# The exponential parameterization of the quark mixing matrix

G. Dattoli<EMAIL_ADDRESS>E. Di Palma<EMAIL_ADDRESS>ENEA - Centro Richerche Frascati, Via Enrico Fermi 45, 00044, Frascati, Rome,
Italy

###### Abstract

We comment on the exponential parameterization of the quark mixing matrix, by
stressing that it naturally incorporates the Cabibbo structure and the
hierarchical features of the Wolfenstein form. We extend our results to the
neutrino mixing and introduce an exponential generator of the tribimaximal
matrix.

Quark mixing matrix, Kobayashi and Wolfenstein matrix , Cabibbo structure

##  I Introduction

The quark mixing matrix can be written in different ways, any of the proposed
forms displays nice features and disadvantages. Whatever form one uses, four
arbitrary parameters and the assumption of its unitarity are necessary to get
physically meaningful results. The models can be roughly grouped in two
categories, the first inspired to Euler like rotation matrices, the second,
containing explicit hierarchical features, employs an expansion, around the
unit matrix, in term of some key parameters. The original Kobayashi and
Maskawa matrixKobayashi  had been written in terms of three mixing angles
$\theta_{1,2,3}$ and one CP violating phase $\delta$. In this parameterization
the first family decouples from the others in the limit $\theta_{1}\rightarrow
0$. The particle data groupChau  chooses a form in which the CP violating term
is appended to the matrix entries responsible for the coupling of the first
and third generations of quark mass eigenstates. Finally
WolfensteinWolfenstein  has proposed a matrix emerging from a kind of
perturbative expansion in terms of the Cabibbo coupling parameter
$\lambda\cong 0.22$Cabibbo . A third model, bridging between the (Kobayashi
,Chau ) and Wolfenstein , is based on the so called exponential
parameterization, which emerges from the request of unitarity, automatically
satisfied by setting Dattoli

$\left.\begin{array}[]{ll}\widehat{V}&=e^{\widehat{A}}\\\
\widehat{A}^{{\dagger}}&=-\widehat{A}\end{array}\right.$ (1)

The second condition in Eq. (1), expressing the anti-hermiticity of the
matrix, is ensured by the following specific choice

$\widehat{A}=\left(\begin{array}[]{ccc}0&\Lambda_{1}&\Lambda_{3}\\\
-\Lambda_{1}&0&\Lambda_{2}\\\ -\Lambda_{3}^{*}&-\Lambda_{2}&0\\\
\end{array}\right)$ (2)

The vanishing of the diagonal entries secures that the matrix $\widehat{V}$ be
unimodular111It would be sufficient to have a matrix with null trace, but for
practical reasons we use the form(2).. The sub-labels 1, 2, 3 determine the
mixing d-s, s-b, d-b respectively, all the entries, except $\Lambda_{3}$, are
real. In the spirit of Wolfeinstein criteria, we use the Cabibbo strength
$\lambda$ as key parameter and make the following identificationsDattoli2

$\left.\begin{array}[]{ll}\Lambda_{1}&=\lambda\\\ \Lambda_{2}&=y\lambda^{2}\\\
\Lambda_{3}&=x\lambda^{3}e^{\imath\delta}\end{array}\right.$ (3)

containing an implicit hierarchical assumption on the coupling between the
different quark families. The vanishing of the $x$, $y$ coefficients allows
the decoupling from the b sector reducing the matrix to the s-d Cabibbo
mixing, namely

$\widehat{V}_{(x,y)\rightarrow 0}=e^{\left(\begin{array}[]{ccc}0&\lambda&0\\\
-\lambda&0&0\\\ 0&0&0\\\
\end{array}\right)}=\left(\begin{array}[]{ccc}\cos(\lambda)&\sin(\lambda)&0\\\
-\sin(\lambda)&\cos(\lambda)&0\\\ 0&0&1\\\ \end{array}\right)$ (4)

It is also to be stressed that $\widehat{V}_{\lambda\rightarrow
0}=\widehat{\mathbb{1}}$, which means that the parameterization in (3)
contains the assumption that the vanishing of the Cabibbo parameter determines
the decoupling of the entire quark matrix. The phase $\delta$ is associated,
as in the particle data group choice, with the smallest coupling term. In this
paper we will see how the exponential parameterization yields a flexible tool
to analyse the quark mixing phenomenology and the relevant consequences.

##  II The matrix A and the Wolfenstein parameterization

We will prove that the quark mixing matrix written as in Eq. (1) naturally
contains the Wolfenstein parameterization and the Euler like forms as well. By
keeping the expansion of the exponential in Eq. (1) up to third order in
$\lambda$, namely

$\widehat{V}=\widehat{\mathbb{1}}+\widehat{A}+\frac{\widehat{A}^{2}}{2}+\frac{\widehat{A}^{3}}{3!}+\frac{\widehat{A}^{4}}{4!}+o(\lambda^{5})$
(5)

we obtain the mixing matrix in the form

$\begin{array}[]{l}\widehat{V}\cong\left(\begin{array}[]{ccc}1-\frac{\lambda^{2}}{2}+\frac{\lambda^{4}}{4!}&\lambda-\frac{\lambda^{3}}{3!}&AF\lambda^{3}\\\
-\lambda+\frac{\lambda^{3}}{3!}&1-\frac{\lambda^{2}}{2}+\frac{\lambda^{4}}{4!}-\frac{(A\lambda^{2})^{2}}{2}&\frac{AB\lambda^{2}}{2}\\\
AG\lambda^{3}&\frac{AC\lambda^{2}}{2}&1-\frac{A^{2}\lambda^{4}}{2}\\\
\end{array}\right)\\\ \\\ A=y,\;F=\rho-\imath\eta,\;G=1-\rho-\imath\eta\\\
B=2-\lambda^{2}(\rho-\frac{1}{6}-\imath\eta),\;B+C=-2\lambda^{2}(\rho-\frac{1}{2})\\\
\rho=\frac{x}{y}\cos(\delta)+\frac{1}{2},\;\eta=-\frac{x}{y}\sin(\delta)\end{array}$
(6)

Eq. (6) is recognized as a Wolfenstein-type parameterization, the Taylor
expansion at higher order can provide more accurate expansion in the Cabibbo
coupling parameter, as we will see in the following. The expansion at the
third order allows a one to one correspondence between the Wolfenstein
parameters and those of the matrix $\widehat{A}$, which can be written in the
form

$\widehat{A}=\left(\begin{array}[]{ccc}0&\lambda&A\lambda^{3}(\rho-\imath\eta-\frac{1}{2})\\\
-\lambda&0&A\lambda^{2}\\\
-A\lambda^{3}(\rho+\imath\eta-\frac{1}{2})&-A\lambda^{2}&0\end{array}\right)$
(7)

Using for $A,\;\rho,\;\eta$ the following values, close to those given in the
literature DataGroup :

$\begin{array}[]{ll}\lambda=0.2272\pm
0.0010,&\;\;\;\;\;A=0.818^{+0.007}_{-0.017}\\\ \\\
\rho=0.221^{+0.064}_{-0.028},&\;\;\;\;\;\eta=0.340^{+0.017}_{-0.045}\end{array}$

we find for $x$ and $\delta$ the following values

$x=-0.359^{+0.049}_{-0.052},\;\;\;\delta=0.883^{+0.145}_{-0.118}$

and we get for the mixing matrix222 This result has been obtained by expanding
the matrix at any arbitrary order, namely
$\widehat{V}=\displaystyle{\sum_{0}^{N}}\frac{A^{n}}{n!}$ and by keeping N=50.
We have not included the errors deriving from the experimental and systematic
uncertainties, the relevant analysis would require extreme care for fitting
the data and such an effort is out of the purposes of the present note.

$|A|=\left(\begin{array}[]{ccc}0.97429&0.22523&3.86\cdot 10^{-3}\\\
0.22512&0.97341&0.04215\\\ 8.10\cdot 10^{-3}&0.04154&0.99910\\\
\end{array}\right)$ (8)

in good agreement with the values reported inDataGroup . Higher orders
expansions will be considered in the forthcoming sections.

##  III The geometrical meaning of the exponential parameterization and the
Euler like forms

We have so far proved that the exponential parameterization of the mixing
matrix has some nice features which makes its use quite interesting. Before
going further let us speculate on the geometrical (physical) meaning of the
matrix $\widehat{A}$, which can be understood as a kind of Hamiltonian ruling
the process of quark mixing. We introduce, therefore, the Schroedinger
equation

$\imath\partial_{\tau}\underline{\psi}=\widehat{H}\underline{\psi}$ (9)

where $\underline{\psi}|_{\tau=0}$ are the quark mass eigenstates, and

$\widehat{H}\propto\imath\widehat{A}$ (10)

Within such a picture the matrix $\widehat{V}$ is the evolution operator
associated with Eq. (9). In the case of vanishing CP phase $\delta\rightarrow
0$, the Hamiltonian in (10) can be written in terms of SO(3) generators,
namely

$\widehat{H}=\lambda R_{1}+y\lambda^{2}R_{2}+x\lambda^{3}R_{3}$ (11)

with

$\begin{array}[]{l}R_{1}=\imath\left(\begin{array}[]{ccc}0&1&0\\\ -1&0&0\\\
0&0&0\\\ \end{array}\right),\;R_{2}=\imath\left(\begin{array}[]{ccc}0&0&0\\\
0&0&1\\\ 0&-1&0\\\ \end{array}\right)\\\ \\\
R_{3}=\imath\left(\begin{array}[]{ccc}0&0&1\\\ 0&0&0\\\ -1&0&0\\\
\end{array}\right)\end{array}$

The Schroedinger equation (9) can, accordingly, be viewed as a vector equation
of the type

$\left.\begin{array}[]{ll}\partial_{\tau}\overrightarrow{Q}=&\overrightarrow{\Omega}\times\overrightarrow{Q}\\\
\\\
\overrightarrow{\Omega}\equiv&\lambda(-y\lambda,x\lambda^{2},-1)\end{array}\right.$
(12)

where $\overrightarrow{Q}\equiv(\psi_{1},\psi_{2},\psi_{3})$ is the vector
associated with the quark field. The problem of the quark mixing is therefore
understood as a rotation, induced by an Euler-like torque equation.
The torque vector $\overrightarrow{\Omega}$ is reported in Fig. (1) along with
the role played by each vector component.

Figure 1: The Quark mixing torque vector

The quark mixing matrix can be written using the Cayley Hamilton theorem
Babusci  (see Sect. IV) as

$\left.\begin{array}[]{l}\widehat{V}=e^{\tau\widehat{A}}|_{\tau=1}=\widehat{\mathbb{1}}+\mbox{sinc}(|\overrightarrow{\Omega}|)\widehat{A}+\displaystyle{\frac{1}{2}(\mbox{sinc}(\frac{|\overrightarrow{\Omega}|}{2}))^{2}\widehat{A}^{2}}\\\
\\\
|\overrightarrow{\Omega}|=\lambda\sqrt{1+y^{2}\lambda^{2}+x^{2}\lambda^{4}}\\\
\\\ \mbox{sinc}(\alpha)=\displaystyle\frac{\sin(\alpha)}{\alpha}\\\
\end{array}\right.$ (13)

Moreover from Eq. (12) the action of the mixing matrix on the initial vector
$\overrightarrow{Q}$ can be specified through the following Rodriguez
rotationBabusci1

$\left.\begin{array}[]{rl}\overrightarrow{Q}=&\cos(|\overrightarrow{\Omega}|)\overrightarrow{Q}_{0}+\sin(|\overrightarrow{\Omega}|)\overrightarrow{n}\times\overrightarrow{Q}_{0}\\\
\\\
&+(1-\cos(|\overrightarrow{\Omega}|))(\overrightarrow{n}\cdot\overrightarrow{Q}_{0})\cdot\overrightarrow{n}\\\
\\\
\overrightarrow{n}=&\frac{\overrightarrow{\Omega}}{|\overrightarrow{\Omega}|}\end{array}\right.$
(14)

The geometrical interpretation is less obvious if we include the CP violating
term. We assume Eq. (11) to be still valid and with a slight abuse of the
notation write

$\left.\begin{array}[]{rll}\overrightarrow{\Omega}&\equiv&\overrightarrow{\Omega}_{1}+\imath\overrightarrow{\Omega}_{2}\\\
\\\
\overrightarrow{\Omega}_{1}&\equiv&(-y\lambda^{2},x\lambda^{3}\cos(\delta),-\lambda)\\\
\\\
\overrightarrow{\Omega}_{2}&\equiv&(0,x\lambda^{3}\sin(\delta),0)\end{array}\right.$
(15)

This assumption contains the bare essence of CP violation from a geometrical
point of view. The vector $\overrightarrow{\Omega}$ splits into a real and
imaginary part, as shown in Fig. (2) where the second component of the torque
vector is composed by two subcomponents:

a)

the coupling vector
$\overrightarrow{\Omega}_{1,3}\equiv(y\lambda^{2},0,-\lambda)$ is the
component of the vector in the 1-3 plane;

b)

the CP violating sector is viewed as the pseudo vector
$(\Omega_{1,3},\mbox{Im}(\Omega_{2}),\mbox{Re}(\Omega_{2}))$.

Figure 2: The real(a) and the imaginary part (b) of the torque vector
$\Omega$.

