# Parametrized topological complexity of sphere bundles

M. Farber  School of Mathematical Sciences, Queen Mary University of London
and  S. Weinberger  Department of Mathematics, University of Chicago

###### Abstract.

Parametrized motion planning algorithms [1] have high degree of flexibility
and universality, they can work under a variety of external conditions, which
are viewed as parameters and form part of the input of the algorithm. In this
paper we analyse the parameterized motion planning problem in the case of
sphere bundles. Our main results provide upper and lower bounds for the
parametrized topological complexity; the upper bounds typically involve
sectional categories of the associated fibrations and the lower bounds are
given in terms of characteristic classes and their properties. We explicitly
compute the parametrized topological complexity in many examples and show that
it may assume arbitrarily large values.

###### 1991 Mathematics Subject Classification:

55M30

M. Farber was partially supported by a grant from the EPSRC

S. Weinberger was partially supported by a grant from the NSF

##  1\. Introduction

The motion planning problem of robotics is one of the central themes which
makes possible autonomous robot motion, see [9]. A motion planning algorithm
takes as input the initial and the desired states of the system and produces
as output a motion of the system starting at the initial and ending at the
desired states. A robot is “told” where it needs to go and the execution of
this task, including selection of a specific route of motion, is made by the
robot itself. In this approach it is understood that the external conditions
(such as the positions of the obstacles and the geometry of the enclosing
domain) are known.

In recent papers [1], [2], motion planning algorithms of a new type were
considered. These are parametrized motion planning algorithms, which, besides
the initial and desired states, take as input the parameters characterising
the external conditions. The output of a parametrized motion planning
algorithm is a continuous motion of the system from the initial to the desired
state respecting the given external conditions. The papers [1, 2] laid out the
new formalism and analysed in full detail the problem of moving a number $n$
of robots in the domain with $m$ unknown obstacles. The authors used
techniques of algebraic topology and were able to find the answer by using a
combination of upper and lower bounds. The lower bounds use the structure of
the cohomology algebras. A brief introduction into the concept of parametrized
topological complexity is given below in §3.

The purpose of this article is to analyse the parametrized topological
complexity of sphere bundles. The Stiefel - Whitney and Euler characteristic
classes play an important role in these estimates. Our main results give upper
and lower bounds for the parametrized topological complexity and we compute
the parametrized topological complexity of a number of examples.

It would be interesting to adopt the theory of weights of cohomology classes
of E. Fadell and S. Husseini [4] with the purpose of strengthening the
cohomological lower bounds in application to the parametrized topological
complexity.

The authors thank the referee for very helpful comments.

##  2\. Sectional category of sphere bundle

In this section we recall some well-known results, see [11], which will be
useful later in this paper.

Let $\xi:E\to B$ be a rank $q$ vector bundle. We shall denote
$q={\rm{rk}}(\xi)$ and shall write $E(\xi)$ instead of $E$ when dealing with
several bundles at once. In this article we shall always assume that vector
bundles are equipped with metric structures, i.e. with continuous scalar
product on fibres.

We shall denote by $\dot{\xi}:\dot{E}\to B$ the associated bundle of
$(q-1)$-dimensional spheres, i.e. $\dot{\xi}=\xi|_{\dot{E}}$. Here
$\dot{E}\subset E$ is the set of vectors of length 1. If $\xi$ is oriented,
its Euler class ${\mathfrak{e}}(\xi)\in H^{q}(B)$ is defined, see [10]. Here
the cohomology is taken with integral coefficients. We shall adopt the
convention of skipping ${\mathbb{Z}}$ from the notations while indicating
explicitly all other coefficient groups.

For a cohomology class $\alpha\not=0$ we shall denote by
${\mathfrak{h}}(\alpha)$ its height, i.e. the largest integer $k$ such that
its $k$-th power is nonzero, $\alpha^{k}\not=0$. We shall also set
${\mathfrak{h}}(\alpha)=0$ for $\alpha=0$.

Recall that the sectional category (or Schwarz genus [11]) of a fibration with
base $B$ is defined as the minimal integer $k\geq 0$ such that there exists an
open cover

$B=U_{0}\cup U_{1}\cup\dots\cup U_{k}$

with the property that over each set $U_{i}$ the fibration admits a continuous
section.

######  Lemma 1.

Let $\xi:E\to B$ be a vector bundle, $q={\rm{rk}}(\xi)$. Then

  1. (A)

The sectional category of the sphere fibration $\dot{\xi}:\dot{E}\to B$
satisfies

${\sf{secat}}(\dot{\xi})\geq{\mathfrak{h}}({\rm w}_{q}(\xi))$

where ${\rm w}_{q}(\xi)\in H^{q}(B;{\mathbb{Z}}_{2})$ is the top Stiefel-
Whitney class of $\xi$.

  2. (B)

If the bundle $\xi$ is orientable then

${\sf{secat}}(\dot{\xi})\geq{\mathfrak{h}}({\mathfrak{e}}(\xi)).$

  3. (C)

Moreover, if $\xi$ is orientable and the base $B$ is a CW-complex whose
dimension satisfies

$\dim B\leq q\cdot{\mathfrak{h}}({\mathfrak{e}}(\xi))+q,$

then

$\displaystyle{\sf{secat}}(\dot{\xi})={\mathfrak{h}}({\mathfrak{e}}(\xi)).$

###### Proof.

(A) First we observe that $\dot{\xi}^{\ast}({\rm w}_{q}(\xi))=0\in
H^{q}(\dot{E};{\mathbb{Z}}_{2})$. Indeed, using functoriality of the Stiefel-
Whitney classes, we see that $\dot{\xi}^{\ast}({\rm w}_{q}(\xi))$ is the top
Stiefel - Whitney class of the induced fibration $\dot{\xi}^{\ast}(\xi)$ over
$\dot{E}(\xi)$. However this fibration admits a nonzero section $s(x)=x$ where
$x\in\dot{E}(\xi)$. The top Stiefel - Whitney class of a vector bundle having
a section vanishes, hence $\xi^{\ast}({\rm w}_{q}(\xi))={\rm
w}_{q}(\dot{\xi}^{\ast}(\xi))=0$.

Finally we apply the general cohomological lower bound for the sectional
category, see [11], Theorem 4; this gives
${\sf{secat}}(\dot{\xi})\geq{\mathfrak{h}}({\rm w}_{q}(\xi))$.

(B) follows similarly.

To prove (C) we apply Theorem 3 of Schwarz [11] which identifies the sectional
category of $\dot{\xi}$ with the smallest number $k$ such that the
$(k+1)$-fold fiberwise join $(\dot{\xi})^{\ast(k+1)}$ admits a continuous
section. Note that $(\dot{\xi})^{\ast(k+1)}$ is fiberwise homeomorphic to the
unit sphere bundle of the vector bundle
$(k+1)\xi=\xi\oplus\xi\oplus\dots\oplus\xi$, the Whitney sum of $k+1$ copies
of $\xi$. The obstructions for a section of $(\dot{\xi})^{\ast(k+1)}$ lie in
the groups

$H^{i}(B;\pi_{i-1}(S^{q(k+1)-1})),\quad i=1,2,\dots.$

The first obstruction (with $i=q(k+1)$) equals

${\mathfrak{e}}(\xi)^{k+1}\,=\,{\mathfrak{e}}((k+1)\xi)\,\,\in H^{q(k+1)}(B).$

Taking $k={\mathfrak{h}}({\mathfrak{e}}(\xi))$ we obtain
${\mathfrak{e}}(\xi)^{k+1}=0$, i.e. the first obstruction vanishes. The
further obstructions (with $i=q(k+1)+j$ where $j=1,\,2,\dots$) also vanish
because of our assumption $\dim B\leq
q\cdot({\mathfrak{h}}({\mathfrak{e}}(\xi))+1)$. This completes the proof. ∎

######  Example 2.

Let $\eta:E\to{\mathbb{CP}}^{n}$ denote the canonical complex line bundle over
${\mathbb{CP}}^{n}$. We shall view $\eta$ as a rank 2 real vector bundle. Its
Euler class ${\mathfrak{e}}(\eta)$ is the generator of
$H^{2}({\mathbb{CP}}^{n})$ and ${\mathfrak{h}}({\mathfrak{e}}(\eta))=n$. Since
$\dim{\mathbb{CP}}^{n}=2n\leq 2\cdot(n+1)$, Lemma 1 (C) applies and gives
${\sf{secat}}(\eta)=n$.

######  Example 3.

Let $\eta:E\to{\mathbb{CP}}^{n}$ be as in the previous Example. For $k\leq n$
consider $\xi_{k}=k\eta=\eta\oplus\eta\oplus\dots\oplus\eta$, the Whitney sum
of $k$ copies of $\eta$. We have ${\rm{rk}}(\xi_{k})=2k$, and
${\mathfrak{e}}(\xi_{k})={\mathfrak{e}}(\eta)^{k}$ implying
${\mathfrak{h}}({\mathfrak{e}}(\xi_{k}))=\lfloor n/k\rfloor$. The inequality
$\dim B=2n\leq 2k\cdot(\lfloor n/k\rfloor+1)$ is satisfied and using Lemma 1
we obtain

(1) $\displaystyle{\sf{secat}}(\xi_{k})\,=\lfloor n/k\rfloor$

for any $k=1,2,\dots,n$. Formula (1) is also true for $k>n$ as then the bundle
$\xi_{k}$ admits a section and hence ${\sf{secat}}(\xi_{k})=0$.

######  Remark 4.

There is a version of statement (C) of Lemma 1 for non-orientable bundles; in
this case the Euler class lies in the cohomology $e(\xi)\in
H^{q}(B;\tilde{\mathbb{Z}})$ with twisted coefficients and its powers
$e(\xi)^{k}$ lie in the groups $H^{kq}(B,(\tilde{\mathbb{Z}})^{\otimes k})$.
It is easy to see that $(\tilde{\mathbb{Z}})^{\otimes k}={\mathbb{Z}}$ for $k$
even and $(\tilde{\mathbb{Z}})^{\otimes k}=\tilde{\mathbb{Z}}$ for $k$ odd.

##  3\. Parametrized topological complexity

In this section we briefly recall the notion of parametrized topological
complexity which was recently introduced in [1], [2]. It is a generalization
of the concept of topological complexity of robot motion planning problem
introduced in [5]; see also [6].

Let $X$ be a path-connected topological space viewed as the space of states of
a mechanical system. The motion planning problem of robotics asks for an
algorithm which takes as input an initial state and a desired state of the
system, and produces as output a continuous motion of the system from the
initial state to the desired state, see [9]. That is, given $(x_{0},x_{1})\in
X\times X$, the algorithm will produce a continuous path $\gamma:I\to X$ with
$\gamma(0)=x_{0}$ and $\gamma(1)=x_{1}$, where $I=[0,1]$ denotes the unit
interval.

Let $X^{I}$ denote the space of all continuous paths in $X$, equipped with the
compact-open topology. The map $\pi:X^{I}\to X\times X$, where
$\pi(\gamma)=(\gamma(0),\gamma(1))$, is a fibration in the sense of Hurewicz.
A solution of the motion planning problem, a motion planning algorithm, is a
section of this fibration, i.e. a map $s:X\times X\to X^{I}$ with $\pi\circ
s={\rm id}_{X\times X}$. If $X$ is not contractible, no section can be
continuous, see [5]. The topological complexity of $X$ is defined to be the
sectional category, or Schwarz genus, of the fibration $\pi:X^{I}\to X\times
X$; notation: ${\sf TC}(X)={\sf{secat}}(\pi)$. In other words, ${\sf TC}(X)$
is the smallest integer $k$ for which there exists an open cover $X\times
X=U_{0}\cup U_{1}\cup\dots\cup U_{k}$ such that the fibration $\pi$ admits a
continuous section $s_{j}:U_{j}\to X^{I}$ for each $j=0,1,\dots,k$.

