# Counting periodic trajectories of Finsler billiards

Pavle V. M. Blagojević  Institut für Mathematik, FU Berlin, Arnimallee 2,
14195 Berlin, Germany<EMAIL_ADDRESS>Mathematički Institut
SANU, Knez Mihailova 36, 11001 Beograd, Serbia<EMAIL_ADDRESS>,
Michael Harrison  Lehigh University, Department of Mathematics, Bethlehem, PA
18015, USA<EMAIL_ADDRESS>,  Serge Tabachnikov  Pennsylvania State
University, Department of Mathematics, University Park, PA 16802, USA
<EMAIL_ADDRESS>and  Günter M. Ziegler  Institut für Mathematik, FU
Berlin, Arnimallee 2, 14195 Berlin, Germany<EMAIL_ADDRESS>
###### Abstract.

We provide lower bounds on the number of periodic Finsler billiard
trajectories inside a quadratically convex smooth closed hypersurface $M$ in a
$d$-dimensional Finsler space with possibly irreversible Finsler metric. An
example of such a system is a billiard in a sufficiently weak magnetic field.
The $r$-periodic Finsler billiard trajectories correspond to $r$-gons
inscribed in $M$ and having extremal Finsler length. The cyclic group
$\mathbb{Z}_{r}$ acts on these extremal polygons, and one counts the
$\mathbb{Z}_{r}$-orbits. Using Morse and Lusternik-Schnirelmann theories, we
prove that if $r\geq 3$ is prime, then the number of $r$-periodic Finsler
billiard trajectories is not less than $(r-1)(d-2)+1$. We also give stronger
lower bounds when $M$ is in general position. The problem of estimating the
number of periodic billiard trajectories from below goes back to Birkhoff. Our
work extends to the Finsler setting the results previously obtained for
Euclidean billiards by Babenko, Farber, Tabachnikov, and Karasev.

P.B. was supported by the DFG via the Collaborative Research Center TRR 109
“Discretization in Geometry and Dynamics”, and by the grant ON 174008 of
Serbian Ministry of Education and Science.

M.H. was supported by NSF grant DMS-1510055

S.T. was supported by NSF grant DMS-1510055 and by the DFG via the
Collaborative Research Center TRR 109 “Discretization in Geometry and
Dynamics”

G.Z. was supported by the DFG via the Collaborative Research Center TRR 109
“Discretization in Geometry and Dynamics”

##  1\. Introduction

The study of mathematical billiards goes back to G. D. Birkhoff who wrote in
[10]:

> … in this problem the formal side, usually so formidable in dynamics, almost
> completely disappears, and only the interesting qualitative questions need
> to be considered.

One of the main motivations for the study of billiards has been their relation
with mathematical physics and statistical mechanics, namely, with the
Boltzmann Ergodic Hypothesis. We refer to the books [14, 31, 40, 42] for
various aspects of mathematical billiards.

###  1.1. Billiards in Euclidean geometry

A Birkhoff billiard table is bounded by a smooth strictly convex closed
hypersurface $M$ in $\mathbb{R}^{d}$. The billiard dynamical system describes
the motion of a free particle inside $M$ with elastic reflection off the
boundary. That is, the point (billiard ball) moves with unit speed along a
straight line until it hits the boundary $M$; at the impact point, the normal
component of the velocity instantaneously changes sign, while the tangential
component remains the same, and the point continues its rectilinear motion. In
dimension two, this is the familiar law of geometrical optics: the angle of
incidence equals the angle of reflection.

An $r$-periodic billiard trajectory in $M$ is an $r$-tuple of points
$(x_{1},\ldots,x_{r})$ of $M$ such that $x_{i}\neq x_{i+1}$ and the billiard
reflection in $M$ takes segment $x_{i-1}x_{i}$ to $x_{i}x_{i+1}$ for
$i=1,\ldots,n$ (as usual, the indices are understood cyclically, that is,
$x_{r+i}=x_{i}$).

The Dihedral group of symmetries of a regular $r$-gon $D_{r}$ acts naturally
on the set of all periodic billiard trajectories of period $r$ by cyclically
permuting the points and reversing the orientation. Thus, when counting
periodic orbits, it is natural to count such dihedral orbits.

######  Problem 1.1.

Let $d\geq 2$ and $r\geq 2$ be integers, and let
$M^{d-1}\subseteq\mathbb{R}^{d}$ be a smooth closed strictly convex
hypersurface. Estimate below the number $N_{E}(M^{d-1},r)$ of equivalence
classes of periodic billiard trajectories of period $r$ inside $M^{d-1}$
modulo the action of the dihedral group $D_{r}$.

The way we have formulated the problem, multiple trajectories are included
into the count, so that for example a 2-periodic trajectory traversed thrice
and a 3-periodic trajectory traversed twice, contribute to the number of
6-periodic trajectories. Of course, this issue is not a concern if $r$ is a
prime.

The first progress in addressing the question posed in Problem 1.1 was made by
Birkhoff in 1927 [10]. He considered the case $d=2$, and proved that there
exist at least two $r$-periodic orbits with every rotation number coprime with
$r$, which implies that $N_{E}(M^{1},r)\geq
2\phi(\lfloor\tfrac{r}{2}\rfloor)$, where $\phi$ is the Euler totient
function. We remark that this lower bound holds for the number of prime
periodic trajectories. Birkhoff deduced his result from Poincaré’s Geometric
Theorem, that Poincaré published without proof shortly before his death and
that Birkhoff proved a year later.

The concept of rotation number is not available in dimensions $d\geq 3$, but
one can use the variational approach to the problem. Periodic billiard
trajectories correspond to the critical points of the length function on
$r$-tuple of points $(x_{1},\ldots,x_{r})$, that is, on $r$-gons inscribed in
$M$:

$L(x_{1},\ldots,x_{r})=\sum_{i=1}^{r}|x_{i}-x_{i+1}|.$ (1)

This function is $D_{r}$-invariant.

The first to address Problem 1.1 for $d=3$ was Babenko [5], whose approach was
based on analyzing critical points of the length function $L$. Although his
paper contained an error, the main idea was rescued and refined by Farber and
Tabachnikov who established several lower bounds for $N_{E}(M^{d-1},r)$.

When dealing with critical points of functions, one applies either the Morse
theory or the Lusternik-Schnirelmann theory. The former usually gives stronger
lower bounds on the number of critical points, but it applies to a more
restrictive set of smooth functions, namely, Morse (or Morse–Bott) functions;
the latter sometimes gives weaker lower bounds on the number of critical
points, but it does not rely on genericity assumptions on the functions
involved.

The variational methods subsequently produced a number of improved lower
bounds in a sequence of papers by a number of authors:

  * $\bullet$

Farber & Tabachnikov [21, Thm. 1 (B), p. 555] and [20, Thm. 3] Let $d\geq 3$
be an integer, let $r\geq 3$ be an odd integer, and let $M^{d-1}$ be a generic
smooth closed strictly convex hypersurface. Then

$N_{E}(M^{d-1},r)\geq(r-1)(d-1).$

This is a Morse-theoretical result, that is why $M$ is assumed to be generic.
The next results are Lusternik-Schnirelmann-theoretical, and they hold without
the genericity assumption.

  * $\bullet$

Farber & Tabachnikov [20, Thm. 1 (A), p. 554] Let $d\geq 4$ be an integer, let
$r\geq 3$ be an odd integer, and let $M^{d-1}$ be a smooth closed strictly
convex hypersurface. Then

$N_{E}(M^{d-1},r)\geq\lfloor\log_{2}(r-1)\rfloor+d-1.$

  * $\bullet$

Farber [19, Thm. 2, p. 589] Let $r\geq 3$ be an odd prime, and let $M^{d-1}$
be a smooth closed strictly convex hypersurface.

    * –

If $d\geq 4$ is even, then

$N_{E}(M^{d-1},r)\geq r.$

    * –

If $d\geq 3$ is odd, then

$N_{E}(M^{d-1},r)\geq\tfrac{r+1}{2}.$

  * $\bullet$

Karasev [29, Thm. 1, p. 424] Let $d\geq 3$, let $r\geq 3$ be prime, and let
$M^{d-1}$ be a smooth closed strictly convex hypersurface. Then

$N_{E}(M^{d-1},r)\geq(r-1)(d-2)+2.$

Let us also mention Mazzucchelli’s paper [33] concerning the multiplicity of
Birkhoff billiard periodic trajectories whose period is a power of a fixed
prime number.

###  1.2. Billiards in Finsler geometry

Our goal is to extend the above described results to periodic trajectories in
Finsler billiards.

From the point of view of physics, Finsler geometry describes the propagation
of light in a medium that is not necessarily homogeneous or isotropic. The
speed of light depends on the point and the direction, and is given by a
smoothly varying norm on the tangent spaces to the medium thought of as a
smooth manifold. We allow these norms to be asymmetric. These norms need not
correspond to inner products, which is the case when the metric is Riemannian.
To quote Chern’s description [13], “Finsler geometry is just Riemannian
geometry without the quadratic restriction.”

The distance $f(A,B)$ between points $A$ and $B$ is defined as the least time
it takes light to travel from $A$ to $B$; in general, $f(A,B)\neq f(B,A)$. The
trajectories of light are Finsler geodesics. See Section 2.1 for precise
definitions.

An example of a Finsler manifold is a Minkowski space, that is, a finite-
dimensional normed space. Another example is a projective metric in a domain
in the projective space, a Finsler metric whose geodesics are straight lines.
Hilbert’s fourth problem asked to describe all projective Finsler metrics (a
projective Riemannian metric is a metric of constant curvature), see, e.g.,
[12].

The first steps in the study of billiards in Finsler geometry were made by
Gutkin and Tabachnikov in [26]. In this paper, the Finsler billiard reflection
is defined (see Section 2.2 below) using the variational approach, and
Minkowski billiards are studied in some detail. Minkowski billiards have close
relations with convex geometry (Mahler’s conjecture) and symplectic topology
(symplectic capacities), see [3, 4].

We consider a domain in a Finsler manifold bounded by a smooth closed
hypersurface $M^{d-1}$, a Finsler billiard table. We assume that $M$ is
strictly convex in the following sense:

  * •

for every pair of points $x,y$ inside $M$, there is a unique geodesic from $x$
to $y$, and a unique geodesic from $y$ to $x$, both contained inside $M$ and
distance-minimizing;

  * •

$M$ is quadratically convex: every geodesic tangent to $M$ has second order
contact with $M$, and not higher order.

In particular, these conditions imply that $M$ is diffeomorphic to the sphere
$S^{d-1}$.

We are interested in periodic Finsler billiard trajectories inside $M$. They
still correspond to critical points of the analog of the length function (1)

$\Lambda(x_{1},\ldots,x_{r})=f(x_{1},x_{2})+f(x_{2},x_{3})+\cdots+f(x_{r},x_{1}),$

however this function has less symmetry than in the Euclidean case: it is
invariant under the cyclic permutations of the points, but not under the
orientation reversal. Thus $\Lambda(x_{1},\ldots,x_{r})$ is
$\mathbb{Z}_{r}$-invariant, but not necessarily $D_{r}$-invariant.
Accordingly, when counting periodic Finsler billiard trajectories, we count
$\mathbb{Z}_{r}$-orbits.

Denote by $N_{F}(M^{d-1},r)$ the number of equivalence classes of periodic
Finsler billiard trajectories of period $r$ inside $M^{d-1}$ modulo the action
of the cyclic group $\mathbb{Z}_{r}$.

###  1.3. Statement of main results

Our main result is as follows.

######  Main Theorem.

Let $d\geq 3$ be an integer and $r\geq 3$ be a prime. Consider Finsler
billiard inside a smooth closed hypersurface $M^{d-1}$, satisfying the above
formulated strict convexity assumptions. Then

  1. (A)

$N_{F}(M^{d-1},r)\geq(r-1)(d-2)+1$.

  2. (B)

For a generic $M$,

    1. (1)

if $d$ is even, then $N_{F}(M^{d-1},r)\geq(r-1)d$;

    2. (2)

if $d$ is odd, then $N_{F}(M^{d-1},r)\geq(r-1)(d-1)$.

The general position assumption in case (B) is the assumption that the length
$\Lambda(x_{1},\ldots,x_{r})$ is a Morse function. The latter condition is
generic in the sense that it holds in an open dense subset of strictly convex
hypersurfaces, considered in the Whitney $C^{\infty}$ topology. For Euclidean
billiards, this is deduced in [21, Lem. 4.4] from an appropriate version of
the multi-jet transversality theorem. A similar argument works in the Finsler
case; we do not elaborate on it here.

######  Remark 1.2.

The rate of growth of the numbers $N_{F}(M^{d-1},r)$, provided by Theorem Main
Theorem, is the same as in the above described results for Euclidean
billiards: it is, roughly, $rd$. Since we count $\mathbb{Z}_{r}$-orbits of
periodic Finsler billiard trajectories, rather than $D_{r}$-orbits, one might
expect the numbers $N_{F}(M^{d-1},r)$ to be about twice as large as the
numbers $N_{E}(M^{d-1},r)$.

For example, consider Euclidean billiard inside a strictly convex closed
smooth hypersurface, and switch on a weak magnetic field. One may expect each
periodic trajectory in absence of the magnetic field to give rise to two
periodic magnetic billiard trajectories, see Figure 1.

Figure 1. A 3-periodic billiard trajectory giving rise to two 3-periodic
magnetic billiard trajectories

In dimension two, this is indeed the case: the Finsler billiard map is an area
preserving twist map, and it has two $r$-periodic trajectories for every
rotation number $k$ coprime with $r$. If the metric is symmetric, then the
orbits corresponding to the rotation numbers $k$ and $r-k$ differ only by the
orientation, and they are counted as one, but in the asymmetric case, these
are indeed different orbits.

We do not know how close the lower bounds of Theorem Main Theorem are to being
sharp. One may expect a notable difference between the reversible and non-
reversible cases. In the related problem of closed geodesics, it is known that
every Riemannian metric on the 2-sphere possesses infinitely many
geometrically distinct closed geodesics [24, 6], but a Finsler metric may have
precisely two distinct prime closed geodesics, as in the well known Katok
example [30] (in which the two closed geodesics are the inverses of each
other). Do similar examples exist for Finsler billiards?

_Acknowledgements._ We are grateful to Sergei Ivanov for useful discussions of
Finsler geometry.

##  2\. From geometry of billiards to topology of cyclic configuration spaces

###  2.1. Introduction to Finsler geometry

We begin with a very brief introduction to Finsler geometry. For a thorough
treatment, see [28, 7, 38].

A Finsler metric on a smooth manifold $U$ is determined by a smooth non-
negative fiberwise-convex Lagrangian function $L:TU\longrightarrow[0,\infty)$,
with the property that on each tangent space $T_{y}U$, $L$ is positively
homogeneous: $L(y,tv)=tL(y,v)$ for non-negative $t$ and positive off the zero
section. The restriction of $L$ to any tangent space $T_{y}U$ gives the
Finsler length of vectors in $T_{y}U$.

The vectors in $T_{y}U$ of unit Finsler length form a strictly convex
hypersurface $I\subset T_{y}U$, called the _indicatrix_ , which plays the role
of the unit sphere in Riemannian geometry. We make the additional assumption
that each indicatrix is quadratically convex. Specifying a smooth field of
indicatrices on $U$ is an equivalent method of defining a Finsler metric on
$U$.