In terms of the Wolfenstein parameters the modulus of the Torque vector can be
written as

$|\overrightarrow{\Omega}|=\sqrt{\lambda^{2}+(A\lambda^{2})^{2}+\left[\left(\rho-\frac{1}{2}\right)^{2}+\eta^{2}\right](A\lambda^{3})^{2}}$
(16)

or as

$\left.\begin{array}[]{l}|\overrightarrow{\Omega}|=\sqrt{\lambda^{2}(1+\tan^{2}(\phi_{A}))+\lambda^{3}\left(\rho-\frac{1}{2}\right)^{2}}\\\
\\\
\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\
\overline{\left[1+\tan^{2}(\delta)\right]\tan^{2}(\phi_{A})}\\\ \\\
\tan(\phi_{A})=A\lambda,\;\;\;\tan(\delta)=\displaystyle{\frac{\eta}{\sqrt{\left(\rho-\frac{1}{2}\right)^{2}}}}\end{array}\right.$
(17)

where $\phi_{A}$ and $\delta$ are indicated in Figs. (2).
The angle $\phi_{A}$ lies in the (1,3) sector and specifies the direction of
the $\overrightarrow{\Omega}$ vector components in this plane. We visualize
the geometric content of our problem as indicated in the second of Figs. (2),
in which the complex vector component lying along the direction of the axis 2
is split into an imaginary and a real part.
In more rigorous mathematical terms we can illustrate the above procedure as
it follows. We first note that

$\widehat{A}=\widehat{A}_{1}+\widehat{A}_{2}$ (18)

with

$\widehat{A}_{2}=\left(\begin{array}[]{ccc}0&0&\Lambda_{3}\\\ 0&0&0\\\
-\Lambda_{3}^{*}&0&0\end{array}\right),\;\;\widehat{A}_{1}=\left(\begin{array}[]{ccc}0&\Lambda_{1}&0\\\
-\Lambda_{1}&0&\Lambda_{2}\\\ 0&-\Lambda_{2}&0\end{array}\right)$

The matrices labelled with 2, 1 are not commuting each other, therefore we
have at the first order in the Zassenhaus disentanglement formula333 The
Zassenhaus formula writes
$e^{\widehat{A}+\widehat{B}}=e^{\widehat{A}}e^{\widehat{B}}\displaystyle{\prod_{n=1}^{\infty}}e^{\widehat{C}_{m}}$
where the operators $\widehat{C}_{m}$ are given in terms of successive
commutators, the first two being
$\widehat{C}_{1}=-\frac{1}{2}[\widehat{A},\widehat{B}]$,
$\widehat{C}_{2}=\frac{1}{3}[\widehat{A},\widehat{B}]+\frac{1}{6}[\widehat{A},[\widehat{A},\widehat{B}]]$
Magnus

$\left.\begin{array}[]{l}\widehat{V}=e^{\widehat{A}_{2}+\widehat{A}_{1}}\cong
e^{\widehat{A}_{2}}e^{\widehat{A}_{1}}e^{\widehat{C}}\\\ \\\
\widehat{C}=-\frac{1}{2}[\widehat{A}_{2},\widehat{A}_{1}]\end{array}\right.$
(19)

Where the (anti-hermitian) matrix $\widehat{C}$ is given by

$\begin{array}[]{rcl}\widehat{C}&=&-\frac{1}{2}\left(\begin{array}[]{ccc}0&-\Lambda_{3}\Lambda_{2}&0\\\
\Lambda_{3}^{*}\Lambda_{2}&0&\Lambda_{1}\Lambda_{3}\\\
0&-\Lambda_{1}\Lambda_{3}^{*}&0\\\ \end{array}\right)=\\\ \\\
&=&-\displaystyle{\frac{\lambda^{4}x}{2}}\left(\begin{array}[]{ccc}0&-\lambda
ye^{\imath\delta}&0\\\ \lambda ye^{-\imath\delta}&0&e^{\imath\delta}\\\
0&e^{-\imath\delta}&0\\\ \end{array}\right)\end{array}$ (20)

Neglecting the matrix $\widehat{C}$, which is of the order $o(\lambda^{4})$,
we find that the CKM matrix can be expressed as

$\widehat{V}\cong e^{\widehat{A}_{2}}e^{\widehat{A}_{1}}$ (21)

With

$e^{\widehat{A}_{2}}=\widehat{V}_{2}=\left(\begin{array}[]{ccc}\cos(|\Lambda_{3}|)&0&\frac{\Lambda_{3}}{|\Lambda_{3}|}\sin(|\Lambda_{3}|)\\\
0&1&0\\\
-\frac{\Lambda_{3}^{*}}{|\Lambda_{3}|}\sin(|\Lambda_{3}|)&0&\cos(|\Lambda_{3}|)\\\
\end{array}\right)$ (22)

and the use of the Cayley Hamilton theorem allows the following (exact) form
of the second exponential

$e^{\widehat{A}_{1}}=\widehat{V}_{1}=C_{0}\widehat{\mathbb{1}}+C_{1}\widehat{A}_{1}+C_{2}\widehat{A}_{1}^{2}$
$\left(\begin{array}[]{c}C_{0}\\\ \\\ C_{1}\\\ \\\ C_{2}\\\
\end{array}\right)=\left(\begin{array}[]{ccc}|\Lambda_{1,2}|^{2}&0&0\\\ \\\
-\Upsilon&\Upsilon&\Upsilon\\\ \\\ \imath&1-\imath&-(1+\imath)\\\
\end{array}\right)\left(\begin{array}[]{c}\displaystyle{\frac{1}{|\Lambda_{1,2}|^{2}}}\\\
\\\ \displaystyle{\frac{e^{\imath|\Lambda_{1,2}|}}{2|\Lambda_{1,2}|^{2}}}\\\
\\\ \displaystyle{\frac{e^{-\imath|\Lambda_{1,2}}|}{2|\Lambda_{1,2}|^{2}}}\\\
\end{array}\right)$ (23)

$\Upsilon=|\Lambda_{1,2}|(1+\imath),\;\;|\Lambda_{1,2}|=\sqrt{\Lambda_{1}^{2}+\Lambda_{2}^{2}}$

The above formulae are a restatement of the tentative geometrical picture of
Fig. (2). The na ve disentanglement has reduced the CKM generation to the
product of two matrices, $\widehat{V}_{1}$ accounting for the mixing, induced
by the vector $\overrightarrow{\Omega}_{1,3}$ , and $\widehat{V}_{2}$
specifying a complex rotation, responsible for the CP violating contributions.
The matrix (21) is an approximation of the exponential form at the order
$o(\lambda^{4})$, but it is not equivalent to Wolfenstein matrix. The matrix
(21), albeit an approximation, since we have neglected higher order
commutators, is unitary at any order in the coupling parameter, while
$\widehat{V}_{W}\widehat{V}_{W}^{{\dagger}}=\widehat{\mathbb{1}}+o(\lambda^{4})$
(where $\widehat{V}_{W}$ is the matrix (6)).
We have stressed that the simple picture in terms of Euler rotation is
hampered by the presence of a complex term, the $\widehat{V}$ matrix cannot be
written in terms of the generators of rotations and indeed we find

$\begin{array}[]{rl}\widehat{V}=&e^{-\imath(\lambda\widehat{R}_{1}+y\lambda^{2}\widehat{R}_{2}+x\lambda^{3}\widehat{T})}\\\
\\\ \widehat{T}=&-\imath\left(\begin{array}[]{ccc}0&0&e^{\imath\delta}\\\
&0&0\\\ -e^{-\imath\delta}&0&0\\\ \end{array}\right)\end{array}$ (24)

The $\widehat{T}$ matrix does not belong to SO(3) and the quark mixing matrix,
written as the product of the exponential matrix correct up to the order
$o(\lambda^{4})$ is

$\widehat{V}=e^{-\imath x\lambda^{3}\widehat{T}}\;e^{-\imath
y\lambda^{2}\widehat{R}_{2}}\;e^{-\imath\lambda^{3}\widehat{R}_{1}}+o(\lambda^{4})$

$\begin{array}[]{rl}\widehat{V}\cong&e^{\left(\begin{array}[]{ccc}0&0&x\lambda^{3}e^{\imath\delta}\\\
0&0&0\\\ -x\lambda^{3}e^{-\imath\delta}&0&0\\\ \end{array}\right)}\\\ \\\
&\;\;\;\;\;e^{\left(\begin{array}[]{ccc}0&0&0\\\ 0&0&y\lambda^{2}\\\
0&-y\lambda^{2}&0\\\
\end{array}\right)}e^{\left(\begin{array}[]{ccc}0&\lambda&0\\\ -\lambda&0&0\\\
0&0&0\\\ \end{array}\right)}=\\\ \\\
=&{\left(\begin{array}[]{ccc}C(x\lambda^{3})&0&e^{\imath\delta}S(x\lambda^{3})\\\
0&1&0\\\ -e^{-\imath\delta}S(x\lambda^{3})&0&C(x\lambda^{3})\\\
\end{array}\right)}\\\ \\\ &\;\;\;{\left(\begin{array}[]{ccc}1&0&0\\\
0&C(y\lambda^{2})&S(y\lambda^{2})\\\ 0&-S(y\lambda^{2})&C(y\lambda^{2})\\\
\end{array}\right)}{\left(\begin{array}[]{ccc}C(\lambda)&S(\lambda)&0\\\
-S(\lambda)&C(\lambda)&0\\\ 0&0&1\\\ \end{array}\right)}\\\ \\\
&C(\phi)=\cos(\phi),\;S(\phi)=\sin(\phi)\end{array}$ (25)

and displays the largely well-known feature that the mixing angles are
proportional to the Cabibbo coupling parameter according to

$\left.\begin{array}[]{ccc}\vartheta_{1,3}&\propto&\lambda^{3}\cong\displaystyle{\sqrt{\frac{m_{d}}{m_{b}}}}\\\
\\\
\vartheta_{2,3}&\propto&\lambda^{2}\cong\displaystyle{\sqrt{\frac{m_{s}}{m_{b}}}}\\\
\\\
\vartheta_{1,2}&\propto&\lambda\cong\displaystyle{\sqrt{\frac{m_{d}}{m_{s}}}}\\\
\\\ \end{array}\right.$ (26)

Furthermore, in full agreement with the particle data group paradigm, we get

$\begin{array}[]{l}e^{\left(\begin{array}[]{ccc}0&0&x\lambda^{3}e^{\imath\delta}\\\
0&0&0\\\ -x\lambda^{3}e^{-\imath\delta}&0&0\\\
\end{array}\right)}=\widehat{U}_{\delta}^{{\dagger}}e^{-\imath
x\lambda^{3}\widehat{R}_{3}}\widehat{U}_{\delta}\\\ \\\
\widehat{U}_{\delta}=e^{\imath\widehat{\delta}},\;\;\mbox{with}\;\;\widehat{\delta}=\left(\begin{array}[]{ccc}0&0&0\\\
0&0&0\\\ 0&0&\delta\\\
\end{array}\right),\;\widehat{R}_{3}=\imath\left(\begin{array}[]{ccc}0&0&1\\\
0&0&0\\\ -1&0&0\\\ \end{array}\right)\end{array}$ (27)

It is also interesting to note that the $\widehat{T}$ matrix can be written as

$\begin{array}[]{l}\widehat{T}=-\imath\cos(\delta)\widehat{R}_{3}+\imath\sin(\delta)\widehat{S}_{3}\\\
\\\ \widehat{S}_{3}=\left(\begin{array}[]{ccc}0&0&1\\\ 0&0&0\\\ 1&0&0\\\
\end{array}\right)\end{array}$ (28)

The na ve disentanglement (order $o(\lambda^{4})$)

$\begin{array}[]{l}\widehat{V}\cong e^{-\imath\widehat{R}}e^{\imath
x\lambda^{3}\sin(\delta)\widehat{S}_{3}}\\\ \\\
\widehat{R}=\lambda\widehat{R}_{1}+y\lambda^{2}\widehat{R}_{2}+x\lambda^{3}\cos(\delta)\widehat{R}_{3}\end{array}$
(29)

Corresponds to the product of two matrices, namely

$\begin{array}[]{l}\widehat{V}\cong\widehat{V}_{R}\widehat{V}_{I}\\\ \\\
e^{\widehat{R}}=\widehat{V}_{R}\\\ \\\
\widehat{V}_{I}=\left(\begin{array}[]{ccc}C(x\lambda^{3}\sin(\delta))&0&\imath
S(x\lambda^{3}\sin(\delta))\\\ 0&1&0\\\ \imath
S(x\lambda^{3}\sin(\delta))&0&C(x\lambda^{3}\sin(\delta))\\\
\end{array}\right)=\\\ \\\
\;\;\;\;=\left(\begin{array}[]{ccc}C(A\lambda^{3}\eta)&0&-\imath
S(A\lambda^{3}\eta)\\\ 0&1&0\\\ -\imath
S(A\lambda^{3}\eta)&0&C(A\lambda^{3}\eta)\\\ \end{array}\right)\end{array}$
(30)

and $\widehat{V}_{R}$ can be written as Eq. 13 with

$\overrightarrow{\Omega}\equiv\lambda(-y\lambda,x\lambda^{2}\cos(\delta),-1)$
(31)

The matrix $\widehat{V}_{I}$ mixes the first and third quark generation mass
eigenstates and is responsible for the CP violation. It is a pseudo rotation
matrix and is generated by a matrix whose determinant is the Jarlskog
invariant Jarlskog , discussed in the forthcoming section.
We have so far shown that the exponential parameterization implicitly contains
Wolfenstein and Euler type forms, in the following sections we will dwell on
its further advantages.

