has a possibility to
control and also charge for the Coverage Enhancement service for individual
subscriptions.

###  II-E Coverage Enhancement for roaming vehicles

Roaming is an important aspect of all automotive services. Vehicles may be
manufactured in one country, sold in another country, and driven in a third
country.

3GPP has standardized the roaming interface between Home Public Land Mobile
Network (H-PLMN) and Visited Public Land Mobile Network (V-PLMN) for Coverage
Enhancement. S6a is the roaming interface between MME in the visited network
and HSS in the home network.

There are a few prerequisites for usage of Coverage Enhancement in the visited
network:

  1. 1.

eNB and MME in the visited network must support CE mode A for UEs of high LTE
UE category.

  2. 2.

MME in the visited network must get the subscription data “Enhanced Coverage
[Not] Allowed” from HSS in the home network for roaming UEs.

There might also be a need for an update of the commercial agreement between
the home network operator and the visited network operator regarding the
roaming UE’s use of Coverage Enhancement in the visited network.

###  II-F End-to-end concept for usage of Coverage Enhancement

Coverage Enhancement may be used when the vehicle is outside normal coverage
and only uses services with low requirements on data rates and latency. On the
other hand, Coverage Enhancement shall not be used when the vehicle uses
services with high requirements on data rates and latency.

In order to use Coverage Enhancement in a proper way, an end-to-end concept
for the system is needed. The system consists of three entities: vehicle,
mobile network, and cloud. It is described in Fig. 4.

Figure 4: End-to-end system using Coverage Enhancement

The services in the vehicle are connected via the mobile network to the OEM
backend servers in the cloud. Proprietary application layer protocols are
often used for these connections.

The vehicle has a Telematics Control Unit (TCU) with a modem and a connection
manager.

The modem is connected to the mobile network via the 3GPP wireless interface
(Uu). The modem establishes and monitors the connection to the mobile network.
It exposes its services via an Application Programming Interface (API). The
API could be based on AT-commands [5], or other types of interfaces.

The connection manager in the TCU establishes and monitors the connection from
the vehicle to the OEM backend servers. In addition, the connection manager
keeps track of whether the UE support CE mode A and whether any service in the
vehicle needs data traffic with high requirements on data rates and latency.

###  II-G End-to-end procedure to enable the use of CE mode A

The end-to-end procedure to enable the use of CE mode A consists of four
steps, see Fig. 5.

Figure 5: End-to-end procedure to enable the use of CE mode A

  1. 1.

The connection manager detects that the vehicle is outside normal coverage.
This could be based on information through the modem API.

  2. 2.

If one or more services in the vehicle need data traffic with high
requirements on data rates and latency, it is not possible to use CE mode A
and the procedure ends here.

  3. 3.

Otherwise, the connection manager updates the UE capabilities in the modem
through the modem API to indicate that CE mode A is supported.

  4. 4.

The connection manager starts an attach procedure in the modem through modem
API333If AT-commands are used as modem API, the “AT+CGATT=1” may be used [5]..

###  II-H End-to-end procedure to disable the use of CE mode A

The end-to-end procedure to disable the use of CE mode A consists of five
steps, see Fig. 6.

Figure 6: End-to-end procedure to disable the use of CE mode A

  1. 1.

A service notifies the connection manager that the service needs data traffic
with high requirements on data rates and latency.

  2. 2.

If the UE capabilities in the modem has no support for CE mode A, data traffic
with high requirements on data rates and latency is permitted and the
procedure ends here.

  3. 3.

Otherwise, the connection manager starts a detach procedure in the modem
through modem API444If AT-commands are used as modem API, the “AT+CGATT=0” may
be used [5].

  4. 4.

The connection manager updates the UE capabilities in the modem through the
modem API to indicate that CE mode A is not supported.

  5. 5.

The connection manager starts an attach procedure in the modem through modem
API.

##  III Conclusion

The paper presented an end-to-end concept for enabling a dynamic use of CE for
vehicles connected to cellular networks in a resource efficient manner. The
proposed framework can be realized with LTE based on 3GPP Release 14
specification, with a potential of around 10 dB coverage improvement for low
data rate applications. The framework presented in the paper can be
generalized to other types of moving IoT devices, for example, drones and
aerial vehicles. The concept can also be extended to 5G IoT devices, when the
CE functionality is fully standardized for 5G in the future 3GPP releases.

## References

  * [1] O. Liberg, M. Sundberg, E. Wang, J. Bergman, and J. Sachs, Cellular Internet of Things, Technologies, Standards and Performance, Academic Press, 2018.
  * [2] E. Dahlman, S. Parkvall, J. Skold, 4G LTE-Advanced Pro and The Road to 5G, Academic Press, 2016.
  * [3] 3GPP TS 36.306, Rel. 14, ”Evolved Universal Terrestrial Radio Access (E-UTRA); User Equipment (UE) radio access capabilities,” 2020\.
  * [4] 3GPP TS 23.401, Rel. 14, ”General Packet Radio Service (GPRS) enhancements for Evolved Universal Terrestrial Radio Access Network (E-UTRAN) access,” 2019\.
  * [5] 3GPP TS 27.007, Rel. 14, ”AT command set for User Equipment (UE),” 2018\.

Stefan Runeson<EMAIL_ADDRESS>is a Senior Specialist Connected
Vehicles at Ericsson Business Area Networks. He received his Master’s degree
in Engineering Physics from Lund University, Sweden. He joined Ericsson in
1997 to work with data communications in mobile phones. Since 2016, he has
been working with the impact of connected vehicles on mobile networks. Since
2017, he is Ericsson’s delegate in the 5G Automotive Association (5GAA).
---
Ali Zaidi<EMAIL_ADDRESS>is a strategic product manager for Cellular
IoT at Ericsson and also serves as the company’s head of IoT Competence. Since
joining Ericsson in 2014, he has been working with technology and business
development of 4G and 5G radio access at Ericsson. He is currently responsible
for LTE-M, URLLC, Industrial IoT, vehicle-to-everything and local industrial
networks. He holds a Ph.D. in telecommunications from KTH Royal Institute of
Technology, Stockholm.
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Ana Cantarero<EMAIL_ADDRESS>received her Master’s degree in
Communications Technology from the University of Ulm, Germany in 2012. In her
position at BMW Group, her field of research is the connected vehicle with
recent emphasis on Vehicle-to-everything (V2X) communication and future
wireless communication technologies. She is currently involved in several
national and European projects related to cellular communication networks
contributing to the groundwork of 5G. Before joining BMW Group, she worked as
Radio Access Network Planning Engineer at Millicom International Cellular and
as Software Developer for High-Frequency 4G and 5G Test Systems at Rohde &
Schwarz.
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