The Finsler metric on $TU$ induces a notion of distance on the base manifold
$U$. The length of a smooth curve $\gamma:[a,b]\longrightarrow U$ is given by
the integral

$\operatorname{Length}(\gamma)=\int_{a}^{b}L(\gamma(t),\gamma^{\prime}(t))\mbox{
d}t.$

The length of $\gamma$ is independent of the parametrization, and a Finsler
geodesic is an extremal of the length functional. In particular, a Finsler
geodesic $\gamma$ satisfies the Euler-Lagrange equations:

$L_{vv}\big{(}\gamma(t),\gamma^{\prime}(t)\big{)}\cdot\gamma^{\prime\prime}(t)+L_{vy}\big{(}\gamma(t),\gamma^{\prime}(t)\big{)}\cdot\gamma^{\prime}(t)=L_{y}\big{(}\gamma(t),\gamma^{\prime}(t)\big{)}.$

In this formula we use a shorthand notation, so that $L_{v}$ is the vector
$(L_{v_{1}},\ldots,L_{v_{n}})$ and $L_{vv}$ is the matrix $(L_{v_{i}v_{j}})$,
etc.

For each pair of points $x$ and $y$ there is a corresponding _Finsler
distance_ $f(x,y)$, equal to the length of the shortest oriented geodesic from
$x$ to $y$. We stress that since geodesics are not necessarily reversible,
this distance function $f$ need not be symmetric, and therefore $f$ is not a
genuine metric on $U$.

The _figuratrix_ $J\subset T_{y}^{*}U$ is the “unit sphere of the cotangent
space,” defined as follows: for a vector $v$ in the indicatrix $I\subset
T_{y}U$, there is a unique covector $D_{v}$, defined by the properties that
$\operatorname{Ker}(D_{v})=T_{v}I$ and $D_{v}(v)=1$. The map
$D:I\longrightarrow T_{y}^{*}U$ which maps $v$ to $D_{v}$ is the _Legendre
transform_ , and the image is the figuratrix $J$. The dual transform
$D^{*}:J\longrightarrow I$ is defined similarly. The Legendre transform is an
involution: the composition of $D$ and $D^{*}$ is the identity map.
Rightfully, the Legendre transform is a smooth bundle map from the indicatrix
bundle to the figuratrix bundle, but we will often use the notation
$D:I\longrightarrow J$ when the basepoint $y$ is understood.

###  2.2. The Finsler billiard map

Let $U$ be a smooth $d$-dimensional Finsler manifold. Let $X\subset U$ be a
compact $d$-dimensional submanifold with boundary $M=\partial X$. We assume
that $X$ and $M$ satisfy the strict convexity assumptions formulated in
Section 1.2. We will refer to $X$ as a _billiard table_.

Let $xy$ and $yz$ be two oriented geodesic segments, where $x,z\in X$ are in
the interior of the billiard table, and $y\in M$ is on the boundary. We say
that $yz$ is the _Finsler billiard reflection_ of $xy$ if $y$ is a critical
point of the distance function
$f(x,\cdot)+f(\cdot,z):M\longrightarrow[0,\infty)$. This relation is _not
symmetric_ , so it does not imply that $yx$ is the billiard reflection of
$zy$.

We describe the reflection law in the context of the Finsler setup following
the treatment in [26]. Although the Finsler metric there is assumed to be
symmetric, the reflection law is the same.

For each $y\in M$, let $p\in J\subset T_{y}^{*}U$ be the _conormal_ , defined
as the unit cotangent vector which vanishes on $T_{y}M$ and is positive on the
outward vectors. Let $xy$ be an incoming geodesic and $yz$ the reflected
outgoing geodesic, corresponding respectively to tangent vectors $u$ and $v$
in $I$. The reflection $xy$ to $yz$ manifests as the following relation in the
cotangent space.

######  Lemma 2.1 (Finsler Billiard Reflection Law).

The covector $D_{u}-D_{v}$ is conormal to $T_{y}M$; in particular
$D_{u}-D_{v}=tp$ for some $t>0$, see Figure 2.

Figure 2. The Finsler billiard reflection in dimension two

###### Proof.

The point $y\in M$ is a relative extremum of the function
$f(x,\cdot)+f(\cdot,z)$, so the differential of this function is conormal to
the tangent hyperplane $T_{y}M$. Let us compute this differential to show that
it is $D_{u}-D_{v}$.

Fix a point $x\in X$, and consider the wave propagation from $x$. Let $c$ be a
non-singular point of the wave front $\mathcal{F}_{t_{0}}$ such that the
oriented geodesic segment $xc$ of length $t_{0}$ is contained in the interior
of $X$. The Finsler length function $f(x,\cdot)$ extends to a smooth function
in a neighborhood of $c$. More precisely, for every point $d$ sufficiently
close to $c$, there exists a $t$ near $t_{0}$ such that $d\in\mathcal{F}_{t}$.
Let $I$ and $J$ represent the indicatrix and figuratrix at $c$, and let $u\in
I$ be the Finsler unit vector along $xc$.

We claim that, at the point $c$, d$f(x,\cdot)=D_{u}$. The wave front
$\mathcal{F}_{t_{0}}$ is a level set of $f(x,\cdot)$. Hence d$f(x,\cdot)$
annihilates the tangent space to $\mathcal{F}_{t_{0}}$. By the Huygens
principle, $D_{u}$ is conormal to this plane, hence proportional to
d$f(x,\cdot)$. It also follows from the Huygens principle that
d$f(x,\cdot)(u)=1$, which is true of $D_{u}$ by definition. Hence
d$f(x,\cdot)=D_{u}$.

Since the Finsler distance function is non-symmetric, we also need to
understand d$f(\cdot,z)$. Consider the oriented geodesic segment $yz$.
Although $zy$ is not necessarily a geodesic with respect to the Finsler
structure $L$, it is a geodesic with respect to the “reverse” Finsler
structure on $M$ defined by the Lagrangian $\bar{L}(v)\coloneqq L(-v)$. For
the corresponding Finsler distance function $\bar{f}$ we have the equality
$\bar{f}(z,y)=f(y,z)$, and for the corresponding Legendre transform $\bar{D}$
we have the equality $\bar{D}_{-v}=-D_{v}$ (by definition of the covector
$D_{v}$). Therefore we have d$f(\cdot,z)=$
d$\bar{f}(z,\cdot)=\bar{D}_{-v}=-D_{v}$, where the middle equality follows
from the paragraph above. This completes the proof. ∎

###  2.3. Magnetic billiards as Finsler billiards

A popular billiard model that has been extensively studied in the last decades
are magnetic billiards [8, 9, 25, 35, 41, 43, 44].

On a Riemannian surface, a magnetic field is given by a function $B(x)$, and
the motion of a charged particle is described by the differential equation
$\ddot{x}=B(x)J\dot{x}$, where $J$ is the rotation of the tangent plane by
$\pi/2$. This equation implies that the speed of the particle remains constant
(the Lorentz force is perpendicular to the direction of motion). In the
Euclidean plane, if the magnetic field is constant, the trajectories are
circles of a fixed radius (Larmor circles).

In general, a magnetic field on a Riemannian manifold $M$ is a closed
differential 2-form $\beta$, and the magnetic flow is the Hamiltonian flow of
the usual Hamiltonian $|p|^{2}/2$ on the cotangent bundle $T^{*}M$ with
respect to the twisted symplectic structure $\omega+\pi^{*}(\beta)$, where
$\omega=dp\wedge dq$ is the standard symplectic form and
$\pi:T^{*}M\longrightarrow M$ is the projection.

Magnetic billiard describes the motion of a charged particle confined to a
domain with elastically reflecting boundary. The reflection law is the same as
for the usual billiards, with zero magnetic field: the angle of incidence
equals the angle of reflection. Following [41], one can interpret magnetic
billiards as Finsler ones.

Let us assume that the magnetic 2-form is exact: $\beta=d\alpha$ for some
differential 1-form on a Riemannian manifold $M^{n}$. Then the magnetic flow
admits a Lagrangian formulation with the Lagrangian function

$\bar{L}(x,v)=\frac{1}{2}|v|^{2}+\alpha(x)(v),$

where $x\in M,v\in T_{x}M$, and $|v|$ is the Riemannian norm of the tangent
vector $v$.

We want to consider the motion with unit speed, that is, to fix an energy
level. Following the Maupertuis principle, replace the Lagrangian $\bar{L}$
with

$L(x,v)=|v|+\alpha(x)(v).$ (2)

Assume that the magnetic field is weak enough so that $L(x,v)>0$ for all non-
zero tangent vectors $v$ and all points $x$, that is, we assume that
$|\alpha(x)|<1$ everywhere. Then formula (2) defines a non-symmetric Finsler
metric.

######  Lemma 2.2 (Magnetic Billiard Reflection Law).

The indicatrix of the magnetic Finsler metric (2) is an ellipsoid of
revolution with a focus at the origin, and the Finsler billiard reflection law
coincides with the usual one: the angle of incidence equals the angle of
reflection.

###### Proof.

Fix a point $x$ and consider the tangent space at this point. The tangent
space is Euclidean, and the indicatrix $I$ is given by the equation
$|v|+\alpha(v)=1$. Let $e$ be the tangent vector dual to the covector
$\alpha$, that is, $\alpha(v)=e\cdot v$. We can choose an orthonormal basis so
that $e=(t,0,\ldots,0)$ with $t<1$.

The equation of the indicatrix is $L(x,v)=1$, or
$x_{1}^{2}+\ldots+x_{n}^{2}=(1-tx_{1})^{2}$, that is,

$(1-t^{2})^{2}\left(x_{1}+\frac{t}{1-t^{2}}\right)^{2}+(1-t^{2})(x_{2}^{2}+\ldots+x_{n}^{2})=1.$

This is the equation of an ellipsoid of revolution obtain by revolving the
ellipse

$\frac{(x_{1}+c)^{2}}{a^{2}}+\frac{x_{2}^{2}}{b^{2}}=1,$

where

$a^{2}=\frac{1}{(1-t^{2})^{2}},\ b^{2}=\frac{1}{(1-t^{2})},\
c=\frac{t}{1-t^{2}},$

and hence $c^{2}=a^{2}-b^{2}$. Therefore this ellipse has a focus at the
origin.

The second statement of the lemma reduces to a known geometric property of
conics depicted in Figure 3, see, e.g., [2]. Compare with the Finsler
reflection law, Figure 2. ∎

Figure 3. Point $A$ is a focus of an ellipse; then $\angle BAC=\angle BAD$

We assume that the convexity conditions of Section 1.2 hold for magnetic
billiards. This implies that the magnetic field is weak enough. For example,
if the billiard table is a planar domain bounded by a smooth strictly convex
curve and the magnetic field is constant, then the Larmor radius is greater
that the greatest radius of curvature of the boundary curve. This is one of
the three regimes in magnetic billiards described in [35], in which “motion is
qualitatively similar to the field-free case,” albeit not time-reversible.

###  2.4. Morse-theoretic approach to Finsler billiards

We now discuss the necessary Finsler geometry and topology to apply results
from Morse and Lusternik-Schnirelmann theories to the problem of periodic
Finsler billiard trajectories. Similar preparation work is easier to do in the
Euclidean case; it was the content of [21, Sec. 4].

The _ordered cyclic configuration space_ of $r$ consecutively distinct points
on $M$ is the space

$\operatorname{G}(M,r)\coloneqq\\{(x_{1},\ldots,x_{r})\in M^{\times
r}:x_{i}\neq x_{i+1}\ \text{for all}\ i\\},$

where by convention $x_{r+1}=x_{1}$. As mentioned in the Introduction, we
consider the _length function_

$\Lambda\colon\operatorname{G}(M,r)\longrightarrow\mathbb{R},\qquad\qquad\Lambda(x_{1},\dots,x_{r})=f(x_{1},x_{2})+f(x_{2},x_{3})+\cdots+f(x_{r},x_{1}).$

The function $\Lambda$ is smooth and $\mathbb{Z}_{r}$-equivariant, but in
contrast with Euclidean billiards, $\Lambda$ need not be $D_{r}$-equivariant.
By definition of the Finsler billiard reflection for geodesic rays (Section
2.2), the $r$-periodic Finsler billiard orbits are precisely the critical
points of $\Lambda$.

We would like to use this fact by applying Morse and Lusternik-Schnirelmann
theories to obtain a lower bound for the number of periodic Finsler billiard
orbits; however, these theories cannot be applied directly because
$\operatorname{G}(M,r)$ is not a closed manifold. We will show that we can
replace $\operatorname{G}(M,r)$, without affecting the topology, by the
following compact manifold with boundary:

$\operatorname{G}_{\varepsilon}(M,r)=\Big{\\{}(x_{1},\dots,x_{r})\in M^{\times
r}\ \bigg{|}\ \prod_{i=1}^{r}f(x_{i},x_{i+1})\geq\varepsilon>0\Big{\\}},$

where again the indices are understood cyclically. Similar to the Euclidean
case [21, Prop. 4.1], we establish the following proposition in the Finsler
setup.

######  Proposition 2.3.

If $\varepsilon>0$ is sufficiently small, then

  1. (1)

$\operatorname{G}_{\varepsilon}(M,r)$ is a smooth manifold with boundary;

  2. (2)

the inclusion
$\operatorname{G}_{\varepsilon}(M,r)\subset\operatorname{G}(M,r)$ is a
$\mathbb{Z}_{r}$-equivariant homotopy equivalence;

  3. (3)

all critical points of
$\Lambda:\operatorname{G}(M,r)\longrightarrow\mathbb{R}$ are contained in the
interior of $\operatorname{G}_{\varepsilon}(M,r)$;

  4. (4)

at every critical point of
$\Lambda\big{|}_{\partial\operatorname{G}_{\varepsilon}(M,r)}$, the
differential d$\Lambda$ is positive on inward vectors.

This proposition makes it possible to apply $\mathbb{Z}_{r}$-equivariant Morse
and Lusternik-Schnirelmann theories to the length function $\Lambda$ on the
manifold with boundary $\operatorname{G}_{\varepsilon}(M,r)$. We shall
restrict ourselves to the case when $r$ is prime. Then the group
$\mathbb{Z}_{r}$ acts freely on $\operatorname{G}_{\varepsilon}(M,r)$, and
$\mathbb{Z}_{r}$-equivariant Morse theory reduces to Morse theory on the
quotient manifold $\operatorname{G}_{\varepsilon}(M,r)/\mathbb{Z}_{r}$.

Due to item (4) of Proposition 2.3, the topological lower bounds on the number
of critical points of $\Lambda$, such as the sum of Betti numbers or the
Lusternik-Schnirelmann category, come from the topology of
$\operatorname{G}_{\varepsilon}(M,r)/\mathbb{Z}_{r}$, and due to item (2), and
the fact that $M$ is topologically the sphere, from the topology of the cyclic
configuration space $\operatorname{G}(S^{d-1},r)/\mathbb{Z}_{r}$. We refer to
[32] for the Morse theory on manifolds with boundary.

###  2.5. Technical bounds for Finsler billiards

We start with a number of technical lemmas working toward the proof of
Proposition 2.3.

The boundary $\partial\operatorname{G}_{\varepsilon}(M,r)$ is a level set of
the smooth function

$F\colon M^{\times r}\longrightarrow\mathbb{R},\qquad\qquad
F(x_{1},\dots,x_{r})=\prod_{i=1}^{r}f(x_{i},x_{i+1})^{2},$

and $F^{-1}(0)=M^{\times r}\setminus\operatorname{G}(M,r)$ is a critical
level. The first two items are therefore a consequence of the following lemma.

