Topology of cyclic configuration spaces and periodic trajectories of multi-dimensional billiards

Michael Farber ^{∗} and Serge Tabachnikov^{∗∗} School of Mathematical Sciences, Tel Aviv University Ramat Aviv, Tel Aviv 69978, Israel and Department of Mathematics, Penn State University University Park, PA 16802, USA e-mail: farber@math.tau.ac.il, tabachni@math.psu.edu

July 17, 2000

Abstract

We give lower bounds on the number of periodic trajectories in strictly convex smooth billiards in R^{m+1} for m ≥ 3. For plane billiards (when m = 1) such bounds were obtained by G. Birkhoff in the 1920’s. Our proof is based on topological methods of calculus of variations – equivariant Morse and Lusternik-Schnirelman theories. We compute the equivariant cohomology ring of the cyclic configuration space of the sphere S^{m}, i.e., the space of n-tuples of points (x_{1}, …, x_{n}), where x_{i} ∈ S^{m} and x_{i} ≠ x_{i+1} for i = 1, …, n.

Keywords: mathematical billiards, Morse and Lusternik-Schnirelman theories, cyclic configuration space, equivariant cohomology

1 Introduction

The billiard dynamical system describes the free motion of a mass-point in a domain in Euclidean space with a reflecting boundary: a point moves along a straight line with unit speed until it hits the boundary, at the impact point the normal component of the velocity instantaneously changes sign while the tangential component remains the same, and the rectilinear motion continues with a new unit velocity. We refer to [16],

[19] for surveys of mathematical billiards.

∗Partially supported by a grant from the Israel Academy of Sciences and Humanities and by the Herman Minkowski Center for Geometry

∗∗Partially supported by an NSF grant.

We address the following problem: how many periodic billiard trajectories are there in a smooth strictly convex domain in R^{m+1}?

For plane billiards (m = 1) this problem was studied by G. Birkhoff in [4]. A closed billiard orbit of a plane billiard is an inscribed plane n-gon whose consecutive sides make equal angles with a closed convex plane curve X. Such a periodic orbit has a rotation number 0 < r < n/2. Birkhoff proved that for every n ≥ 2 and r ≤ n/2, coprime with n, there exist at least two distinct n-periodic billiard trajectories with the rotation number r – see [4]. From contemporary view point, this result follows from the theory of area-preserving twist maps – see, e.g., [3] for a survey.

In the present paper we use a well-known variational reduction of the periodic billiard trajectories problem which is based on the observation that these trajectories are critical points of the perimeter length functional on the variety of inscribed polygons. This allows to apply topological methods (Morse and Lusternik-Schnirelman theories) and reduces the problem to studying the topology of the cyclic configuration space. Inscribed n-gons are in one-to-one correspondence with sequences

(x_{1}, x_{2}, …, x_{n}) ∈ X^{×n} = X × … × X,

such that

x_{1} ≠ x_{2}, x_{2} ≠ x_{3}, …, x_{n} ≠ x_{1};

here X ⊂ R^{m+1} denotes the boundary of the billiard domain, topologically, the sphere. The variety of all such configurations is the cyclic configuration space. Our main effort in this paper is in studying the cohomology ring of this configuration space. Note that the length functional, considered as a map X^{×n} → R on the total Cartesian power of X, fails to be smooth at the points with x_{i} = x_{i+1} for some i. This explains why the cyclic configuration space is the natural topological object related to the billiard problem.

The dihedral group D_{n} acts on the cyclic configuration space; its action is generated by the cyclic permutation and the reflection

(x_{1}, x_{2}, …, x_{n}) ↦ (x_{2}, x_{3}, …, x_{n}, x_{1}), (x_{1}, x_{2}, …, x_{n}) ↦ (x_{n}, x_{n−1}, …, x_{1}).

Two n-periodic billiard trajectories will be considered the same if they belong to the same orbit of D_{n}. In particular, the estimates in Theorem 1 below concern the number of distinct D_{n}-orbits in the cyclic configuration space.

Our main result is as follows.

Theorem 1 Let X ⊂ R^{m+1} be a smooth strictly convex hypersurface, where m ≥ 3. Fix an odd number n ≥ 3. Then

(A) The number of distinct D_{n}-orbits of n-periodic billiard trajectories inside X is not less than

[log_{2}(n − 1)] + m.  (1)

(B) For a generic X, the number of distinct D_{n}-orbits of n-periodic billiard trajectories inside X is not less than

(n − 1)m.  (2)

Square brackets in (1) denote the floor function, i,e., the largest integer not exceeding log_{2}(n − 1).

We deduce statement (A) of Theorem 1 from Lusternik-Schnirelman theory and statement (B) from Morse theory; this explains the genericity assumption in (B) (precise definition of the needed genericity is given in Section 4).

We conjecture that the estimate in case (A) can be improved to be n + m − 2.

It is important to note that if the period n in Theorem 1 is not prime then n-periodic billiard trajectories, whose existence is asserted by this theorem, may be multiple ones; for example, instead of a genuine 9-periodic orbit one might have a 3-periodic orbit, traversed three times.

To the best of our knowledge, the only previous attempt to extend Birkhoff’s results to multi-dimensional setting was made by I. Babenko in [1]. He applied the variational approach to periodic trajectories of convex billiards in 3-dimensional space. Unfortunately, the paper [1] contains an error (cf. Remark 3.4), and the situation with m = 2 remains unclear. We would like to emphasize that ideas and results from [1] are substantially used in the present paper.