##  IV The Cayley Hamilton Theorem and The Quark mixing matrix

The exponential matrix (1) can be treated in different ways.
We have already shown that the use of a Taylor expansion leads to a
Wolfenstein form, which preserves the unitarity of $\widehat{V}$ at the
expansion order (the mixing matrix in Eq. (6) is unitary at the order
$o(\lambda^{4})$).
The method of the exponential disentanglement can be used too and such a
procedure allows an interesting geometrical picture of the mixing dynamics and
albeit an approximation in the Cabibbo coupling parameter, the mixing matrix
written as in Eq. (17) preserves the unitarity at any order in $\lambda$, as
discussed more accurately in the concluding remarks.
The matrix $\widehat{V}$ can, however, be written in an exact form using the
Cayley Hamilton theorem, by setting

$\widehat{V}=C_{0}\widehat{\mathbb{1}}+C_{1}\widehat{A}+C_{2}\widehat{A}^{2}$
(32)

where

$e^{\varepsilon_{j}}=C_{0}+\varepsilon_{j}C_{1}+\varepsilon_{j}^{2}C_{2},\;\;\mbox{with}\;\;j=1,2,3$
(33)

With $\varepsilon_{j}$ being the roots associated with the characteristic
equation of the matrix $\widehat{A}$, namely

$\varepsilon_{j}^{3}+|\overrightarrow{\Omega}|^{2}\varepsilon_{j}+\imath\triangle=0$
(34)

where

$\left.\begin{array}[]{l}\triangle=2xy\lambda^{6}\sin(\delta)=-2A\eta\lambda^{6}\\\
\\\
|\overrightarrow{\Omega}|=\lambda\sqrt{1+y^{2}\lambda^{2}+x^{2}\lambda^{4}}=\\\
\\\
=\lambda\sqrt{1+(A\lambda)^{2}+(A^{2}\lambda^{4})\left[\left(\rho-\frac{1}{2}\right)^{2}+\eta^{2}\right]}\end{array}\right.$
(35)

$\imath\triangle$ is the determinant of the matrix $\widehat{A}$.
A little bit of algebra yields to define the $C_{i}\;\;(i=0,1,2)$ coefficients
as the product of two matrix

$\left(\begin{array}[]{c}C_{0}\\\ \\\ C_{1}\\\ \\\
C_{2}\end{array}\right)=\left(\begin{array}[]{ccc}\varepsilon_{2}\varepsilon_{3}&\varepsilon_{1}\varepsilon_{3}&\varepsilon_{1}\varepsilon_{2}\\\
\\\
-(\varepsilon_{2}+\varepsilon_{3})&-(\varepsilon_{1}+\varepsilon_{3})&-(\varepsilon_{1}+\varepsilon_{2})\\\
\\\ 1&1&1\end{array}\right)\\\
\left(\begin{array}[]{c}\displaystyle{\frac{e^{\varepsilon_{1}}}{(\varepsilon_{2}-\varepsilon_{1})(\varepsilon_{3}-\varepsilon_{1})}}\\\
\\\
\displaystyle{\frac{e^{\varepsilon_{2}}}{(\varepsilon_{1}-\varepsilon_{2})(\varepsilon_{3}-\varepsilon_{2})}}\\\
\\\
\displaystyle{\frac{e^{\varepsilon_{3}}}{(\varepsilon_{1}-\varepsilon_{3})(\varepsilon_{2}-\varepsilon_{3})}}\end{array}\right)$
(36)

Eq. (23) (along with Eqs. (27)) is the most general form of the quark mixing
matrix which can be derived from an exponential parameterization, it is exact
but not easy to remember.
Let us now give an idea of the orders of the numerical values characterizing
the various quantities entering the above equations. The use of the previously
quoted values for the Wolfenstein parameters lead to the following evaluations
for the solution of Eq. (34)

$\begin{array}[]{l}\varepsilon_{1}\cong-0.23171\imath\\\ \\\
\varepsilon_{2}\cong 0.00117\imath\cdot\\\ \\\ \varepsilon_{2}\cong
0.23054\imath\cdot\end{array}$ (37)

It is worth stressing that the matrix $\widehat{D}$ provides the diagonal
forms of either $\widehat{V}$ and $\widehat{A}$. It follows therefore that the
two matrices have the same eigenvectors. They can be determined using
$\widehat{A}$ instead of $\widehat{V}$, because the procedure is significantly
simpler. We find that the eigenvalues are in the form

$|j>=\left(\begin{array}[]{c}1\\\ \\\
-\varepsilon_{j}-xy\lambda^{5}e^{-\imath\delta}\\\ \\\
-y\lambda^{3}-\varepsilon_{j}x\lambda^{3}e^{-\imath\delta}\\\ \\\
\end{array}\right)$ (38)

$\widehat{V}=\left(\begin{array}[]{cc}C(\lambda)+\frac{A^{2}\lambda^{6}}{4!}\Phi&S(\lambda)-\frac{A^{2}\lambda^{5}}{2}(\Pi^{*}-\frac{1}{6})\\\
\\\
-S(\lambda)-\frac{A^{2}\lambda^{5}}{2}(\Pi-\frac{5}{6})&C(\lambda)-(A\lambda^{2})^{2}\left[\frac{1}{2}-\frac{\lambda^{2}}{3}\left(\frac{1}{4}-\imath\eta\right)\right]\\\
\\\
-A\lambda^{3}\left[\left(\Pi-1\right)-\frac{\lambda^{2}}{6}\left(\Pi-\frac{3}{4}\right)\right]&-S(A\lambda^{2})+\frac{A\lambda^{4}}{4!}\left(\lambda^{2}(\Pi-\frac{7}{10})-12(\Pi-\frac{2}{3})\right)\end{array}\right.\\\
\left.\begin{array}[]{c}A\lambda^{3}\left[\Pi^{*}-\frac{\lambda^{2}}{6}\left(\Pi^{*}-\frac{1}{4}\right)\right]\\\
\\\
S(A\lambda^{2})+\frac{A\lambda^{4}}{4!}\left(\lambda^{2}(\Pi^{*}-\frac{3}{10})-12(\Pi^{*}-\frac{1}{3})\right)\\\
\\\ 1-\frac{(A\lambda^{2})^{2}}{2}+\frac{A^{2}\lambda^{6}}{4!}\Phi\\\ \\\
\end{array}\right)$ (39)

$\begin{array}[]{l}\Phi=-12(\rho^{2}+\eta^{2})-8\imath\eta+12\rho-2;\;\;\Pi=\rho+\imath\eta;\;\;\Pi^{*}=\rho-\imath\eta\end{array}$

It is worth mentioning the companion matrix associated with the characteristic
equation (30) Companionmatrix , which writes

$C_{A}=\left(\begin{array}[]{ccc}0&\;\;0&-\varepsilon_{1}\varepsilon_{2}\varepsilon_{3}\\\
\\\
1&\;\;0&-(\varepsilon_{1}\varepsilon_{2}+\varepsilon_{2}\varepsilon_{3}+\varepsilon_{1}\varepsilon_{3})\\\
\\\ 0&\;\;1&-(\varepsilon_{1}+\varepsilon_{2}+\varepsilon_{3})\\\ \\\
\end{array}\right)$ (40)

It is accordingly expressed in terms of three invariants444A $3\times 3$
matrix has three invariants given by its determinant, its trace and by the sum
of the determinants of its minors, namely

$\left.\begin{array}[]{l}\displaystyle\varepsilon_{1}\varepsilon_{2}\varepsilon_{3}=-2\imath
x\lambda^{6}\sin(\delta)\\\ \\\
\varepsilon_{1}^{2}+\varepsilon_{2}^{2}+\varepsilon_{3}^{2}=|\overrightarrow{\Omega}|^{2}\\\
\\\ \varepsilon_{1}+\varepsilon_{2}+\varepsilon_{3}=0\end{array}\right.$ (41)

the first of which is the Jarlskog invariant, a measure of the amount of CP
violations, emerging in quite a natural way in the present analysis.

##  V Concluding remarks

We have shown that the exponential parameterization interpolates between
Wolfestein and Euler like forms and could provide a useful and flexible tool
of analysis. Its approximations in terms of the Cabibbo coupling can be either
expressed as Taylor expansions or as unitarity preserving forms bassed on the
Zassenhaus formula.
The Taylor expansion does not meet too much aesthetical criteria, but it can
usefully be exploited to get higher order approximations of Wolfenstein type
parameterizations an example is shown below, where we report the na ve
expansion of the exponential matrix up to the order $o(\lambda^{7})$.
We have reported the matrix (39) (where $C(\lambda),\;S(\lambda)$ denote the
expansion of cosine and sine up to the order $o(\lambda^{7})$) for comparison
purposes with other forms available in literature. The accuracy of this last
matrix is one part over $10^{9}$ and can therefore considered exact for any
expansion purposes.
The extension of the CKM matrix to higher dimensions by the use of the
exponential matrix method is not complicated. In the case of four quark
generations, we define the matrix containing 2 CP violating phases, appended
to the smallest coupling terms. We have furthermore assumed that the coupling
strengths to the fourth family be of the order $\lambda^{3+n},\;\;n=1,2,3$.

$A=\left(\begin{array}[]{cccc}0&\lambda&e^{\imath\delta_{1}}x\lambda^{3}&e^{\imath\delta_{2}}z\lambda^{6}\\\
-\lambda&0&y\lambda^{2}&p\lambda^{5}\\\
-e^{-\imath\delta_{1}}x\lambda^{3}&-y\lambda^{2}&0&u\lambda^{4}\\\
-e^{-\imath\delta_{2}}z\lambda^{6}&-p\lambda^{5}&-u\lambda^{4}&0\end{array}\right)$
(42)

The relevant Wolfenstein like approximation of the mixing matrix is reported
in (43).

$\widehat{V}=\left(\begin{array}[]{cccc}C(\lambda)+\frac{A^{2}\lambda^{6}}{4!}\Phi&S(\lambda)-\frac{A^{2}\lambda^{5}}{2}(\Pi_{1}^{*}-\frac{1}{6})\\\
\\\
-S(\lambda)-\frac{A^{2}\lambda^{5}}{2}(\Pi_{1}-\frac{5}{6})&C(\lambda)-(A\lambda^{2})^{2}\left[\frac{1}{2}-\frac{\lambda^{2}}{3}\left(\frac{1}{4}-\imath\eta\right)\right]\\\
\\\
-A\lambda^{3}\left[\left(\Pi_{1}-1\right)-\frac{\lambda^{2}}{6}\left(\Pi_{1}-\frac{3}{4}\right)\right]&-S(A\lambda^{2})+\frac{A\lambda^{4}}{4!}\left(\lambda^{2}(\Pi_{1}-\frac{7}{10})-12(\Pi_{1}-\frac{2}{3})\right)\\\
\\\
-\lambda^{6}p(\Pi_{2}-1)&-\lambda^{5}\left(-p+Au\frac{\lambda}{2}\right)\end{array}\right.\\\
\\\
\left.\begin{array}[]{cc}A\lambda^{3}\left[\Pi^{*}_{1}-\frac{\lambda^{2}}{6}\left(\Pi^{*}_{1}-\frac{1}{4}\right)\right]&-\Pi^{*}_{2}p\lambda^{6}\\\
\\\
S(A\lambda^{2})+\frac{A\lambda^{4}}{4!}\left(\lambda^{2}(\Pi^{*}_{1}-\frac{3}{10})-12(\Pi^{*}_{1}-\frac{1}{3})\right)&\lambda^{5}\left(p+Au\frac{\lambda}{2}\right)\\\
\\\
1-\frac{(A\lambda^{2})^{2}}{2}+\frac{A^{2}\lambda^{6}}{4!}\Phi&u\lambda^{4}\\\
\\\ -u\lambda^{4}&1\end{array}\right)$ (43)

$\begin{array}[]{l}\Phi_{1}=\rho_{1}+\imath\eta_{1}\;\;\mbox{with}\;\;\displaystyle{\rho_{1}=\frac{x}{y}\cos(\delta_{1})+\frac{1}{2},\;\;\eta_{1}=-\frac{x}{y}\sin(\delta_{1})};\;\;\Phi_{2}=\rho_{2}+\imath\eta_{2}\;\;\mbox{with}\;\;\displaystyle{\rho_{2}=\frac{z}{p}\cos(\delta_{2})+\frac{1}{2},\;\;\eta_{2}=-\frac{z}{p}\sin(\delta_{2})}\\\
\Phi_{j}^{*}=\rho_{j}-\imath\eta_{j}\;\;j=1,2\end{array}$

Furtheremore the invariants (4 for a $4\times 4$ matrix), obtained directly
from (42) read

$\begin{array}[]{rl}J_{2}=&\lambda^{2}f(1,y\lambda,x\lambda^{2})+\lambda^{8}f(u,p\lambda,z\lambda^{2})\\\
&\mbox{where}\;\;f(a,b,c)=a^{2}+b^{2}+c^{2}\\\ \\\
J_{3}=&2\imath\lambda^{6}\left[xy\sin(\delta_{1})+zp\lambda^{6}\sin(\delta_{2})\right]=\\\
=&xy\lambda^{6}(e^{\imath\delta_{1}}-e^{-\imath\delta_{1}})+zp\lambda^{12}(e^{\imath\delta_{2}}-e^{-\imath\delta_{2}})\\\
\\\
J_{4}=&u^{2}\lambda^{10}+[2uyz\cos(\delta_{2})-2pux\cos(\delta_{1})]\lambda^{13}+\\\
&[p^{2}x^{2}-px^{2}ye^{\imath(\delta_{2}-\delta_{1})}-pxyze^{-\imath(\delta_{2}-\delta_{1})}+xy^{2}z]\lambda^{16}\end{array}$

the first invariant, associated with the trace of A, is zero.
It is evident that the $J_{2}$ and $J_{3}$ invariants are just a
generalization of those reported in Eq. (32) while the fourth is completely
new being associated to the full determinant of the matrix. We have reported
this example to show the flexibility of the method it is however evident that
the detection of CP violating effects due to the new phase require an accuracy
at least of the order $\lambda^{6}$.
Before concluding the paper we will address the problems associated with the
exponential forms of the neutrino mixing matrix, which have also been
discussed in Dattoli3 , where the lepton-quark complementarity Minakata  has
been reformulated by noting that the relevant rotation occur around axes
forming an angle of $45^{0}$. The present experimental data seem to favor the
tribimaximal (TBM) form Harrison  therefore the neutrino mixing matrix reads