######  Lemma 2.4.

There exists a constant $\delta>0$ such that the interval $(0,\delta)$
consists of regular values of $F$.

We offer some geometric intuition for this statement. As indicated above, the
critical points of the length function $\Lambda$ are precisely the periodic
Finsler billiard orbits. Similarly, we may think of the critical points of the
function $F$ as the periodic orbits of some “unusual” billiard trajectory,
which we will call the $F$-_billiard_ trajectory, and for which the reflection
law is given by Lemma 2.7 below. In this terminology, Lemma 2.4 claims the
existence of $\delta$ such that any $r$-periodic $F$-billiard orbit
$(x_{1},\dots,x_{r})$ satisfies $F(x_{1},\dots,x_{r})\geq\delta$.

Assume that no such $\delta$ exists. Then we can find a closed $F$-billiard
orbit such that one of the edge lengths $\ell_{i}\coloneqq f(x_{i-1},x_{i})$
is arbitrarily small. The contradiction will arise in Section 2.7, where we
show the following two statements:

  1. (1)

If one edge of a closed $F$-orbit is “arbitrarily short,” then _all_ of the
edges are “arbitrarily short.”

  2. (2)

A closed $F$-orbit cannot have all edges “arbitrarily short.”

We will not explicitly define _arbitrarily short_ , but a suitable quantity
could be determined in terms of the period $r$ and certain “curvature”
quantities, which depend on the Finsler geometry of $M$ and on the geometry of
the indicatrix bundle. A similar shortness statement was proven in the
Euclidean case [21]; however, those curvature estimates rely on symmetry of
the inner product, symmetry of the unit tangent spheres, and also some
trigonometry; none of which we have at our disposal in the Finsler setting. We
treat these subtleties in the following Sections 2.5.1–2.5.3.

####  2.5.1. A “curvature” bound for Finsler manifolds

In the Finsler setting, we would like to formalize the intuitive idea that “a
geodesic segment $yz$ is short if and only if the corresponding tangent vector
at $y$ is almost tangent to $M$.”

This is easy to establish in the Euclidean case [21] as follows. The boundary
$M$ of the billiard table is a strictly convex hypersurface in Euclidean space
$\mathbb{R}^{d}$. By strict convexity of $M$, there exist positive numbers
$\rho<R$ such that for every $y\in M$, there exist two spheres tangent to $y$,
of radii $\rho$ and $R$, which are, respectively, contained in $M$ and
containing $M$. Let $n$ be the outward unit normal at $y$ and let $v\in
T_{y}\mathbb{R}^{d}$ be a tangent vector with $\langle v,n\rangle<0$, and
follow the geodesic from $y$ in the direction of $v$ until colliding again
with the boundary, say at $z\in M$. Then the numbers $\rho$ and $R$ satisfy

$\rho<\frac{|y-z|}{-2\langle v,n\rangle}<R.$

In particular, _the measurements_ $g(v)\coloneqq|y-z|$ _and_ $-p(v)=-\langle
v,n\rangle$ _are of the same order_. We will use the notation $g\sim p$ for
such a statement.

Now let $U$ be a Finsler manifold and $M$ a smooth, closed hypersurface,
quadratically convex with respect to the Finsler geodesics. For $y\in M$, let
$p\in J$ represent the unit covector which is conormal to $M$ at $y$ and
positive on outward vectors. Given $v\in I\subset T_{y}U$ such that $p(v)<0$,
let $z\in M$ be the first collision with $M$ of the geodesic ray emanating
from $y$ in the direction $v$. Define $g(v)=f(y,z)$.

Similarly, given $u\in I\subset T_{y}U$ such that $p(u)>0$, let $x\in M$ be
the point such that the oriented geodesic ray $xy$ enters $y$ with direction
$u$. Define $g(u)=f(x,y)$.

######  Remark 2.5.

To verify that such a point $x$ exists, note that the oriented ray $xy$ is a
geodesic if and only if the oriented ray $yx$ is a geodesic with respect to
the “reverse” Finsler structure defined by the Lagrangian $\bar{L}(u)\coloneqq
L(-u)$. The point $x$ can be defined as the first collision with $M$ of the
reverse geodesic ray emanating from $y$ in the direction $-u$.

######  Lemma 2.6.

There exist positive constants $\rho<R$ such that, for all $y\in M$, all $v\in
I\subset T_{y}U$ with $p(v)<0$, and all $u\in I\subset T_{y}U$ with $p(u)>0$,
we have

$\rho<\frac{f(y,z)}{-p(v)}<R\ \ \ \ \ \mbox{ and }\ \ \ \ \
\rho<\frac{f(x,y)}{p(u)}<R.$

Equivalently, the functions $g$ and $p$ have the same order, $g\sim p$.

The following proof is due to Sergei Ivanov.

###### Proof.

We will show the left inequalities, corresponding to vectors $v$ emanating
from $y$. The right inequalities follow similarly.

By compactness of $M$, it suffices to show existence for a single point $y\in
M$. In fact, it is enough to show that the lemma holds near $y$: that is, for
$v\in I\subset T_{y}U$ with $p(v)$ near $0$. Indeed, for all $v$ such that
$p(v)$ is sufficiently away from $0$, the quantity $\frac{f(y,z)}{-p(v)}$ is
bounded away from zero.

Consider smooth coordinates $(y_{1},\dots,y_{d})$ near $y$, such that a
neighborhood of $y$ in $M$ is given by the equation $y_{d}=0$. Let
$\gamma(t)=(\gamma_{1}(t),\dots,\gamma_{d}(t))$ be the unit-speed geodesic
passing through $y$ at time $0$ and with unit tangent vector $v\in I$ (see
Figure 4, left).

Figure 4. (Left): The unit speed geodesic
$\gamma(t)=(\gamma_{1}(t),\dots,\gamma_{d}(t))$ passes through $y\in M$ at
time $0$ with tangent vector $v$ and again at $z\in M$ at time $t_{0}\sim
p(v)$. (Right): The $d^{\mbox{\scriptsize th}}$ component of $\gamma$, shown
to the second order.

In case $v\in I\cap T_{y}M$, we have $\gamma_{d}(0)=0$ and
$\gamma_{d}^{\prime}(0)=0$, so by quadratic convexity,
$\gamma_{d}^{\prime\prime}(0)$ is bounded away from $0$ for all $v\in I\cap
T_{y}M$. Therefore, for $v$ almost tangent to $M$ (i.e., with $p(v)<0$
sufficiently near $0$), we have $\gamma_{d}^{\prime\prime}(0)\sim 1$. In
addition, for $v$ almost tangent to $M$, we have
$p(v)\sim\gamma_{d}^{\prime}(0)<0$.

Now writing $\gamma_{d}(t)$ up to second order yields

$\gamma_{d}(t)=\gamma_{d}(0)+\gamma_{d}^{\prime}(0)t+\frac{1}{2}\gamma_{d}^{\prime\prime}(0)t^{2},$

as depicted in Figure 4, right. Thus $\gamma_{d}(t)$ will meet zero again at a
time

$t_{0}=-2\frac{\gamma_{d}^{\prime}(0)}{\gamma_{d}^{\prime\prime}(0)}\sim
p(v).$

Since $t_{0}$ is the length $f(y,z)=g(v)$, we conclude that $g\sim p$,
completing the proof. ∎

####  2.5.2. Reflection law for the $F$-billiard

Just as the critical points of the length function $\Lambda$ correspond to
periodic orbits of the Finsler billiard, we think of the critical points of
$F$ as the periodic orbits of some “unusual” billiard trajectory determined by
the function $F$. Suppose that $x,y,z\in M$ are points such that the oriented
geodesic segment $yz$ is the $F$-reflection of the oriented geodesic segment
$xy$. Let $u,v\in I\subset T_{y}U$ be the tangent vectors which correspond,
respectively, to $xy$ and $yz$. Let $p\in J\subset T^{*}_{y}U$ represent the
outward-pointing unit conormal, and let $n=D^{*}_{p}$ represent the outward
pointing unit normal. Then the $F$-billiard has the following reflection law.

######  Lemma 2.7.

The $F$-billiard reflection law is given by the following cotangent relation:

$\frac{D_{u}}{f(x,y)}-\frac{D_{v}}{f(y,z)}=tp,\ t>0.$

###### Proof.

The point $y\in M$ is a relative extremum of the function $F(\cdot)\coloneqq
f(x,\cdot)^{2}f(\cdot,z)^{2}$, so the differential of this function at $y$ is
conormal to the tangent hyperplane $T_{y}M$. Therefore the differential of the
function

$\frac{1}{2}\ln F(\cdot)=\ln f(x,\cdot)+\ln f(\cdot,z)$

at $y$ is also conormal to the tangent hyperplane $T_{y}M$. We compute

$\frac{1}{2}\mbox{d}\big{(}\ln F(\cdot)\big{)}=\mbox{d}\big{(}\ln
f(x,\cdot)+\ln
f(\cdot,z)\big{)}=\frac{\mbox{d}f(x,\cdot)}{f(x,\cdot)}+\frac{\mbox{d}f(\cdot,z)}{f(\cdot,z)}.$

Recall from the proof of Lemma 2.1 that at the point $y$, d$f(x,\cdot)=D_{u}$
and d$f(\cdot,z)=-D_{v}$. Therefore, at the point $y$, we have

$\frac{D_{u}}{f(x,y)}-\frac{D_{v}}{f(y,z)}=tp,$

as desired. ∎

We will also make use of the following consequence.

######  Lemma 2.8.

If $v\in I\subset T_{y}M$ is the $F$-reflection of $u\in I\subset T_{y}M$,
then either the linear subspaces $T_{u}I,T_{v}I,T_{y}M\subset T_{y}U$ are
equal, or they intersect in a subspace of codimension $2$.

###### Proof.

By the reflection law, the covectors $D_{u}$, $D_{v}$, and $p$ are linearly
dependent, hence span a subspace of at most two dimensions. Therefore, the
intersection of their kernels, $T_{u}I\cap T_{v}I\cap T_{y}M$, has codimension
at most two. ∎

Given $u\in I\subset T_{y}U$ with $p(u)>0$, let $v$ be the $F$-reflection of
$u$. Suppose that $u\neq n$, so that the hyperplanes $T_{u}I$, $T_{v}I$, and
$T_{y}M$ are not parallel. Then, from Lemma 2.8, the intersection of the
linear hyperplanes $T_{u}I\cap T_{v}I$ is a codimension $1$ subspace of
$T_{y}M$, hence there is a unique vector $w=w(u)\in T_{y}M\cap I$ for which
the kernel of $D_{w}$ contains $T_{u}I\cap T_{v}I$ and such that $D_{u}(w)>0$.
Now applying the (cotangent) relation of Lemma 2.7 to the vector $w$ yields:

$\displaystyle\frac{D_{u}(w)}{f(x,y)}=\frac{D_{v}(w)}{f(y,z)}.$ (3)

Note that $D_{v}(w)>0$, since the denominators are positive and $D_{u}(w)>0$
by assumption.

####  2.5.3. Global indicatrix bound

Since the Finsler norm need not arise from an inner product, it is useful to
develop some method of comparing two vectors $u,v\in I$. One non-symmetric
idea is to apply the Legendre transform to obtain a covector $D_{u}$ which
then acts on $v$. We have a general lower bound by compactness and a specific
upper bound by strict convexity of the indicatrices.

######  Lemma 2.9.

There exists a positive constant $K\geq 1$ such that, for every triple $y\in
M$, $u,v\in I\subset T_{y}U$, one has $-K\leq D_{u}(v)\leq 1$. In particular,
$D_{u}(v)=1$ if and only if $u=v$.

We will also require a more refined bound. For $y\in M$, consider $u\in
I\subset T_{y}U$ such that $p(u)>0$ and $u\neq n$. Let $v\in I\subset T_{y}U$
be the $F$-reflection of $u$ and let $w$ be as described in Section 2.5.2.
Although $w=w(u)$ is not defined when $u=n$, any sequence
$u_{i}\longrightarrow n$ has $D_{u_{i}}(w_{i})\longrightarrow 0$ and
$D_{v_{i}}(w_{i})\longrightarrow 0$, so that the maps $u\longmapsto D_{u}(w)$
and $u\longmapsto D_{v}(w)$ extend continuously to $u=n$.

######  Lemma 2.10.

There exists a positive constant $k<1$ such that for all $y\in M$ and all
$u\in I\subset T_{y}U$ with $p(u)>0$,

$k<D_{u}(w)+p(u)\ \ \ \ \ \mbox{ and }\ \ \ \ \ k<D_{v}(w)-p(v).$

###### Proof.

The numbers $D_{u}(w)$, $D_{v}(w)$, $p(u)$, and $-p(v)$ are all positive. The
first two are bounded away from $0$ except when $u$ is almost parallel to $n$,
and the last two are bounded away from $0$ except when $u$ is almost tangent
to $M$. ∎

###  2.6. The $F$-billiard reflection preserves shortness and almost-tangency

With all of the technical bounds obtained in Section 2.4, we are ready to
study the $F$-billiard reflection in more detail. We show that short geodesics
$F$-reflect to short geodesics, and that almost-tangent vectors $F$-reflect to
almost-tangent vectors.

In particular, suppose that $u\in I\subset T_{y}U$, let $v\in I\subset T_{y}U$
be the $F$-reflection of $u$, and let $x$ and $z$ be points in $M$ such that
$u$ and $v$ correspond, respectively, to the oriented geodesic segments $xy$
and $yz$. In this sense, we may consider $v$, $x$, and $z$ as functions of
$u$. Let $p\in J\subset T_{y}^{*}U$ be the outward pointing unit conormal and
let $n=D^{*}_{p}$ be the outward pointing unit normal. Our goal is to show the
following:

$p(u)\sim f(x,y)\sim f(y,z)\sim p(v).$

Note that the outer equivalences were already established in Lemma 2.6.

We make use of the positive constants $\rho$ and $R$ defined by Lemma 2.6, as
well as the positive constants $K$ and $k$ determined in Lemmas 2.9 and 2.10.

######  Lemma 2.11.

If $v\in I\subset T_{y}U$ is the $F$-reflection of $u\in I\subset T_{y}U$,
with $p(u)>0$, then

$\frac{k\rho}{K(\rho+R)}\leq\frac{f(x,y)}{f(y,z)}\leq\frac{K(\rho+R)}{k\rho}.$

In particular, the measurements $f(x,y)$ and $f(y,z)$ are equivalent:
$f(x,y)\sim f(y,z)$.

###### Proof.

We prove by contradiction. Assume that the left inequality fails, so that

$\displaystyle\frac{f(x,y)}{f(y,z)}<\frac{k\rho}{K(\rho+R)}.$ (4)

Then the $F$-reflection law applied to $w$ (3), combined with (4) and then
Lemma 2.9, yields

$\displaystyle
D_{u}(w)=\frac{f(x,y)}{f(y,z)}D_{v}(w)<\frac{k\rho}{K(\rho+R)}D_{v}(w)\leq\frac{k\rho}{\rho+R}.$
(5)

On the other hand, Lemma 2.6, combined with (4) and then Lemma 2.9, yields

$\displaystyle p(u)<-\frac{kR}{K(\rho+R)}p(v)<\frac{kR}{\rho+R}.$ (6)

But the sum of (5) and (6) gives a contradiction; the right side is equal to
$k$, while the left side is greater than $k$ by Lemma 2.10.

The right inequality of Lemma 2.11 can be shown similarly. ∎

######  Lemma 2.12.

The $F$-billiard reflection $u\longmapsto v$ can be extended continuously to
$I\cap T_{y}M$. Moreover, this extension is the identity map on $I\cap
T_{y}M$.