In the parametrized setting developed in [1], one assumes that the motion of
the system is constrained by external conditions, such as obstacles or
variable geometry of the containing domain. The initial and terminal states of
the system, as well as the motion between them, must live under the same
external conditions.

This situation is modelled by a fibration $p:E\to B$, with path-connected
fibers, where the base $B$ is a topological space encoding the variety of
external conditions. For $b\in B$, the fiber $X_{b}=p^{-1}(b)$ is viewed as
the space of achievable configurations of the system given the constraints
imposed by $b$. A parametrized motion planning algorithm takes as input
initial and terminal states of the system (consistent with external conditions
$b$), and produces a continuous path between them, achievable under external
conditions $b$. The initial and terminal points, as well as the path between
them, all lie within the same fiber $X_{b}$.

To define the parametrized topological complexity of the fibration $p:E\to B$
one needs to introduce the associated fibration $\Pi:E^{I}_{B}\to
E\times_{B}E,$ where $E\times_{B}E$ is the space of all pairs of
configurations lying in the same fiber of $p$, while $E_{B}^{I}$ stands for
the space of continuous paths in $E$ lying in a single fiber of $p$; the map
$\Pi$ sends a path to its endpoints.

######  Definition 5.

The parametrized topological complexity of the fibration $p:E\to B$ is defined
as the sectional category of the fibration

(2) $\displaystyle\Pi:E_{B}^{I}\to
E\times_{B}E,\quad\Pi(\gamma)=(\gamma(0),\gamma(1)).$

In more detail,

${\sf TC}[p:E\to B]:={\sf{secat}}(\Pi:E_{B}^{I}\to E\times_{B}E)$

is the minimal integer $k$ such that $E\times_{B}E$ admits an open cover
$E\times_{B}E=U_{0}\cup U_{1}\cup\dots\cup U_{k}$ with the property that each
set $U_{i}$ admits a continuous section of $\Pi$, where $i=0,1,\dots,k$.

Note that $\Pi:E_{B}^{I}\to E\times_{B}E$ is a Hurewicz fibration assuming
that $p:E\to B$ is a Hurewicz fibration, see [2], Proposition 2.1.

If $B^{\prime}\subset B$ is a subset and $E^{\prime}=p^{-1}(B^{\prime})\subset
E$, then the topological complexity ${\sf TC}[p^{\prime}:E^{\prime}\to
B^{\prime}]$ of the restricted fibration (where $p^{\prime}=p|_{E^{\prime}}$)
clearly satisfies

${\sf TC}[p^{\prime}:E^{\prime}\to B^{\prime}]\leq{\sf TC}[p:E\to B].$

In particular, we obtain the inequality

(3) $\displaystyle{\sf TC}(X)\leq{\sf TC}[p:E\to B]$

where $X$ is the fibre of $p:E\to B$.

######  Lemma 6.

Let $p:E\to B$ be a locally trivial fibration with fibre $X$. (A) If ${\sf
TC}[p:E\to B]=0$ then $X$ is contractible. (B) Conversely, if the the fibre
$X$ is contractible and the base $B$ is paracompact then there exists a
globally defined continuous parametrized motion planning algorithm
$s:E\times_{B}E\to E^{I}_{B}$ and therefore ${\sf TC}[p:E\to B]=0$.

###### Proof.

If ${\sf TC}[p:E\to B]=0$ then ${\sf TC}(X)=0$ because of (3). By Theorem 1
from [5] this implies that $X$ is contractible. To prove (B) we shall apply
Corollary 3.2 from the paper of A. Dold [3]. It implies that a locally trivial
fibre bundle $p:E\to B$ with paracompact base and contractible fibre is
shrinkable; this means that there exists a continuous section $\sigma:B\to E$
and a homotopy $H:E\times I\to E$ such that for any $e\in E$ one has
$H(e,0)=e$, $H(e,1)=\sigma p(e)$ and $p(H(e,t))=p(e)$. We may define the
section $s:E\times_{B}E\to E^{I}_{B}$ by the formula

(7) $\displaystyle
s(e,e^{\prime})(t)=\left\\{\begin{array}[]{lll}H(e,2t),&\mbox{for}&0\leq t\leq
1/2,\\\ \\\ H(e^{\prime},2-2t),&\mbox{for}&1/2\leq t\leq 1,\end{array}\right.$

where $(e,e^{\prime})\in E\times_{B}E$ and $t\in I$. Since $H(e,1)=\sigma
p(e)=\sigma p(e^{\prime})=H(e^{\prime},1)$, we see that both parts of the
formula (7) match and hence $s$ is continuous. We clearly have
$s(e,e^{\prime})(0)=e$ and $s(e,e^{\prime})(1)=e^{\prime}$. Besides,
$p(s(e,e^{\prime})(t))=p(e)=p(e^{\prime})$, i.e. $s$ is a continuous
parametrized motion planning algorithm. ∎

Next we mention the upper and lower bounds for the parametrized topological
complexity established in [1].

######  Proposition 7 (Proposition 7.2 from [1]).

Let $p:E\rightarrow B$ be a locally trivial fibration with fiber $X$, where
the spaces $E,B,X$ are CW-complexes. Assume that the fiber $X$ is r-connected.
Then

(8) $\displaystyle{\sf TC}[p:E\to B]<\frac{2\dim X+\dim B+1}{r+1}.$

######  Proposition 8 (Proposition 7.3 from [1]).

Let $p:E\rightarrow B$ be a fibration with path-connected fiber. Consider the
diagonal map $\Delta:E\rightarrow E\times_{B}E$, where $\Delta(e)=(e,e)$. Then
the parametrized topological complexity ${\sf TC}[p:E\rightarrow B]$ is
greater than or equal to the cup-length of the kernel
$\ker[\Delta^{\ast}:H^{\ast}(E\times_{B}E;R)\rightarrow H^{\ast}(E;R)]$, where
$R$ is an arbitrary coefficient ring.

######  Proposition 9.

[Proposition 4.7 from [1]] If $p:E\rightarrow B$ is a locally trivial
fibration, and the spaces $E$ and $B$ are metrizable separable ANRs, then in
Definition 5, instead of open covers one may use arbitrary covers of
$E\times_{B}E$ or, equivalently, arbitrary partitions

$E\times_{B}E=F_{0}\sqcup F_{1}\sqcup\cdot\cdot\cdot\sqcup F_{k},\quad
F_{i}\cap F_{j}=\emptyset,\quad i\not=j$

admitting continuous sections $s_{i}:F_{i}\rightarrow E_{B}^{I}$, where
$i=0,1,...,k.$

######  Example 10.

As an illustration consider the canonical complex line bundle
$\eta:E\to{\mathbb{CP}}^{n}=B$ viewed as a real rank 2 vector bundle. The unit
sphere bundle $\dot{\eta}:\dot{E}(\eta)\to{\mathbb{CP}}^{n}$ is a principal
$S^{1}$-bundle, its total space $\dot{E}(\eta)$ is the sphere $S^{2n-1}$, the
set of unit vectors $z\in{\mathbb{C}}^{n}$. The unit circle
$S^{1}\subset{\mathbb{C}}$ acts by multiplication, this action is free and the
quotient is ${\mathbb{CP}}^{n}$. We claim that

(9) $\displaystyle{\sf TC}[\dot{\eta}:\dot{E}(\eta)\to{\mathbb{CP}}^{n}]=1.$

Indeed, using (3) we get ${\sf
TC}[\dot{\eta}:\dot{E}(\eta)\to{\mathbb{CP}}^{n}]\geq{\sf TC}(S^{1})=1$. To
obtain the inverse inequality we consider the following partition

$\dot{E}(\eta)\times_{B}\dot{E}(\eta)=F_{0}\sqcup F_{1}$

where $F_{0}$ is the set of all pairs $(z_{1},z_{2})$ of unit vectors
$z_{1},z_{2}\in S^{2n-1}$ lying in the same fibre but not antipodal, i.e.
$z_{1}\not=-z_{2}$; the set $F_{1}$ is the set of antipodal pairs $(z,-z)$.
For $(z_{1},z_{2})\in F_{0}$ we can write $z_{2}/z_{1}=e^{i\phi}$ where
$\phi\in(-\pi,\pi)$ and a continuous section $s_{0}$ of the fibration (2) over
$F_{0}$ can be defined as follows:

$\displaystyle s_{0}(z_{1},z_{2})(t)\,=\,e^{i\phi t}z_{1},\quad t\in[0,1].$

On the other hand, over $F_{1}$ we can define a continuous section $s_{1}$
where

(10) $\displaystyle s_{1}(z,-z)(t)\,=\,e^{i\pi t}\cdot z,\quad t\in[0,1].$

This proves (9).

This example is a special case of a more general statement that the
parametrized topological complexity of any principal bundle equals the
Lusternik - Schnirelmann category of the fibre, see Proposition 4.3 in [1].

The main result of [1] is the computation of the parametrized topological
complexity of the Fadell - Neuwirth fibration which, in term of robotics, can
be understood as the complexity of controlling multiple robots in the presence
of multiple movable obstacles.

##  4\. The cup-length associated with a section

Material of this section will play an auxiliary role in the sequel. We shall
use notations introduced in the beginning of §2.

Let $\xi:E\to B$ be an oriented vector bundle of rank $q\geq 2$ equipped with
scalar product structure $\langle\,,\,\rangle$. As above, let
$\dot{\xi}:\dot{E}\to B$ denote the unit sphere bundle; its fibre is
homeomorphic to $S^{q-1}$. In this section we shall assume that the fibration
$\dot{\xi}:\dot{E}\to B$ has a continuous section $s:B\to\dot{E}$ and our goal
will be to identify the kernel

(11) $\ker[s^{\ast}:H^{\ast}(\dot{E})\to H^{\ast}(B)]$

and its cup-length, i.e. the length of the longest nontrivial products of
elements of this kernel. This result will be used in the following sections to
estimate the parametrized topological complexity from below.

We shall use the following remark. The oriented sphere fibration
$\dot{\xi}:\dot{E}\to B$ admits a cohomological extension of the fibre (see
[12], chapter 5, §7) if and only if its Euler class vanishes,
${\mathfrak{e}}(\xi)=0$. In particular, any oriented sphere fibration with
trivial Euler class ${\mathfrak{e}}(\xi)=0$ satisfies the conclusion of the
Leray - Hirsch theorem, see [12].

Let $\dot{F}\subset\dot{E}$ denote the set $\\{e\in\dot{E};\,e\perp
s(\xi(e))\\}$; it is the set of unit vectors perpendicular to the section. The
projection

$\eta:\,\dot{F}\to B,\quad\eta=\dot{\xi}|_{\dot{F}}$

is an oriented bundle of $(q-2)$-dimensional spheres. Let
${\mathfrak{e}}(\eta)\,\in\,H^{q-1}(B)$ denote the Euler class of $\eta$. The
mod-2 reduction of the class ${\mathfrak{e}}(\eta)$ equals the Stiefel -
Whitney class ${\rm w}_{q-1}(\xi)\in H^{q-1}(B;{\mathbb{Z}}_{2})$ of $\xi$.