$\widehat{U}=\left(\begin{array}[]{ccc}\displaystyle\sqrt{\frac{2}{3}}&\displaystyle\frac{1}{\sqrt{3}}&0\\\
\\\
\displaystyle-\frac{1}{\sqrt{6}}&\displaystyle\frac{1}{\sqrt{3}}&\displaystyle\frac{1}{\sqrt{2}}\\\
\\\
\displaystyle\frac{1}{\sqrt{6}}&\displaystyle-\frac{1}{\sqrt{3}}&\displaystyle\frac{1}{\sqrt{2}}\\\
\\\ \end{array}\right)$ (44)

If we assume that also this form is generated by an exponential matrix (with
all real entries) according to

$\widehat{U}=e^{\widehat{B}};\;\;\;\widehat{B}=\left(\begin{array}[]{ccc}0&\;\;\alpha&\;\;\beta\\\
\\\ -\alpha&\;\;0&\;\;\gamma\\\ \\\ -\beta&\;\;-\gamma&\;\;0\\\ \\\
\end{array}\right)$ (45)

We obtain the following correspondence between the entries of the
$\widehat{B}$ matrix and those of the TBM form

$\begin{array}[]{l}\widehat{B}=\alpha\left(\begin{array}[]{ccc}0&\;\;1&\;\;\displaystyle-\frac{1}{\sqrt{2}+1}\\\
\\\ -1&\;\;0&\;\;\frac{\sqrt{3}+\sqrt{2}}{\sqrt{2}+1}\\\ \\\
\frac{1}{\sqrt{2}+1}&\;\;-\frac{\sqrt{3}+\sqrt{2}}{\sqrt{2}+1}&\;\;0\\\ \\\
\end{array}\right)\\\ \\\
\alpha=\displaystyle{2\frac{\sqrt{2\sqrt{2}+3}}{\sqrt{2\sqrt{2}+2\sqrt{6}+9}}\frac{1}{\sin\left(\frac{\sqrt{3-\left(\sqrt{\frac{2}{3}}+\frac{1}{\sqrt{3}}+\frac{1}{\sqrt{2}}\right)}}{2}\right)}}\end{array}$
(46)

The values of the entries of the TBM matrix do not allow the interpretation of
the neutrino mixing matrix as an expansion around the unit, notwithstanding it
is possible to get a better agreement with experimental by making an
appropriate expansions around the matrix $\widehat{B}$ and then around the
TBM, as it will be shown in a dedicated paper.
In this paper we have provided an extensive account of the possibilities
offered by the exponential form of the CKM matrix, which looks like a
prototype from which all the other forms can be derived, we hope that our
suggestions provide a useful tool in the relevant applications.

###### Acknowledgements.

The authors are deeply indebted to Dr. D. Babusci for stimulating discussions
and comments during any stage of the paper.

## References

  * (1) M. Kobayashi, T. Maskawa, Prog. Theor. Phys. 49, 652 (1973)
  * (2) L.L. Chau, W. Y. Keung, Phys. Rev. Lett. 53, 1802 (1984)
  * (3) L. L. Wolfenstein, Phys. Rev. Lett. 51, 1945 (1983)
  * (4) N. Cabibbo, Phys. Rev. Lett. 10, 531 (1963)
  * (5) G. Dattoli, K. Zhukowky, Eur. Phys. J. C. 50, 817 (2007) and references therein for earlier works on this subject
  * (6) G. Dattoli, K. Zhukowsky, Eur. Phys. C 52, 591 (2007)
  * (7) Particle Data Group, W. M. et al. J. Phys. G, Nucl. Part. Phys. 331, 1, (2005)
  * (8) D. Babusci, G. Dattoli and M. Del Franco, Lectures on Mathematical Methods For Physics. Thecnical Report 58 ENEA (2010)
  * (9) D. Babusci, G. Dattoli and E. Sabia, J. Math. Phys. 3 , P110601(2011)
  * (10) W. Magnus, Commun. Pure Appl. Math. 7, 649 (1954)
F. Cassas, A. Murua and Mladen Nadinic, Efficient Computation of the
Zassenhaus formula, [arXiv:math-ph/1204.0389v2], 15 June 2012

  * (11) C. Jarlskog, Phys. Rev. Lett. 55, 1039 (1985)
  * (12) K. Fujii, H. Oike,[arXiv:quant-ph/0604115vi]
  * (13) G. Dattoli, K. Zhukowsky, Eur. Phys. C 55, 547 (2008)
  * (14) H. Minakata, A. Y. Smirnov, Phys. Rev. D 70, 073009 (2004)
M. Raidal, Phys. Rev. Lett. 93, 161801 (2004)

  * (15) P. F. Harrison, D. H. Perkins and W. G. Scott, Physics Letters B 530: 167 (2002) [arXiv:hep-ph/0202074]


J=\int_{\hat{\mathcal{Q}}_{j}}(u-\hat{k}_{j})_{-}^{p}|\nabla\Phi_{j}|^{p}\,dx\,dt\quad\text{and}\quad\hat{K}=\operatorname*{ess\,sup}_{t\in\Gamma_{j}}\int_{\hat{B}_{j}}(u-\hat{k}_{j})_{-}^{2}\,dx.$

Due to (5.19), from (5.8), we recall that
$(u-\hat{k}_{j})_{-}\leq(\mu^{-}+M-\lambda^{-})_{+}:=L$ in
$\mathbb{R}^{N}\times(0,T)$. Estimate of $J$: From the proof of (5.9), we have

(5.22)
$\begin{split}J&=\int_{\hat{\mathcal{Q}}_{j}}(u-\hat{k}_{j})_{-}^{p}|\nabla\Phi_{j}|^{p}\,dx\,dt\leq
C\frac{2^{jp}}{r^{p}}L^{p}|A_{j}|,\end{split}$

for some constant $C=C(N,p)>0$.
Estimate of $I$ and $\hat{K}$: Let $\xi_{j}(x,t)=\xi_{j}(x)$ be a time
independent smooth function with compact support in $B_{j}$ such that
$0\leq\xi_{j}\leq 1$, $|\nabla\xi_{j}|\leq C\frac{2^{j}}{r}$ in
$\mathcal{Q}_{j}$,
$\mathrm{dist}(\mathrm{supp}\,\xi_{j},\mathbb{R}^{N}\setminus B_{j})\geq
2^{-j-1}r$ and $\xi_{j}\equiv 1$ in $\hat{B}_{j}$ for some constant
$C=C(N,p,s)>0$. Therefore, $\partial_{t}\xi_{j}=0$. Also, since
$\hat{k}_{j}<\mu^{-}+M$, due to the hypothesis $u(\cdot,t_{0})\geq\mu^{-}+M$
a.e. in $B_{r}(x_{0})$, we deduce that $(u-\hat{k}_{j})_{-}(\cdot,t_{0})=0$
a.e. in $B_{r}(x_{0})$. Noting these facts along with $g\equiv 0$ in
$\Omega_{T}\times\mathbb{R}$ and $(u-k_{j})_{-}\geq(u-\hat{k}_{j})_{-}$, by
Lemma 2.10, Remark 3.5 and Lemma 3.3, we obtain

(5.23)
$\begin{split}\hat{K}+I&=\operatorname*{ess\,sup}_{\Gamma_{j}}\int_{\hat{B}_{j}}(u-\hat{k}_{j})_{-}^{2}\,dx+\int_{\Gamma_{j}}\int_{\hat{B}_{j}}|\nabla(u-\hat{k}_{j})_{-}|^{p}\,dx\,dt\leq
J_{1}+J_{2}+J_{3},\end{split}$

where

$\begin{split}J_{1}&=C\Bigg{(}\int_{\Gamma_{j}}\int_{B_{j}}\int_{B_{j}}{\max\\{(u-k_{j})_{-}(x,t),(u-k_{j})_{-}(y,t)\\}^{p}|\xi_{j}(x,t)-\xi_{j}(y,t)|^{p}}\,d\mu\,dt,\\\
J_{2}&=C\int_{\Gamma_{j}}\int_{B_{j}}(u-k_{j})_{-}^{p}|\nabla\xi_{j}|^{p}\,dx\,dt\quad\text{and}\\\
J_{3}&=\operatorname*{ess\,sup}_{(x,t)\in\mathrm{supp}\,\xi_{j},\,t\in\Gamma_{j}}\int_{{\mathbb{R}^{N}\setminus
B_{j}}}{\frac{(u-k_{j})_{-}(y,t)^{p-1}}{|x-y|^{N+ps}}}\,dy\int_{\Gamma_{j}}\int_{B_{j}}(u-k_{j})_{-}\xi_{j}^{p}\,dx\,dt,\end{split}$

for some positive constant $C=C(N,p,s,\Lambda,C_{1},C_{2},C_{3},C_{4})$. From
(5.12), (5.13) and (5.15), it follows that

(5.24) $\hat{K}+I\leq J_{1}+J_{2}+J_{3}\leq
C\frac{2^{j(N+p)}}{r^{p}}L^{p}|A_{j}|,$

for some positive constant $C=C(N,p,s,\Lambda,C_{1},C_{2},C_{3},C_{4})$. Now
employing (5.22) and (5.24) in (5.21), by setting
$Y_{j}=\frac{|A_{j}|}{|\mathcal{Q}_{j}|}$ we obtain

(5.25) $Y_{j+1}\leq C\frac{(\theta
L^{N+p})^{\frac{1}{N+2}}}{(1-a)M}2^{j((N+p)^{2}+1)}Y_{j}^{1+\frac{1}{N+2}},$

for some positive constant $C=C(N,p,s,\Lambda,C_{1},C_{2},C_{3},C_{4})$.
Letting

$d_{0}=\frac{CL^{\frac{N+p}{N+2}}}{(1-a)M},\quad
b=2^{(N+p)^{2}+1},\quad\delta_{2}=\delta_{1}=\delta=\frac{1}{N+2}\quad\text{and}\quad
K=\frac{d_{0}\,\theta^{\delta}}{2}$

in Lemma 2.11, we have $\lim_{j\to\infty}Y_{j}\to 0$, if
$Y_{0}\leq\nu=(2K)^{-\frac{1}{\delta}}b^{-\frac{1}{\delta^{2}}}$. Let
$\beta\in(0,1)$, then choosing
$\theta=\beta\,d_{0}^{-\frac{1}{\delta}}\,b^{-\frac{1}{\delta^{2}}}$, which
depends on $a,M,\mu^{-},\lambda^{-},N,p,s,\Lambda,C_{1},C_{2},C_{3},C_{4}$, we
get $\nu=\beta^{-1}>1$. Hence the fact that $Y_{0}\leq 1$ and thus Lemma 2.11
imply that $\lim_{j\to\infty}Y_{j}\to 0$. Therefore, we have

$u\geq\mu^{-}+aM\text{ a.e. in
}\mathcal{Q}_{\frac{3r}{4},\theta}^{+}(x_{0},t_{0}).$