We would like to mention, in passing, the case of 2-periodic billiard trajectories. These are the diameters of the billiard hypersurface X ⊂ R^{m+1}, that is, the chords, perpendicular to the hypersurface at both ends. It is well known that the least number of diameters is m + 1, and this estimate is sharp as the example of a generic ellipsoid shows.

The arguments that prove Theorem 1 apply to a much wider class of billiards. Namely, let X be a smooth strictly geodesically convex billiard hypersurface in a Riemannian manifold such that there exists a unique geodesic line through every two points of X. For example, this condition holds when the ambient manifold is a spherical or hyperbolic space or, more generally, a Hadamard space.

Theorem 2 For X as in the preceding paragraph, both statements of Theorem 1 hold true, provided dim X ≥ 3.

The content of the paper is, briefly, as follows. In Section 2 we construct a general spectral sequence that computes the cohomology ring of the cyclic configuration space of a smooth manifold. In Section 3 we use it to compute the cohomology ring of the cyclic configuration space of the sphere. Section 4 is Morse theoretical: we describe the relations (mostly known) between the information about the periodic billiard trajectories and the topology of the cyclic configuration space of the billiard hypersurface. In the last Section 5 we compute the D_{n}-equivariant cohomology of the cyclic configuration space of the sphere; the computation makes use of the topological results from Section 3 and the Morse-theoretical ones from Section 4. We finish the paper with a proof of our main result, Theorems 1 and 2.

Acknowledgments. It is a pleasure to acknowledge the hospitality of Max-Planck-Institut in Bonn where the authors started to work on this paper. Many thanks to Burt Totaro for patiently explaining to us his paper [20] and to Ivan Babenko for discussions on his paper [1].

2 Cyclic configuration space of a smooth manifold

Let X be a smooth manifold. Consider the configurations of ordered n-tuples (x_{1}, x_{2}, …, x_{n}) of points of X satisfying:

x_{i} ≠ x_{i+1} for i = 1, 2, …, n.  (3)

Here and elsewhere below we understand the indices cyclically, that is n + i = i (in particular, (3) contains the requirement x_{n} ≠ x_{1}). The space of all such configurations will be called the cyclic configuration space and denoted by G(X, n). The dihedral group D_{n} acts naturally on G(X, n) (cf. above); this action is free if n is an odd prime.

Example 2.1 The simplest example of a cyclic configuration space is provided by G(S^{1}, n), the cyclic configuration space of a circle, which plays an important role in Birkhoff’s theory of convex plane billiards [4]. A configuration (x_{1}, x_{2}, …, x_{n}) ∈ G(S^{1}, n) can be uniquely described by the initial point x_{1} ∈ S^{1} and by the angles 0 < ϕ_{i} < 1, where i = 1, 2, …, n − 1, assuming that the sum ϕ_{1} + ϕ_{2} + … + ϕ_{n−1} is not an integer. Namely, given x_{1} and ϕ_{1}, …, ϕ_{n−1} we set

x_{2} = x_{1} exp(2πiϕ_{1}), x_{3} = x_{2} exp(2πiϕ_{2}), …, x_{n} = x_{n−1} exp(2πiϕ_{n−1}).

Here we identify S^{1} with the unit circle in C. The hyperplanes

ϕ_{1} + ϕ_{2} + … + ϕ_{n−1} = r, where r = 1, 2, …, n − 2,

divide the cube (0, 1)^{n−1} into domains N_{1}, N_{2}, …, N_{n−1}, where

N_{r} = {(ϕ_{1}, ϕ_{2}, …, ϕ_{n−1}) : r − 1 < ∑_{i=1}^{n−1} ϕ_{i} < r}.

Each N_{r} is homeomorphic to R^{n−1}. It follows that the cyclic configuration space G(S^{1}, n) is homeomorphic to a disjoint union of S^{1} × N_{r}, where r = 1, 2, …, n − 1. Note that the number r, determining the connected component of a given configuration, is precisely the rotation number. Let us also indicate how the dihedral group D_{n} acts on G(S^{1}, n). Given a point (x_{1}, ϕ_{1}, …, ϕ_{n−1}) ∈ S^{1} × N_{r}, the cyclic permutation takes it to

(x_{1} exp(2πiϕ_{1}), ϕ_{2}, …, ϕ_{n−1}, r − ∑_{i=1}^{n−1} ϕ_{i}).

The reflection maps it to

(x_{1}, 1 − r + ∑_{i=1}^{n−1} ϕ_{i}, 1 − ϕ_{n−1}, 1 − ϕ_{n−2}, …, 1 − ϕ_{2}).

Hence the cyclic permutation preserves the rotation number, and the reflection maps configurations with rotation number r to configurations with rotation number n − r.

The cyclic configuration space G(X, n) is not to be confused with the usual configuration space

F (X, n) = {(x_{1}, …, x_{n}) ∈ X^{×n} : x_{i} ≠ x_{j} for i ≠ j}.