######  Theorem 11.

The cup-length of the kernel (11) equals
${\mathfrak{h}}({\mathfrak{e}}(\eta))+1$ where
${\mathfrak{h}}({\mathfrak{e}}(\eta))$ denotes the height of the Euler class
${\mathfrak{e}}(\eta)\in H^{q-1}(B)$.

###### Proof.

Let $U\in H^{q-1}(\dot{E})$ denote a fundamental class: for any $b\in B$ the
restriction $U|_{\dot{E}_{b}}$ is the fundamental class of the fibre
$\dot{E}_{b}$. By the Leray - Hirsch theorem every cohomology class in
$H^{\ast}(\dot{E})$ has a unique representation in the form

$\xi^{\ast}(u)+\xi^{\ast}(v)\smile U,$

where $u,v\in H^{\ast}(B)$.

Let $W,W^{\prime}\subset\dot{E}$ denote the following subsets:

$W=\\{e\in\dot{E};\langle e,s(\xi(e))\rangle\geq 0\\}\quad\mbox{and}\quad
W^{\prime}=\\{e\in\dot{E};\langle e,s(\xi(e))\rangle\leq 0\\}.$

Clearly $W\cup W^{\prime}=\dot{E}$ and $W\cap W^{\prime}=\dot{F}$. One can
identify $W$ with the unit disc bundle of the sphere bundle $\dot{F}$.
Therefore the quotient $\dot{E}/W^{\prime}=W/\dot{F}$ can be naturally
identified with the Thom space of the fibration $\eta$.

Next we observe that the fundamental class $U\in H^{q-1}(\dot{E})$ can be
chosen such that $U|_{W^{\prime}}=0$. Indeed, starting with an arbitrary
choice $U^{\prime}$ we can replace it by $U=U^{\prime}-\xi^{\ast}(x)$ where
$x\in H^{q-1}(B)$ is such that
$(\xi|_{W^{\prime}})^{\ast}(x)=U^{\prime}|_{W^{\prime}}$. Here we use the
observation that $\xi|_{W^{\prime}}:W^{\prime}\to B$ is a homotopy equivalence
and hence the class $x$ mentioned above exists and is unique. With this choice
clearly $U|_{W^{\prime}}=0$.

Once the fundamental class $U$ satisfies $U|_{W^{\prime}}=0$ we have the
following formulae which fully describe the multiplicative structure of
$H^{\ast}(\dot{E})$:

(12) $\displaystyle s^{\ast}(U)={\mathfrak{e}}(\eta)\,\in\,H^{q-1}(B)$

and

(13) $\displaystyle U\smile U=\xi^{\ast}({\mathfrak{e}}(\eta)))\smile
U\,\in\,H^{2(q-1)}(\dot{E}).$

To prove (13) we note that $U|_{W}=0$ implies that the class $U$ can be
refined to a relative class $\tilde{U}\in
H^{q-1}(\dot{E},W^{\prime})=H^{q-1}(\dot{E}/W^{\prime})=H^{q-1}(W/\dot{F})$.
We already mentioned that the quotient $\dot{E}/W^{\prime}$ can be identified
with the Thom space of the vector bundle having $\eta$ as its unit sphere
bundle. Examining the long exact sequence in cohomology

$\dots\to
H^{q-2}(\dot{E})\stackrel{{\scriptstyle\simeq}}{{\to}}H^{q-2}(W^{\prime})\to
H^{q-1}(\dot{E},W^{\prime})\to H^{q-1}(\dot{E})$

we see that the refinement $\tilde{U}$ is unique and coincides with the Thom
class. Now, by the definition (see [10], §9), we have

(14)
$\displaystyle{\mathfrak{e}}(\eta)\,=\,s^{\ast}(\tilde{U}|_{W})\,=\,s^{\ast}(U),$

which proves (12). From (14) we also obtain

$\tilde{U}\smile\tilde{U}=(\tilde{U}|_{W})\smile\tilde{U}=\xi^{\ast}({\mathfrak{e}}(\eta))\smile\tilde{U}\,\,\in\,\,H^{2(q-1)}(\dot{E},W^{\prime}).$

Applying the restriction homomorphism $H^{2(q-1)}(\dot{E},W^{\prime})\to
H^{2(q-1)}(\dot{E})$ to both sides of this equality gives (13).

Note that the order of the factors in the RHS of formula (13) is irrelevant:
if $q$ is odd then the classes commute and for $q$ is even the Euler class
${\mathfrak{e}}(\eta)$ has order two.

Consider now an arbitrary class $x\in H^{\ast}(\dot{E})$ satisfying
$s^{\ast}(x)=0$. We can write

$x=\xi^{\ast}(\alpha)+\xi^{\ast}(\beta)\smile
U\quad\mbox{with}\quad\alpha,\,\beta\in H^{\ast}(B).$

Applying $s^{\ast}$ and using (12) we get

(15) $\displaystyle
s^{\ast}(x)\,=\,0\,=\,\alpha+\beta\smile{\mathfrak{e}}(\eta).$

Conversely, any two classes $\alpha,\,\beta$ satisfying (15) produce a class
$x=\xi^{\ast}(\alpha)+\xi^{\ast}(\beta)\smile U$ lying in the kernel of
$s^{\ast}$. A particular choice $\alpha=-{\mathfrak{e}}(\eta)$ and $\beta=1$
gives the class

$x_{0}=U-\xi^{\ast}({\mathfrak{e}}(\eta))\,\in H^{q-1}(\dot{E}).$

Using (13) we have

$x_{0}^{2}=U^{2}-2\xi^{\ast}({\mathfrak{e}}(\eta))\smile
U+\xi^{\ast}({\mathfrak{e}}(\eta))^{2}=-\xi^{\ast}({\mathfrak{e}}(\eta))\smile
x_{0}$

and we obtain by induction

$\displaystyle x_{0}^{n}$ $\displaystyle=$
$\displaystyle(-1)^{n-1}\xi^{\ast}({\mathfrak{e}}(\eta)^{n-1})\smile x_{0}$
$\displaystyle=$
$\displaystyle(-1)^{n}\xi^{\ast}({\mathfrak{e}}(\eta)^{n})+(-1)^{n-1}\xi^{\ast}({\mathfrak{e}}(\eta))^{n-1}\smile
U.$

For $n=h({\mathfrak{e}}(\eta))+1$ the class $x_{0}^{n}$ equals
$(-1)^{n-1}\xi^{\ast}({\mathfrak{e}}(\eta))^{n-1}\smile U$ and is obviously
nonzero (as follows from the Leray - Hirsch theorem). This implies that the
cup-length of the kernel $\ker s^{\ast}$ is at least
${\mathfrak{h}}({\mathfrak{e}}(\eta))+1$.

If $x=\xi^{\ast}(\alpha)+\xi^{\ast}(\beta)\smile U\,\in\,H^{\ast}(\dot{E})$ is
an arbitrary class with $s^{\ast}(x)=0$ then
$\alpha=-\beta\smile{\mathfrak{e}}(\eta)$ (see above) and thus
$x=\xi^{\ast}(\beta)\smile x_{0}$. In other words, the kernel $\ker
s^{\ast}\subset H^{\ast}(\dot{E})$ is the principal ideal generated by the
class $x_{0}$. We see that the cup-length of the kernel equals the highest
nonzero power of $x_{0}$ which, as we have shown above, is
${\mathfrak{h}}({\mathfrak{e}}(\eta))+1$. ∎

The following Corollary is an analogue of Theorem 11 where we use
$\mathbb{Z}_{2}$ coefficients. The role of the Euler class plays the top
Stiefel - Whitney class of the bundle $\eta$ of vectors orthogonal to the
section. The advantage of this statement is that the answer is given in terms
of the original bundle $\xi$ and its characteristic class ${\rm
w}_{q-1}(\xi)\in H^{q-1}(B;\mathbb{Z}_{2})$.

######  Corollary 12.

Let $\xi:E\to B$ be a rank $q\geq 2$ vector bundle (not necessarily
orientable). Let $s:B\to\dot{E}$ be a continuous section of the unit sphere
bundle. Then the cup-length of the kernel
$\ker[s^{\ast}:H^{\ast}(\dot{E};{\mathbb{Z}}_{2})\to
H^{\ast}(B;{\mathbb{Z}}_{2})]$ equals ${\mathfrak{h}}({\rm w}_{q-1}(\xi))+1$.

###### Proof.

One repeats the arguments of the proof of Theorem 11 replacing the integer
coefficients by ${\mathbb{Z}}_{2}$. The bundle $\eta:\,F\to
B,\quad\eta=\xi|_{F}$ is the bundle of vectors orthogonal to the section, i.e.
$F=\\{e\in E;\,e\perp s(\xi(e))\\}$ and the arguments of the proof of Theorem
11 show that the the kernel
$\ker[s^{\ast}:H^{\ast}(\dot{E};{\mathbb{Z}}_{2})\to
H^{\ast}(B;{\mathbb{Z}}_{2})]$ is the principal ideal generated by the class
$x_{0}=U-\xi^{\ast}({\rm w}_{q-1}(\eta))$. The height of $x_{0}$ equals one
plus the height of the class ${\rm w}_{q-1}(\eta)$. However,
$\xi=\eta\oplus\epsilon$ where $\epsilon$ is the trivial line bundle
determined by the section and therefore ${\rm w}_{q-1}(\eta)={\rm
w}_{q-1}(\xi)$ and the result follows. ∎

##  5\. Parametrized topological complexity of sphere bundles

Let $\xi:E\to B$ be an oriented rank $q\geq 2$ vector bundle equipped with
fibrewise scalar product. Let $\dot{\xi}:\dot{E}\to B$ denote the unit sphere
bundle; its fibre is the sphere of dimension $q-1$. Our goal is to estimate
the parametrized topological complexity ${\sf TC}[\dot{\xi}:\dot{E}\to B]$. By
Proposition 7 we have an upper bound

(17) $\displaystyle{\sf TC}[\dot{\xi}:\dot{E}\to B]<2+\frac{\dim B+1}{q-1}.$

To state our result, consider the bundle

(18) $\displaystyle\ddot{\xi}:\ddot{E}\to\dot{E},$

where $\ddot{E}\subset\dot{E}\times_{B}\dot{E}$ is the space of pairs of
mutually orthogonal unit vectors $(x,y)\in\dot{E}\times_{B}\dot{E}$, $x\perp
y$. The projection $\ddot{\xi}$ acts as $\ddot{\xi}(x,y)=x.$ The map (18) is
an oriented locally trivial fibration with fibre sphere of dimension $q-2$.
Consider its Euler class

(19) $\displaystyle{\mathfrak{e}}(\ddot{\xi})\in H^{q-1}(\dot{E}).$

An obvious property of the class ${\mathfrak{e}}(\ddot{\xi})$ is that for any
point $b\in B$ the restriction ${\mathfrak{e}}(\ddot{\xi})|_{\dot{E}_{b}}$ is
the Euler class of the tangent bundle of the sphere $\dot{E}_{b}$.

######  Remark 13.

A section $s$ of bundle (18) associates with a unit vector $e\in\dot{E}(\xi)$
a unit vector $s(e)$ which is perpendicular to $e$ and the integer
${\sf{secat}}(\ddot{\xi})$ is a measure of complexity of construction such a
section $s$ globally, i.e. over all $\dot{E}$. In particular,
${\sf{secat}}(\ddot{\xi})=0$ if the vector bundle $\xi$ admits a complex
structure: in this case one can define the section by $s(e)=\sqrt{-1}\cdot e$.