###### Proof.

This lemma clearly holds for the ordinary Finsler billiard reflection law in
Lemma 2.1. In the $F$-billiard case, by Lemmas 2.6 and 2.11, we have $p(u)\sim
f(x,y)\sim f(y,z)\sim p(v)$, so if a sequence $u_{n}\subset I$ tends to
$T_{y}M$, so does the sequence of $F$-reflections $v_{n}$.

From Lemma 2.8, we have $\operatorname{Ker}D_{u_{n}}\cap
T_{y}M=\operatorname{Ker}D_{v_{n}}\cap T_{y}M$ for every $n$. Therefore, if
$u_{n}\longrightarrow u\in I\cap T_{y}M$, then
$\lim\big{(}\operatorname{Ker}D_{u_{n}}\big{)}\cap T_{y}M$ is some codimension
1 subspace $W\subset T_{y}M$, and we must have $W=\operatorname{Ker}D_{v}\cap
T_{y}M$ for any continuously-defined $F$-reflection $v$ of $u$. Since we also
must have $v\in I\cap T_{y}M$, there are only two candidates for $v$: $u$
itself and the “opposite” vector $u^{\prime}$, for which
$D_{u}(u^{\prime})<0$. Now, by definition of the unit vector $w_{n}\in T_{y}M$
(see Section 2.5.2), $u=\lim w_{n}$. We have $D_{v_{n}}(w_{n})>0$ for all $n$,
so $D_{v}(u)$ must be nonnegative. Hence $v=u$. ∎

We are now ready to prove Lemma 2.4.

###  2.7. Nonexistence of short periodic $F$-billiard trajectories

###### Proof of Lemma 2.4.

Assume that $(x_{1},\dots,x_{r})$ is an $r$-periodic $F$-billiard orbit. Let
$u_{i},v_{i}\in I\subset T_{x_{i}}U$ represent the tangent vectors which
correspond, respectively, to the oriented geodesic segments $x_{i-1}x_{i}$ and
$x_{i}x_{i+1}$. Let $p_{i}\in J\subset T_{x_{i}}^{*}U$ be the outward pointing
unit conormal and let $n_{i}=D^{*}_{p_{i}}$ be the outward pointing unit
normal. Let $\ell_{i}=f(x_{i-1},x_{i})$. We aim to show that no $\ell_{i}$ can
be arbitrarily small. From Lemma 2.11 we obtain the edge comparison for any
$i$ and $j$:

$\frac{\ell_{i}}{\ell_{j}}<\left(\frac{K(\rho+R)}{k\rho}\right)^{r}.$

In particular, this establishes that $\ell_{i}\sim\ell_{j}$, so that if
$\ell_{i}$ is small for any $i$, every $\ell_{j}$ is also small.

Now, seeking a contradiction, we assume that a critical $r$-gon
$(x_{1},\dots,x_{r})$ has all edges arbitrarily short. Then each $u_{i}$ is
almost tangent, so by Lemma 2.12, $u_{i}\sim v_{i}$ in any auxiliary metric on
$M$. In particular, for any auxiliary Euclidean metric on $M$,
$(x_{1},\dots,x_{r})$ is an $r$-gon with all exterior angles small. The
contradiction will arise once we show the following statement which is a
discrete analog of the theorem that the total curvature of a spacial closed
curve is not less that $2\pi$ (see, e.g., [22]):

_The sum of the exterior angles of an $r$-gon in Euclidean space
$\mathbb{R}^{q}$ is at least $2\pi$._

Let $a_{1},\dots,a_{r}$ be oriented vectors representing the edges of the
polygon. The Gauss map sends these vectors to some points $A_{1},\dots,A_{r}$
on $S^{q-1}$, and the great circle segment connecting $A_{i}$ and $A_{i+1}$
has length equal to the exterior angle at the vertex connecting $a_{i}$ and
$a_{i+1}$ (here the indices are cyclic). Thus it is enough to show that the
perimeter of this spherical polygon $(A_{1},\dots,A_{r})$ is at least $2\pi$.
By the Crofton formula, for this it is enough to show that the polygon
intersects every great sphere $S^{q-2}$.

Choose a great sphere $S=S^{q-2}\subset S^{q-1}$ and let
$P=\mathbb{R}^{q-1}\subset\mathbb{R}^{q}$ be the corresponding hyperplane. If
some vector $a_{i}$ is contained in $P$, then $S$ contains $A_{i}$. Otherwise,
translate $P\subset\mathbb{R}^{q}$ so that it sits as a supporting hyperplane
to the polygon $(a_{1},\dots,a_{r})$, say at the vertex connecting edge
$a_{i}$ to edge $a_{i+1}$. In this case the points $A_{i}$ and $A_{i+1}$ are
separated by the great sphere $S$, so the great circle segment connecting
$A_{i}$ to $A_{i+1}$ must intersect $S$. ∎

###  2.8. Critical points of the restricted length function

With the proof of Lemma 2.4 complete, we turn our attention to the fourth item
of Proposition 2.3, that if some $r$-gon
$x=(x_{1},\dots,x_{r})\in\partial\operatorname{G}_{\varepsilon}(M,r)$ is a
critical point of
$\Lambda\big{|}_{\partial\operatorname{G}_{\varepsilon}(M,r)}$, then
d$\Lambda_{x}$ is positive on inward vectors. Since
$\partial\operatorname{G}_{\varepsilon}(M,r)$ is a level set of $F$, the
differential d$F$ vanishes precisely on vectors tangent to
$\partial\operatorname{G}_{\varepsilon}(M,r)$. Then, since
d$F_{x}=t$d$\Lambda_{x}$ on $T_{x}\operatorname{G}_{\varepsilon}(M,r)$, the
sign of d$\Lambda_{x}$ is the same for all inward pointing tangent vectors at
point $x$. Therefore, it is enough to show that d$\Lambda_{x}$ is positive on
a single inward vector $V\in T_{x}\operatorname{G}_{\varepsilon}(M,r)$.

For the remainder of this section we let $x=(x_{1},\dots,x_{r})$ be a critical
point as discussed above, and we let $u_{i},v_{i}\in T_{x_{i}}U$ be unit
vectors tangent to the geodesic rays $x_{i-1}x_{i}$ and $x_{i}x_{i+1}$. We
will drop the subscript on the differentials d$\Lambda$ and d$F$, since we
will only be discussing them at the point $x$. We will continue to use the
notation $n_{i}$ for the outward-pointing unit normal vector at $x_{i}$ and
$p_{i}=D_{n_{i}}$. We will use $\ell_{i}$ to represent the Finsler distance
$f(x_{i-1},x_{i})$.

We recall the differentials:

$\mbox{d}\Lambda=(D_{u_{1}}-D_{v_{1}},\dots,D_{u_{r}}-D_{v_{r}}),$

and

$\frac{1}{2}\mbox{d}(\ln
F)=\left(\frac{D_{u_{1}}}{\ell_{1}}-\frac{D_{v_{1}}}{\ell_{2}},\dots,\frac{D_{u_{r}}}{\ell_{r}}-\frac{D_{v_{r}}}{\ell_{1}}\right).$

To show that there exists $V\in T_{x}\operatorname{G}_{\varepsilon}(M,r)$ for
which both differentials are positive, it is enough to find a vector $W\in
T_{x_{i}}M$, for some $i$, such that $D_{u_{i}}(W)>0$ and $D_{v_{i}}(W)<0$.
Indeed, we can then let $V\in T_{x}\operatorname{G}_{\varepsilon}(M,r)$ be the
vector whose only nonzero component is $W$.

Assume that such a vector $W$ does not exist at the vertex $x_{i}$. Then
$\operatorname{ker}(D_{u_{i}})\cap
T_{x_{i}}M=\operatorname{ker}(D_{v_{i}})\cap T_{x_{i}}M$, hence the covectors
$D_{u}$ and $D_{v}$ are positive-proportional when restricted to $T_{x_{i}}M$.
That is, there exist $a_{i}$ and $b_{i}$, both positive or both negative, such
that either

$a_{i}D_{u_{i}}-b_{i}D_{v_{i}}=0\ \ {\rm or}\ \
a_{i}D_{u_{i}}-b_{i}D_{v_{i}}=p_{i}.$ (7)

The former is impossible since it implies that $u_{i}=v_{i}$.

In the latter case, the three covectors $D_{u_{i}}$, $D_{v_{i}}$, and $p_{i}$
span a $2$-plane $P$, hence there exists a unique vector $w_{i}\in
T_{x_{i}}M\cap I$ such that $D_{w_{i}}\in P$ and $D_{u_{i}}(w_{i})>0$. In the
next lemma we develop a universal constant, which allows us to compare
$D_{u_{i}}$ with $p_{i}(u_{i})$ and $D_{v_{i}}$ with $p_{i}(v_{i})$, for any
$u_{i}$ and $v_{i}$ satisfying a reflection law as in (7). We observe the
similarities in (7) to both the Finsler reflection law and the $F$-reflection
law, and we note that $w_{i}$ is an analogue of the vector $w$ introduced
after Lemma 2.8 for the $F$-billiard reflection.

######  Lemma 2.13.

There exist constants $C>0$ and $m\in\mathbb{N}$, such that, for all $x\in M$,
and for all $w\in T_{x}M\cap I$, if
$P=\operatorname{Span}\left\\{p,D_{w}\right\\}$, and if $u\in I$ is any vector
with $D_{u}\in P$ and $D_{u}(w)>0$, then

$\frac{1}{C}p(u)^{2m}\leq 1-D_{u}(w)\leq Cp(u)^{2m}.$

###### Proof.

By compactness of $M$ and $T_{x}M\cap I$ it is enough to show the existence of
$C$ at a single point $x$ and for a single $2$-plane $P$. Let $\Gamma\subset
I$ be the open curve consisting of those points $u$ described in the statement
of the lemma. We claim that $p|_{\Gamma}\colon\Gamma\longrightarrow\mathbb{R}$
is injective. Indeed, suppose that $p(u)=p(v)$ for $u,v\in\Gamma$. We may
write $aD_{u}-bD_{v}=p$; here $a$ and $b$ have the same sign, since $D_{u}(w)$
and $D_{v}(w)$ are both positive and $p(w)=0$. Applying this relation to $u$
and $v$ yields

$a-bD_{v}(u)=p(u)=p(v)=aD_{u}(v)-b,$

therefore $a(1-D_{u}(v))=-b(1-D_{v}(u))$, contradicting that $a$ and $b$ have
the same sign.

Now consider the map $h\colon p(\Gamma)\longrightarrow\mathbb{R}$ given by
$p(u)\longmapsto D_{u}(w)$. Then $h(0)=D_{w}(w)=1$, and $h^{\prime}(0)=0$
since $D_{u}(w)$ is maximized at $u=w$. Therefore the first nonzero derivative
of $h$ has even order and is negative; in particular, $1-D_{u}(w)\sim
p(u)^{2m}$ for some $m$. ∎

It follows from quadratic convexity of the indicatrix that $m=1$, but we omit
the details here. We are most interested in the following form of Lemma 2.13.

######  Corollary 2.14.

For any $x\in M$ and any $u,v\in I\subset T_{x}U$ such that $p(u)>0$,
$p(v)<0$, and $aD_{u}-bD_{v}=p$, where $a$ and $b$ have the same sign, the
following inequality holds:

$\frac{p(v)^{2m}}{C^{2}p(u)^{2m}}\leq\frac{1-D_{v}(w)}{1-D_{u}(w)}\leq
C^{2}\frac{p(v)^{2m}}{p(u)^{2m}},$

where $w\in T_{x}M$ is the unique unit vector with
$D_{w}\in\operatorname{Span}\left\\{D_{u},D_{v}\right\\}$ and $D_{u}(w)>0$.

We will show the existence of an appropriate vector $W$ in two separate cases:
the first, when all side lengths of the $r$-gon are sufficiently short; and
the second, when there is at least one long side. The following lemma treats
the first case.

######  Lemma 2.15.

There exists $\eta>0$ such that, if $p_{i}(u_{i})<\eta$ and
$-p_{i}(v_{i})<\eta$ for all $i\in\left\\{1,\dots,r\right\\}$, then there
exists an index $i$ and a vector $W\in T_{x_{i}}M$, such that $D_{u_{i}}(W)>0$
and $D_{v_{i}}(W)<0$.

###### Proof.

Assume that no such $W$ exists, then second equation (7), with $a_{i}$ and
$b_{i}$ both positive or both negative, holds at every vertex $i$. We will use
the “all small” hypothesis of the lemma to arrive at a contradiction. We first
focus on a single vertex and drop the subscript $i$. Apply (7) to vectors $u$
and $v$ to obtain

$\displaystyle a-bD_{v}(u)$ $\displaystyle=p(u),$ (8) $\displaystyle
aD_{u}(v)-b$ $\displaystyle=p(v).$ (9)

Suppose that $\eta=\frac{\delta}{2+2K}$, where $\delta>0$ is a small
unspecified number and $K\geq 1$ is the constant determined in Lemma 2.9. We
claim that $D_{u}(v)$ and $D_{v}(u)$ cannot be negative.

First, if $a$ and $b$ are negative, then $D_{u}(v)$ and $D_{v}(u)$ are both
positive because $p(u)>0,\ p(v)<0$. Otherwise, if $a$ and $b$ are positive,
and also if $D_{v}(u)$ is negative, then

$\displaystyle a$ $\displaystyle=p(u)+bD_{v}(u)<p(u),$ $\displaystyle b$
$\displaystyle=-p(v)+aD_{u}(v)<-p(v)+p(u)D_{u}(v)<-p(v)+p(u),$

and so $a$ and $b$ are both less than $2\eta$. The same bounds hold if
$D_{u}(v)$ is negative.

Now apply (7) to the vector $n$ to obtain

$1=aD_{u}(n)-bD_{v}(n)<a-bD_{v}(n)<2\eta+bK<2\eta(1+K)=\delta,$

contradicting the assumption that $D_{v}(u)$ or $D_{u}(v)$ is negative. Now,
solve for $a$ and $b$ using equations (8) and (9) to obtain

$\displaystyle a$
$\displaystyle=\frac{p(u)-p(v)D_{v}(u)}{1-D_{u}(v)D_{v}(u)}<\frac{2\eta}{1-D_{u}(v)D_{v}(u)},$
$\displaystyle b$
$\displaystyle=\frac{-p(v)+p(u)D_{u}(v)}{1-D_{u}(v)D_{v}(u)}<\frac{2\eta}{1-D_{u}(v)D_{v}(u)}.$

Again we apply (7) to the vector $n$ to obtain

$1=aD_{u}(n)-bD_{v}(n)<\frac{2\eta(1+K)}{1-D_{u}(v)D_{v}(u)}=\frac{\delta}{1-D_{u}(v)D_{v}(u)}.$

Therefore, $1-D_{u}(v)D_{v}(u)=\delta$, and so both $D_{u}(v)\geq 1-\delta$
and $D_{v}(u)\geq 1-\delta$. It follows that $u$ and $v$ are sufficiently
close.

But this analysis holds at every vertex $i$, contradicting the existence of a
large exterior angle in an auxiliary Euclidean metric. ∎

We are now equipped to prove Proposition 2.3.