Hence the result follows. ∎

## References

  * [1] Agnid Banerjee, Prashanta Garain, and Juha Kinnunen.  Some local properties of subsolutons and supersolutions for a doubly nonlinear nonlocal parabolic $p$-Laplace equation.  arXiv e-prints, page arXiv:2010.05727, October 2020.
  * [2] Agnid Banerjee, Prashanta Garain, and Juha Kinnunen.  Lower semicontinuity and pointwise behavior of supersolutions for some doubly nonlinear nonlocal parabolic $p$-Laplace equations.  arXiv e-prints, page arXiv:2101.10042, January 2021.
  * [3] Martin T. Barlow, Richard F. Bass, Zhen-Qing Chen, and Moritz Kassmann.  Non-local Dirichlet forms and symmetric jump processes.  Trans. Amer. Math. Soc., 361(4):1963–1999, 2009.
  * [4] M. Belloni, V. Ferone, and B. Kawohl.  Isoperimetric inequalities, Wulff shape and related questions for strongly nonlinear elliptic operators.  volume 54, pages 771–783. 2003.  Special issue dedicated to Lawrence E. Payne.
  * [5] Marino Belloni and Bernd Kawohl.  The pseudo-$p$-Laplace eigenvalue problem and viscosity solutions as $p\to\infty$.  ESAIM Control Optim. Calc. Var., 10(1):28–52, 2004.
  * [6] Stefano Biagi, Serena Dipierro, Enrico Valdinoci, and Eugenio Vecchi.  Mixed local and nonlocal elliptic operators: regularity and maximum principles.  arXiv e-prints, page arXiv:2005.06907, May 2020.
  * [7] Stefano Biagi, Serena Dipierro, Enrico Valdinoci, and Eugenio Vecchi.  Semilinear elliptic equations involving mixed local and nonlocal operators.  arXiv e-prints, page arXiv:2006.05830, June 2020.
  * [8] Stefano Biagi, Dimitri Mugnai, and Eugenio Vecchi.  Global boundedness and maximum principle for a Brezis-Oswald approach to mixed local and nonlocal operators.  arXiv e-prints, page arXiv:2103.11382, March 2021.
  * [9] Lorenzo Brasco and Erik Lindgren.  Higher Sobolev regularity for the fractional $p$-Laplace equation in the superquadratic case.  Adv. Math., 304:300–354, 2017.
  * [10] Lorenzo Brasco, Erik Lindgren, and Armin Schikorra.  Higher Hölder regularity for the fractional $p$-Laplacian in the superquadratic case.  Adv. Math., 338:782–846, 2018.
  * [11] Lorenzo Brasco, Erik Lindgren, and Martin Strömqvist.  Continuity of solutions to a nonlinear fractional diffusion equation.  arXiv e-prints, page arXiv:1907.00910, July 2019.
  * [12] S. Buccheri, J. V. da Silva, and L. H. de Miranda.  A System of Local/Nonlocal $p$-Laplacians: The Eigenvalue Problem and Its Asymptotic Limit as $p\to\infty$.  arXiv e-prints, page arXiv:2001.05985, January 2020.
  * [13] Verena Bögelein, Frank Duzaar, and Naian Liao.  On the Hölder regularity of signed solutions to a doubly nonlinear equation.  arXiv e-prints, page arXiv:2003.04158, March 2020.
  * [14] Zhen-Qing Chen, Panki Kim, Renming Song, and Zoran Vondraček.  Boundary Harnack principle for $\Delta+\Delta^{\alpha/2}$.  Trans. Amer. Math. Soc., 364(8):4169–4205, 2012.
  * [15] Zhen-Qing Chen and Takashi Kumagai.  A priori Hölder estimate, parabolic Harnack principle and heat kernel estimates for diffusions with jumps.  Rev. Mat. Iberoam., 26(2):551–589, 2010.
  * [16] Agnese Di Castro, Tuomo Kuusi, and Giampiero Palatucci.  Local behavior of fractional $p$-minimizers.  Ann. Inst. H. Poincaré Anal. Non Linéaire, 33(5):1279–1299, 2016.
  * [17] João Vitor da Silva and Ariel M. Salort.  A limiting problem for local/non-local $p$-Laplacians with concave-convex nonlinearities.  Z. Angew. Math. Phys., 71(6):Paper No. 191, 27, 2020.
  * [18] Eleonora Di Nezza, Giampiero Palatucci, and Enrico Valdinoci.  Hitchhiker’s guide to the fractional Sobolev spaces.  Bull. Sci. Math., 136(5):521–573, 2012.
  * [19] Emmanuele DiBenedetto.  Degenerate parabolic equations.  Universitext. Springer-Verlag, New York, 1993.
  * [20] Emmanuele DiBenedetto, Ugo Gianazza, and Vincenzo Vespri.  Harnack’s inequality for degenerate and singular parabolic equations.  Springer Monographs in Mathematics. Springer, New York, 2012.
  * [21] Mengyao Ding, Chao Zhang, and Shulin Zhou.  Local boundedness and Hölder continuity for the parabolic fractional $p$-Laplace equations.  Calc. Var. Partial Differential Equations, 60(1):Paper No. 38, 45, 2021.
  * [22] Serena Dipierro, Edoardo Proietti Lippi, and Enrico Valdinoci.  Linear theory for a mixed operator with Neumann conditions.  arXiv e-prints, page arXiv:2006.03850, June 2020.
  * [23] Serena Dipierro, Edoardo Proietti Lippi, and Enrico Valdinoci.  (Non)local logistic equations with Neumann conditions.  arXiv e-prints, page arXiv:2101.02315, January 2021.
  * [24] Serena Dipierro, Xavier Ros-Oton, Joaquim Serra, and Enrico Valdinoci.  Non-symmetric stable operators: regularity theory and integration by parts.  arXiv e-prints, page arXiv:2012.04833, December 2020.
  * [25] Serena Dipierro and Enrico Valdinoci.  Description of an ecological niche for a mixed local/nonlocal dispersal: an evolution equation and a new Neumann condition arising from the superposition of Brownian and Lévy processes.  arXiv e-prints, page arXiv:2104.11398, April 2021.
  * [26] Lawrence C. Evans.  Partial differential equations, volume 19 of Graduate Studies in Mathematics.  American Mathematical Society, Providence, RI, 1998.
  * [27] Mohammud Foondun.  Heat kernel estimates and Harnack inequalities for some Dirichlet forms with non-local part.  Electron. J. Probab., 14:no. 11, 314–340, 2009.
  * [28] Prashanta Garain and Juha Kinnunen.  On the regularity theory for mixed local and nonlocal quasilinear elliptic equations.  arXiv e-prints, page arXiv:2102.13365, February 2021.
  * [29] Prashanta Garain and Juha Kinnunen.  Weak Harnack inequality for a mixed local and nonlocal parabolic equation.  arXiv e-prints, page arXiv:2105.15016, May 2021.
  * [30] Prashanta Garain and Alexander Ukhlov.  Mixed local and nonlocal Sobolev inequalities with extremal and associated quasilinear singular elliptic problems.  arXiv e-prints, page arXiv:2106.04458, June 2021.
  * [31] Ky Ho and Inbo Sim.  Corrigendum to “Existence and some properties of solutions for degenerate elliptic equations with exponent variable” [Nonlinear Anal. 98 (2014) 146–164] [ MR3158451].  Nonlinear Anal., 128:423–426, 2015.
  * [32] Moritz Kassmann.  A new formulation of Harnack’s inequality for nonlocal operators.  C. R. Math. Acad. Sci. Paris, 349(11-12):637–640, 2011.
  * [33] Juha Kinnunen and Peter Lindqvist.  Pointwise behaviour of semicontinuous supersolutions to a quasilinear parabolic equation.  Ann. Mat. Pura Appl. (4), 185(3):411–435, 2006.
  * [34] Tuomo Kuusi.  Lower semicontinuity of weak supersolutions to nonlinear parabolic equations.  Differential Integral Equations, 22(11-12):1211–1222, 2009.
  * [35] Naian Liao.  Regularity of weak supersolutions to elliptic and parabolic equations: lower semicontinuity and pointwise behavior.  J. Math. Pures Appl. (9), (to appear), page arXiv:2011.03944, November 2020.
  * [36] Jan Malý and William P. Ziemer.  Fine regularity of solutions of elliptic partial differential equations, volume 51 of Mathematical Surveys and Monographs.  American Mathematical Society, Providence, RI, 1997.
  * [37] José M. Mazón, Julio D. Rossi and Julián Toledo.  Fractional $p$-Laplacian evolution equations.  J. Math. Pures Appl. (9) 105:810–844, 2016.
  * [38] Bin Shang, Yuzhou Fang and Chao Zhang.  Regularity theory for mixed local and nonlocal parabolic $p$-Laplace equations.  arXiv e-prints, page arXiv:2107.09825, July 2021.
  * [39] Martin Strömqvist.  Local boundedness of solutions to non-local parabolic equations modeled on the fractional $p$-Laplacian.  J. Differential Equations, 266(12):7948–7979, 2019.
  * [40] Juan Luis Vázquez.  The evolution fractional $p$-Laplacian equation in $\mathbb{R}^{N}$. Fundamental solution and asymptotic behaviour.  Nonlinear Anal., 199: 32 pp. 2020.
  * [41] Chao Xia.  On a class of anisotropic problems, Dissertation zur Erlangung des Doktorgrades der Fakultät Mathematik und Physik der Albert-Ludwigs-Universität tFreiburg, 2012. https://freidok.uni-freiburg.de/fedora/objects/freidok:8693/datastreams/FILE1/content.


C_{0}(\widetilde{X})\ignorespaces\ignorespaces\ignorespaces\ignorespaces\
}$$\textstyle{\ 0\ ,}$

where $C_{n}(\widetilde{X})$ is a $\mathbb{Z}[\pi]$-module, and $d_{n}\colon
C_{n}(\widetilde{X})\longrightarrow C_{n-1}(\widetilde{X})$ is a
$\mathbb{Z}[\pi]$-module homomorphism, for every $n\in\mathbb{Z}$. The
cohomology $H^{*}(X;\mathcal{L})$ of $X$ with local coefficients in
$\mathcal{L}$ is cohomology of the cochain complex

$\textstyle{\cdots\ }$$\textstyle{\
\ignorespaces\ignorespaces\ignorespaces\ignorespaces\hom_{\mathbb{Z}[\pi]}(C_{n+1}(\widetilde{X});\mathcal{L})\
}$$\textstyle{\
\ignorespaces\ignorespaces\ignorespaces\ignorespaces\hom_{\mathbb{Z}[\pi]}(C_{n}(\widetilde{X});\mathcal{L})}$$\textstyle{\ignorespaces\ignorespaces\ignorespaces\ignorespaces\hom_{\mathbb{Z}[\pi]}(C_{n-1}(\widetilde{X});\mathcal{L})}$$\textstyle{\cdots\
}$$\textstyle{\
\ignorespaces\ignorespaces\ignorespaces\ignorespaces\hom_{\mathbb{Z}[\pi]}(C_{1}(\widetilde{X});\mathcal{L})\
}$$\textstyle{\
\ignorespaces\ignorespaces\ignorespaces\ignorespaces\hom_{\mathbb{Z}[\pi]}(C_{0}(\widetilde{X});\mathcal{L})\
}$$\textstyle{\ \ignorespaces\ignorespaces\ignorespaces\ignorespaces 0\
.\qquad\qquad\qquad}$

This means that

$H^{n}(X;\mathcal{M}):=\frac{\ker\big{(}\hom_{\mathbb{Z}[\pi]}(C_{n}(\widetilde{X});\mathcal{L}))\longrightarrow\hom_{\mathbb{Z}[\pi]}(C_{n+1}(\widetilde{X});\mathcal{L})\big{)}}{\operatorname{im}\big{(}\hom_{\mathbb{Z}[\pi]}(C_{n-1}(\widetilde{X});\mathcal{L})\longrightarrow\hom_{\mathbb{Z}[\pi]}(C_{n}(\widetilde{X});\mathcal{L})\big{)}}\
.$

For $d=2$ the universal cover of the projective space $\mathbb{R}{\rm P}^{1}$
is the real line $\mathbb{R}^{1}$, and the fundamental group
$\pi=\pi_{1}(\mathbb{R}{\rm P}^{1})=\langle t\rangle$ is the infinite cyclic
group. An associated $\mathbb{Z}[\pi]$-CW-complex of the universal cover
$\widetilde{\mathbb{R}{\rm P}^{1}}=\mathbb{R}^{1}$ is defined as follows: The
$0$-cells are all integers $\\{x_{0}^{i}:=\\{i\\}:i\in\mathbb{Z}\\}$, while
the $1$-cells are intervals $\\{x_{1}^{i}:=[i,i+1]:i\in\mathbb{Z}\\}$. The
action of the generator $t$ of the fundamental group on the cells of the
$\mathbb{Z}[\pi]$-CW-complex is given by $t\cdot x_{j}^{i}=x_{j}^{i+1}$ for
$j\in\\{0,1\\}$. Hence, the induced cellular chain complex of
$\mathbb{Z}[\pi]$-modules is given by

$C_{0}(\widetilde{\mathbb{R}{\rm
P}^{1}})=\mathbb{Z}[\mathbb{Z}]=_{\mathbb{Z}[\mathbb{Z}]\text{-module}}\langle
x_{0}^{1}\rangle\qquad\text{and}\qquad C_{1}(\widetilde{\mathbb{R}{\rm
P}^{1}})=\mathbb{Z}[\mathbb{Z}]=_{\mathbb{Z}[\mathbb{Z}]\text{-module}}\langle
x_{1}^{1}\rangle$

where the only non-trivial boundary homomorphism $d_{1}$ is defined on the
generator of $C_{1}(\widetilde{\mathbb{R}{\rm P}^{1}})$ by
$d_{1}(x_{1}^{1}):=x_{0}^{1}-x_{0}^{0}=(t-1)\cdot x_{0}^{0}$. Thus, the
cellular chain complex we consider in this case is

$\textstyle{0\ignorespaces\ignorespaces\ignorespaces\ignorespaces\
}$$\textstyle{\
\mathbb{Z}[\mathbb{Z}]\ignorespaces\ignorespaces\ignorespaces\ignorespaces\
}$$\scriptstyle{(t-1)\cdot}$$\textstyle{\
\mathbb{Z}[\mathbb{Z}]\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\
0.}$

After applying the functor $\hom_{\mathbb{Z}[\mathbb{Z}]}(\cdot,\mathcal{M})$
we get the cochain complex

$\textstyle{0\ }$$\textstyle{\
\ignorespaces\ignorespaces\ignorespaces\ignorespaces\mathcal{M}\
}$$\textstyle{\
\ignorespaces\ignorespaces\ignorespaces\ignorespaces\mathcal{M}\
}$$\scriptstyle{(t-1)\cdot}$$\textstyle{\
\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ 0.}$

If we recall that additively $\mathcal{M}=\mathbb{F}_{2}\oplus\mathbb{F}_{2}$
by direct inspection we see that

$H^{r}(\mathbb{R}{\rm
P}^{1};\mathcal{M})\cong\begin{cases}\ker\big{(}\mathcal{M}\overset{(t-1)\cdot}{\longrightarrow}\mathcal{M}\big{)}\cong\mathbb{F}_{2},&r=0,\\\
\mathcal{M}/\operatorname{im}\big{(}\mathcal{M}\overset{(t-1)\cdot}{\longrightarrow}\mathcal{M}\big{)}\cong\mathbb{F}_{2},&r=1,\\\
0,&\text{otherwise}.\end{cases}$

For further use we denote the generator of the group $H^{1}(\mathbb{R}{\rm
P}^{d-1};\mathcal{M})$ by $z_{1}$.