######  Theorem 14.

One has the estimates

(20) $\displaystyle{\mathfrak{h}}({\mathfrak{e}}(\ddot{\xi}))+1\,\leq\,{\sf
TC}[\dot{\xi}:\dot{E}\to B]\leq{\sf{secat}}(\ddot{\xi})+1.$

Moreover, if $B$ is a CW-complex satisfying $\dim
B\leq(q-1)\cdot{\mathfrak{h}}({\mathfrak{e}}(\ddot{\xi}))$ then

(21) $\displaystyle{\sf TC}[\dot{\xi}:\dot{E}\to
B]\,=\,{\mathfrak{h}}({\mathfrak{e}}(\ddot{\xi}))+1\,=\,{\sf{secat}}(\ddot{\xi})+1.$

###### Proof.

Consider the diagonal map $\Delta:\dot{E}\to\dot{E}\times_{B}\dot{E}$ and
apply Proposition 8; we obtain that the parametrized topological complexity
${\sf TC}[\dot{\xi}:\dot{E}\to B]$ is greater than or equal to the cup-length
of the kernel $\ker[\Delta^{\ast}:H^{\ast}(\dot{E}\times_{B}\dot{E})\to
H^{\ast}(\dot{E})]$. However, $\Delta$ is a section of the sphere fibration
$\dot{E}\times_{B}\dot{E}\to\dot{E}$ given by projection on the first vector;
hence we may apply Theorem 11 which describes the cup-length of the kernel of
the induced map. The bundle of vectors perpendicular to the section is exactly
the bundle $\ddot{\xi}$. By Theorem 11 the cup-length of the kernel
$\ker[\Delta^{\ast}:H^{\ast}(\dot{E}\times_{B}\dot{E})\to H^{\ast}(\dot{E})]$
equals ${\mathfrak{h}}({\mathfrak{e}}(\ddot{\xi}))+1$. This gives the lower
bound in (20).

To prove the right inequality in (20) consider the set
$U\subset\dot{E}\times_{B}\dot{E}$ consisting of pairs
$(e,e^{\prime})\in\dot{E}\times_{B}\dot{E}$ with $e\not=-e^{\prime}$. Over
$U$, we can define a continuous motion planning algorithm
$s:U\to\dot{E}_{B}^{I}$ by setting

(22) $\displaystyle
s(e,e^{\prime})(t)=\frac{(1-t)e+te^{\prime}}{||(1-t)e+te^{\prime}||},\quad
t\in[0,1].$

In view of Proposition 9 it remains to construct a motion planning algorithm
over the complementary set
$V=\\{(e,-e);\,e\in\dot{E}\\}\subset\dot{E}\times_{B}\dot{E}.$ Denote by
$p_{1},p_{2}:V\to\dot{E}$ the projections $p_{1}(e,-e)=e$ and
$p_{2}(e,-e)=-e.$

Consider again the bundle $\ddot{\xi}:\ddot{E}\to\dot{E}$ and suppose that
$A\subset\dot{E}$ is a subset such that the bundle $\ddot{\xi}$ admits a
continuous section $s_{A}:A\to\ddot{E}$ over $A$. Using this section we may
construct a section $s^{\prime}_{A}$ of the fibration

$\Pi:\dot{E}_{B}^{I}\to\dot{E}\times_{B}\dot{E}$

over the set $p_{1}^{-1}(A)$ as follows:

(23) $\displaystyle s^{\prime}_{A}(e,-e)(t)\,=\,\cos\left(t\pi\right)\cdot
e\,+\,\sin\left(t\pi\right)\cdot s_{A}(e),\quad t\in[0,1].$

Let $\dot{E}=A_{0}\cup A_{1}\cup\dots\cup A_{k}$ be an open covering, where
$k={\sf{secat}}(\ddot{\xi})$, with the property that
$\ddot{\xi}:\ddot{E}\to\dot{E}$ admits a continuous section over each $A_{i}$.
Then the sets $p_{1}^{-1}(A_{i})$ cover $V$ and over each of these sets the
fibration $\Pi$ admits a continuous section. Thus we get an inequality ${\sf
TC}[\dot{\xi}:\dot{E}\to B]\leq{\sf{secat}}(\ddot{\xi})+1$.

Finally we apply Lemma 1 which claims that the sectional category of
$\ddot{\xi}$ equals ${\mathfrak{h}}({\mathfrak{e}}(\ddot{\xi}))$ under an
additional assumption that
$\dim\dot{E}\leq(q-1)\cdot{\mathfrak{h}}({\mathfrak{e}}(\ddot{\xi}))+(q-1)$
which is equivalent to $\dim
B\leq(q-1)\cdot{\mathfrak{h}}({\mathfrak{e}}(\ddot{\xi}))$. Hence under this
assumption we obtain ${\sf TC}[\dot{\xi}:\dot{E}\to
B]\,=\,{\mathfrak{h}}({\mathfrak{e}}(\ddot{\xi}))+1\,=\,{\sf{secat}}(\ddot{\xi})+1$.
This completes the proof. ∎

######  Corollary 15.

For a vector bundle $\xi:E\to B$ satisfying ${\sf{secat}}(\ddot{\xi})=0$ one
has ${\sf TC}[\dot{\xi}:\dot{E}\to B]=1.$

###### Proof.

The inequality (20) gives ${\sf TC}[\dot{\xi}:\dot{E}\to B]\leq 1.$ On the
other hand, ${\sf TC}[\dot{\xi}:\dot{E}\to B]\geq{\sf TC}(S^{2r-1})=1$ by (3).
∎

######  Example 16.

Consider the canonical rank 2 bundle $\xi$ over ${\mathbb{CP}}^{n}$ as in
Example 10. In this case $\dot{E}(\xi)=S^{2n-1}$ and the bundle
$\ddot{\xi}:\ddot{E}\to\dot{E}$ is the trivial bundle with fibre $S^{0}$, i.e.

${\sf{secat}}(\ddot{\xi})=0={\mathfrak{h}}({\mathfrak{e}}(\ddot{\xi})).$

We obtain from (20) that ${\sf TC}[\dot{\xi}:S^{2n-1}\to{\mathbb{CP}}^{n}]=1$
confirming the result of Example 10.

Generalising Example 16 we may state:

######  Corollary 17.

For any vector bundle $\xi:E\to B$ of even rank ${\rm{rk}}(\xi)=2r$ admitting
a complex structure, one has

${\sf TC}[\dot{\xi}:\dot{E}\to B]=1={\sf TC}(S^{2r-1}).$

###### Proof.

In this case ${\sf{secat}}(\ddot{\xi})=0$ (by Remark 13) and the result
follows from Corollary 15. ∎

######  Remark 18.

Introducing the bundle $\ddot{\xi}$ over $\dot{E}$ and using its sectional
category to estimate the parametrized topological complexity we made an
approximation of the space of paths on the sphere connecting a pair of
antipodal points by the sphere of one dimension below. This sphere is however
only the first term in the James’ construction $JS^{q-2}$, see [8], which
gives a CW complex having the homotopy type of this space of paths.

Note that for $q$ even the Euler class ${\mathfrak{e}}(\ddot{\xi})\in
H^{q-1}(\dot{E})$ has order 2, i.e. $2\cdot{\mathfrak{e}}(\ddot{\xi})=0$. We
shall focus below on the case when $q$ odd. Compared with Theorem 14,
Corollary 19 stated below has the advantage of dealing with cohomology of the
base $B$.

######  Corollary 19.

For $q\geq 3$ odd, let $\eta:E(\eta)\to B$ be an oriented vector bundle of
rank $q-1$. Let $\xi=\eta\oplus\epsilon$ be the sum where $\epsilon$ is the
trivial line bundle over $B$. Then one has

(24) $\displaystyle{\sf TC}[\dot{\xi}:\dot{E}(\xi)\to
B]\geq{\mathfrak{h}}({\mathfrak{e}}(\eta))+1.$

Moreover, if the height ${\mathfrak{h}}({\mathfrak{e}}(\eta))$ is even and the
integral cohomology of the base $B$ in dimension
$(q-1)\cdot{\mathfrak{h}}({\mathfrak{e}}(\eta))$ has no 2-torsion then

(25) $\displaystyle{\sf TC}[\dot{\xi}:\dot{E}(\xi)\to
B]\geq{\mathfrak{h}}({\mathfrak{e}}(\eta))+2;$

###### Proof.

Consider the Euler class ${\mathfrak{e}}(\ddot{\xi})\in
H^{q-1}(\dot{E}(\xi))$. Applying the Leray - Hirsch theorem we see that any
class in $H^{q-1}(\dot{E}(\xi))$ has a unique representation as
$\dot{\xi}^{\ast}(a)+bU$ where $U\in H^{q-1}(\dot{E})$ is a fundamental class,
$a\in H^{q-1}(B)$ and $b\in{\mathbb{Z}}$. Let $s:B\to\dot{E}(\xi)$ be the
section determined by the trivial summand $\epsilon$. We showed in the proof
of Theorem 11 that the fundamental class $U$ can be chosen so that

(26) $\displaystyle s^{\ast}(U)={\mathfrak{e}}(\eta),$

see formula (12). Note that

(27) $\displaystyle s^{\ast}(\ddot{\xi})=\eta.$

Besides,

(28)
$\displaystyle{\mathfrak{e}}(\ddot{\xi})=\dot{\xi}^{\ast}(a)+2U,\quad\mbox{for
some class}\quad a\in H^{q-1}(B).$

Indeed, the class ${\mathfrak{e}}(\ddot{\xi})$ restricted to each fibre
$\dot{E}_{b}(\xi)$ equals twice the fundamental class of the sphere
$\dot{E}_{b}(\xi)\simeq S^{q-1}$ (here we use our assumption that $q$ is odd,
and hence the Euler characteristic of $S^{q-1}$ equals 2). Applying $s^{\ast}$
to both sides of equation (28) we find
$s^{\ast}({\mathfrak{e}}(\ddot{\xi}))={\mathfrak{e}}(s^{\ast}(\ddot{\xi}))={\mathfrak{e}}(\eta)$,
and $s^{\ast}(\dot{\xi}^{\ast}(a))=a$ which together with (26) give
$a=-{\mathfrak{e}}(\eta).$ Therefore we have

(29)
$\displaystyle{\mathfrak{e}}(\ddot{\xi})=-\dot{\xi}^{\ast}({\mathfrak{e}}(\eta))+2U.$

Using $U^{2}=\dot{\xi}^{\ast}({\mathfrak{e}}(\eta))\smile U$ (see (13)) we
find
${\mathfrak{e}}(\ddot{\xi})^{2}=\dot{\xi}^{\ast}({\mathfrak{e}}(\eta)^{2})$
and therefore the even and odd powers of the class
${\mathfrak{e}}(\ddot{\xi})$ are as follows

(30)
$\displaystyle{\mathfrak{e}}(\ddot{\xi})^{2n}=\dot{\xi}^{\ast}({\mathfrak{e}}(\eta)^{2n})$

and

(31)
$\displaystyle{\mathfrak{e}}(\ddot{\xi})^{2n+1}=-\dot{\xi}^{\ast}({\mathfrak{e}}(\eta)^{2n+1})+2\dot{\xi}^{\ast}({\mathfrak{e}}(\eta)^{2n})\smile
U.$

From formulae (30) and (31) we see that the height
${\mathfrak{h}}({\mathfrak{e}}(\ddot{\xi}))$ either equals to
${\mathfrak{h}}({\mathfrak{e}}(\eta))$ or it equals
${\mathfrak{h}}({\mathfrak{e}}(\eta))+1$; the second possibility happens iff
${\mathfrak{h}}({\mathfrak{e}}(\eta))$ is even and the group
$H^{(q-1){\mathfrak{h}}({\mathfrak{e}}(\eta))}(B)$ has no 2-torsion.