###  2.9. The proof of Proposition 2.3

###### Proof of Proposition 2.3.

The first two items follow immediately from Lemma 2.4. The argument for the
third item is similar. In particular, we apply the same logic of Lemma 2.4 to
the length function $\Lambda$, instead of the function $F$, to obtain a
similar statement: if one edge of an $r$-periodic Finsler billiard trajectory
is short, then so are all its edges. Therefore there exists some
$\varepsilon>0$ such that all critical points of $\Lambda$ are contained in
the interior of $\operatorname{G}_{\varepsilon}(M,r)$.

Let $\eta$ be a fixed number satisfying Lemma 2.15, and let $R$ and $\rho$ be
the constants from Lemma 2.6. To satisfy the fourth item of the proposition,
choose $\varepsilon$ small enough so that
$\eta>\frac{A}{\rho}\varepsilon^{\frac{1}{2r}}$, where $A>0$ is a constant we
will specify shortly. Assume that all lengths $\ell_{i}$ satisfy
$\ell_{i}<\eta\rho$. Then, by Lemma 2.6, all corresponding $u_{i}$ and $v_{i}$
are $\eta$-tangent, so the conditions of Lemma 2.15 are satisfied, confirming
the existence of an appropriate vector $W$. Otherwise, there exists an index
$j$ such that $\ell_{j}\geq\eta\rho>A\varepsilon^{\frac{1}{2r}}$. Using the
fact that the product of the squared lengths is $\varepsilon$, we write

$\varepsilon^{\frac{1}{2}}=\prod_{1\leq i\leq
r}\ell_{i}>A\varepsilon^{\frac{1}{2r}}\prod_{i\neq j}\ell_{i}.$

Therefore, there exists an index $k$ such that
$\ell_{k}<\varepsilon^{\frac{1}{2r}}A^{-\frac{1}{r-1}}$. It follows that

$A^{\frac{r}{r-1}}<\frac{\ell_{j}}{\ell_{k}}=\frac{\ell_{j}}{\ell_{j-1}}\cdot\frac{\ell_{j-1}}{\ell_{j-2}}\cdots\frac{\ell_{k+1}}{\ell_{k}},$

where the indices are understood cyclically. There are at most $r-1$ factors
on the right side, therefore, there exists some index $i$ such that

$A^{\frac{r}{(r-1)^{2}}}<\frac{\ell_{i+1}}{\ell_{i}}.$

Now let $A=(\frac{R}{\rho})^{\frac{(r-1)^{2}}{r}}\cdot
C^{\frac{(r-1)^{2}}{mr}}$, where $C$ and $m$ are the constants determined in
Lemma 2.13. Then we have

$\frac{R}{\rho}\cdot
C^{\frac{1}{m}}<\frac{\ell_{i+1}}{\ell_{i}}\leq-\frac{R}{\rho}\frac{p_{i}(v_{i})}{p_{i}(u_{i})},$

where the second inequality is Lemma 2.6. It follows that

$\displaystyle 1<\frac{p_{i}(v_{i})^{2m}}{C^{2}p_{i}(u_{i})^{2m}}.$ (10)

Assume there is no vector $W\in T_{x_{i}}M$ such that $D_{u_{i}}(W)>0$ and
$D_{v_{i}}(W)<0$, since otherwise, we are done. Then the hypotheses of
Corollary 2.14 hold, and we write

$\frac{1-D_{v_{i}}(w)}{1-D_{u_{i}}(w)}\geq\frac{p_{i}(v_{i})^{2m}}{C^{2}p_{i}(u_{i})^{2m}}>1,$

where the second inequality is (10). Therefore $D_{u_{i}}(w)>D_{v_{i}}(w)$,
and since $\ell_{i+1}>\ell_{i}$, we also have
$D_{u_{i}}(w)\ell_{i+1}>D_{v_{i}}(w)\ell_{i}$. Thus the vector $V\in
T_{x}\operatorname{G}_{\varepsilon}(M,r)$, whose only nonzero component is
$w$, satisfies d$\Lambda(V)>0$ and d$F(V)>0$, as desired. ∎

##  3\. Topology of cyclic configuration spaces

Let $r\geq 2$ be an integer, and let $M$ be a topological space. The ordered
configuration space of $r$ pairwise distinct points on $M$ is the space

$\operatorname{F}(M,r)\coloneqq\\{(x_{1},\ldots,x_{r})\in M^{r}:x_{i}\neq
x_{j}\ \text{for all}\ i\neq j\\}.$

The symmetric group $\mathfrak{S}_{r}$ acts (from the left) on
$\operatorname{F}(M,r)$ by permuting the points, that is, for
$\pi\in\mathfrak{S}_{r}$

$\pi\cdot(x_{1},\ldots,x_{r})=(x_{\pi(1)},\ldots,x_{\pi(r)}).$

The unlabeled configuration space of $r$ pairwise distinct points on $M$ is
the quotient space $\operatorname{F}(M,r)/\mathfrak{S}_{r}$. We refer to F.
Cohen [15] and Fadell & Husseini [18] for background on configuration spaces.

Let us repeat that the ordered cyclic configuration space of $r$ consecutively
distinct points on $M$ is the space

$\operatorname{G}(M,r)\coloneqq\\{(x_{1},\ldots,x_{r})\in M^{r}:x_{i}\neq
x_{i+1}\ \text{for all}\ i\\},$

where by convention $x_{r+1}=x_{1}$. Clearly,
$\operatorname{F}(M,r)\subseteq\operatorname{G}(M,r)\subseteq M^{r}$. The
Dihedral group $D_{r}=\langle a,b:a^{r}=b^{2}=1,ab=ba^{r-1}\rangle$ acts
naturally on $\operatorname{G}(M,r)$ by

$\displaystyle a\cdot(x_{1},x_{2},\ldots,x_{r-1},x_{r})$ $\displaystyle=$
$\displaystyle(x_{r},x_{1},\ldots,x_{r-2},x_{r-1}),$ $\displaystyle
b\cdot(x_{1},x_{2},\ldots,x_{r-1},x_{r})$ $\displaystyle=$
$\displaystyle(x_{r},x_{r-1},\ldots,x_{2},x_{1}).$

On the other hand, due to the geometric restriction coming from the Finsler
distance being not symmetric, we consider only the action of the cyclic
subgroup $\mathbb{Z}_{r}=\langle a\rangle$ on $\operatorname{G}(M,r)$. Thus,
in this paper, the unlabeled cyclic configuration space of $r$ consecutively
distinct points on $M$ is the quotient space
$\operatorname{G}(M,r)/\mathbb{Z}_{r}$.

In this section we study the topology of the unlabeled cyclic configuration
space $\operatorname{G}(S^{d-1},r)/\mathbb{Z}_{r}$ for $r$ a prime. First,
using an appropriate spectral sequence, we determine the cohomology of the
unlabeled configuration space with coefficients in the field $\mathbb{F}_{r}$.

######  Theorem 3.1.

Let $r\geq 3$ be a prime.

  1. (1)

Let $d$ be an even integer, and let

$A\coloneqq\\{\ell(d-2),\ell(d-2)+1:1\leq\ell\leq r-2\\}\qquad\text{and}\qquad
B\coloneqq\\{0,1,\ldots,(r-1)(d-2)+1\\}{\setminus}A.$

Then

$H^{n}(\operatorname{G}(S^{d-1},r)/\mathbb{Z}_{r};\mathbb{F}_{r})=\begin{cases}\mathbb{F}_{r}\oplus\mathbb{F}_{r},&n\in
A,\\\ \mathbb{F}_{r},&n\in B,\\\ 0,&\text{\rm otherwise}.\end{cases}$

  2. (2)

Let $d$ be an odd integer, and let

$C\coloneqq\\{2\ell(d-2),2\ell(d-2)+1:1\leq\ell\leq\tfrac{r-3}{2}\\}\qquad\text{and}\qquad
D\coloneqq\\{0,1,\ldots,(r-1)(d-2)+1\\}{\setminus}C.$

Then

$H^{n}(\operatorname{G}(S^{d-1},r)/\mathbb{Z}_{r};\mathbb{F}_{r})=\begin{cases}\mathbb{F}_{r}\oplus\mathbb{F}_{r},&n\in
C,\\\ \mathbb{F}_{r},&n\in D,\\\ 0,&\text{\rm otherwise}.\end{cases}$

Second, using the same spectral sequence we derive the following estimate of
the Lusternik–Schnirelmann category, which can be also deduced from the work
of Karasev [29, Thm. 7].

######  Theorem 3.2.

Let $d\geq 3$ be an integer, and let $r\geq 3$ be a prime. Then the
Lusternik–Schnirelmann category of the unlabeled cyclic configuration space
$\operatorname{G}(S^{d-1},r)/\mathbb{Z}_{r}$ is bounded from below as follows:

$\operatorname{cat}(\operatorname{G}(S^{d-1},r)/\mathbb{Z}_{r})\geq(r-1)(d-2)+1.$

###  3.1. Setting up a spectral sequence

Now we set up a spectral sequence that converges to the cohomology of the
unlabeled cyclic configuration space
$H^{*}(\operatorname{G}(S^{d-1},r)/\mathbb{Z}_{r};\mathbb{F}_{r})$. We use the
fact that the group $\mathbb{Z}_{r}$ acts freely on the cyclic configuration
space $\operatorname{G}(S^{d-1},r)$.

Thus, since $\mathbb{Z}_{r}$ acts freely on $\operatorname{G}(S^{d-1},r)$, the
Borel construction
$\mathrm{E}(\mathbb{Z}_{r})\times_{\mathbb{Z}_{r}}\operatorname{G}(S^{d-1},r)$
and the quotient space $\operatorname{G}(S^{d-1},r)/\mathbb{Z}_{r}$ are
homotopy equivalent. Indeed, the map
$\mathrm{E}(\mathbb{Z}_{r})\times_{\mathbb{Z}_{r}}\operatorname{G}(S^{d-1},r)\longrightarrow\operatorname{G}(S^{d-1},r)/\mathbb{Z}_{r}$,
induced by the $\mathbb{Z}_{r}$-equivariant projection on the second factor
$\mathrm{E}(\mathbb{Z}_{r})\times\operatorname{G}(S^{d-1},r)\longrightarrow\operatorname{G}(S^{d-1},r)$,
is a fibration with a contractible fiber $\mathrm{E}(\mathbb{Z}_{r})$, and
therefore a homotopy equivalence. Hence,

$H^{*}(\operatorname{G}(S^{d-1},r)/\mathbb{Z}_{r};\mathbb{F}_{r})\cong
H^{*}(\mathrm{E}(\mathbb{Z}_{r})\times_{\mathbb{Z}_{r}}\operatorname{G}(S^{d-1},r);\mathbb{F}_{r}),$

and so we compute the cohomology of the Borel construction instead. An
advantage of the Borel construction is that it is the total space of the
following fibration

$\textstyle{\operatorname{G}(S^{d-1},r)\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\mathrm{E}(\mathbb{Z}_{r})\times_{\mathbb{Z}_{r}}\operatorname{G}(S^{d-1},r)\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\pi_{G}}$$\textstyle{\mathrm{B}(\mathbb{Z}_{r}),}$
(11)

where $\pi_{G}$ is induced by the $\mathbb{Z}_{r}$-equivariant projection on
the first factor
$\mathrm{E}(\mathbb{Z}_{r})\times\operatorname{G}(S^{d-1},r)\longrightarrow\mathrm{E}(\mathbb{Z}_{r})$.

The Serre spectral sequence induced by the fibration (11) has the $E_{2}$-term
given by

$E_{2}^{p,q}(\mathrm{E}(\mathbb{Z}_{r})\times_{\mathbb{Z}_{r}}\operatorname{G}(S^{d-1},r))=\mathcal{H}^{p}(\mathrm{B}(\mathbb{Z}_{r});H^{q}(\operatorname{G}(S^{d-1},r);\mathbb{F}_{r}))\cong
H^{p}(\mathbb{Z}_{r};H^{q}(\operatorname{G}(S^{d-1},r);\mathbb{F}_{r})).$ (12)

Here $\mathcal{H}^{*}(\cdot)$ indicates that the we have the cohomology with
local coefficients; for more details consult for example [27, Sec. 3.H]. The
local system is determined by the action of the fundamental group of the base
space $\pi_{1}(\mathrm{B}(\mathbb{Z}_{r}))\cong\mathbb{Z}_{r}$ on the
cohomology of a fiber of the fibration (11). The second notation assumes that
the coefficients we have for the group cohomology of $\mathbb{Z}_{r}$ are
given in the $\mathbb{Z}_{r}$-module
$H^{q}(\operatorname{G}(S^{d-1},r);\mathbb{F}_{r})$. In the case when
$\mathbb{F}_{r}$ is a trivial $\mathbb{Z}_{r}$-module we set
$H^{*}(\mathbb{Z}_{r};\mathbb{F}_{r})=\mathbb{F}_{r}[t]\otimes\Lambda(e)$
where $\deg(t)=2$, $\deg(e)=1$, and $\Lambda(\cdot)$ denotes the exterior
algebra.

All the cohomologies we work with are $\mathbb{F}_{r}$ vector spaces and
therefore the Serre spectral sequence induced by the fibration (11) converges
to the cohomology
$H^{*}(\mathrm{E}(\mathbb{Z}_{r})\times_{\mathbb{Z}_{r}}\operatorname{G}(S^{d-1},r);\mathbb{F}_{r})$
as a vector space, that is

$H^{n}(\operatorname{G}(S^{d-1},r)/\mathbb{Z}_{r};\mathbb{F}_{r})\cong
H^{n}(\mathrm{E}(\mathbb{Z}_{r})\times_{\mathbb{Z}_{r}}\operatorname{G}(S^{d-1},r);\mathbb{F}_{r})\cong\bigoplus_{p+q=n}E^{p,q}_{\infty},$

for every $n\in\mathbb{N}$. In turn, since
$\operatorname{G}(S^{d-1},r)/\mathbb{Z}_{r}$ is an open $(r(d-1))$-manifold
and has no cohomology in dimensions $\geq r(d-1)$, we have that
$E^{p,q}_{\infty}=0$ for all $p+q\geq r(d-1)$. For more details on Serre
spectral sequences consult for example [34, Ch. 5–6] or [23, Ch. 3].

The first ingredient in the computation of the $E_{2}$-term of the spectral
sequence (12) is a description of the cohomology of the fiber
$H^{q}(\operatorname{G}(S^{d-1},r);\mathbb{F}_{r})$. For that we use the
results of Farber [19, Thm. 18 and Thm. 19].

######  Proposition 3.3.

Let $d\geq 3$ be an integer, and let $r$ be an odd prime.

  1. (1)

If $d$ is even, then the cohomology ring
$H^{*}(\operatorname{G}(S^{d-1},r);\mathbb{F}_{r})$ of the ordered cyclic
configuration space $\operatorname{G}(S^{d-1},r)$ is generated by the elements

$\alpha,\ \beta_{1},\ \beta_{2},\ \ldots,\ \beta_{r-2},$

of degrees

$\deg(\alpha)=d-1,\ \deg(\beta_{1})=d-2,\ \deg(\beta_{2})=2(d-2),\ \ldots,\
\deg(\beta_{r-2})=(r-2)(d-2),$

subject to the relations

$\alpha^{2}=0,\qquad\qquad\beta_{i}\beta_{j}=\begin{cases}\tfrac{(i+j)!}{i!\,j!}\beta_{i+j},&\text{for
}i+j\leq r-2,\\\ 0,&\text{otherwise},\end{cases}$

for $1\leq i\leq j\leq r-2$. In particular, for every $1\leq k\leq r-2$ we
have that $\beta_{k}=a_{k}\cdot\beta_{1}^{k}$ for some
$a_{k}\in\mathbb{F}_{r}{\setminus}\\{0\\}$.