Now, let $d\geq 3$ be an integer. Then the universal cover of the projective
space $\mathbb{R}{\rm P}^{d-1}$ is the sphere $S^{d-1}$, and the fundamental
group is $\pi:=\pi_{1}(\mathbb{R}{\rm P}^{d-1})\cong\mathbb{Z}_{2}=\langle
t\rangle$. We associate an $\mathbb{Z}[\pi]$-CW-complex to the universal cover
$\widetilde{\mathbb{R}{\rm P}^{d-1}}=S^{d-1}$ as follows: In each dimension
$i$, where $0\leq i\leq d-1$, there are two cells $x_{i}^{0}$ and $x_{i}^{1}$.
The generator $t$ of the fundamental group $\pi$ acts on the cells by $t\cdot
x_{j}^{0}=x_{j}^{1}$ and $t\cdot x_{j}^{1}=x_{j}^{0}$ for $0\leq j\leq d-1$.
Thus, the induced cellular chain complex of $\mathbb{Z}[\pi]$-modules is given
by

$C_{n}(\widetilde{\mathbb{R}{\rm
P}^{d-1}})=\mathbb{Z}[\mathbb{Z}_{2}]=_{\mathbb{Z}[\mathbb{Z}_{2}]\text{-module}}\langle
x_{n}^{0}\rangle,\qquad\qquad\text{for }0\leq n\leq d-1,$

where the boundary homomorphism $d_{n}\colon C_{n}(\widetilde{\mathbb{R}{\rm
P}^{d-1}})\longrightarrow C_{n-1}(\widetilde{\mathbb{R}{\rm P}^{d-1}})$ on the
generator $x_{n}^{0}$ is

$\displaystyle d_{n}(x_{n}^{0})$
$\displaystyle:=x_{n-1}^{1}-x_{n-1}^{0}=(t-1)\cdot
x_{n-1}^{0},\qquad\qquad\text{for }1\leq n\leq d-1\text{ odd},$ $\displaystyle
d_{n}(x_{n}^{0})$ $\displaystyle:=x_{n-1}^{1}+x_{n-1}^{0}=(t+1)\cdot
x_{n-1}^{0},\qquad\qquad\text{for }1\leq n\leq d-1\text{ even},$

and otherwise zero. The cellular chain complex we obtained in this case is

$\textstyle{0\ignorespaces\ignorespaces\ignorespaces\ignorespaces\
}$$\textstyle{\
\mathbb{Z}[\mathbb{Z}_{2}]\ignorespaces\ignorespaces\ignorespaces\ignorespaces\
}$$\scriptstyle{(t+1)\cdot}$$\textstyle{\
\mathbb{Z}[\mathbb{Z}_{2}]\ignorespaces\ignorespaces\ignorespaces\ignorespaces\
}$$\scriptstyle{(t-1)\cdot}$$\textstyle{\
\cdots\ignorespaces\ignorespaces\ignorespaces\ignorespaces\
}$$\scriptstyle{(t+1)\cdot}$$\textstyle{\
\mathbb{Z}[\mathbb{Z}_{2}]\ignorespaces\ignorespaces\ignorespaces\ignorespaces\
}$$\scriptstyle{(t-1)\cdot}$$\textstyle{\
\mathbb{Z}[\mathbb{Z}_{2}]\ignorespaces\ignorespaces\ignorespaces\ignorespaces\
}$$\textstyle{\ 0,}$

when $d$ is odd, and

$\textstyle{0\ignorespaces\ignorespaces\ignorespaces\ignorespaces\
}$$\textstyle{\
\mathbb{Z}[\mathbb{Z}_{2}]\ignorespaces\ignorespaces\ignorespaces\ignorespaces\
}$$\scriptstyle{(t-1)\cdot}$$\textstyle{\
\mathbb{Z}[\mathbb{Z}_{2}]\ignorespaces\ignorespaces\ignorespaces\ignorespaces\
}$$\scriptstyle{(t+1)\cdot}$$\textstyle{\
\cdots\ignorespaces\ignorespaces\ignorespaces\ignorespaces\
}$$\scriptstyle{(t+1)\cdot}$$\textstyle{\
\mathbb{Z}[\mathbb{Z}_{2}]\ignorespaces\ignorespaces\ignorespaces\ignorespaces\
}$$\scriptstyle{(t-1)\cdot}$$\textstyle{\
\mathbb{Z}[\mathbb{Z}_{2}]\ignorespaces\ignorespaces\ignorespaces\ignorespaces\
}$$\textstyle{\ 0,}$

when $d$ is even.

Applying the functor $\hom_{\mathbb{Z}[\mathbb{Z}_{2}]}(\cdot,\mathcal{M})$ we
get the cochain complex which is isomorphic to the following cochain complex

$\textstyle{0\ }$$\textstyle{\
\ignorespaces\ignorespaces\ignorespaces\ignorespaces\mathcal{M}\
}$$\textstyle{\
\ignorespaces\ignorespaces\ignorespaces\ignorespaces\mathcal{M}\
}$$\scriptstyle{(t+1)\cdot}$$\textstyle{\
\ignorespaces\ignorespaces\ignorespaces\ignorespaces\cdots\
}$$\scriptstyle{(t-1)\cdot}$$\textstyle{\
\ignorespaces\ignorespaces\ignorespaces\ignorespaces\mathcal{M}\
}$$\scriptstyle{(t+1)\cdot}$$\textstyle{\
\ignorespaces\ignorespaces\ignorespaces\ignorespaces\mathcal{M}\
}$$\scriptstyle{(t-1)\cdot}$$\textstyle{\
\ignorespaces\ignorespaces\ignorespaces\ignorespaces 0,}$

when $d$ is odd, and

$\textstyle{0\ }$$\textstyle{\
\ignorespaces\ignorespaces\ignorespaces\ignorespaces\mathcal{M}\
}$$\textstyle{\
\ignorespaces\ignorespaces\ignorespaces\ignorespaces\mathcal{M}\
}$$\scriptstyle{(t-1)\cdot}$$\textstyle{\
\ignorespaces\ignorespaces\ignorespaces\ignorespaces\cdots\
}$$\scriptstyle{(t+1)\cdot}$$\textstyle{\
\ignorespaces\ignorespaces\ignorespaces\ignorespaces\mathcal{M}\
}$$\scriptstyle{(t+1)\cdot}$$\textstyle{\
\ignorespaces\ignorespaces\ignorespaces\ignorespaces\mathcal{M}\
}$$\scriptstyle{(t-1)\cdot}$$\textstyle{\
\ignorespaces\ignorespaces\ignorespaces\ignorespaces 0,}$

when $d$ is even. Since $\mathcal{M}=\mathbb{F}_{2}\oplus\mathbb{F}_{2}$, and
the homomorphisms
$\mathcal{M}\overset{(t-1)\cdot}{\longrightarrow}\mathcal{M}$ and
$\mathcal{M}\overset{(t+1)\cdot}{\longrightarrow}\mathcal{M}$ coincide on
$\mathcal{M}$, we conclude that

$H^{r}(\mathbb{R}{\rm
P}^{d-1};\mathcal{M})\cong\begin{cases}\ker\big{(}\mathcal{M}\overset{(t-1)\cdot}{\longrightarrow}\mathcal{M}\big{)}\cong\mathbb{F}_{2},&r=0,\\\
\mathcal{M}/\operatorname{im}\big{(}\mathcal{M}\overset{(t-1)\cdot}{\longrightarrow}\mathcal{M}\big{)}\cong\mathbb{F}_{2},&r=d-1,\\\
0,&\text{otherwise}.\end{cases}$

We denote the generator of the group $H^{d-1}(\mathbb{R}{\rm
P}^{d-1};\mathcal{M})$ by $z_{d-1}$.

####  7.4.5. Homology of the real projective space with local coefficients

For $d\geq 2$, along the lines of the previous section, we compute the
homology of the projective space $H_{*}(\mathbb{R}{\rm P}^{d-1};\mathcal{M})$
where the local coefficient system $\mathcal{M}$ is additively
$\mathbb{F}_{2}\oplus\mathbb{F}_{2}$, and the action of
$\pi_{1}(\mathbb{R}{\rm P}^{d-1})=\langle t\rangle$ on $\mathcal{M}$ is given
by $t\cdot(a_{1},a_{2})=(a_{2},a_{1})$ where
$(a_{1},a_{2})\in\mathbb{F}_{2}\oplus\mathbb{F}_{2}$.

Like in the case of cohomology we consider two separate cases: $d=2$ when
$\pi_{1}(\mathbb{R}{\rm P}^{d-1})\cong\mathbb{Z}$, and $d\geq 3$ when
$\pi_{1}(\mathbb{R}{\rm P}^{d-1})\cong\mathbb{Z}_{2}$. First we recall the
definition of the homology with local coefficients, consult for example [60,
Sec. 3.H].

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

$\textstyle{\cdots\ \ignorespaces\ignorespaces\ignorespaces\ignorespaces\
}$$\textstyle{\
C_{n+1}(\widetilde{X})\ignorespaces\ignorespaces\ignorespaces\ignorespaces\
}$$\scriptstyle{d_{n+1}}$$\textstyle{\
C_{n}(\widetilde{X})\ignorespaces\ignorespaces\ignorespaces\ignorespaces\
}$$\scriptstyle{d_{n}}$$\textstyle{\
C_{n-1}(\widetilde{X})\ignorespaces\ignorespaces\ignorespaces\ignorespaces\
}$$\scriptstyle{d_{n-1}}$$\textstyle{\ \cdots}$$\textstyle{\cdots\
\ignorespaces\ignorespaces\ignorespaces\ignorespaces\
}$$\scriptstyle{d_{2}}$$\textstyle{\
C_{1}(\widetilde{X})\ignorespaces\ignorespaces\ignorespaces\ignorespaces\
}$$\scriptstyle{d_{1}}$$\textstyle{\
C_{0}(\widetilde{X})\ignorespaces\ignorespaces\ignorespaces\ignorespaces\
}$$\textstyle{\ 0\ ,}$

where $C_{n}(\widetilde{X})$ is a $\mathbb{Z}[\pi]$-module, and $d_{n}\colon
C_{n}(\widetilde{X})\longrightarrow C_{n-1}(\widetilde{X})$ is a
$\mathbb{Z}[\pi]$-module homomorphism, for every $n\in\mathbb{N}$.

The homology $H_{*}(X;\mathcal{L})$ of $X$ with local coefficients in
$\mathcal{L}$ is homology of the chain complex

$\textstyle{\cdots\ignorespaces\ignorespaces\ignorespaces\ignorespaces\
}$$\textstyle{\
C_{n+1}(\widetilde{X})\otimes_{\mathbb{Z}[\pi]}\mathcal{L}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\
}$$\textstyle{\
C_{n}(\widetilde{X})\otimes_{\mathbb{Z}[\pi]}\mathcal{L}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\
}$$\textstyle{\
C_{n-1}(\widetilde{X})\otimes_{\mathbb{Z}[\pi]}\mathcal{L}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\
}$$\textstyle{\
\cdots}$$\textstyle{\cdots\ignorespaces\ignorespaces\ignorespaces\ignorespaces\
}$$\textstyle{\
C_{1}(\widetilde{X})\otimes_{\mathbb{Z}[\pi]}\mathcal{L}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\
}$$\textstyle{\
C_{0}(\widetilde{X})\otimes_{\mathbb{Z}[\pi]}\mathcal{L}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\
}$$\textstyle{\ 0\ .\qquad\qquad}$

This means that

$H_{n}(X;\mathcal{M}):=\frac{\ker\big{(}C_{n}(\widetilde{X})\otimes_{\mathbb{Z}[\pi]}\mathcal{L}\longrightarrow
C_{n-1}(\widetilde{X})\otimes_{\mathbb{Z}[\pi]}\mathcal{L}\big{)}}{\operatorname{im}\big{(}C_{n+1}(\widetilde{X})\otimes_{\mathbb{Z}[\pi]}\mathcal{L}\longrightarrow
C_{n}(\widetilde{X})\otimes_{\mathbb{Z}[\pi]}\mathcal{L}\big{)}}\ .$

First we consider the case $d=2$. The universal cover of the projective space
$\mathbb{R}{\rm P}^{1}$ is the real line $\mathbb{R}^{1}$, and the fundamental
group $\pi=\pi_{1}(\mathbb{R}{\rm P}^{1})=\langle t\rangle$ is the infinite
cyclic group. An associated $\mathbb{Z}[\pi]$-CW-complex of the universal
cover $\widetilde{\mathbb{R}{\rm P}^{1}}=\mathbb{R}^{1}$ is defined as
follows: The $0$-cells are all integers
$\\{x_{0}^{i}:=\\{i\\}:i\in\mathbb{Z}\\}$, while the $1$-cells are intervals
$\\{x_{1}^{i}:=[i,i+1]:i\in\mathbb{Z}\\}$. The action of the generator $t$ of
the fundamental group on the cells of the $\mathbb{Z}[\pi]$-CW-complex is
given by $t\cdot x_{j}^{i}=x_{j}^{i+1}$ for $j\in\\{0,1\\}$. Hence, the
induced cellular chain complex of $\mathbb{Z}[\pi]$-modules is given by

$C_{0}(\widetilde{\mathbb{R}{\rm
P}^{1}})=\mathbb{Z}[\mathbb{Z}]=_{\mathbb{Z}[\mathbb{Z}]\text{-module}}\langle
x_{0}^{1}\rangle\qquad\text{and}\qquad C_{1}(\widetilde{\mathbb{R}{\rm
P}^{1}})=\mathbb{Z}[\mathbb{Z}]=_{\mathbb{Z}[\mathbb{Z}]\text{-module}}\langle
x_{1}^{1}\rangle$

where the only non-trivial boundary homomorphism $d_{1}$ is defined on the
generator of $C_{1}(\widetilde{\mathbb{R}{\rm P}^{1}})$ by
$d_{1}(x_{1}^{1}):=x_{0}^{1}-x_{0}^{0}=(t-1)\cdot x_{0}^{0}$. Thus, the
cellular chain complex we consider in this case is

$\textstyle{0\ignorespaces\ignorespaces\ignorespaces\ignorespaces\
}$$\textstyle{\
\mathbb{Z}[\mathbb{Z}]\ignorespaces\ignorespaces\ignorespaces\ignorespaces\
}$$\scriptstyle{(t-1)\cdot}$$\textstyle{\
\mathbb{Z}[\mathbb{Z}]\ignorespaces\ignorespaces\ignorespaces\ignorespaces\
}$$\textstyle{\ 0.}$

After applying the functor $\cdot\otimes_{\mathbb{Z}[\mathbb{Z}]}\mathcal{M}$
we get the chain complex

$\textstyle{0\ignorespaces\ignorespaces\ignorespaces\ignorespaces\
}$$\textstyle{\
\mathcal{M}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\
}$$\scriptstyle{(t-1)\cdot}$$\textstyle{\
\mathcal{M}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\
}$$\textstyle{\ 0.}$

Since additively $\mathcal{M}=\mathbb{F}_{2}\oplus\mathbb{F}_{2}$ by direct
inspection we see that

$H_{r}(\mathbb{R}{\rm
P}^{1};\mathcal{M})\cong\begin{cases}\mathcal{M}/\operatorname{im}\big{(}\mathcal{M}\overset{(t-1)\cdot}{\longrightarrow}\mathcal{M}\big{)}\cong\mathbb{F}_{2},&r=0,\\\
\ker\big{(}\mathcal{M}\overset{(t-1)\cdot}{\longrightarrow}\mathcal{M}\big{)}\cong\mathbb{F}_{2},&r=1,\\\
0,&\text{otherwise}.\end{cases}$

For further use we denote the generator of the group $H_{1}(\mathbb{R}{\rm
P}^{d-1};\mathcal{M})$ by $h_{1}$.