Applying Theorem 14 completes the proof. ∎

######  Example 20.

Consider the situation of Corollary 19 in the case when
$\eta:E(\eta)\to{\mathbb{CP}}^{n}$ is the canonical bundle over the complex
projective space as in Example 10. Taking $\xi=\eta\oplus\epsilon$ we have
${\rm{rk}}(\xi)=q=3$ is odd and ${\mathfrak{h}}({\mathfrak{e}}(\eta))=n$. By
Corollary 19 we get ${\sf TC}[\dot{\xi}:\dot{E}(\xi)\to{\mathbb{CP}}^{n}]\geq
n+1$ and moreover for $n$ even ${\sf
TC}[\dot{\xi}:\dot{E}(\xi)\to{\mathbb{CP}}^{n}]\geq n+2$. On the other hand,
the upper bound (17) gives ${\sf
TC}[\dot{\xi}:\dot{E}(\xi)\to{\mathbb{CP}}^{n}]\leq n+2$. Thus, we see that

${\sf TC}[\dot{\xi}:\dot{E}(\xi)\to{\mathbb{CP}}^{n}]=n+2$

for all even $n$. In particular, we see that the parametrized topological
complexity of sphere bundles can be arbitrarily large. This contrasts the
situation with the usual (i.e. unparametrized) topological complexity which
takes the values $1$ and $2$ only for spheres.

Finally we describe an explicit parametrized motion planning algorithm having
complexity $n+2$ for the unit sphere bundle associated with the vector bundle
$\xi=\eta\oplus\epsilon$ over $B={\mathbb{CP}}^{n}$ as considered in Example
20. We shall describe a partition

(32) $\displaystyle\dot{E}(\xi)\times_{B}\dot{E}(\xi)\,=\,F_{0}\sqcup
F_{1}\sqcup\dots\sqcup F_{n+2}$

and a continuous section $s_{i}$ of the fibration

$\Pi:\dot{E}(\xi)^{I}_{B}\,\to\,\dot{E}(\xi)\times_{B}\dot{E}(\xi)$

over each of the sets $F_{i}$ where $i=0,1,\dots,n+2$.

The set $F_{0}\subset\dot{E}(\xi)\times_{B}\dot{E}(\xi)$ will be defined as
the set of pairs $(x,y)\in\dot{E}(\xi)\times_{B}\dot{E}(\xi)$ with $x\not=-y$.
The section $s_{0}$ over $F_{0}$ can be defined by formula (22).

The unit sphere bundle of the trivial summand $\epsilon$ gives the sections
$\pm\sigma:B\to\dot{E}(\epsilon)\subset\dot{E}(\xi)$. Let
$\dot{E}(\xi)^{\ast}$ denote the complement $\dot{E}(\xi)-\dot{E}(\epsilon)$.
We define the set $F_{1}\subset\dot{E}(\xi)\times_{B}\dot{E}(\xi)$ to be the
set of all pairs $(x,-x)$ with $x\in\dot{E}(\xi)^{\ast}$. Let ${\rm
pr}:\dot{E}(\xi)^{\ast}\to\dot{E}(\eta)\subset\dot{E}(\xi)$ denote the
retraction given by the formula

(33) $\displaystyle{\rm pr}(x)\,=\,\frac{x-\langle
x,\sigma(b)\rangle\cdot\sigma(b)}{||x-\langle
x,\sigma(b)\rangle\cdot\sigma(b)||}\quad\mbox{where}\quad b=\xi(x).$

Here the symbol $\langle\,,\,\rangle$ denotes scalar product in the fibre. The
deformation

$\alpha_{t}(x)\,=\,\frac{x-t\cdot\langle
x,\sigma(b)\rangle\cdot\sigma(b)}{||x-t\cdot\langle
x,\sigma(b)\rangle\cdot\sigma(b)||}\quad\mbox{where}\quad
b=\xi(x),\,t\in[0,1],$

satisfies $\alpha_{0}(x)=x$ and $\alpha_{1}(x)={\rm pr}(x)$. The homotopy
$t\mapsto(\alpha_{t}(x),\alpha_{t}(-x))$ deforms the initial pair $(x,-x)$ to
a pair of antipodal points lying in the equatorial sphere
$\dot{E}(\eta)\subset\dot{E}(\xi)$. Note that the circle $S^{1}$ acts freely
on $\dot{E}(\eta)$, see Example 10. We may define a continuous section $s_{1}$
over $F_{1}$ by setting $s_{1}(x,-x)(t)$ to be the concatenation of the
following three paths: (a) the deformation $\alpha_{t}(x)$, (b) the section
(10) of Example 10, and (c) the reverse of the deformation $\alpha_{t}(-x)$.
In more detail,

$s_{1}(x,-x)(t)=\left\\{\begin{array}[]{lll}\alpha_{3t}(x),&\mbox{for}&t\in[0,1/3],\\\
\\\ e^{i\pi(3t-1)}\cdot{\rm pr}(x),&\mbox{for}&t\in[1/3,2/3],\\\ \\\
\alpha_{3(1-t)}(-x),&\mbox{for}&t\in[2/3,1].\end{array}\right.$

Finally we define the sections $s_{i}:F_{i}\to\dot{E}(\xi)^{I}_{B}$ for
$i=2,3,\dots,n+2$ as follows. The base $B={\mathbb{CP}}^{n}$ has the well-
known cell decomposition ${\mathbb{CP}}^{n}=e^{0}\sqcup e^{2}\sqcup\dots\sqcup
e^{2n}$ with a single cell $e^{2i}$ in each even dimension $2i\leq 2n$. For
$i=2,3,\dots,n+2$, let $F_{i}$ denote the set of pairs $(x,-x)$ with
$x=\pm\sigma(b)$ for $b=\xi(x)$ lying in the cell $e^{2(i-2)}$. Since the cell
$e^{2(i-2)}$ is contractible, the bundle $\eta$ admits a continuous section
$\phi_{i}$ over $e^{2(i-2)}$. Hence we may define the section
$s_{i}:F_{i}\to\dot{E}(\xi)^{I}_{B}$ by the formula

$s_{i}(x,-x)(t)=\cos(\pi t)\cdot x+\sin(\pi t)\cdot\phi_{i}(\xi(x)),$

similarly to (23). Here we assume that the Euclidean structure on the vector
bundle $\xi$ is the orthogonal sum of the Euclidean structures of $\eta$ and
$\epsilon$.

We conclude the paper with the following observations.

Below we always assume that the base $B$ is an ANR.

######  Lemma 21.

Let $\xi:E\to B$ be a vector bundle such that $\xi=\eta\oplus\tau$ and
${\rm{rk}}(\tau)\geq 2$. Then

(34) $\displaystyle\hskip
28.45274pt{\sf{secat}}(\ddot{\xi}:\ddot{E}(\xi)\to\dot{E}(\xi))\,\leq\,{\sf{secat}}(\ddot{\tau}:\ddot{E}(\tau)\to\dot{E}(\tau))+{\sf{secat}}(\dot{\tau}:\dot{E}(\tau)\to
B)+1$

and consequently

(35) $\displaystyle{\sf TC}[\dot{\xi}:\dot{E}\to
B]\leq{\sf{secat}}(\ddot{\tau}:\ddot{E}(\tau)\to\dot{E}(\tau))+{\sf{secat}}(\dot{\tau}:\dot{E}(\tau)\to
B)+2.$

###### Proof.

It is enough to prove the inequality (34) as (35) follows from (34) and
Theorem 14.

If $\xi=\eta\oplus\tau$ then for any point of the base $b\in B$ we have
$E(\xi)_{b}=E(\eta)_{b}\oplus E(\tau)_{b}$. The scalar product can be chosen
so that the spaces $E(\eta)_{b},\,E(\tau)_{b}\subset E(\xi)_{b}$ are mutually
orthogonal. We shall denote by $P^{\eta}_{b}$ and $P^{\tau}_{b}$ the
orthogonal projections of $E(\xi)_{b}$ onto $E(\eta)_{b}$ and $E(\tau)_{b}$
correspondingly.

Let $k$ denote ${\sf{secat}}(\ddot{\tau}:\ddot{E}(\tau)\to\dot{E}(\tau))$ and
let $\ell$ denote ${\sf{secat}}(\dot{\tau}:\dot{E}(\tau)\to B)$. Let
$\dot{E}(\tau)=G_{0}\sqcup G_{1}\sqcup\dots\sqcup G_{k}$ be a partition such
that for each $j=0,\dots,k$ there exists a continuous map
$\sigma_{j}:G_{j}\to\dot{E}(\tau)$ with the property that for every $e\in
G_{j}$ the vectors $e$ and $\sigma_{j}(e)$ lie in the same fibre and are
perpendicular to each other. Besides, let $B_{0}\sqcup B_{1}\sqcup\dots\sqcup
B_{\ell}=B$ be a partition of the base $B$ with continuous sections
$\nu_{i}:B_{i}\to\dot{E}(\tau)$, where $i=0,\dots,\ell$.

We want to show that $\dot{E}(\xi)$ can be partitioned as

$\dot{E}(\xi)=F_{0}\sqcup F_{1}\sqcup\dots\sqcup F_{k}\sqcup
F_{k+1}\sqcup\dots\sqcup F_{k+\ell+1}$

such that there exist continuous sections $s_{i}:F_{i}\to E(\ddot{\xi})$ of
the fibration $\ddot{\xi}$. In view of the result of J. M. Garcia-Calcines
[7], this is equivalent to ${\sf{secat}}(\ddot{\xi})\leq k+\ell+1$.

For $i=0,1,\dots,k$ we set

$F_{i}=\\{e\in\dot{E}(\xi);P_{\xi(e)}^{\tau}(e)\not=0,\,\mbox{and}\,||P_{\xi(e)}^{\tau}(e)||^{-1}\cdot
P_{\xi(e)}^{\tau}(e)\in G_{i}\\}.$

and for $i=k+1,\dots,k+\ell+1$, we set

$F_{i}=\\{e\in\dot{E}(\xi);\,P_{\xi(e)}^{\tau}(e)=0\,\,\mbox{and}\,\,\xi(e)\in
B_{i-k-1}\\}.$

Next we construct the continuous sections $s_{i}:F_{i}\to E(\ddot{\xi})$.

For $i=0,1,\dots,k$, given a unit vector $e\in F_{i}$, consider
$e^{\prime}=P_{\xi(e)}^{\tau}(e)$ which is a nonzero vector of $E(\tau)_{b}$,
where $b=\xi(e)$. Then $e^{\prime\prime}=||e^{\prime}||^{-1}\cdot e^{\prime}$
is a unit vector and $\sigma_{i}(e^{\prime\prime})\in\dot{E}(\tau)_{b}$ is a
unit vector satisfying $\sigma_{i}(e^{\prime\prime})\perp e^{\prime}$. In
particular we see that the unit vectors $e,\,\sigma_{i}(e^{\prime\prime})\in
E(\xi)_{b}$ are linearly independent. Hence the unit vector

$e^{\prime\prime\prime}=||\sigma_{i}(e^{\prime\prime})-\langle\sigma_{i}(e^{\prime\prime}),e\rangle\cdot
e||^{-1}\cdot(\sigma_{i}(e^{\prime\prime})-\langle\sigma_{i}(e^{\prime\prime}),e\rangle\cdot
e)$

is perpendicular to $e$ and depends continuously on $e$. Thus, we may define
the section $s_{i}$ by setting $s_{i}(e)=(e,e^{\prime\prime\prime})$.