  2. (2)

If $d$ is odd, then the cohomology ring
$H^{*}(\operatorname{G}(S^{d-1},r);\mathbb{F}_{r})$ of the ordered cyclic
configuration space $\operatorname{G}(S^{d-1},r)$ is generated by the elements

$\gamma,\ \delta_{1},\ \delta_{2},\ \ldots,\ \delta_{\tfrac{r-3}{2}},$

of degrees

$\deg(\gamma)=2(d-2)+1,\ \deg(\delta_{1})=1\cdot 2(d-2),\
\deg(\delta_{2})=2\cdot 2(d-2),\ \ldots,\
\deg(\delta_{\tfrac{r-3}{2}})=\tfrac{r-3}{2}\cdot 2(d-2),$

subject to the relations

$\gamma^{2}=0,\qquad\qquad\delta_{i}\delta_{j}=\begin{cases}\tfrac{(i+j)!}{i!\,j!}\delta_{i+j},&\text{for
}i+j\leq\tfrac{r-3}{2},\\\ 0,&\text{otherwise},\end{cases}$

for $1\leq i\leq j\leq\tfrac{r-3}{2}$. In particular, for every $1\leq
k\leq\tfrac{r-3}{2}$ we have that $\delta_{k}=b_{k}\cdot\delta_{1}^{k}$ for
some $b_{k}\in\mathbb{F}_{r}{\setminus}\\{0\\}$.

Transforming the previous information on the cohomology ring of the ordered
cyclic configuration space $\operatorname{G}(S^{d-1},r)$ into an additive
language we get the following corollary.

######  Corollary 3.4.

Let $d\geq 3$ be an integer, and let $r$ be an odd prime.

  1. (1)

If $d$ is even and $A=\\{\ell(d-2),\ell(d-2)+1:1\leq\ell\leq
r-2\\}\cup\\{0,(r-1)(d-2)+1\\}$, then

$H^{n}(\operatorname{G}(S^{d-1},r);\mathbb{F}_{r})=\begin{cases}\mathbb{F}_{r},&n\in
A,\\\ 0,&\text{\rm otherwise}.\end{cases}$

  2. (2)

If $d$ is odd and
$B=\\{2\ell(d-2),2\ell(d-2)+1:1\leq\ell\leq\tfrac{r-3}{2}\\}\cup\\{0,(r-1)(d-2)+1\\}$,
then

$H^{n}(\operatorname{G}(S^{d-1},r);\mathbb{F}_{r})=\begin{cases}\mathbb{F}_{r},&n\in
B,\\\ 0,&\text{\rm otherwise}.\end{cases}$

From Proposition 3.3 and its consequence Corollary 3.4 we can detect the
action of $\pi_{1}(\mathrm{B}(\mathbb{Z}_{r}))\cong\mathbb{Z}_{r}$ on the
cohomology of the ordered cyclic configuration space and compute the
$E_{2}$-term of the Serre spectral sequence (12); for an illustration of the
$E_{2}$-term in the case of even $d$ see Figure 5.

Figure 5. The $E_{2}$-term of the Serre spectral sequence (12) in the case
when $d\geq 4$ is even

######  Corollary 3.5.

Let $d\geq 3$ be an integer, and let $r$ be an odd prime. The $E_{2}$-term of
the Serre spectral sequence (12) has the following description.

  1. (1)

If $d$ is even and $A=\\{\ell(d-2),\ell(d-2)+1:1\leq\ell\leq
r-2\\}\cup\\{0,(r-1)(d-2)+1\\}$, then

$\qquad E_{2}^{p,q}\cong
H^{p}(\mathbb{Z}_{r};H^{q}(\operatorname{G}(S^{d-1},r);\mathbb{F}_{r}))\cong\\\
H^{p}(\mathbb{Z}_{r};\mathbb{F}_{r})\otimes
H^{q}(\operatorname{G}(S^{d-1},r);\mathbb{F}_{r})\cong\begin{cases}H^{p}(\mathbb{Z}_{r};\mathbb{F}_{r}),&n\in
A,\\\ 0,&\text{\rm otherwise}.\end{cases}\qquad$

  2. (2)

If $d$ is odd and
$B=\\{2\ell(d-2),2\ell(d-2)+1:1\leq\ell\leq\tfrac{r-3}{2}\\}\cup\\{0,(r-1)(d-2)+1\\}$,
then

$\qquad E_{2}^{p,q}\cong
H^{p}(\mathbb{Z}_{r};H^{q}(\operatorname{G}(S^{d-1},r);\mathbb{F}_{r}))\cong\\\
H^{p}(\mathbb{Z}_{r};\mathbb{F}_{r})\otimes
H^{q}(\operatorname{G}(S^{d-1},r);\mathbb{F}_{r})\cong\begin{cases}H^{p}(\mathbb{Z}_{r};\mathbb{F}_{r}),&n\in
B,\\\ 0,&\text{\rm otherwise}.\end{cases}\qquad$

###### Proof.

For the fibration (11), the cohomology group of the fiber
$H^{q}(\operatorname{G}(S^{d-1},r);\mathbb{F}_{r})$ is isomorphic either to
$0$ or to $\mathbb{F}_{r}$. Thus the group
$\pi_{1}(\mathrm{B}(\mathbb{Z}_{r}))\cong\mathbb{Z}_{r}$ can only act
trivially on $H^{q}(\operatorname{G}(S^{d-1},r);\mathbb{F}_{r})$: Indeed, if
$f\colon\mathbb{F}_{r}\longrightarrow\mathbb{F}_{r}$ is a linear map induced
by the action of a generator of $\mathbb{Z}_{r}$ and $f(1)=a\in\mathbb{F}_{r}$
then by Fermat’s little theorem
$1=\operatorname{id}_{\mathbb{F}_{r}}(1)=f^{r}(1)=a^{r}=a$ and so
$f=\operatorname{id}_{\mathbb{F}_{r}}$. Consequently, the cohomology
$H^{q}(\operatorname{G}(S^{d-1},r);\mathbb{F}_{r})$ is a trivial
$\mathbb{Z}_{r}$-module for every $q$, and the statement of the corollary
follows. ∎

######  Remark 3.6.

It is important to point out that the previous corollary also implies that the
differentials of the Serre spectral sequence (12) are not only
$H^{*}(\mathbb{Z}_{r};\mathbb{F}_{r})$-module morphisms, but even more, they
satisfy the Leibniz rule; see [34, Def. 1.6 and Prop. 5.6].

###  3.2. Computing the spectral sequence

In this section, before making explicit computation, we consider the ordered
configuration space $\operatorname{F}(\mathbb{R}^{d-1},r)$ as a subset of the
ordered cyclic configuration space $\operatorname{G}(S^{d-1},r)$. The
$\mathbb{Z}_{r}$-equivariant inclusion
$i\colon\operatorname{F}(\mathbb{R}^{d-1},r)\longrightarrow\operatorname{G}(S^{d-1},r)$
induces the following morphism of Borel construction fibrations:

$\textstyle{\mathrm{E}(\mathbb{Z}_{r})\times_{\mathbb{Z}_{r}}\operatorname{F}(\mathbb{R}^{d-1},r)\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\pi_{F}}$$\scriptstyle{\operatorname{id}\times_{\mathbb{Z}_{r}}i}$$\textstyle{\mathrm{E}(\mathbb{Z}_{r})\times_{\mathbb{Z}_{r}}\operatorname{G}(S^{d-1},r)\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\pi_{G}}$$\textstyle{\mathrm{B}(\mathbb{Z}_{r})\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\operatorname{id}}$$\textstyle{\mathrm{B}(\mathbb{Z}_{r}).}$
(13)

The morphism of fibrations (13) induces a morphism of associated Serre
spectral sequences

$\textstyle{E^{*,*}_{*}(\operatorname{id}\times_{\mathbb{Z}_{r}}i)\colon
E^{*,*}_{*}(\mathrm{E}(\mathbb{Z}_{r})\times_{\mathbb{Z}_{r}}\operatorname{F}(\mathbb{R}^{d-1},r))}$$\textstyle{E^{*,*}_{*}(\mathrm{E}(\mathbb{Z}_{r})\times_{\mathbb{Z}_{r}}\operatorname{G}(S^{d-1},r))\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$
(14)

which is the identity on the zero row of the $E_{2}$-term, that is
$E^{*,0}_{2}(\operatorname{id}\times_{\mathbb{Z}_{r}}i)=\operatorname{id}$.

The Serre spectral sequence
$E^{*,*}_{*}(\mathrm{E}(\mathbb{Z}_{r})\times_{\mathbb{Z}_{r}}\operatorname{F}(\mathbb{R}^{d-1},r))$
has been completely described first by F. Cohen in his seminal paper [15, Sec.
8–10], and much later, from the point of view of the equivariant
Goresky–MacPherson formula, in [11, Thm. 6.1]. For an illustration of the
$E_{2}$-term of this spectral sequence see Figure 6. In particular, we will
use the following facts proved in [15, 11].

######  Proposition 3.7.

Let $d\geq 3$ be an integer, and let $r$ be an odd prime. The Serre spectral
sequence associated to the Borel construction fibration

$\textstyle{\operatorname{F}(\mathbb{R}^{d-1},r)\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\mathrm{E}(\mathbb{Z}_{r})\times_{\mathbb{Z}_{r}}\operatorname{F}(\mathbb{R}^{d-1},r)\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\pi_{F}}$$\textstyle{\mathrm{B}(\mathbb{Z}_{r}),}$
(15)

with the $E_{2}$-term

$E^{p,q}_{2}=\mathcal{H}^{p}(\mathrm{B}(\mathbb{Z}_{r});H^{q}(\operatorname{F}(\mathbb{R}^{d-1},r);\mathbb{F}_{r}))\cong
H^{p}(\mathbb{Z}_{r};H^{q}(\operatorname{F}(\mathbb{R}^{d-1},r);\mathbb{F}_{r}))$

has to following properties:

  1. (1)

For all $p\geq 1$ and $1\leq q\leq(r-1)(d-2)-1$, we have that

$E^{p,q}_{2}(\mathrm{E}(\mathbb{Z}_{r})\times_{\mathbb{Z}_{r}}\operatorname{F}(\mathbb{R}^{d-1},r))=E^{p,q}_{\infty}(\mathrm{E}(\mathbb{Z}_{r})\times_{\mathbb{Z}_{r}}\operatorname{F}(\mathbb{R}^{d-1},r))=0.$

  2. (2)

For all $p\in\mathbb{Z}$ and $q\geq(r-1)(d-2)+1$, we have that

$E^{p,q}_{2}(\mathrm{E}(\mathbb{Z}_{r})\times_{\mathbb{Z}_{r}}\operatorname{F}(\mathbb{R}^{d-1},r))=E^{p,q}_{\infty}(\mathrm{E}(\mathbb{Z}_{r})\times_{\mathbb{Z}_{r}}\operatorname{F}(\mathbb{R}^{d-1},r))=0.$

  3. (3)

For all $p\geq 0$ and $2\leq s\leq(r-1)(d-2)$, the differential

$\partial_{s}\colon
E_{s}^{p,s-1}(\mathrm{E}(\mathbb{Z}_{r})\times_{\mathbb{Z}_{r}}\operatorname{F}(\mathbb{R}^{d-1},r))\longrightarrow
E_{s}^{p+s,0}(\mathrm{E}(\mathbb{Z}_{r})\times_{\mathbb{Z}_{r}}\operatorname{F}(\mathbb{R}^{d},r))$

vanishes. Consequently, for $0\leq p\leq(r-1)(d-2)$, we have that

$H^{p}(\mathbb{Z}_{r};\mathbb{F}_{r})=E^{p,0}_{2}(\mathrm{E}(\mathbb{Z}_{r})\times_{\mathbb{Z}_{r}}\operatorname{F}(\mathbb{R}^{d-1},r))=E_{\infty}^{p,0}(\mathrm{E}(\mathbb{Z}_{r})\times_{\mathbb{Z}_{r}}\operatorname{F}(\mathbb{R}^{d-1},r)).$

  4. (4)

The only non-zero differential of the spectral sequence is

$\partial_{(d-2)(r-1)+1}\colon\\\
E_{(d-2)(r-1)+1}^{p,(d-2)(r-1)}(\mathrm{E}(\mathbb{Z}_{r})\times_{\mathbb{Z}_{r}}\operatorname{F}(\mathbb{R}^{d-1},r))\longrightarrow
E_{(d-2)(r-1)+1}^{p+(d-2)(r-1)+1,0}(\mathrm{E}(\mathbb{Z}_{r})\times_{\mathbb{Z}_{r}}\operatorname{F}(\mathbb{R}^{d-1},r)),\qquad$

and is an isomorphism for all $p\geq 0$. Consequently, for all
$p\geq(r-1)(d-2)+1$, one has

$E_{(d-2)(r-1)+2}^{p,0}(\mathrm{E}(\mathbb{Z}_{r})\times_{\mathbb{Z}_{r}}\operatorname{F}(\mathbb{R}^{d-1},r))\cong
E_{\infty}^{p,0}(\mathrm{E}(\mathbb{Z}_{r})\times_{\mathbb{Z}_{r}}\operatorname{F}(\mathbb{R}^{d-1},r))=0.$

Figure 6. The $E_{2}=E_{(r-1)(d-2)}$-term of the Serre spectral sequence of
the fibration (15)

From the existence of the morphism
$E^{*,*}_{*}(\operatorname{id}\times_{\mathbb{Z}_{r}}i)$ of the Serre spectral
sequences (14), the fact that
$E^{*,0}_{2}(\operatorname{id}\times_{\mathbb{Z}_{r}}i)$ is the identity, and
the following equality from Proposition 3.7:

$E_{\infty}^{p,0}(\mathrm{E}(\mathbb{Z}_{r})\times_{\mathbb{Z}_{r}}\operatorname{F}(\mathbb{R}^{d-1},r))=E^{p,0}_{2}(\mathrm{E}(\mathbb{Z}_{r})\times_{\mathbb{Z}_{r}}\operatorname{F}(\mathbb{R}^{d-1},r))=H^{p}(\mathbb{Z}_{r};\mathbb{F}_{r}),$

which holds for $0\leq p\leq(r-1)(d-2)$, we deduce an important property of
the Serre spectral sequence
$E_{*}^{*,*}(\mathrm{E}(\mathbb{Z}_{r})\times_{\mathbb{Z}_{r}}\operatorname{G}(S^{d-1},r))$
stated in the next corollary.

######  Corollary 3.8.

Let $d\geq 3$ be an integer, and let $r$ be an odd prime. For $0\leq
p\leq(r-1)(d-2)$, one has

$E_{\infty}^{p,0}(\mathrm{E}(\mathbb{Z}_{r})\times_{\mathbb{Z}_{r}}\operatorname{G}(S^{d-1},r))\cong
E^{p,0}_{2}(\mathrm{E}(\mathbb{Z}_{r})\times_{\mathbb{Z}_{r}}\operatorname{G}(S^{d-1},r))\cong
H^{p}(\mathbb{Z}_{r};\mathbb{F}_{r}).$

In particular, all the differentials

$\partial_{s}\colon
E_{s}^{p,s-1}(\mathrm{E}(\mathbb{Z}_{r})\times_{\mathbb{Z}_{r}}\operatorname{G}(S^{d-1},r))\longrightarrow
E_{s}^{p+s,0}(\mathrm{E}(\mathbb{Z}_{r})\times_{\mathbb{Z}_{r}}\operatorname{G}(S^{d-1},r))$

vanish for $p\geq 0$ and $2\leq s\leq(r-1)(d-2)$.