Recall a description of the cohomology ring H^{∗}(F (R^{m}, n); Z), obtained by F. Cohen – see [10], [11]. The space F (R^{m}, 2) is homotopy equivalent to m − 1-dimensional sphere; denote by ω its top-dimensional cohomology class. Let p_{ij} be the projection F (R^{m}, n) → F (R^{m}, 2) on the i-th and j-th components, and let G_{i,j} = p_{ij}^{∗}(ω). Then the ring H^{∗}(F (R^{m}, n); Z), where m > 1, is the graded-commutative algebra over Z with generators

G_{i,j}, 1 ≤ i, j ≤ n, i ≠ j, deg G_{i,j} = m − 1

and relations

(a) G_{i,j} = (−1)^{m}G_{j,i},

(b) G_{i,j}^{2} = 0,

(c) G_{i,j}G_{i,k} + G_{j,k}G_{j,i} + G_{k,i}G_{k,j} = 0 for i, j, k distinct.

An additive basis for H^{r(m−1)}(F (R^{m}, n); Z) is given by monomials G_{i_{1},j_{1}}G_{i_{2},j_{2}} … G_{i_{r},j_{r}} with i_{1} < … < i_{r} and i_{k} < j_{k} for all k.

We start the study of topology of cyclic configuration spaces with the space G(R^{m}, n). The following proposition is an analog of Cohen’s result.

Proposition 2.2 For m > 1 the ring H^{∗}(G(R^{m}, n); Z) is a graded commutative algebra over Z with generators

s_{1}, s_{2}, …, s_{n}, with deg s_{i} = m − 1

and the relations

s_{1}^{2} = s_{2}^{2} = … = s_{n}^{2} = 0,
s_{1}s_{2} … s_{n−1} + ϵs_{2}s_{3} … s_{n} + ϵ^{2}s_{3}s_{4} … s_{n}s_{1} + … + ϵ^{n−1}s_{n}s_{1} … s_{n−2} = 0,  (4)

where ϵ = (−1)^{(m−1)(n−1)}. If ϕ denotes the inclusion of F (R^{m}, n) into G(R^{m}, n), then ϕ^{∗}(s_{i}) = G_{i,i+1}.

Proof. In general, the cohomology groups of the complement R^{N} −∪V_{i} of a collection of affine subspaces V_{i} ⊂ R^{N}, where i = 1, …, k, is described in [15]. Recall this description, following [21], chapter 3, §6. For I ⊂ {1, …, k} let

V_{I} = ∩_{i∈I}V_{i}.

Two subsets I, J ⊂ {1, …, k} are equivalent, I ∼ J, if V_{I} = V_{J}. Each equivalence class has a unique maximal element. Given such a maximal element I, consider the standard chain complex of the simplex Δ_{I} spanned by I. Factorize this complex by the subcomplex generated by the faces Δ_{J} with J ⊂ I, J ≁ I, and denote the resulting quotient complex by C(I). Then the cohomology of the complement R^{N} −∪V_{i} is given by

H̃^{q}(R^{N} − ∪V_{i}; Z) = ⊕H̃_{N−dim V_{I}−q−1}(C(I); Z),

the sum taken over all the equivalence classes with maximal elements I, such that V_{I} are nonempty.

In our situation, R^{N} = (R^{m})^{×n}, and V_{i} = {x_{i} = x_{i+1}} for i = 1, 2, …, n. If |I| ≤ n − 2 then the equivalence class of I consists of one element only; on the other hand, all subsets of cardinality n − 1 are equivalent to each other and to {1, …, n}. Therefore, if |I| ≤ n − 2 then

H̃_{∗}(C(I); Z) = H_{∗}(Δ_{I}, ∂Δ_{I}; Z) = H̃_{∗}(S^{|I|−1}; Z),

and each subset I with s = |I| ≤ n − 2 makes a contribution of a copy of Z to H^{s(m−1)}(G(R^{m}, n); Z). Similarly, if I = {1, …, n} then

H̃_{∗}(C(I); Z) = H_{∗}(Δ_{I}, sk_{n−2}(Δ_{I}); Z) = H̃_{∗}(∨_{i=1}^{n−1}S_{i}^{n−1}; Z),

(where sk_{n−2}(Δ_{I}) denotes the (n − 2)-dimensional skeleton of the simplex Δ_{I}) and I makes a contribution of Z^{n−1} to H^{(n−1)(m−1)}(G(R^{m}, n); Z). This implies that H^{∗}(G(R^{m}, n); Z) is free abelian of rank

rk H^{s(m−1)}(G(R^{m}, n); Z) = \{ \begin{matrix}\binom{n}{s}, & for 0 ≤ s ≤ n − 2, \\ n − 1, & for s = n − 1 \end{matrix}

Hence, the Poincaré polynomial of the cyclic configuration space G(R^{m}, n) equals

(t^{m−1} + 1)^{n} − t^{(n−1)(m−1)} − t^{n(m−1)}.

For i = 1, …, n denote by s_{i} the generators of H^{m−1}(G(R^{m}, n); Z) coming from 1-element sets {i}. It is clear that s_{i} is the pull-back of the top-dimensional class of G(R^{m}, 2) = F (R^{m}, 2) ≈ S^{m−1} under the projection p_{i,i+1}. Therefore s_{i}^{2} = 0 and ϕ^{∗}(s_{i}) = G_{i,i+1}.

It remains to show that an additive basis for H^{r(m−1)}(G(R^{m}, n); Z) with r ≤ n − 1, is given by the monomials s_{i_{1}} … s_{i_{r}}, where i_{1} < … < i_{r}, with one additional relation (4).