Next we describe the sections $s_{i}$ over the sets $F_{i}$ where
$i=k+1,\dots,k+\ell+1$. If $e\in F_{i}$ then $P_{b}^{\tau}(e)=0$ and
$b=\xi(e)$ lies in $B_{i-k-1}$. In other words, $e\in\dot{E}(\eta)$ and
$b=\xi(e)\in B_{i-k-1}$. The section $\nu_{i-k-1}:B_{i-k-1}\to\dot{E}(\tau)$
defines a unit vector $\nu_{i-k-1}(b)\in\dot{E}(\tau)\subset\dot{E}(\xi)$
which is perpendicular to $e$. Therefore we may define the section $s_{i}$ of
$\ddot{\xi}$ over $F_{i}$ by the formula $s_{i}(e)=(e,\nu_{i-k-1}(e)).$

∎

######  Corollary 22.

Let $\xi:E\to B$ be a vector bundle such that $\xi=\eta\oplus\tau$ where
$\tau:E(\tau)\to B$ admits a complex structure and has a nowhere zero
continuous section. Then ${\sf TC}[\dot{\xi}:\dot{E}\to B]\leq 2$.

###### Proof.

This follows from Remark 13 and Lemma 21. ∎

######  Corollary 23.

Let $\xi:E\to B$ be a vector bundle admitting two continuous linearly
independent nowhere zero sections. Then ${\sf TC}[\dot{\xi}:\dot{E}\to B]\leq
2$.

###### Proof.

This reduces to the previous Corollary with $\tau$ the trivial bundle of rank
2. ∎

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# Two fields quintessential Higgs Inflation

Mehdi Es-haghi  eshaghi249(AT)gmail.com Ahmad Sheykhi
asheykhi(AT)shirazu.ac.ir Physics Department and Biruni Observatory, Shiraz
University, Shiraz 71454, Iran

###### Abstract

We study a two-field quintessential inflation model with non-minimal coupling
term where an exponential quintessence potential coupled to Higgs potential
from inflationary era. This non-minimal two-field model is shown to provide
very good consistency to the recent cosmological observation. Although, the
quintessence field is sub-dominant during inflation, it can remove Higgs
instability in the early universe which is a serious problem. Also,
considering quintessence from early time helps overcoming its initial
condition problem.

##  I Introduction

Cosmologists consider two accelerating eras for cosmic expansion history. The
first is inflationary era in the early universe Guth  which was driven by a
scalar field, called inflaton, with a nearly flat potential and negative
pressure. Based on the latest Planck data Aghanim:2018eyx ; Akrami:2018odb ,
the complicated inflationary models with multiple fields and non-canonical
ones have been eliminated. In exchange, single field inflationary potentials
with plateau-like shapes are the most favored models by Cosmic Microwave
Background (CMB) data.

The second is the late time accelerating era, discovered during 1998
Riess:1998cb . The most straightforward justification for this accelerating
expansion is an infinitesimal positive cosmological constant $\Lambda$ which
maybe introduced via the vacuum state of a scalar field with non-zero constant
energy $V_{0}$ called dark energy. Dark energy which is thought to be
homogeneous and has very low density and negative pressure, makes up about
70$\%$ of the universe compounds.

Following the strange nature of dark energy and considering the similarity of
this component with inflaton field, an alternative to cosmological constant
for explanation of the late time accelerated expansion is a dynamical scalar
field with infinitesimal energy density, called quintessence, which its slow-
rolling evolution at the late time can accelerate the universe Copeland:1997et
. This scalar field avoids the extreme fine-tuning of the cosmological
constant Tsujikawa:2013fta . Although, $\Lambda$CDM model with dark energy
equation of state $\omega_{\Lambda}=-1$ satisfies the cosmological data very
well and we may not need any other explanation for dark energy, but if future
cosmological experiments show a deviation from $\omega_{\Lambda}=-1$ for dark
energy equation of state, then quintessence may help us to explain this
deviation Akrami:2017cir .

In both inflation and quintessence models, potential energy gives the dominant
contribution to the total energy density of a scalar field which slowly rolls
down its almost flat potential. As, quintessence has its own tuning problems,
namely its initial conditions, explaining both inflation and dark energy in a
unified scenario with a single dynamical scalar field, named single field
quintessential inflation, may overcome this difficulty Peebles:1998qn ;
Dimopoulos:2017zvq . In this scenario, the single scalar field needs two
plateau shoulders with an extremely large difference in their heights. So, it
needs that the scalar field rolls down from its inflationary plateau to its
quintessential ones very very slowly without oscillating around the potential
minimum. It means that the reheating of the universe must have happened at the
end of inflation via a mechanism, called gravitational reheating Ford ;
Peebles:1998qn , instead of inflaton field decay. Indeed, through a mechanism,
called kination, almost all of inflaton potential energy becomes converted to
its kinetic ones to prevent the field decays completely and by rolling the
field towards its large negative values, it freezes at some $\phi_{F}$. After
the densities of radiation and cold dark matter (CDM) become sufficiently
small, the field starts rolling down again and acts as quintessence field
Akrami:2017cir ; Dimopoulos:2017zvq .

Another scenario, called two-field quintessential inflation, assumes that from
early times, there is a two-dimensional potential energy in which one field is
responsible for dark energy, while inflation is driven by another field
Akrami:2017cir . During inflation, the inflaton field has main role in the
evolution of the universe, while the role of quintessence field is
insignificant. When the inflation ends, the inflaton field falls down to the
quintessence valley. In this case, the reheating occurs due to oscillations of
the inflaton field near the minimum point of its potential. Because of the
tiny slope of the potential along the quintessence field axis, the
quintessence field does not start rolling until the density of products of
reheating (radiation and matter) decrease significantly.

A general theory of inflation in which a single inflaton field couples non-
minimally to gravity has been extensively studied during past years
Futamase:1987ua . One of the favored inflationary models which satisfys recent
CMB data well Akrami:2018odb  is Higgs inflation in which Standard Model Higgs
boson with non-minimal coupling to gravity has the role of inflaton scalar
field Salopek ; Bezrukov:2007ep . A characteristic feature of this model is
that the only scalar field which is discovered yet is Higgs boson
Chatrchyan:2012ufa . However, quantum corrections during inflation make self-
interacting coupling constant of Higgs boson negative at an energy scale
around $10^{11}$GeV which may leads to serious problem, called Higgs
instability Bezrukov:2007ep ; Han:2018yrk . One proposal for overcoming this
problem is a model of a specific coupling between Higgs boson and quintessence
field Denef:2018etk ; Han:2018yrk . The interesting feature of this proposal
is that the constraints come from the stability of electroweak vacuum in
inflationary era and the observations for dark energy are in agreement with
the proposed Swampland Conjecture Brennan:2017rbf . In this paper, we aim to
describe a new model of two-field quintessential inflation including Higgs and
quintessence fields which have a trilinear coupling with each other during
inflation era and in later times and Higgs field has also a non-minimal
coupling to gravity throughout inflation.

In this work, we consider a two-field quintessential inflation model where a
quintessence potential coupled to the Higgs potential from inflationary era.
These Higgs and quintessence fields have non-minimally and minimally coupling
to gravity, respectively. In the following, we study the behaviors of their
coupled potential in the Einstein frame and during the inflationary era.
Finally, we derive the cosmological parameters of the model and compare them
with the recent cosmological data.

##  II The model

In Jordan frame, the generalized action with two fields, non-minimally coupled
to gravity, is as follow

$\displaystyle S_{J}=\int
d^{4}x\sqrt{-g}\left[\frac{1}{2}M_{P}^{2}~{}\Omega^{2}(\phi_{1},\phi_{2})R-\frac{1}{2}\partial_{\mu}\phi_{1}\,\partial^{\mu}\phi_{1}-\frac{1}{2}\partial_{\mu}\phi_{2}\,\partial^{\mu}\phi_{2}-V(\phi_{1},\phi_{2})\right]~{},$
(1)

where $R$ is the Ricci scalar and $\Omega^{2}(\phi_{1},\phi_{2})$ is the
conformal transformation factor defined as

$\displaystyle\Omega^{2}(\phi_{1},\phi_{2})=1+f(\phi_{1},\phi_{2})~{},$ (2)

where $f(\phi_{1},\phi_{2})$ is usually a polynomial function of $\phi_{1}$
and $\phi_{2}$ and also includes non-minimal coupling constants $\xi_{1}$ and
$\xi_{2}$. This metric has signature $(-,+,+,+)$.

In this paper, we consider $V(\phi_{1},\phi_{2})$ in the form of a proposed
two-field potential in Denef:2018etk , in which there is a trilinear coupling
between the Higgs boson and the dilaton particle as a quintessence field. This
potential is a special combination of the well-known Higgs boson potential
$V_{H}(h)$ and a familiar exponential quintessence potential. Of course in
this case, the quintessence field $\phi$ is decoupled from all other Standard
Model fields.

The Higgs boson potential is

$V_{H}(h)=\frac{\lambda}{4}(h^{2}-\nu^{2})^{2},$ (3)

where $\lambda$ is the self-interacting coupling constant of the Higgs field
and $\nu\approx 246$ GeV is the Higgs vacuum expectation value.

The above-mentioned exponential quintessence potential is as

$V_{Q}(\phi)=\Lambda\>e^{-\delta\phi/M_{P}}~{}~{}~{}(\delta>0),$ (4)

where $\Lambda$ is the present value of dark energy density and
$\delta\equiv\lvert M_{P}\nabla_{\phi}V_{Q}/V_{Q}\rvert$. For present
accelerating universe, $\lvert\nabla V_{Q}\rvert_{today}=\Lambda\sim
10^{-120}$ in Planck units. The scale factor of the universe in this model
grows as $t^{2/\delta^{2}}$ and therefore for having an accelerating universe,
it needs $\delta<\sqrt{2}$ Kallosh:2003mt . On the other hand, in recent
years, some efforts in string theory to find suitable cosmological models led
to obtain two so-called Swampland criteria which gave testable predictions
about dark energy. During these efforts, Vafa et al. found that specific
quintessence models can satisfy these bounds Agrawal:2018own . Based on
Swampland criterias and current observational constrains on dark energy,
marginalized upper bounds on $\delta$ gives $\delta<0.6~{}$, $\delta<1.35$
with 68% CL (and 95% CL) and 99.7% CL, respectively Agrawal:2018own ;
Heisenberg:2018yae . Then, by comparing the above potential to the data,
Akrami et al. Showed that models with $\delta\gtrsim 1.02$ were ruled out by
the data at the 99.7% CL Akrami:2018ylq .

It should be mentioned that the above exponential quintessence model cannot
alone act as a single field quintessential inflationary model, i.e. it cannot
describe inflation and quintessence, simultaneously. Because, it supports
inflation for $\delta\ll 1$, but then inflation never ends Akrami:2017cir .
Therefore, one should consider another adequate field act as the inflaton.