Combining the previous fact about the Serre spectral sequence
$E_{*}^{*,*}(\mathrm{E}(\mathbb{Z}_{r})\times_{\mathbb{Z}_{r}}\operatorname{G}(S^{d-1},r))$
with the observation that
$E_{\infty}^{p,q}(\mathrm{E}(\mathbb{Z}_{r})\times_{\mathbb{Z}_{r}}\operatorname{G}(S^{d-1},r))=0$
for all $p+q\geq r(d-1)$, we have that for some $s\geq(r-1)(d-2)+1$ the
differential

$\partial_{s}\colon
E_{s}^{p,s-1}(\mathrm{E}(\mathbb{Z}_{r})\times_{\mathbb{Z}_{r}}\operatorname{G}(S^{d-1},r))\longrightarrow
E_{s}^{p+s,0}(\mathrm{E}(\mathbb{Z}_{r})\times_{\mathbb{Z}_{r}}\operatorname{G}(S^{d-1},r))$

does not vanish. In particular,

$E_{\infty}^{p,0}(\mathrm{E}(\mathbb{Z}_{r})\times_{\mathbb{Z}_{r}}\operatorname{G}(S^{d-1},r))=0$

for all $p\geq r(d-1)$.

Now the properties of the Serre spectral sequence
$E_{*}^{*,*}(\mathrm{E}(\mathbb{Z}_{r})\times_{\mathbb{Z}_{r}}\operatorname{G}(S^{d-1},r))$,
derived so far, will be used to completely compute it. The computation
proceeds in two separate steps, depending on the parity of $d$.

####  3.2.1. $d$ is an even integer, $d\geq 4$

The Serre spectral sequence
$E^{*,*}_{*}(\mathrm{E}(\mathbb{Z}_{r})\times_{\mathbb{Z}_{r}}\operatorname{G}(S^{d-1},r))$
has a multiplicative structure, and differentials satisfy the Leibniz rule.
Thus, to compute the spectral sequence completely it suffices to determine
values of the differentials only on the generators of the cohomology ring of
the fiber. In this particular case, we need to evaluate the differentials on
the elements $1\otimes\alpha,1\otimes\beta_{1},\ldots,1\otimes\beta_{r-1}$. It
is important to notice that

$(1\otimes\beta_{1})^{i}=1\otimes\beta_{1}^{i}=a_{i}\cdot(1\otimes\beta_{i})$

for $1\leq i\leq r-1$ and some $a_{i}\in\mathbb{F}_{r}{\setminus}\\{0\\}$.

Figure 7. For $d\geq 4$ even the $E_{2}$-term with differentials of the Serre
spectral sequence (12)

Consider the differential $\partial_{2}$. The degrees of the generators
$\beta_{1},\ldots,\beta_{r-1}$ are $(d-2),\ldots,(r-2)(d-2)$, respectively.
Since $H^{i}(\operatorname{G}(S^{d-1},r);\mathbb{F}_{r})=0$ vanishes for
$i\in\\{(d-2)-1,\ldots,(r-2)(d-2)-1\\}$, one has

$\partial_{2}(1\otimes\beta_{1})=\partial_{2}(1\otimes\beta_{2})=\cdots=\partial_{2}(1\otimes\beta_{r-1})=0.$
(16)

Thus, to completely determine the second differential we need to find the
value

$\partial_{2}(1\otimes\alpha)\in
E^{2,d-2}_{2}(\mathrm{E}(\mathbb{Z}_{r})\times_{\mathbb{Z}_{r}}\operatorname{G}(S^{d-1},r)).$

Let us assume that $\partial_{2}(1\otimes\alpha)=0$. Then, due to the
multiplicative property of the Serre spectral sequence, we have that the
second differential $\partial_{2}$ vanishes, implying that

$E^{*,*}_{2}(\mathrm{E}(\mathbb{Z}_{r})\times_{\mathbb{Z}_{r}}\operatorname{G}(S^{d-1},r))\cong
E^{*,*}_{3}(\mathrm{E}(\mathbb{Z}_{r})\times_{\mathbb{Z}_{r}}\operatorname{G}(S^{d-1},r)).$

The vanishing of cohomology groups
$H^{*}(\operatorname{G}(S^{d-1},r);\mathbb{F}_{r})$ in appropriate dimensions
implies that the next possible non-zero differential is $\partial_{d-1}$. More
precisely, $\partial_{d-1}(1\otimes\beta_{1})$ might be non-zero while we know
that $\partial_{d-1}(1\otimes\alpha)=0$. Since $d-1\leq(r-1)(d-2)$, using
Corollary 3.8, we get that $\partial_{d-1}(1\otimes\beta_{1})=0$, and
consequently the differential $\partial_{d-1}$ vanishes. For the next
differential we know that $\partial_{d}(1\otimes\beta_{1})=0$, and since
$d\leq(r-1)(d-2)$ from Corollary 3.8, we have that
$\partial_{d}(1\otimes\alpha)=0$. Hence, the differential $\partial_{d}$ also
vanishes. In particular this means that

$E^{*,*}_{2}(\mathrm{E}(\mathbb{Z}_{r})\times_{\mathbb{Z}_{r}}\operatorname{G}(S^{d-1},r))\cong\
\cdots\ \cong
E^{*,*}_{d+1}(\mathrm{E}(\mathbb{Z}_{r})\times_{\mathbb{Z}_{r}}\operatorname{G}(S^{d-1},r)).$

Since for $s\geq d+1$ all differentials $\partial_{s}(1\otimes\beta_{1})$ and
$\partial_{s}(1\otimes\alpha)$ are zero, the multiplicative property of the
Serre spectral sequence implies that all the differentials $\partial_{s}$
vanish. Thus

$E^{*,*}_{2}(\mathrm{E}(\mathbb{Z}_{r})\times_{\mathbb{Z}_{r}}\operatorname{G}(S^{d-1},r))\cong
E^{*,*}_{\infty}(\mathrm{E}(\mathbb{Z}_{r})\times_{\mathbb{Z}_{r}}\operatorname{G}(S^{d-1},r)).$

This is a contradiction to the fact, observed after Corollary 3.8, that at
least one of the differentials

$\partial_{s}\colon
E_{s}^{p,s-1}(\mathrm{E}(\mathbb{Z}_{r})\times_{\mathbb{Z}_{r}}\operatorname{G}(S^{d-1},r))\longrightarrow
E_{s}^{p+s,0}(\mathrm{E}(\mathbb{Z}_{r})\times_{\mathbb{Z}_{r}}\operatorname{G}(S^{d-1},r)),$

for $s\geq(r-1)(d-2)+1$, does not vanish. Therefore,
$\partial_{2}(1\otimes\alpha)\neq 0$.

Set $\partial_{2}(1\otimes\alpha)=a\cdot(t\otimes\beta_{1})$ where
$a\in\mathbb{F}_{r}{\setminus}\\{0\\}$; for illustration see Figure 7. The
multiplicative property of the Serre spectral sequence, combined with (16),
yields that

$\partial_{2}(1\otimes\alpha\beta_{i})=\partial_{2}((1\otimes\alpha)(1\otimes\beta_{i}))=\partial_{2}(1\otimes\alpha)(1\otimes\beta_{i})+(-1)^{d-1}(1\otimes\alpha)\partial_{2}(1\otimes\beta_{i})=\\\
\partial_{2}(1\otimes\alpha)(1\otimes\beta_{i})=a\cdot(t\otimes\beta_{1})(1\otimes\beta_{i})=\begin{cases}\big{(}a\tfrac{(i+1)!}{i!1!}\big{)}\cdot(t\otimes\beta_{i+1}),&1\leq
i\leq r-3,\\\ 0,&i=r-2.\end{cases}$

[ht!]

Figure 8. For $d\geq 4$ even the $E_{3}$-term of the Serre spectral sequence
(12)

Thus the $E_{3}$-term, illustrated in Figure 8, is given by

$E^{p,q}_{3}(\mathrm{E}(\mathbb{Z}_{r})\times_{\mathbb{Z}_{r}}\operatorname{G}(S^{d-1},r))\cong\\\
\begin{cases}H^{p}(\mathbb{Z}_{r};\mathbb{F}_{r}),&q\in\\{0,(r-1)(d-2)+1\\},\\\
\mathbb{F}_{r},&p\in\\{0,1\\}\text{ and
}q\in\\{(d-2),2(d-2),\ldots,(r-2)(d-2)\\},\\\ 0,&\text{otherwise}.\end{cases}$

Moreover the multiplicative property of the Serre spectral sequence implies
that all the differentials $\partial_{3},\ldots,\partial_{(r-1)(d-2)+1}$
vanish. Therefore

$E^{*,*}_{3}(\mathrm{E}(\mathbb{Z}_{r})\times_{\mathbb{Z}_{r}}\operatorname{G}(S^{d-1},r))\cong\
\cdots\ \cong
E^{*,*}_{(r-1)(d-2)+2}(\mathrm{E}(\mathbb{Z}_{r})\times_{\mathbb{Z}_{r}}\operatorname{G}(S^{d-1},r)).$

The differential $\partial_{(r-1)(d-2)+2}$ is the only remaining differential
that can be non-zero. Since

$E^{p,0}_{2}(\mathrm{E}(\mathbb{Z}_{r})\times_{\mathbb{Z}_{r}}\operatorname{G}(S^{d-1},r))\cong\
\cdots\ \cong
E^{p,0}_{(r-1)(d-2)+2}(\mathrm{E}(\mathbb{Z}_{r})\times_{\mathbb{Z}_{r}}\operatorname{G}(S^{d-1},r)),$

and we know that

$E_{2}^{p,0}(\mathrm{E}(\mathbb{Z}_{r})\times_{\mathbb{Z}_{r}}\operatorname{G}(S^{d-1},r))\neq
0=E_{\infty}^{p,0}(\mathrm{E}(\mathbb{Z}_{r})\times_{\mathbb{Z}_{r}}\operatorname{G}(S^{d-1},r))$

for all $p\geq r(d-1)$, we conclude that $\partial_{(r-1)(d-2)+2}\neq 0$. The
multiplicative property of the Serre spectral sequence again yields that
$\partial_{(r-1)(d-2)+2}$ is completely determined by its image at
$1\otimes\alpha\beta_{r-2}$, which has to be non-zero. Set

$\partial_{(r-1)(d-2)+2}(1\otimes\alpha\beta_{r-2})=b\cdot\big{(}t^{\tfrac{(r-1)(d-2)+2}{2}}\otimes
1\big{)}$

for some $b\in\mathbb{F}_{r}{\setminus}\\{0\\}$, as illustrated in Figure 9.
Consequently, the differentials

$\partial_{(r-1)(d-2)+2}\colon
E_{s}^{p,(r-1)(d-2)+1}(\mathrm{E}(\mathbb{Z}_{r})\times_{\mathbb{Z}_{r}}\operatorname{G}(S^{d-1},r))\longrightarrow
E_{s}^{p+(r-1)(d-2)+2,0}(\mathrm{E}(\mathbb{Z}_{r})\times_{\mathbb{Z}_{r}}\operatorname{G}(S^{d-1},r))$

are isomorphisms for all $p\geq 0$.

Figure 9. For $d\geq 4$ even the differential $\partial_{(r-1)(d-2)+2}$ of the
Serre spectral sequence (12)

Hence, the $E_{\infty}$-term is given by

$E^{p,q}_{\infty}(\mathrm{E}(\mathbb{Z}_{r})\times_{\mathbb{Z}_{r}}\operatorname{G}(S^{d-1},r))\cong\\\
\begin{cases}H^{p}(\mathbb{Z}_{r};\mathbb{F}_{r}),&q=0\text{ and }0\leq
p\leq(r-1)(d-2)+1,\\\ \mathbb{F}_{r},&p\in\\{0,1\\}\text{ and
}q\in\\{(d-2),2(d-2),\ldots,(r-2)(d-2)\\},\\\ 0,&\text{otherwise},\end{cases}$
(17)

and illustrated in Figure 10. Moreover, we have obtained that the map
$\pi_{G}^{*}$ induced by the projection map of the fibration (11) in
cohomology

$\pi_{G}^{*}\colon H^{p}(\mathbb{Z}_{r};\mathbb{F}_{r})\longrightarrow
H^{p}(\mathrm{E}(\mathbb{Z}_{r})\times_{\mathbb{Z}_{r}}\operatorname{G}(S^{d-1},r);\mathbb{F}_{r})$
(18)

is an injection for $0\leq p\leq(r-1)(d-2)+1$.

Figure 10. For $d\geq 4$ even the $E_{\infty}$-term of the Serre spectral
sequence (12)

####  3.2.2. $d$ is an odd integer, $d\geq 3$

The Serre spectral sequence
$E^{*,*}_{*}(\mathrm{E}(\mathbb{Z}_{r})\times_{\mathbb{Z}_{r}}\operatorname{G}(S^{d-1},r))$
has a multiplicative structure. Hence, to compute it completely, we will
determine values of the differentials on the generators of the cohomology ring
of the fiber. In this particular case, we need to evaluate the differentials
on the elements $\gamma,\delta_{1},\ldots,\delta_{\tfrac{r-3}{2}}$. Observe
that

$(1\otimes\delta_{1})^{i}=1\otimes\delta_{1}^{i}=a_{i}\cdot(1\otimes\delta_{i})$

for $1\leq i\leq\tfrac{r-3}{2}$ and some
$a_{i}\in\mathbb{F}_{r}{\setminus}\\{0\\}$.