Denote by S = S^{m−1} ⊂ R^{m} the unit sphere and let S^{×(n−1)} be the Cartesian power of S. For any p = 1, 2, …, n we construct an embedding

g_{p} : S^{×(n−1)} → G(R^{m}, n), (y_{1}, y_{2}, …, y_{n−1}) ↦ (x_{1}, x_{2}, …, x_{n}),

where

x_{p} = 0,
x_{p+1} = y_{1},
x_{p+2} = y_{1} + y_{2},
…
x_{n} = y_{1} + y_{2} + … + y_{n−p}
x_{1} = y_{1} + y_{2} + … + y_{n−p} + y_{n−p+1}
…
x_{p−2} = y_{1} + y_{2} + … + y_{n−2}
x_{p−1} = y_{1} + y_{2} + … + y_{n−2} + ny_{n−1}.

If \overline{s}_{i} ∈ H^{m−1}(S^{×(n−1)}; Z) denotes the obvious generator corresponding to the i-th factor (where i = 1, 2, …, n − 1), we have

g_{p}^{∗}(s_{i}) = \{ \begin{matrix}\overline{s}_{n−p+1+i}, for 1 ≤ i ≤ p − 2, \\ \overline{s}_{i−p+1}, for p ≤ i ≤ n, \\ (−1)^{m}\overline{s}_{n−1}, for i = p − 1. \end{matrix}

All but the last relation being obvious, let us explain the relation g_{p}^{∗}(s_{p−1}) = (−1)^{m}\overline{s}_{n−1}. Consider the homotopy F_{t} : S^{×(n−1)} → S, where t ∈ [0, 1], given by

F_{t}(y_{1}, …, y_{n−1}) = \frac{ty_{1} + … + ty_{n−2} + ny_{n−1}}{||ty_{1} + … + ty_{n−2} + ny_{n−1}||}, y_{i} ∈ S.

It is clear from the definitions that F_{1}^{∗}(v) = (−1)^{m}g_{p}^{∗}(s_{p−1}), where v ∈ H^{m−1}(S; Z) is the generator. On the other hand, the map F_{0} is just the projection (y_{1}, …, y_{n−1}) ↦ y_{n−1} and hence F_{0}^{∗}(v) = \overline{s}_{n−1}.

Let a_{i} ∈ H^{(n−1)(m−1)}(G(R^{m}, n); Z) denote the product s_{i+1}s_{i+2} … s_{n}s_{1} … s_{i−1}. Then we obtain

g_{p}^{∗}(a_{i}) = \{ \begin{matrix}\overline{s}_{1}\overline{s}_{2} … \overline{s}_{n−1}, for i = p − 1, \\ −ϵ\overline{s}_{1}\overline{s}_{2} … \overline{s}_{n−1}, for i = p − 2, \\ 0, for i ≠ p − 2, p − 1, \end{matrix}

where ϵ = (−1)^{(m−1)(n−1)}. Varying p, this shows that all a_{i} are nonzero and also that for any nontrivial relation ∑ β_{i}a_{i} in H^{(n−1)(m−1)}(G(R^{m}, n); Z) one must have β_{i+1} = ϵβ_{i}. In other words, any relation of degree n−1 between the classes s_{i} must be a consequence of (4). The rank calculation above shows that at least one nontrivial relation between the classes a_{1}, …, a_{n} exists, and so (4) holds.

Let us finally show that there exist no nontrivial relations between s_{1}, …, s_{n} of degree r < n − 1. Assume that there exists such a relation x containing a monomial s_{i_{1}}s_{i_{2}} … s_{i_{r}}. Choose a set of indices {j_{1}, …, j_{n−r−1}}, disjoint from the set {i_{1}, …, i_{r}}, and let y = s_{j_{1}} … s_{j_{n−r−1}}. Then xy is a relation of degree n − 1, containing neither of the terms a_{j_{1}}, …, a_{j_{n}−r−1}; we have shown above that it is impossible. This completes the proof. □

Remark 2.3 The cyclic configuration space G(R^{m}, 3) coincides with F (R^{m}, 3). In this case relation (4) turns into F. Cohen’s relation (c) in the cohomology H^{∗}(F (R^{m}, 3); Z) after substituting s_{i} = G_{i,i+1}.

Next, we construct a spectral sequence computing the cohomology of the cyclic configuration space G(X, n), where X is an arbitrary smooth orientable manifold. This spectral sequence is an analog of the one constructed by Totaro in [20] for the configuration space F (X, n).

Denote by p_{j} : X^{×n} → X the projection on the j-th component and by q_{j} = p_{j,j+1} the projection X^{×n} → X^{×2} on the j-th and j+1-th components. Also, let Δ ∈ H^{m}(X^{×2}; k) denote the cohomology class of the diagonal; here m = dim X and k is a field.

Theorem 3 Let X be a smooth orientable manifold of dimension m > 1; let k be a field.

(A) There exists a spectral sequence of bigraded differential algebras over k which converges to H^{∗}(G(X, n); k) whose E_{2}-term is the quotient of the bigraded commutative algebra

H^{∗}(X^{×n}; k) ⊗ H^{∗}(G(R^{m}, n); k),

where H^{p}(X^{×n}; k) has bidegree (p, 0) and H^{q}(G(R^{m}, n); k) has bidegree (0, q), by the relations

p_{i}^{∗}(v)s_{i} = p_{i+1}^{∗}(v)s_{i}, i = 1, …, n,

where v ∈ H^{∗}(X; k) is an arbitrary class.