When $d\geq 3$ the universal cover of the projective space $\mathbb{R}{\rm
P}^{d-1}$ is the sphere $S^{d-1}$, and the fundamental group is
$\pi:=\pi_{1}(\mathbb{R}{\rm P}^{d-1})\cong\mathbb{Z}_{2}=\langle t\rangle$.
An $\mathbb{Z}[\pi]$-CW-complex is associated to the universal cover
$\widetilde{\mathbb{R}{\rm P}^{d-1}}=S^{d-1}$ as follows: In each dimension
$i$, where $0\leq i\leq d-1$, there are two cells $x_{i}^{0}$ and $x_{i}^{1}$.
The generator $t$ of the fundamental group $\pi$ acts on the cells by $t\cdot
x_{j}^{0}=x_{j}^{1}$ and $t\cdot x_{j}^{1}=x_{j}^{0}$ for $0\leq j\leq d-1$.
Thus, the induced cellular chain complex of $\mathbb{Z}[\pi]$-modules is given
by

$C_{n}(\widetilde{\mathbb{R}{\rm
P}^{d-1}})=\mathbb{Z}[\mathbb{Z}_{2}]=_{\mathbb{Z}[\mathbb{Z}_{2}]\text{-module}}\langle
x_{n}^{0}\rangle,\qquad\qquad\text{for }0\leq n\leq d-1,$

where the boundary homomorphism $d_{n}\colon C_{n}(\widetilde{\mathbb{R}{\rm
P}^{d-1}})\longrightarrow C_{n-1}(\widetilde{\mathbb{R}{\rm P}^{d-1}})$ on the
generator $x_{n}^{0}$ is

$\displaystyle d_{n}(x_{n}^{0})$
$\displaystyle:=x_{n-1}^{1}-x_{n-1}^{0}=(t-1)\cdot
x_{n-1}^{0},\qquad\qquad\text{for }1\leq n\leq d-1\text{ odd},$ $\displaystyle
d_{n}(x_{n}^{0})$ $\displaystyle:=x_{n-1}^{1}+x_{n-1}^{0}=(t+1)\cdot
x_{n-1}^{0},\qquad\qquad\text{for }1\leq n\leq d-1\text{ even},$

and otherwise zero. The cellular chain complex we obtained in this case is

$\textstyle{0\ignorespaces\ignorespaces\ignorespaces\ignorespaces\
}$$\textstyle{\
\mathbb{Z}[\mathbb{Z}_{2}]\ignorespaces\ignorespaces\ignorespaces\ignorespaces\
}$$\scriptstyle{(t+1)\cdot}$$\textstyle{\
\mathbb{Z}[\mathbb{Z}_{2}]\ignorespaces\ignorespaces\ignorespaces\ignorespaces\
}$$\scriptstyle{(t-1)\cdot}$$\textstyle{\
\cdots\ignorespaces\ignorespaces\ignorespaces\ignorespaces\
}$$\scriptstyle{(t+1)\cdot}$$\textstyle{\
\mathbb{Z}[\mathbb{Z}_{2}]\ignorespaces\ignorespaces\ignorespaces\ignorespaces\
}$$\scriptstyle{(t-1)\cdot}$$\textstyle{\
\mathbb{Z}[\mathbb{Z}_{2}]\ignorespaces\ignorespaces\ignorespaces\ignorespaces\
}$$\textstyle{\ 0,}$

when $d$ is odd, and

$\textstyle{0\ignorespaces\ignorespaces\ignorespaces\ignorespaces\
}$$\textstyle{\
\mathbb{Z}[\mathbb{Z}_{2}]\ignorespaces\ignorespaces\ignorespaces\ignorespaces\
}$$\scriptstyle{(t-1)\cdot}$$\textstyle{\
\mathbb{Z}[\mathbb{Z}_{2}]\ignorespaces\ignorespaces\ignorespaces\ignorespaces\
}$$\scriptstyle{(t+1)\cdot}$$\textstyle{\
\cdots\ignorespaces\ignorespaces\ignorespaces\ignorespaces\
}$$\scriptstyle{(t+1)\cdot}$$\textstyle{\
\mathbb{Z}[\mathbb{Z}_{2}]\ignorespaces\ignorespaces\ignorespaces\ignorespaces\
}$$\scriptstyle{(t-1)\cdot}$$\textstyle{\
\mathbb{Z}[\mathbb{Z}_{2}]\ignorespaces\ignorespaces\ignorespaces\ignorespaces\
}$$\textstyle{\ 0,}$

when $d$ is even.

Applying the functor $\cdot\otimes_{\mathbb{Z}[\mathbb{Z}]}\mathcal{M}$ yields
a chain complex isomorphic to the chain complex:

$\textstyle{0\ignorespaces\ignorespaces\ignorespaces\ignorespaces\
}$$\textstyle{\
\mathcal{M}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\
}$$\scriptstyle{(t+1)\cdot}$$\textstyle{\
\mathcal{M}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\
}$$\scriptstyle{(t-1)\cdot}$$\textstyle{\
\cdots\ignorespaces\ignorespaces\ignorespaces\ignorespaces\
}$$\scriptstyle{(t+1)\cdot}$$\textstyle{\
\mathcal{M}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\
}$$\scriptstyle{(t-1)\cdot}$$\textstyle{\
\mathcal{M}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\
}$$\textstyle{\ 0,}$

when $d$ is odd, and

$\textstyle{0\ignorespaces\ignorespaces\ignorespaces\ignorespaces\
}$$\textstyle{\
\mathcal{M}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\
}$$\scriptstyle{(t-1)\cdot}$$\textstyle{\
\mathcal{M}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\
}$$\scriptstyle{(t+1)\cdot}$$\textstyle{\
\cdots\ignorespaces\ignorespaces\ignorespaces\ignorespaces\
}$$\scriptstyle{(t+1)\cdot}$$\textstyle{\
\mathcal{M}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\
}$$\scriptstyle{(t-1)\cdot}$$\textstyle{\
\mathcal{M}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\
}$$\textstyle{\ 0,}$

when $d$ is even. Since $\mathcal{M}=\mathbb{F}_{2}\oplus\mathbb{F}_{2}$, and
the homomorphisms
$\mathcal{M}\overset{(t-1)\cdot}{\longrightarrow}\mathcal{M}$ and
$\mathcal{M}\overset{(t+1)\cdot}{\longrightarrow}\mathcal{M}$ coincide on
$\mathcal{M}$, we have that

$H_{r}(\mathbb{R}{\rm
P}^{d-1};\mathcal{M})\cong\begin{cases}\mathcal{M}/\operatorname{im}\big{(}\mathcal{M}\overset{(t-1)\cdot}{\longrightarrow}\mathcal{M}\big{)}\cong\mathbb{F}_{2},&r=0,\\\
\ker\big{(}\mathcal{M}\overset{(t-1)\cdot}{\longrightarrow}\mathcal{M}\big{)}\cong\mathbb{F}_{2},&r=d-1,\\\
0,&\text{otherwise}.\end{cases}$

We denote the generator of the group $H_{d-1}(\mathbb{R}{\rm
P}^{d-1};\mathcal{M})$ by $h_{d-1}$, and the generator of the group
$H_{0}(\mathbb{R}{\rm P}^{d-1};\mathcal{M})$ by $h_{0}$.