Similarly to Section 3.2.1, we consider first the values of the differential
$\partial_{2}$ on the generators. Since the generator $\delta_{1}$ is of
degree $2$ and $H^{1}(\operatorname{G}(S^{d-1},r);\mathbb{F}_{r})=0$ we have
that $\partial_{2}(1\otimes\delta_{1})=0$. Further, the Leibniz rule implies
that

$\partial_{2}(1\otimes\delta_{i})=a_{i}\cdot\partial_{2}((1\otimes\delta_{1})^{i})=(ia_{i})\cdot\partial_{2}(1\otimes\delta_{1})=0$

for $1\leq i\leq\tfrac{r-3}{2}$ and some
$a_{i}\in\mathbb{F}_{r}{\setminus}\\{0\\}$. Thus, in order to determine the
second differential $\partial_{2}$, we need to determine

$\partial_{2}(1\otimes\gamma)\in
E^{2,2(d-2)}_{2}(\mathrm{E}(\mathbb{Z}_{r})\times_{\mathbb{Z}_{r}}\operatorname{G}(S^{d-1},r)).$

Assume that $\partial_{2}(1\otimes\gamma)=0$. Then the multiplicative property
of the Serre spectral sequence implies that the second differential
$\partial_{2}$ gives

$E^{*,*}_{2}(\mathrm{E}(\mathbb{Z}_{r})\times_{\mathbb{Z}_{r}}\operatorname{G}(S^{d-1},r))\cong
E^{*,*}_{3}(\mathrm{E}(\mathbb{Z}_{r})\times_{\mathbb{Z}_{r}}\operatorname{G}(S^{d-1},r)).$

The vanishing of cohomology groups
$H^{*}(\operatorname{G}(S^{d-1},r);\mathbb{F}_{r})$ in appropriate dimensions
implies that the next possible non-zero differential is $\partial_{2(d-2)+1}$.
More precisely, $\partial_{2(d-2)+1}(1\otimes\delta_{1})$ might be non-zero,
while we know that $\partial_{2(d-2)+1}(1\otimes\gamma)=0$. Since
$2(d-2)+1\leq(r-1)(d-2)$ from Corollary 3.8, we have that
$\partial_{2(d-2)+1}(1\otimes\delta_{1})=0$. Consequently, the differential
$\partial_{2(d-2)+1}$ vanishes. About the next differential we know that
$\partial_{2(d-2)+2}(1\otimes\delta_{1})=0$. Since $2(d-2)+2\leq(r-1)(d-2)$,
again Corollary 3.8 implies that $\partial_{2(d-2)+2}(1\otimes\gamma)=0$.
Therefore, the differential $\partial_{2(d-2)+2}$ also vanishes. This means
that

$E^{*,*}_{2}(\mathrm{E}(\mathbb{Z}_{r})\times_{\mathbb{Z}_{r}}\operatorname{G}(S^{d-1},r))\cong\
\cdots\ \cong
E^{*,*}_{2(d-2)+3}(\mathrm{E}(\mathbb{Z}_{r})\times_{\mathbb{Z}_{r}}\operatorname{G}(S^{d-1},r)).$

For $s\geq 2(d-2)+3$, all differentials $\partial_{s}(1\otimes\delta_{1})$ and
$\partial_{s}(1\otimes\gamma)$ are zero. Hence, the multiplicative property of
the Serre spectral sequence implies that, for $s\geq 2(d-2)+3$, all the
differentials $\partial_{s}$ vanish. Thus

$E^{*,*}_{2}(\mathrm{E}(\mathbb{Z}_{r})\times_{\mathbb{Z}_{r}}\operatorname{G}(S^{d-1},r))\cong
E^{*,*}_{\infty}(\mathrm{E}(\mathbb{Z}_{r})\times_{\mathbb{Z}_{r}}\operatorname{G}(S^{d-1},r)),$

which is a contradiction to the fact, observed after Corollary 3.8, that at
least one of the differentials

$\partial_{s}\colon
E_{s}^{p,s-1}(\mathrm{E}(\mathbb{Z}_{r})\times_{\mathbb{Z}_{r}}\operatorname{G}(S^{d-1},r))\longrightarrow
E_{s}^{p+s,0}(\mathrm{E}(\mathbb{Z}_{r})\times_{\mathbb{Z}_{r}}\operatorname{G}(S^{d-1},r)),$

for $s\geq(r-1)(d-2)+1$, does not vanish. Hence,
$\partial_{2}(1\otimes\gamma)\neq 0$.

Now set $\partial_{2}(1\otimes\gamma)=a\cdot(t\otimes\delta_{1})$ where
$a\in\mathbb{F}_{r}{\setminus}\\{0\\}$. The multiplicative property yields

$\partial_{2}(1\otimes\gamma\delta_{i})=\partial_{2}((1\otimes\gamma)(1\otimes\delta_{i}))=\partial_{2}(1\otimes\gamma)(1\otimes\delta_{i})+(-1)^{d-1}(1\otimes\gamma)\partial_{2}(1\otimes\delta_{i})=\\\
\partial_{2}(1\otimes\gamma)(1\otimes\delta_{i})=a\cdot(t\otimes\gamma_{1})(1\otimes\gamma_{i})=\begin{cases}\big{(}a\tfrac{(i+1)!}{i!1!}\big{)}\cdot(t\otimes\gamma_{i+1}),&1\leq
i\leq\tfrac{r-3}{2},\\\ 0,&i=r-2.\end{cases}$

Consequently, the $E_{3}$-term is

$E^{p,q}_{3}(\mathrm{E}(\mathbb{Z}_{r})\times_{\mathbb{Z}_{r}}\operatorname{G}(S^{d-1},r))\cong\\\
\begin{cases}H^{p}(\mathbb{Z}_{r};\mathbb{F}_{r}),&q\in\\{0,(r-1)(d-2)+1\\},\\\
\mathbb{F}_{r},&p\in\\{0,1\\}\text{ and
}q\in\\{2(d-2),4(d-2),\ldots,\tfrac{(r-3)}{2}2(d-2)\\},\\\
0,&\text{otherwise}.\end{cases}$

Further, the multiplicative property implies that all the differentials
$\partial_{3},\ldots,\partial_{(r-1)(d-2)+1}$ vanish. Thus,

$E^{*,*}_{3}(\mathrm{E}(\mathbb{Z}_{r})\times_{\mathbb{Z}_{r}}\operatorname{G}(S^{d-1},r))\cong\
\cdots\ \cong
E^{*,*}_{(r-1)(d-2)+2}(\mathrm{E}(\mathbb{Z}_{r})\times_{\mathbb{Z}_{r}}\operatorname{G}(S^{d-1},r)).$

The differential $\partial_{(r-1)(d-2)+2}$ is the last remaining differential
that can be non-zero. Since for all $p$

$E^{p,0}_{2}(\mathrm{E}(\mathbb{Z}_{r})\times_{\mathbb{Z}_{r}}\operatorname{G}(S^{d-1},r))\cong\
\cdots\ \cong
E^{p,0}_{(r-1)(d-2)+2}(\mathrm{E}(\mathbb{Z}_{r})\times_{\mathbb{Z}_{r}}\operatorname{G}(S^{d-1},r)),$

and we know that

$E_{2}^{p,0}(\mathrm{E}(\mathbb{Z}_{r})\times_{\mathbb{Z}_{r}}\operatorname{G}(S^{d-1},r))\neq
0=E_{\infty}^{p,0}(\mathrm{E}(\mathbb{Z}_{r})\times_{\mathbb{Z}_{r}}\operatorname{G}(S^{d-1},r))$

for $p\geq r(d-1)$, we have that indeed $\partial_{(r-1)(d-2)+2}\neq 0$. The
multiplicative property again yields that $\partial_{(r-1)(d-2)+2}$ is
determined by its image at $1\otimes\gamma\delta_{\tfrac{r-3}{2}}$ and has to
be non-zero. Set

$\partial_{(r-1)(d-2)+2}(1\otimes\gamma\delta_{\tfrac{r-3}{2}})=b\cdot\big{(}t^{\tfrac{(r-1)(d-2)+2}{2}}\otimes
1\big{)}$

for some $b\in\mathbb{F}_{r}{\setminus}\\{0\\}$. Thus, the homomorphisms

$\partial_{(r-1)(d-2)+2}\colon
E_{s}^{p,(r-1)(d-2)+1}(\mathrm{E}(\mathbb{Z}_{r})\times_{\mathbb{Z}_{r}}\operatorname{G}(S^{d-1},r))\longrightarrow
E_{s}^{p+(r-1)(d-2)+2,0}(\mathrm{E}(\mathbb{Z}_{r})\times_{\mathbb{Z}_{r}}\operatorname{G}(S^{d-1},r))$

are isomorphism for all $p\geq 0$. Hence, the $E_{\infty}$-term is given by

$E^{p,q}_{\infty}(\mathrm{E}(\mathbb{Z}_{r})\times_{\mathbb{Z}_{r}}\operatorname{G}(S^{d-1},r))\cong\\\
\begin{cases}H^{p}(\mathbb{Z}_{r};\mathbb{F}_{r}),&q=0\text{ and }0\leq
p\leq(r-1)(d-2)+1,\\\ \mathbb{F}_{r},&p\in\\{0,1\\}\text{ and
}q\in\\{2(d-2),4(d-2),\ldots,\tfrac{(r-3)}{2}2(d-2)\\},\\\
0,&\text{otherwise},\end{cases}$ (19)

Furthermore, we have obtained again that the map $\pi_{G}^{*}$, induced by the
projection map of the fibration (11) in cohomology

$\pi_{G}^{*}\colon H^{p}(\mathbb{Z}_{r};\mathbb{F}_{r})\longrightarrow
H^{p}(\mathrm{E}(\mathbb{Z}_{r})\times_{\mathbb{Z}_{r}}\operatorname{G}(S^{d-1},r);\mathbb{F}_{r}),$
(20)

is an injection for $0\leq p\leq(r-1)(d-2)+1$.

###  3.3. Proof of Theorem 3.1

Recall that for every $n\in\mathbb{N}$ we have isomorphisms of vector spaces

$H^{n}(\operatorname{G}(S^{d-1},r)/\mathbb{Z}_{r};\mathbb{F}_{r})\cong
H^{n}(\mathrm{E}(\mathbb{Z}_{r})\times_{\mathbb{Z}_{r}}\operatorname{G}(S^{d-1},r);\mathbb{F}_{r})\cong\bigoplus_{p+q=n}E^{p,q}_{\infty}.$

Since $H^{p}(\mathbb{Z}_{r};\mathbb{F}_{r})=\mathbb{F}_{r}$ for all $p$, the
relations (17) for the case of $d\geq 4$ even, and (19) for the case of $d\geq
3$ odd, imply the statement of the theorem.∎

###  3.4. Proof of Theorem 3.2

In order to use the results of Section 3.2 for a proof of an estimate on the
Lusternik–Schnirelmann category of the unlabeled cyclic configuration space
$\operatorname{G}(S^{d-1},r)/\mathbb{Z}_{r}$, we first recall some basic
notions and results concerning the Lusternik–Schnirelmann category.

The _Lusternik–Schnirelmann category_ of a topological space $X$, denoted by
$\operatorname{cat}(X)$, is the smallest integer $k$ for which the space $X$
can be covered by $k+1$ open subsets $U_{1},U_{2},\ldots,U_{k+1}$ with the
property that all inclusions $U_{i}\longrightarrow X$ are nullhomotopic. Some
key properties of the Lusternik–Schnirelmann category that we will use are
given in the next lemma, see, e.g., [16].

######  Lemma 3.9.

  1. (1)

If $X$ is homotopy equivalent to $Y$, then
$\operatorname{cat}(X)=\operatorname{cat}(Y)$.

  2. (2)

If $p\colon X\longrightarrow Y$ is a covering, then
$\operatorname{cat}(X)\leq\operatorname{cat}(Y)$.

  3. (3)

If $X$ is a $(k-1)$-connected $CW$-complex, then
$\operatorname{cat}(X)\leq\tfrac{1}{k}\dim(X)$.

Let $X$ be a topological space, and let $R$ be a commutative ring with unit.
We need the notion of the _category weight_ of an element $u\in H^{*}(X;R)$.
Originally introduced by Fadell and Husseini [17], we use the homotopy
invariant version of this notion due to Rudyak [37] and Strom [39]. See [16,
Sec. 2.7, p. 62; Sec. 8.3, p. 240] for details.

In the next lemma we list properties of the category weight that we will use
in the proof of Theorem 3.2. For more details on this lemma consult for
example [16, Prop. 8.22, p. 242–243, p. 259] and [36, Prop 2.2(3)].

######  Lemma 3.10.

Let $R$ be a commutative ring with unit.

  1. (1)

If $0\neq u\in H^{\ell}(X;R)$, then
$\operatorname{wgt}(u)\leq\operatorname{cat}(X)$.

  2. (2)

Let $f:X\longrightarrow Y$ be a continuous map, and let $u\in H^{\ell}(Y;R)$.
If $0\neq f^{*}(u)\in H^{\ell}(X;R)$, then
$\operatorname{wgt}(u)\leq\operatorname{wgt}(f^{*}(u))$.

  3. (3)

If $G$ is a finite group and $0\neq u\in H^{\ell}(\mathrm{B}G;R)$, then
$\ell=\operatorname{wgt}(u)$.

In the following we combine previously established results to give a proof of
Theorem 3.2. Recall that the group $\mathbb{Z}_{r}$ acts freely on the cyclic
configuration space $\operatorname{G}(S^{d-1},r)$ and consider the following
diagram, which commutes up to a homotopy

$\textstyle{\mathrm{E}(\mathbb{Z}_{r})\times_{\mathbb{Z}_{r}}\operatorname{G}(S^{d-1},r)\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{p}$$\scriptstyle{\pi_{G}}$$\textstyle{\operatorname{G}(S^{d-1},r)/\mathbb{Z}_{r}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{c}$$\textstyle{\mathrm{B}(\mathbb{Z}_{r}),}$
(21)

where the map $p$ is induced by the projection on the second factor
$\mathrm{E}(\mathbb{Z}_{r})\times\operatorname{G}(S^{d-1},r)\longrightarrow\operatorname{G}(S^{d-1},r)$
and is a homotopy equivalence, the map $\pi_{G}$ is the projection in the
Borel construction fibration (11) induced by the projection on the first
factor
$\mathrm{E}(\mathbb{Z}_{r})\times\operatorname{G}(S^{d-1},r)\longrightarrow\mathrm{E}(\mathbb{Z}_{r})$,
and $c$ is a classifying map associated to the free $\mathbb{Z}_{r}$ action on
$\operatorname{G}(S^{d-1},r)$. Uniqueness of the classifying map up to a
homotopy implies that the diagram (21) commutes up to a homotopy; for
background see for example [1, Sec. II.1].

Applying the cohomology functor $H^{*}(\,\cdot\,;\mathbb{F}_{r})$ to the
diagram (21), we get the following diagram of abelian groups that commutes up
to an isomorphism

$\textstyle{H^{*}(\mathrm{E}(\mathbb{Z}_{r})\times_{\mathbb{Z}_{r}}\operatorname{G}(S^{d-1},r);\mathbb{F}_{r})}$$\textstyle{\ignorespaces\ignorespaces\ignorespaces\ignorespaces
H^{*}(\operatorname{G}(S^{d-1},r)/\mathbb{Z}_{r})}$$\scriptstyle{p^{*}}$$\textstyle{\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces
H^{*}(\mathrm{B}(\mathbb{Z}_{r});\mathbb{F}_{r})\cong
H^{*}(\mathbb{Z}_{r};\mathbb{F}_{r}).}$$\scriptstyle{c^{*}}$$\scriptstyle{\pi_{G}^{*}}$
(22)

Let

$u\coloneqq t^{\tfrac{(r-1)(d-2)}{2}}e\in
H^{(r-1)(d-2)+1}(\mathbb{Z}_{r};\mathbb{F}_{r}).$

Then, according to (18) and (20), we have that $\pi_{G}^{*}(u)\neq 0$.
Consequently, from diagram (22), we get $c^{*}(u)\neq 0$. Now Lemma 3.10
yields that

$\operatorname{wgt}(u)=(r-1)(d-2)+1,\qquad\operatorname{wgt}(u)\leq\operatorname{wgt}(c^{*}(u)),\qquad\operatorname{wgt}(c^{*}(u))\leq\operatorname{cat}(\operatorname{G}(S^{d-1},r)/\mathbb{Z}_{r}).$

Hence,

$\operatorname{cat}(\operatorname{G}(S^{d-1},r)/\mathbb{Z}_{r})\geq(r-1)(d-2)+1,$

and the proof of Theorem 3.2 is complete.∎

##  4\. Proofs of the main results

According to Section 2.4, the number of $\mathbb{Z}_{r}$-orbits of the
critical points of the length function on the cyclic configuration space
$\operatorname{G}(M,r)$ is bounded below by the category of the space
$\operatorname{G}(S^{d-1},r)/\mathbb{Z}_{r}$ (the Lusternik–Schnirelmann
theory) or, in general position, by the sum of Betti numbers of
$\operatorname{G}(S^{d-1},r)/\mathbb{Z}_{r}$ (the Morse theory). Thus item (A)
of Theorem Main Theorem follows from Theorem 3.2, and item (B) follows from
Theorem 3.1.

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