(B) The dihedral group D_{n} acts on all terms of the spectral sequence E_{r}, r ≥ 2. The group D_{n} commutes with the differentials and its action on E_{r+1} is determined by its action on E_{r}; the action of D_{n} on H^{∗}(G(X, n); k) is compatible with the action on E_{∞}. The action of the dihedral group D_{n} on the initial term E_{2} is given by the action of D_{n} on H^{∗}(X^{×n}; k), induced by its obvious action on X^{×n}, and by

τ^{∗}(s_{i+1}) = (−1)^{m}s_{τ(i)}, τ^{∗}(s_{1}) = (−1)^{m}s_{τ(n)},

where τ ∈ D_{n} is a reflection and i = 1, …, n − 1.

(C) The first non-trivial differential is d_{m}, where m = dim X. It is defined by the formulas

d_{m}s_{i} = q_{i}^{∗}(Δ) and d_{m}(H^{∗}(X^{×n}; k)) = 0.

Here is a comment concerning statement (B). We identify the dihedral group D_{n} with a subgroup of permutations of the set {1, 2, ….n}. An element τ ∈ D_{n} is a reflection if it reverses the cyclic order of the set {1, 2, …, n}. In other words, τ ∈ D_{n} is a reflection if for any i = 1, 2, …, n − 1 holds

τ(i + 1) = τ(i) − 1 mod n and τ(1) = τ(n) − 1 mod n.

D_{n} contains n reflections, which generate D_{n}. Hence, knowing the action of reflections completely describes the action of D_{n} on E_{2}.

Proof. The proof is a modification of the arguments by B. Totaro in [20].

Consider the inclusion ψ : G(X, n) → X^{×n} and the Leray spectral sequence of the continuous map ψ:

E_{2}^{p,q} = H^{p}(X^{×n}; R^{q}ψ_{∗}k) ⇒ H^{p+q}(G(X, n); k)

where R^{q}ψ_{∗}k is the sheaf on X^{×n} associated with the presheaf

U ↦ H^{q}(U ∩ G(X, n); k).

We will consider partitions of the set of indices {1, …, n} into intervals, that is, subsets of the form {i, i + 1, i + 2, …, i + j}; as usual, the indices are understood cyclically. For example, the following are interval partitions of the set {1, 2, 3, 4, 5}:

{1, 2} ∪ {3, 4, 5}, {2} ∪ {3} ∪ {4, 5, 1}.

Let J be such a partition; denote by X_{J} the subset of X^{×n} consisting of configurations c = (x_{1}, x_{2}, …, x_{n}) ∈ X^{×n} satisfying the conditions x_{i} = x_{j} if i and j lie in the same interval of the partition J. Given two interval partitions I and J, we say that J refines I and write I ≺ J if the intervals of I are unions of the intervals of J. Denote by |J| the number of intervals in the partition J. X_{J} is naturally identified with the Cartesian power X^{×|J|}. The case |J| = 1 corresponds to the deepest diagonal {(x, x, …, x)} ⊂ X^{×n}. Note that I ≺ J implies X_{I} ⊂ X_{J} and |I| ≤ |J|.

Denote by D(X, n) the subset of X^{×n} satisfying the conditions x_{i} ≠ x_{i+1} for i = 1, …, n − 1, but not necessarily for i = n. It is clear that D(R^{m}, n) is homotopy equivalent to the product (S^{m−1})^{×(n−1)} and so its cohomology admits a description similar to Proposition 2.2, but without relation (4).

Let J be a partition of {1, 2, …, n} on intervals of lengths j_{1}, ..., j_{r} as above, and let c be a configuration with c ∈ X_{J} such that c does not belong to any X_{I} if J refines I. We claim that the stalk of the sheaf R^{q}ψ_{∗}k at c equals

(R^{q}ψ_{∗}k)_{c} = \{ \begin{matrix}H^{q}(D(R^{m}, j_{1}) × … × D(R^{m}, j_{r}); k) if r ≥ 2 \\ H^{q}(G(R^{m}, n); k) if r = 1. \end{matrix}

Indeed, by definition, this stalk is H^{q}(U ∩ G(X, n); k), where U is a small open ball around c. If c = (x_{1}, x_{2}, … x_{n}) then we may choose points y_{1}, y_{2}, …, y_{r} ∈ X, one for each interval of J, so that x_{i} = y_{j_{s}} if i belongs to the j_{s}-th interval. Let U_{j} ⊂ X be a small open neighborhood of y_{j}, so that each U_{j} is diffeomorphic to R^{m} and the sets U_{j} and U_{j′} are disjoint when the points y_{j} and y_{j′} are distinct. Then we may take U = U_{1}^{×j_{1}} × U_{2}^{×j_{2}} × … × U_{r}^{×j_{r}}, and our claim follows.

In particular, we see that R^{q}ψ_{∗}k vanishes unless q is a multiple of m − 1.

If |J| ≥ 2 then

(R^{∗}ψ_{∗}k)_{c} = H^{∗}((S^{m−1})^{×(n−|J|)}; k),

and hence

dim (R^{s(m−1)}ψ_{∗}k)_{c} = \binom{n − |J|}{s}.