## References

  * [1] Alejandro Adem, John Maginnis, and R. James Milgram, _Symmetric invariants and cohomology of groups_ , Math. Ann. 287 (1990), no. 3, 391–411.
  * [2] Alejandro Adem and James R. Milgram, _Cohomology of Finite Groups_ , 2nd ed., Grundlehren der Mathematischen Wissenschaften, vol. 309, Springer-Verlag, Berlin, 2004.
  * [3] Vladimir Igorevich Arnol’d, _A remark on the branching of hyperelliptic integrals as functions of the parameters_ , Funkcional. Anal. i Priložen. 2 (1968), no. 3, 1–3.
  * [4] by same author, _Singularities of smooth mappings_ , Uspehi Mat. Nauk 23 (1968), no. 1, 3–44.
  * [5] by same author, _The cohomology ring of the group of dyed braids_ , Mat. Zametki 5 (1969), 227–231.
  * [6] by same author, _Certain topological invariants of algebrac functions_ , Trudy Moskov. Mat. Obšč. 21 (1970), 27–46.
  * [7] Emil Artin, _Theorie der Zöpfe_ , Abh. Math. Sem. Univ. Hamburg 4 (1925), no. 1, 47–72. MR 3069440
  * [8] by same author, _Theory of braids_ , Ann. of Math. (2) 48 (1947), 101–126.
  * [9] by same author, _Braids and permutations_ , Ann. of Math. (2) 48 (1947), 643–649.
  * [10] Michael F. Atiyah, _Power operations in $K$-theory_, Quart. J. Math. Oxford Ser. (2) 17 (1966), 165–193.
  * [11] Djordje Baralić, Pavle V. M. Blagojević, Roman Karasev, and Aleksandar Vučić, _Index of Grassmann manifolds and orthogonal shadows_ , Forum Math. 30 (2018), no. 6, 1539–1572.
  * [12] Joan S. Birman, _Braids, links, and mapping class groups_ , Princeton University Press, Princeton, N.J.; University of Tokyo Press, Tokyo, 1974, Annals of Mathematics Studies, No. 82.
  * [13] Anders Björner and Günter M. Ziegler, _Combinatorial stratification of complex arrangements_ , J. American Math. Soc. 5 (1992), no. 1, 105–149.
  * [14] Pavle V. M. Blagojević, Frederick R. Cohen, Wolfgang Lück, and Günter M. Ziegler, _On complex highly regular embeddings and the extended Vassiliev conjecture_ , Int. Math. Res. Not. IMRN (2016), no. 20, 6151–6199.
  * [15] Pavle V. M. Blagojević, Wolfgang Lück, and Günter M. Ziegler, _On highly regular embeddings_ , Trans. Amer. Math. Soc. 368 (2016), no. 4, 2891–2912.
  * [16] Pavle V. M. Blagojević and Günter M. Ziegler, _Convex equipartitions via equivariant obstruction theory_ , Israel J. Math. 200 (2014), no. 1, 49–77.
  * [17] Michael J. Boardman and Rainer M. Vogt, _Homotopy Invariant Algebraic Structures on Topological Spaces_ , Lecture Notes in Mathematics, vol. 347, Springer-Verlag, Berlin-New York, 1973.
  * [18] Carl-Friedrich Bödigheimer, _Stable splittings of mapping spaces_ , Algebraic Topology (Seattle, Wash., 1985), Lecture Notes in Mathematics, vol. 1286, Springer, Berlin, 1987, pp. 174–187.
  * [19] Vladimir Grigorévich Boltjanskiĭ, Sergeĭ Sergeevich Ryškov, and Yu. A. Šaškin, _On $k$-regular imbeddings and their application to the theory of approximation of functions_, Amer. Math. Soc. Transl. (2) 28 (1963), 211–219.
  * [20] Karol Borsuk, _On the $k$-independent subsets of the Euclidean space and of the Hilbert space_, Bull. Acad. Polon. Sci. Cl. III. 5 (1957), 351–356, XXIX.
  * [21] Glen E. Bredon, _Topology and Geometry_ , Graduate Texts in Math., vol. 139, Springer, New York, 1993.
  * [22] William Browder, _Homology operations and loop spaces_ , Illinois J. Math. 4 (1960), 347–357.
  * [23] Kenneth S. Brown, _Cohomology of Groups_ , Graduate Texts in Math., vol. 87, Springer-Verlag, New York, 1994.
  * [24] Jarosław Buczyński, Tadeusz Januszkiewicz, Joachim Jelisiejew, and Mateusz Michałek, _Constructions of $k$-regular maps using finite local schemes_, J. Eur. Math. Soc. (JEMS) 21 (2019), no. 6, 1775–1808.
  * [25] Carl-Friedrich Bödigheimer and Frederick R. Cohen, _Rational cohomology of configuration spaces of surfaces_ , Algebraic topology and transformation groups (Göttingen, 1987), Lecture Notes in Math., vol. 1361, Springer, Berlin, 1988, pp. 7–13.
  * [26] Carl-Friedrich Bödigheimer, Frederick R. Cohen, and Laurence R. Taylor, _On the homology of configuration spaces_ , Topology 28 (1989), no. 1, 111–123.
  * [27] Eddy H. E. A. Campbell and Ian P. Hughes, _The ring of upper triangular invariants as a module over the Dickson invariants_ , Math. Annalen 306 (1996), no. 3, 429–443.
  * [28] Jeffrey Caruso, Frederick R. Cohen, Peter J. May, and Laurence R. Taylor, _James maps, Segal maps, and the Kahn-Priddy theorem_ , Trans. Amer. Math. Soc. 281 (1984), no. 1, 243–283.
  * [29] Michael E. Chisholm, _$k$ -regular mappings of $2^{n}$-dimensional Euclidean space_, Proc. Amer. Math. Soc. 74 (1979), no. 1, 187–190.
  * [30] Wei-Liang Chow, _On the algebraical braid group_ , Ann. of Math. (2) 49 (1948), 654–658.
  * [31] Frederick R. Cohen, _Cohomology of braid spaces_ , Bull. Amer. Math. Soc. 79 (1973), 763–766.
  * [32] by same author, _Homology of $\Omega^{(n+1)}\Sigma^{(n+1)}X$ and $C_{(n+1)}X,\,n>0$_, Bull. Amer. Math. Soc. 79 (1973), 1236–1241 (1974).
  * [33] by same author, _The homology of $\mathcal{C}_{n+1}$-spaces, $n\geq 0$_, in: “The Homology of Iterated Loop Spaces”, Lecture Notes in Mathematics, vol. 533, Springer-Verlag, Heidelberg, 1976, pp. 207–351.
  * [34] by same author, _The unstable decomposition of $\Omega^{2}\Sigma^{2}X$ and its applications_, Math. Z. 182 (1983), no. 4, 553–568.
  * [35] Frederick R. Cohen, Ralph L. Cohen, Nicholas J. Kuhn, and Joseph A. Neisendorfer, _Bundles over configuration spaces_ , Pacific J. Math. 104 (1983), no. 1, 47–54.
  * [36] Frederick R. Cohen and David Handel, _$k$ -regular embeddings of the plane_, Proc. Amer. Math. Soc. 72 (1978), no. 1, 201–204.
  * [37] Frederick R. Cohen, Thomas J. Lada, and J. Peter May, _The homology of iterated loop spaces_ , Lecture Notes in Mathematics, Vol. 533, Springer-Verlag, Berlin-New York, 1976.
  * [38] Frederick R. Cohen, Mark E. Mahowald, and James R. Milgram, _The stable decomposition for the double loop space of a sphere_ , Algebraic and Geometric Topology (Proc. Sympos. Pure Math., Stanford Univ., Stanford, Calif., 1976), Part 2, Proc. Sympos. Pure Math., XXXII, Amer. Math. Soc., Providence, R.I., 1978, pp. 225–228.
  * [39] Frederick R. Cohen, Peter J. May, and Laurence R. Taylor, _Splitting of certain spaces $CX$_, Math. Proc. Camb. Philos. Soc. 84 (1978), 465–496.
  * [40] by same author, _Splitting of some more spaces_ , Math. Proc. Camb. Philos. Soc. 86 (1979), 227–236.
  * [41] by same author, _James maps and $E_{n}$ ring spaces_, Trans. Amer. Math. Soc. 281 (1984), no. 1, 285–295.
  * [42] Michael C. Crabb, _The topological Tverberg theorem and related topics_ , J. Fixed Point Theory Appl. 12 (2012), no. 1-2, 1–25.
  * [43] Leonard Eugene Dickson, _A fundamental system of invariants of the general modular linear group with a solution of the form problem_ , Trans. Amer. Math. Soc. 12 (1911), no. 1, 75–98.
  * [44] Tammo tom Dieck, _Transformation Groups_ , Studies in Mathematics, vol. 8, Walter de Gruyter, Berlin, 1987.
  * [45] Eldon Dyer and Richard K. Lashof, _Homology of iterated loop spaces_ , Amer. J. Math. 84 (1962), 35–88.
  * [46] Edward Fadell, _Homotopy groups of configuration spaces and the string problem of Dirac_ , Duke Math. J. 29 (1962), 231–242.
  * [47] Edward Fadell and Lee Neuwirth, _Configuration spaces_ , Math. Scand. 10 (1962), 111–118.
  * [48] Edward Fadell and James Van Buskirk, _On the braid groups of $E^{2}$ and $S^{2}$_, Bull. Amer. Math. Soc. 67 (1961), 211–213.
  * [49] Edward R. Fadell and Sufian Y. Husseini, _Geometry and topology of configuration spaces_ , Springer Monographs in Mathematics, Springer-Verlag, Berlin, 2001.
  * [50] Yves Félix and Jean-Claude Thomas, _Rational Betti numbers of configuration spaces_ , Topology Appl. 102 (2000), no. 2, 139–149.
  * [51] Mark Feshbach, _The mod 2 cohomology rings of the symmetric groups and invariants_ , Topology 41 (2002), no. 1, 57–84.
  * [52] Ralph Hartzler Fox and Lee Neuwirth, _The braid groups_ , Math. Scand. 10 (1962), 119–126.
  * [53] Dmitry Borisovich Fuks, _Cohomology of the braid group ${\rm mod}\ 2$_, Funkcional. Anal. i Priložen. 4 (1970), no. 2, 62–73.
  * [54] Mohammad Ghomi and Sergei Tabachnikov, _Totally skew embeddings of manifolds_ , Math. Z. 258 (2008), no. 3, 499–512.
  * [55] Chad Giusti, Paolo Salvatore, and Dev Sinha, _The mod-2 cohomology rings of symmetric groups_ , J. Topol. 5 (2012), no. 1, 169–198.
  * [56] Mikhail L. Gromov, _Isoperimetry of waists and concentration of maps_ , Geom. Funct. Anal. 13 (2003), no. 1, 178–215.
  * [57] David Handel, _Obstructions to $3$-regular embeddings_, Houston J. Math. 5 (1979), no. 3, 339–343.
  * [58] by same author, _$2k$ -regular maps on smooth manifolds_, Proc. Amer. Math. Soc. 124 (1996), no. 5, 1609–1613.
  * [59] David Handel and Jack Segal, _On $k$-regular embeddings of spaces in Euclidean space_, Fund. Math. 106 (1980), no. 3, 231–237.
  * [60] Allen Hatcher, _Algebraic Topology_ , Cambridge University Press, Cambridge, England, 2002.
  * [61] Nguyên Hũ’u Viêt Hu’ng, _The mod $2$ cohomology algebras of symmetric groups_, Acta Math. Vietnam. 6 (1981), no. 2, 41–48 (1982).
  * [62] by same author, _The mod $2$ equivariant cohomology algebras of configuration spaces_, Acta Math. Vietnam. 7 (1982), no. 1, 95–100.
  * [63] by same author, _The modulo $2$ cohomology algebras of symmetric groups_, Japan. J. Math. (N.S.) 13 (1987), no. 1, 169–208.
  * [64] by same author, _The mod $2$ equivariant cohomology algebras of configuration spaces_, Pac. J. Math. 143 (1990), no. 2, 251–286.
  * [65] Adolf Hurwitz, _Ueber Riemann’sche Flächen mit gegebenen Verzweigungspunkten_ , Math. Ann. 39 (1891), no. 1, 1–60.
  * [66] Daniel S. Kahn and Stewart B. Priddy, _The transfer and stable homotopy theory_ , Math. Proc. Cambridge Philos. Soc. 83 (1978), no. 1, 103–111.
  * [67] Roman Karasev, Alfredo Hubard, and Boris Aronov, _Convex equipartitions: the spicy chicken theorem_ , Geom. Dedicata 170 (2014), 263–279.
  * [68] Roman Karasev and Peter Landweber, _Estimating the higher symmetric topological complexity of spheres_ , Algebr. Geom. Topol. 12 (2012), no. 1, 75–94.
  * [69] Roman Karasev and Alexey Volovikov, _Waist of the sphere for maps to manifolds_ , Topology Appl. 160 (2013), no. 13, 1592–1602.
  * [70] Tatsuji Kudo and Shôrô Araki, _Topology of $H_{n}$-spaces and $H$-squaring operations_, Mem. Fac. Sci. Kyūsyū Univ. Ser. A. 10 (1956), 85–120.
  * [71] Ian J. Leary, _On the integral cohomology of wreath products_ , J. Algebra 198 (1997), no. 1, 184–239.
  * [72] Ib Madsen and James R. Milgram, _The Classifying Spaces for Surgery and Cobordism of Manifolds_ , Annals of Mathematics Studies, vol. 92, Princeton University Press, Princeton, NJ, 1979.
  * [73] Benjamin M. Mann, _The cohomology of the symmetric groups_ , Trans. Amer. Math. Soc. 242 (1978), 157–184.
  * [74] Benjamin Matschke, _A parametrized version of the Borsuk-Ulam-Bourgin-Yang-Volovikov theorem_ , J. Topol. Anal. 6 (2014), no. 2, 263–280.
  * [75] J. Peter May, _A general algebraic approach to Steenrod operations_ , The Steenrod Algebra and its Applications (Proc. Conf. to Celebrate N. E. Steenrod’s Sixtieth Birthday, Battelle Memorial Inst., Columbus, Ohio, 1970), Lecture Notes in Mathematics, vol. 168, Springer, Berlin, 1970, pp. 153–231.
  * [76] by same author, _The geometry of iterated loop spaces_ , Lectures Notes in Mathematics, vol. 271, Springer-Verlag, Berlin, New York, 1972.
  * [77] by same author, _The homology of $E_{\infty}$ spaces_, in: “The Homology of Iterated Loop Spaces” (Heidelberg), Lecture Notes in Mathematics, vol. 533, Springer-Verlag, 1976, pp. 1–68.
  * [78] Dusa McDuff, _Configuration spaces of positive and negative particles_ , Topology 14 (1975), 91–107.
  * [79] Dusa McDuff [printed: MacDuff], _Configuration spaces_ , $K$-theory and operator algebras (Proc. Conf., Univ. Georgia, Athens, Ga., 1975), 1977, pp. 88–95. Lecture Notes in Math., Vol. 575.
  * [80] John W. Milnor and James D. Stasheff, _Characteristic Classes_ , Annals of Mathematics Studies, vol. 76, Princeton University Press, Princeton, NJ, 1974\.
  * [81] Robert E. Mosher and Martin C. Tangora, _Cohomology Operations and Applications in Homotopy Theory_ , Harper & Row, Publishers, New York-London, 1968.
  * [82] Huỳnh Mùi, _Modular invariant theory and cohomology algebras of symmetric groups_ , J. Fac. Sci. Univ. Tokyo Sect. IA Math. 22 (1975), no. 3, 319–369.
  * [83] by same author, _Duality in the infinite symmetric products_ , Acta Math. Vietnam. 5 (1980), no. 1, 100–149 (1981).
  * [84] James R. Munkres, _Elements of algebraic topology_ , Addison-Wesley Publishing Company, Menlo Park, CA, 1984.
  * [85] Tokusi Nakamura, _On cohomology operations_ , Japan. J. Math. 33 (1963), 93–145.
  * [86] Minoru Nakaoka, _Decomposition theorem for homology groups of symmetric groups_ , Ann. of Math. (2) 71 (1960), 16–42.
  * [87] by same author, _Homology of the infinite symmetric group_ , Ann. of Math. (2) 73 (1961), 229–257.
  * [88] by same author, _Note on cohomology algebras of symmetric groups_ , J. Math. Osaka City Univ. 13 (1962), 45–55.
  * [89] Daniel Quillen, _The Adams conjecture_ , Topology 10 (1971), 67–80.
  * [90] by same author, _The spectrum of an equivariant cohomology ring. I, II_ , Ann. of Math. (2) 94 (1971), 549–572; ibid. (2) 94 (1971), 573–602.
  * [91] Daniel Quillen and Boris B. Venkov, _Cohomology of finite groups and elementary abelian subgroups_ , Topology 11 (1972), 317–318.
  * [92] Graeme Segal, _Configuration-spaces and iterated loop-spaces_ , Invent. Math. 21 (1973), 213–221.
  * [93] Paul A. Smith, _Fixed-point theorems for periodic transformations_ , Amer. J. Math. 63 (1941), 1–8.
  * [94] Victor P. Snaith, _A stable decomposition of $\Omega^{n}S^{n}X$_, J. London Math. Soc. (2) 7 (1974), 577–583.
  * [95] Norman E. Steenrod, _Cohomology Operations (Lectures by N. E. Steenrod written and revised by D. B. A. Epstein)_ , Annals of Mathematics Studies, vol. 50, Princeton University Press, Princeton NJ, 1962.
  * [96] Gordana Stojanović, _Embeddings with multiple regularity_ , Geom. Dedicata 123 (2006), 1–10.
  * [97] Aleksander Nikolaevič Varčenko, _The branching of multiple integrals which depend on parameters_ , Funkcional. Anal. i Priložen 3 (1969), no. 4, 79–80.
  * [98] Victor A. Vassiliev, _Braid group cohomologies and algorithm complexity_ , Funct. Anal. Appl. 22 (1988), no. 3, 182–190.
  * [99] by same author, _Complements of Discriminants of Smooth Maps: Topology and Applications_ , Translations of Mathematical Monographs, vol. 98, American Mathematical Society, Providence, RI, 1992, Translated from the Russian by B. Goldfarb.
  * [100] by same author, _On $r$-neighborly submanifolds in $\mathbb{R}^{N}$_, Topological Methods in Nonlinear Analysis. Journal of the Juliusz Schauder Center 11 (1998), 273–281.
  * [101] Clarence Wilkerson, _A primer on the Dickson invariants_ , Proceedings of the Northwestern Homotopy Theory Conference (Evanston, Ill., 1982), Contemp. Math., vol. 19, Amer. Math. Soc., Providence, RI, 1983, pp. 421–434.
  * [102] Oscar Zariski, _The topological discriminant group of a Riemann surface of genus $p$_, Amer. J. Math. 59 (1937), no. 2, 335–358.