Labeling the fields $\phi_{1}=h$ and $\phi_{2}=\phi$ as the Higgs and the
quintessence fields, respectively, our selected two-field potential
$V(\phi_{1},\phi_{2})$ in this paper is the combined form of (3) and (4)
introduced in Denef:2018etk  as

$V(\phi_{1},\phi_{2})=V(\phi,h)=e^{-\delta(\phi-\phi_{0})/M_{P}}(V_{H}(h)+\Lambda)~{},$
(5)

where $\phi_{0}=\phi_{today}$.

For obtaining a canonical form for the action (1) to satisfy recent CMB data,
it needs moving to the Einstein frame. For this purpose, one may consider a
conformal transformation, defined as

$\hat{g}_{\mu\nu}=\Omega^{2}\,g_{\mu\nu}~{}.$ (6)

Supposing that the quintessence has minimal coupling to gravity, we define
$\xi_{1}=\xi$ and $\xi_{2}=0$. So for a choice of quadratic scalar field, we
simplify (2) as

$\displaystyle\Omega^{2}=1+f(h)=1+\xi\frac{h^{2}}{M_{P}^{2}}~{},$ (7)

which leads to a non-minimal kinetic term for Higgs field in the Einstein
frame action. Therefore, it is helpful to make a change to the new scalar
field $\chi$ with

$\dfrac{d\chi}{dh}=\sqrt{\dfrac{1}{\Omega^{2}}+6(\dfrac{\Omega^{\prime}}{\Omega})^{2}}~{},$
(8)

where $\chi$ is the canonically normalized form of h. This change gives the
following well-known form for the action (1) in the Einstein frame
Starobinsky:2001xq

$S_{E}=\int
d^{4}x\sqrt{-\hat{g}}\left[\frac{1}{2}M_{P}^{2}\hat{R}-\frac{1}{2}\partial_{\mu}\chi\,\partial^{\mu}\chi-\frac{1}{2}e^{2b}\partial_{\mu}\phi\,\partial^{\mu}\phi-\hat{V}(\chi,\phi)\right]~{},$
(9)

with

$\displaystyle\hat{V}(\chi,\phi)=\frac{V(h(\chi),\phi)}{\Omega^{4}}~{},$ (10)
$\displaystyle b=-\frac{1}{2}\ln(\Omega^{2})~{}.$ (11)

The conformal factor $\Omega^{4}$, gives a plateau-like shape to the potential
(5) for large field values and therefore the cosmological parameters of (10)
are consistent with CMB data well.

To study the behavior of the potential (10) for $\xi\gg 1$, we consider two
general regions in the following; super-Planckian region $h\gg
M_{P}/\sqrt{\xi}$ and sub-Planckian one $h\ll M_{P}/\sqrt{\xi}$.

###  II.1 Inflationary era

For $h\gg M_{P}/\sqrt{\xi}$, i.e. during inflation, integrating both sides of
(8) gives

$\displaystyle
h=\frac{M_{P}}{\sqrt{\xi}}\exp\left(\frac{\chi}{\sqrt{6}M_{P}}\right)~{}.$
(12)

Putting (5), (7) and (12) in (10) and ignoring the infinitesimal constants
$\nu$ and $\Lambda$ during inflation, one finds a two dimensional potential,
we call quintessential Higgs inflation

$\hat{V}(\phi,\chi)=\frac{\lambda}{4}\frac{M_{P}^{4}}{\xi^{2}}e^{-\delta(\phi-\phi_{0})/M_{P}}\left(1+e^{-\sqrt{\frac{2}{3}}\chi/M_{P}}\right)^{-2}=V(\phi)\hat{V}(\chi)~{},$
(13)

where

$\displaystyle V(\phi)=e^{-\delta(\phi-\phi_{0})/M_{P}}~{},\>\>\>\>\>\
\hat{V}(\chi)=\frac{\lambda}{4}\frac{M_{P}^{4}}{\xi^{2}}\left(1+e^{-\sqrt{\frac{2}{3}}\chi/M_{P}}\right)^{-2}~{}.$
(14)

The potential (13) is shown in Fig. 1. Without coefficient
$e^{-\delta(\phi-\phi_{0})/M_{P}}$, (13) is reduced to ordinary Higgs
inflation potential in Bezrukov:2007ep  which suffers from Higgs instability,
since the quantum fluctuations of the Higgs field may have exceeded the
instability scale. If one defines a new self coupling for this potential as
$\lambda^{\prime}=\lambda e^{-\delta(\phi-\phi_{0})/M_{P}}$, the coefficient
$e^{-\delta(\phi-\phi_{0})/M_{P}}\gtrsim 1.08$ gives stability to the Higgs
potential during inflation Han:2018yrk .

Figure 1: Two-field quentessential Higgs potential.

The dynamics of the two scalar fields are described by the Klein-Gordon and
Friedmann equations respectively

$\displaystyle\ddot{\chi}+3H\dot{\chi}+\hat{V}_{,\chi}=b_{,\chi}e^{2b}\dot{\phi}^{2}~{},$
(15)
$\displaystyle\ddot{\phi}+(3H+2b_{,\chi}\dot{\chi})\dot{\phi}+e^{-2b}\hat{V}_{,\phi}=0~{},$
(16) $\displaystyle
H^{2}=\frac{1}{3M_{P}^{2}}\left(\frac{1}{2}\dot{\chi}^{2}+\frac{1}{2}e^{2b}\dot{\phi}^{2}+\hat{V}\right)~{},$
(17)
$\displaystyle\dot{H}=-\frac{1}{2M_{P}^{2}}(\dot{\chi}^{2}+e^{2b}\dot{\phi}^{2})~{}.$
(18)

where the dot and the subscript comma denote partial derivative with respect
to time and to scalar fields, respectively.

For having an epoch of accelerating expansion, we need a nearly constant
Hubble parameter, H with a slow-roll evolution, i.e. $\epsilon_{H}\ll 1$ with

$\displaystyle\epsilon_{H}\equiv-\frac{\dot{H}}{H^{2}}~{},$ (19)

called Hubble slow-roll parameter. On the other hand, the potential slow-roll
parameters of the two fields are defined as

$\displaystyle\epsilon_{\chi}\equiv\frac{M_{P}^{2}}{2}\left(\frac{\hat{V}_{,\chi}}{\hat{V}}\right)^{2},\>\>\>\>\>\eta_{\chi}\equiv
M_{P}^{2}\frac{\hat{V}_{,\chi\chi}}{\hat{V}}~{},$ (20)
$\displaystyle\epsilon_{\phi}\equiv\frac{M_{P}^{2}}{2}\left(\frac{\hat{V}_{,\phi}}{\hat{V}}\right)^{2}e^{-2b},\>\>\>\>\>\eta_{\phi}\equiv
M_{P}^{2}\frac{\hat{V}_{,\phi\phi}}{\hat{V}}e^{-2b}~{}.$ (21)

Although, models include single canonical scalar fields need a nearly flat
potential in order to give accelerated expansion and also satisfy CMB data,
multiple field models can also give acceleration, even in the presence of
steep potential. In other words, in the slow-roll regime for multi-field case,
$\epsilon_{H}\ll 1$ but not necessarily $\epsilon_{V}\ll 1$.

Under the assumption of slow-roll conditions, (15)-(18) simplify as

$\displaystyle 3H\dot{\chi}+\hat{V}_{,\chi}=0~{},$ (22) $\displaystyle
3H\dot{\phi}+e^{-2b}\hat{V}_{,\phi}=0~{},$ (23) $\displaystyle
H^{2}=\frac{1}{3M_{P}^{2}}\hat{V}~{},$ (24)
$\displaystyle\dot{H}=-\frac{1}{2M_{P}^{2}}(\dot{\chi}^{2}+e^{2b}\dot{\phi}^{2})~{}.$
(25)

To identify the inflaton field direction in field space during inflation, it
is conventional to define a new field, called adiabatic field $\sigma$, as

$\displaystyle\dot{\sigma}^{2}=\mathcal{G}_{ij}\dot{\varphi}^{i}\dot{\varphi}^{j}~{},$
(26)

where $\mathcal{G}_{ij}$ is the field-space metric. This field represents the
path length along the clasical background trajectory, while fluctuations
orthogonal to the background trajectory, showing non-adiabatic perturbations,
are represented by another field, called entropy field s Gordon:2000hv . In
another word, one can decompose an arbitrary perturbation in multi-field
models into an adiabatic $\delta\sigma$ and an entropy $\delta s$ component.
Following Gordon:2000hv , the adiabatic and entropy fields of our model, are
defined as

$\displaystyle\delta\sigma=\cos\theta~{}\delta\chi+\sin\theta~{}e^{b}\delta\phi~{},$
(27) $\displaystyle\delta
s=\cos\theta~{}e^{b}\delta\chi-\sin\theta~{}\delta\phi,$ (28)

where $\theta$ is the angle of the tangent to the background trajectory and

$\displaystyle\cos\theta=\frac{\dot{\chi}}{\sqrt{\dot{\chi}^{2}+e^{2b}\dot{\phi}^{2}}}~{},$
(29)
$\displaystyle\sin\theta=\frac{e^{b}\dot{\phi}}{\sqrt{\dot{\chi}^{2}+e^{2b}\dot{\phi}^{2}}}~{}.$
(30)

Therefore, the Klein-Gordon and Friedmann equations respectively are re-
defined as

$\displaystyle\ddot{\sigma}+3H\dot{\sigma}+\hat{V}_{,\sigma}=0~{},$ (31)
$\displaystyle
H^{2}=\frac{1}{3M_{P}^{2}}\left(\frac{1}{2}\dot{\sigma}^{2}+\hat{V}\right)~{},$
(32) $\displaystyle\dot{H}=-\frac{1}{2M_{P}^{2}}\dot{\sigma}^{2}~{},$ (33)

where

$\displaystyle\hat{V}_{,\sigma}=\cos\theta~{}\hat{V}_{,\chi}+\sin\theta~{}e^{-b}\hat{V}_{,\phi}~{},$
(34) $\displaystyle\dot{\sigma}^{2}=\dot{\chi}^{2}+e^{2b}\dot{\phi}^{2}~{}.$
(35)

On the other hand, substitution of (32) and (33) into (19) and using slow-roll
conditions gives

$\displaystyle\epsilon_{H}=\frac{3}{2}\frac{\dot{\sigma}^{2}}{\hat{V}}=\frac{3}{2}\frac{\dot{\chi}^{2}+e^{2b}\dot{\phi}^{2}}{\hat{V}}~{}.$
(36)

Then, substituting $\dot{\chi}$ and $\dot{\phi}$ from (22) and (23) into (36)
and using (20) and (21), one obtains the relation between the Hubble and
potential slow-roll parameters of this work as

$\displaystyle\epsilon_{H}=\epsilon_{\chi}+e^{-2b}\epsilon_{\phi}~{}.$ (37)

Putting (13) in (20), the inflationary slow-roll parameters of the model can
be calculated as

$\displaystyle\epsilon_{\chi}=\frac{4}{3}(1-e^{\sqrt{\frac{2}{3}}\frac{\chi}{M_{P}}})^{2}~{},$
(38)
$\displaystyle\eta_{\chi}=\frac{4}{3}(1-e^{\sqrt{\frac{2}{3}}\frac{\chi}{M_{P}}})^{2}(2-e^{\sqrt{\frac{2}{3}}\frac{\chi}{M_{P}}})~{},$
(39)

The number of e-folds to the end of inflation, $N_{k}$ is related to the
Hubble parameter as $dN_{k}=Hdt$. Therefore, in the case of the separable
potential (13) and by the help of (22)-(24), one can find $N_{k}$ as a
function of $\chi_{k}$ or $\phi_{k}$ as