Note that there is a canonical top-dimensional class of dimension (m − 1)(n − |J|), which is uniquely determined by a choice of an orientation of the sphere S^{m−1}.

If |J| = 1 (which means that X_{J} is the deepest diagonal) then, by Proposition 2.2,

dim (R^{r(m−1)}ψ_{∗}k)_{c} = \{ \begin{matrix}\binom{n}{r} for r ≤ n − 2, \\ n − 1 for r = n − 1, \end{matrix}

and hence there exists (n−1)-dimensional space of top-dimensional cohomology classes; they have dimension (n − 1)(m − 1). Note that the sheaf R^{(n−1)(m−1)}ψ_{∗}k is a constant sheaf k^{n−1} with support on the deepest diagonal X_{J}. This follows repeating the arguments of Totaro [20], p. 1062. The fact that this sheaf is locally constant is easy since this sheaf can be represented as sheaf of cohomology of fibers of a fibration over X_{J}. A further statement that this sheaf is constant (i.e., has a trivial monodromy) follows from explicit calculation of the cohomology H^{∗}(G(R^{m}, n); k) (which serves as the cohomology of the fiber), since we may explicitly label the cohomology classes by polynomials in s_{1}, …, s_{n}.

We wish to have a similar description for the sheaves R^{r(m−1)}ψ_{∗}k with r < n − 1. We will show that they can be represented as direct sums of constant sheaves supported on different diagonals X_{J}. For an interval partition J of {1, 2, …, n} with |J| > 1 denote by ε_{J} the constant sheaf with stalk k and support X_{J}. We claim that for r < n − 1 the sheaf R^{r(m−1)}ψ_{∗}k is isomorphic to the direct sum of sheaves

R^{r(m−1)}ψ_{∗}k ≃ ⨁_{|J|=n−r} ε_{J},  (5)

the sum taken over all interval partitions J with |J| = n − r.

To prove the above claim we first make the following remark. Let I be an interval partition of {1, 2, …, n} into intervals of length i_{1}, i_{2}, …, i_{s}, where s = |I| > 1. Then for any interval partition J into intervals of length j_{1}, j_{2}, …, j_{n−r}, such that I ≺ J, we have the canonical inclusion

ν_{JI} : D(R^{m}, i_{1}) × … × D(R^{m}, i_{s}) → D(R^{m}, j_{1}) × … × D(R^{m}, j_{n−r}).

Note that the target of ν_{JI} has a canonical r(m − 1)-dimensional cohomology class (cf. above). The induced map on r(m − 1)-dimensional cohomology with k coefficients

ν_{JI}^{∗} : H^{r(m−1)}(D(R^{m}, j_{1}) × … × D(R^{m}, j_{n−r})) → H^{r(m−1)}(D(R^{m}, i_{1}) × … × D(R^{m}, i_{s}))

is a monomorphism. Let z_{JI} denote the image of the top-dimensional generator. Then we observe (similarly to Lemma 3 in [20]) that for a fixed I the classes {z_{JI}} form a linear basis of the cohomology H^{r(m−1)}(D(R^{m}, i_{1}) × … × D(R^{m}, i_{s}); k), where J runs over all partitions with I ≺ J and |J| = n − r.

A similar statement holds in the case |I| = 1. Namely, for any interval partition J into intervals of length j_{1}, j_{2}, …, j_{n−r}, where n−r > 1, we have the canonical inclusion

ν_{JI} : G(R^{m}, n) → D(R^{m}, j_{1}) × … × D(R^{m}, j_{n−r});

denote by z_{JI} the image of the top-dimensional generator under the induced map on the cohomology, z_{JI} ∈ H^{r(m−1)}(G(R^{m}, n); k). Then the set {z_{JI}} (where J runs over all partitions with |J| = n − r) forms a linear basis of H^{r(m−1)}(G(R^{m}, n); k). This follows from the explicit calculation of the cohomology of G(R^{m}, n), given in Proposition 2.2.

Given an interval partition J on intervals of length j_{1}, …, j_{n−r}, where n − r > 1, consider the following commutative diagram

G(X, n)  → X^{×n}
↓  ↓ id
D(X, j_{1}) × … × D(X, j_{n−r}) →^{g_{J}} X^{×n}

formed by the natural inclusions. Define a sheaf ε′_{J} over X^{×n} by

ε′_{J} = R^{r(m−1)}g_{J}_{∗}(k).

We want to show that ε′_{J} is isomorphic to ε_{J}, defined above. First, ε′_{J} vanishes outside X_{J} (since we are considering the cohomology of the top dimension). Let U be a small open neighborhood of a point c ∈ X_{J} ⊂ X^{×n}, such that U = ∏ U_{i}, where all U_{i} are small open disks and U_{i} = U_{j} if i and j lie in the same interval of J. Then

ε′_{J}(U) = H^{r(m−1)}(D(U_{i_{1}}, j_{1}) × … × D(U_{i_{n−r}}, j_{n−r}); k) = k.

Hence ε′_{J} is a constant sheaf with stalk k supported on X_{J}, i.e., ε′_{J} ≃ ε_{J}.

The commutative diagram above gives a map of sheaves ε_{J} → R^{r(m−1)}ψ_{∗}(k) and we obtain a map of sheaves

⨁_{|J|=n−r} ε_{J} → R^{r(m−1)}ψ_{∗}(k),

which, as we have seen above, is an isomorphism on stalks; hence it is an isomorphism, and the claim (5) follows.

We arrive at the following description of the term E_{2} of the spectral sequence:

E_{2}^{p,r(m−1)} = \{ \begin{matrix}H^{p}(X_{J_{0}}; k) ⊗ k^{n−1}, for r = n − 1, where |J_{0}| = 1, \\ ⨁_{|I|=n−r} H^{p}(X_{I}; k), for r < n − 1 \end{matrix}

We will now identify this description with the one given in the statement of the theorem. Consider first the case r < n−1. Assign to a monomial s_{i_{1}} … s_{i_{r}} with i_{1} < i_{2} < … < i_{r} the equivalence relation on the set of indices {1, …, n} generated by the relations:

i_{1} ∼ i_{1} + 1, i_{2} ∼ i_{2} + 1, …, i_{r} ∼ i_{r} + 1.

This equivalence relation defines a partition I of the set {1, …, n} on n − r intervals. In view of the relations p_{i}^{∗}(v)s_{i} = p_{i+1}^{∗}(v)s_{i}, the term H^{p}(X^{×n}; k)s_{i_{1}} … s_{i_{r}} is isomorphic to H^{p}(X_{I}; k), and we are done in this case. Consider now the case r = n−1. There are n−1 linearly independent monomials of degree n−1 in classes s_{1}, …, s_{n}, and for every such monomial the corresponding interval partition equals J_{0} = {1, …, n}. Hence the image of H^{p}(X^{×n}; k) ⊗ H^{(n−1)(m−1)}(G(R^{m}, n); k) is isomorphic to H^{p}(X_{J_{0}}; k) ⊗ k^{n−1} after imposing the relations p_{i}^{∗}(v)s_{i} = p_{i+1}^{∗}(v)s_{i}, as stated.

Statement (B) follows since the dihedral group D_{n} acts naturally on G(X, n) and on X^{×n} and the inclusion ψ : G(X, n) → X^{×n} is D_{n}-equivariant. Hence, D_{n} acts on the Leray spectral sequence of map ψ.

To find the action of D_{n} on the generators s_{1}, …, s_{n}, consider an embedding R^{m} → X (for example, the interior of a small disk) and the induced embedding of the cyclic configuration spaces ϕ : G(R^{m}, n) → G(X, n); clearly, ϕ is D_{n}-equivariant. ϕ determines a D_{n}-equivariant homomorphism of the spectral sequence of Theorem 3 for X into a similar spectral sequence for R^{m}. This homomorphism is an isomorphism on E_{2}^{0,∗}. Therefore, it is enough to find the action of D_{n} on the cohomology classes s_{i} ∈ H^{m−1}(G(R^{m}, n); k). Each class s_{i} is the pullback of the fundamental class of S^{m−1} under the map

G(R^{m}, n) → S^{m−1}, (x_{1}, …, x_{n}) ↦ \frac{x_{i+1} − x_{i}}{|x_{i+1} − x_{i}|}.

If τ ∈ D_{n} is a reflection then the class τ^{∗}s_{i} is the pullback of the fundamental class of S^{m−1} under the map

(x_{1}, …, x_{n}) ↦ \frac{x_{j−1} − x_{j}}{|x_{j−1} − x_{j}|}, where τ(j) = i.

Hence it follows that

τ^{∗}s_{i} = (−1)^{m}s_{j−1} = (−1)^{m}s_{τ^{−1}(i)−1} = (−1)^{m}s_{τ(i)−1},

as stated.

Now we consider the differentials of the spectral sequence. The first m − 2 of them, d_{2}, …, d_{m−1}, vanish by the dimension considerations. To find d_{m} consider the inclusion ϕ : F (X, n) ↪ G(X, n). This inclusion induces a homomorphism of the spectral sequence for G(X, n) to that for F (X, n), constructed in [20]. In the latter spectral sequence the E_{2}-term is the quotient of the graded commutative algebra

H^{∗}(X^{×n}; k) ⊗ H^{∗}(F (R^{m}, n); k)

modulo the relations

p_{i}^{∗}(v)G_{ij} = p_{j}^{∗}(v)G_{ij} for i ≠ j and v ∈ H^{∗}(X; k),

and the first non-trivial differential acts as follows: d_{m}G_{ij} = p_{ij}^{∗}Δ. According to Proposition 2.2, ϕ^{∗}(s_{i}) = G_{i,i+1}, and, obviously, ϕ^{∗} is identical on H^{∗}(X^{×n}; k). This implies that the differential d_{m} of our spectral sequence acts by d_{m}(s_{i}) = q_{i}^{∗}(Δ), as stated in the theorem. □

Remark 2.4 Without assumption m > 1 the arguments of the proof involving the top-dimensional cohomology classes do not work. In fact, Theorem 3 fails for X = S^{1} (see Example 2.1 above and Remark 3.2 below). The assumption m > 1 should be also added to Theorem 1 of Totaro [20].

3 Cyclic configuration space of the sphere

In this section we will use Theorem 3 to calculate the cohomology ring of G(S^{m}, n) with coefficients in Z_{2}.

Theorem 4 Let m > 2. The cohomology ring H^{∗}(G(S^{m}, n); Z_{2}) is multiplicatively generated by cohomology classes

σ_{i} ∈ H^{i(m−1)}(G(S^{m}, n); Z_{2}) where i = 1, 2, … and u ∈ H^{m}(G(S^{m}, n); Z_{2}).