$N_{k}=\frac{1}{M_{P}^{2}}\int^{\chi_{k}}_{\chi_{end}}\frac{\hat{V}(\chi)}{\hat{V}(\chi)_{,\chi}}d\chi=\frac{1}{M_{P}^{2}}\int^{\phi_{k}}_{\phi_{end}}\frac{V(\phi)}{V(\phi)_{,\phi}}e^{2b}d\phi~{},$
(40)

Substituting (38) in (40) and integrating, one obtains $N(\chi_{k})$ as

$N(\chi_{k})=\frac{\sqrt{6}}{4M_{P}}(\chi_{k}-\chi_{\textrm{end}})+\frac{3}{4}\left(e^{\sqrt{\frac{2}{3}}\frac{\chi_{k}}{M_{P}}}-e^{\sqrt{\frac{2}{3}}\frac{\chi_{end}}{M_{P}}}\right)~{}.$
(41)

This equation can be inverted by the use of the lower Lambert function to
obtain $\chi_{k}(N)$ as

$\chi_{k}(N)\simeq\sqrt{\frac{3}{2}}M_{P}\ln\left[\frac{4}{3}N-\sqrt{\frac{2}{3}}\frac{\chi_{end}}{M_{P}}+e^{\sqrt{\frac{2}{3}}\frac{\chi_{end}}{M_{P}}}\right]~{}.$
(42)

On the other hand, $\phi_{k}(N)$ can be also determined by the equation of
motion, (23) as

$\phi_{k}(N)\simeq-
M_{P}^{2}\int^{N}_{0}e^{-2b}\frac{V(\phi)_{,\phi}}{V(\phi)}dN\simeq
M_{P}~{}\delta\int^{N}_{0}(1+e^{\sqrt{\frac{2}{3}}\frac{\chi_{k}(N)}{M_{P}}})dN~{}.$
(43)

By substituting (42) in (43), the above integration gives

$\phi_{k}(N)\simeq
M_{P}~{}\delta\left[\left(1-\sqrt{\frac{2}{3}}\frac{\chi_{end}}{M_{P}}+e^{\sqrt{\frac{2}{3}}\frac{\chi_{end}}{M_{P}}}\right)N+\frac{2}{3}N^{2}\right]+\phi_{end}~{}.$
(44)

For calculating the value of $N_{k}$ from (41), one needs $\chi_{end}$ and
$\chi_{k}$.

As inflation ends by the breakdown of slow-roll condition, i.e.
$\epsilon_{\chi}(\chi_{end})\simeq 1$, one can find

$\displaystyle\chi_{end}\simeq 0.94\>M_{P}~{}.$ (45)

In order to determine $\chi_{k}$, one need to consider the connection between
the times of horizon crossing and re-entering to horizon of the cosmological
observable scales Liddle:2003as

$\displaystyle\frac{k}{a_{0}H_{0}}=\frac{a_{k}H_{k}}{a_{0}H_{0}}=e^{-N_{k}}\frac{a_{end}}{a_{re}}\frac{a_{re}}{a_{eq}}\frac{H_{k}}{H_{\textrm{eq}}}\frac{a_{\textrm{eq}}H_{\textrm{eq}}}{a_{0}H_{0}}~{},$
(46)

where the comoving wave number $k$ equals the Hubble scale $a_{k}H_{k}$ and
the subscripts refer to different eras, including the horizon exit ($k$),
reheating (re), radiation-matter equality (eq) and the present time (0).
Accepting the assumption of entropy conservation between the end of reheating
and today and using the slow-roll approximation in which $H^{2}_{k}\simeq
V_{k}/3M_{\textrm{Pl}}^{2}$, one can obtain (Martin:2010, ; Akrami:2018odb, )

$\displaystyle
N_{k}=67-\ln\left(\frac{k}{a_{0}H_{0}}\right)+\frac{1}{4}\ln\left(\frac{V_{k}^{2}}{M_{\textrm{Pl}}^{4}}\rho_{\textrm{end}}\right)+\frac{1-3\omega_{\textrm{int}}}{12(1+\omega_{\textrm{int}})}\ln\left(\frac{\rho_{\textrm{re}}}{\rho_{\textrm{end}}}\right)-\frac{1}{12}\ln
g_{\textrm{re}}~{},$ (47)

where $V_{k}$ is the potential energy when $k$ leaves the Hubble horizon
during inflation, $\rho_{\textrm{end}}$ and $\rho_{\textrm{re}}$ are the
energy densities at the end of inflation and reheating, respectively,
$\omega_{\textrm{int}}$ is the e-fold average of the equation of state between
the end of inflation and the end of reheating and $g_{\textrm{re}}$ is the
number of effective bosonic degrees of freedom at the end of reheating. To
find $\chi_{k}$, one should first determine the right hand side terms of (47),
one by one.

In the third term of (47), $V_{k}$ is dependent on the value of
$\lambda/\xi^{2}$ and $e^{-\delta(\phi-\phi_{0})/M_{P}}$. The value of
$\lambda/\xi^{2}$ can be calculated through the normalization of the power
spectrum Baumann:2009ds

$\displaystyle\Delta_{\zeta}^{2}=\frac{k^{3}}{2\pi^{2}}P_{\zeta}(k)=\frac{1}{24\pi^{2}M_{P}^{4}}\frac{\hat{V}_{k}}{\epsilon_{H}}~{}.$
(48)

Putting (13) in (48) and doing some calculations, we find

$\frac{\lambda}{\xi^{2}}=96\>\pi^{2}\Delta_{\zeta}^{2}\>\epsilon_{H}\>e^{\delta(\phi-\phi_{0})/M_{P}}\>\left(1-e^{-\sqrt{\frac{2}{3}}\frac{\chi_{k}}{M_{P}}}\right)^{-2}.$
(49)

On the other hand, if one studies the evolution of the quintessence field
between the end of reheating and the EW symmetry breaking by combining the
Friedmann equation and the equation of motion as in $\phi$ Han:2018yrk , for
the permitted range $0.35<\delta<1.02$, one finds

$\displaystyle 1.08<e^{-\delta(\phi-\phi_{0})/M_{P}}<2.25~{}.$ (50)

The energy density of reheating $\rho_{re}$ is related to the reheating
temperature, $T_{re}$ through

$\displaystyle\rho_{\textrm{re}}=\frac{\pi^{2}}{30}g_{\textrm{re}}T_{\textrm{re}}^{4}~{},$
(51)

and $\rho_{end}$ depends on the potential energy at the end of inflation
$V_{end}$ via

$\displaystyle\rho_{\textrm{end}}=\frac{3}{2}V_{\textrm{end}}~{},$ (52)

obtained by setting $\omega_{end}=-1/3$. It is well known that $\omega_{int}$
of the reheating phase for large field models is given by Turner

$\omega_{int}=\dfrac{p-2}{p+2}~{},$ (53)

where $p$ is the power of the inflaton field in the corresponding potential
when it oscillates around its minimum. In the case of the potential (58) which
is proportional to $h^{4}$, (53) gives $\omega_{int}=1/3$ and as a result, the
fourth term on right hand side of (47) vanishes at $\omega_{int}=1/3$.
Therefore, deriving $\chi_{k}$ is independent of reheating temperature.

Considering $k=0.002$ Mp$c^{-1}$, $g_{re}=1000$, $\Delta_{\zeta}^{2}=2\times
10^{-9}$ Aghanim:2018eyx , $\omega_{int}=1/3$ and $\chi_{end}\simeq
0.94\>M_{P}$, and replacing the above results in (47) we find

$\displaystyle\chi_{k}=5.38~{}M_{P}~{},~{}~{}~{}~{}~{}N_{k}=61.7~{}~{}~{}~{}~{}~{}for~{}~{}~{}~{}~{}\delta=1.02~{}~{}~{}~{}~{}and~{}~{}~{}~{}~{}e^{-\delta(\phi-\phi_{0})/M_{P}}=2.25$
(54)

and

$\displaystyle\chi_{k}=5.37~{}M_{P}~{},~{}~{}~{}~{}~{}N_{k}=61.2~{}~{}~{}~{}~{}~{}for~{}~{}~{}~{}~{}\delta=0.35~{}~{}~{}~{}~{}and~{}~{}~{}~{}~{}e^{-\delta(\phi-\phi_{0})/M_{P}}=1.08$
(55)

Comparing these result to the single field Higgs inflation with $N_{k}=58.7$,
we obtain a shift in the number of e-folds $\Delta N_{k}\simeq 3$ for our two-
field model.

###  II.2 Reheating

For $h\ll M_{P}/\sqrt{\xi}$, the conformal transformation factor
$\Omega^{2}\rightarrow 1$. For detailed analysis, one can divide this region
to two epochs. Defining a critical value Bezrukov:2008ut

$\displaystyle X_{cr}=\sqrt{\frac{2}{3}}\frac{M_{P}}{\xi}~{},$ (56)

and then consider an intermediate regime between inflation and radiation eras,
called reheating in which $X_{cr}<h\ll M_{P}/\sqrt{\xi}$ and the Higgs field
oscillates around its potential minima to reheat the universe. In this epoch,
one obtains

$\displaystyle\chi\simeq\sqrt{\frac{3}{2}}\>\frac{\xi}{M_{P}}h^{2}~{},$ (57)

and the reheating approximated potential of quintessential Higgs model is

$\displaystyle\hat{V}(\chi,\phi)=\frac{1}{2}\>\omega^{2}\chi^{2}~{},$ (58)

where

$\displaystyle\omega^{2}\equiv\dfrac{\lambda}{3}\>e^{-\delta(\phi-\phi_{0})}\>\frac{M_{P}^{2}}{\xi^{2}}~{}.$
(59)

The behavior of the potential near its minima is shown in Fig. 2. As the value
of $e^{-\delta(\phi-\phi_{0})}\sim 1$, therefore, this term does not affect on
the value of the oscillation frequency $\omega$ of the reheating and also on
its temperature.

Figure 2: The behavior of the potential near its minima.

In the second epoch and for small values of $h<X_{cr}\ll M_{P}/\sqrt{\xi}$ ,
we are in the radiation-dominated epoch. For this period, $\chi\simeq h$ the
related two-field potential which is the initial quintessential Higgs
potential (5), transmits its energy to relativistic particles via coherent
oscillations of the Higgs field, rapidly. One could suppose that the the ratio
of the energy density relative to the critical energy density of the model at
the EW symmetry breaking scale changes from $\lambda\nu^{4}/\rho_{cr}$ to
$\Omega_{\Lambda_{0}}$ rapidly and then the evolution of this two-field
potential after coherent oscillation can be described by an attractor
solution, which is ultra-slow-roll in the radiation-dominated and matter-
dominated eras. Finally, in the the dark energy-dominated universe, the
quintessence field undergoes the slow-roll evolution.

##  III Conclusion

In this work, we studied a new two-field model for inflation and drak energy
as quintessential inflation. The quintessence field was coupled with Higgs
field from inflationary era and the Higgs field had non-minimal coupling with
gravity in the Jordan frame. Moving to the Einstein frame through conformal
transformation, we found a plateau-like two-filed potential for inflation
which its cosmological parameters had very good consistency with recent
cosmological data.

Furthermore, this model removes the instability problem of single field Higgs
model during inflation and also satisfies the Swampland criterias, well.

## Acknowledgement

M.E would like to thank Y. Akrami, M. Sasaki, A. Hebecker and A. Linde for
various comments and useful discussions. This project funded by Iran Science
Elites Federation.

## References

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