These classes satisfy the following relations:

(i) σ_{i} = 0 for i ≥ n − 1;

(ii) σ_{i}σ_{j} = \binom{i + j}{i}σ_{i+j};

(iii) u^{2} = 0.

The Poincaré polynomial of the space G(S^{m}, n) with coefficients in Z_{2} equals

\frac{(t^{m} + 1)(t^{(n−1)(m−1)} − 1)}{(t^{m−1} − 1)},

and the sum of Betti numbers is 2(n − 1). The dihedral group D_{n} acts trivially on H^{∗}(G(S^{m}, n); Z_{2}).

Remark 3.1 Relation (ii) implies σ_{i}^{2} = 0, since the binomial coefficient \binom{2i}{i} is always even.

Remark 3.2 The space G(S^{1}, n) consists of n − 1 connected components and each is homotopy equivalent to S^{1}. Comparing with Theorem 4, we see that this theorem gives a correct additive structure of H^{∗}(G(S^{1}, n); Z_{2}), although the multiplicative structure for m = 1 is different. In fact, for any zero-dimensional cohomology class x with Z_{2} coefficients one has x^{2} = x, and so for m = 1 the classes σ_{i} with σ_{i}^{2} = 0 do not exist. Nor is the action of the dihedral group D_{n} on H^{∗}(G(S^{1}, n); Z_{2}) trivial since the reflection changes the rotation number (as we mentioned above) and so it acts nontrivially on H^{0}(G(S^{1}); Z_{2}).

Remark 3.3 Most of the statements of Theorem 4 hold true for m = 2. In particular, for m = 2, the Poincaré polynomial remains the same as stated in Theorem 4. There are only two distinctions. First, for m = 2, relation (ii) should be replaced by a more general one

(ii’) σ_{i}σ_{j} = \binom{i + j}{i}σ_{i+j} + ε_{i,j}σ_{i+j−2}u, where ε_{i,j} ∈ Z_{2};

Secondly, our methods do not prove that the dihedral group D_{n} acts trivially on H^{∗}(G(S^{2}, n); Z_{2}); instead we have a weaker statement that the cyclic group Z_{n} ⊂ D_{n} acts trivially on H^{∗}(G(S^{2}, n); Z_{2}), if n is odd.

Remark 3.4 Paper [1] contains an error. Proposition 3.3 of [1] claims that G(S^{2}, n) is simply connected and all further computations in [1] depend on this claim. In fact, the space G(S^{2}, n) is simply connected only for n = 1 and n = 2, as follows from our previous remark 3.3.

Proof. In what follows we will omit the coefficients Z_{2} from the notation. We will apply the spectral sequence of Theorem 3 in the case when X = S^{m}. Let us describe its E_{2}-term.

Let v ∈ H^{m}(S^{m}) be the generator. For i = 1, 2, …, n let u_{i} ∈ H^{m}((S^{m})^{×n}) denote the class u_{i} = 1 × … × v × 1 × … × 1 (with v at the i-th position). The classes u_{1}, …, u_{n} generate the cohomology ring H^{∗}((S^{m})^{×n}); they commute and satisfy the relations u_{i}^{2} = 0.

The E_{2} = E_{m}-term of the spectral sequence of Theorem 3 is the quotient of the bigraded commutative differential algebra Z_{2}[s_{1}, …, s_{n}, u_{1}, …, u_{n}], where each s_{i} has bidegree (0, m − 1) and each u_{i} has bidegree (m, 0), by the ideal generated by the relations:

(a) s_{i}^{2} = 0 for i = 1, 2, …, n;

(b) σ_{n−1}(s) = s_{1}s_{2} … s_{n−1} + s_{2}s_{3} … s_{n} + s_{3}s_{4} … s_{n}s_{1} + … + s_{n}s_{1} … s_{n−2} = 0;

(c) u_{i}^{2} = 0 for i = 1, 2, …, n;

(d) (u_{i} + u_{i+1})s_{i} = 0 for i = 1, 2, …, n.

Recall that we understand indices cyclically; for example, (d) contains the relation (u_{1} + u_{n})s_{n} = 0.

The first nontrivial differential is d_{m} and it acts as follows

d_{m}s_{i} = u_{i} + u_{i+1}, d_{m}u_{i} = 0, i = 1, ..., n.

Introduce new variables: v_{i} = u_{i} + u_{i+1}, where i = 1, 2, …, n and u = u_{1}. In the new variables the differential algebra (E_{m}, d_{m}) can be understood as the quotient of the polynomial ring Z_{2}[s_{1}, …, s_{n}, v_{1}, …, v_{n}, u] modulo the ideal generated by the relations (a), (b) and the following relations (c’), (d’) and (e):

[figure: 1; u_{i}; u_{i}u_{j}; s_{i}u_{j}u_{k}; s_{i}; s_{i}u_{j}; s_{i}s_{j}; s_{i}s_{j}u_{k}; m; 2m; …; 0; m − 1; 2(m − 1); ⋮; d_{m}; d_{m}; d_{m}; d_{m}]

Figure 1: Term E_{m} of the spectral sequence

(c’) v_{i}^{2} = 0 for i = 1, 2, …, n and u^{2} = 0;

(d’) v_{i}s_{i} = 0 for i = 1, 2, …, n;

(e) ∑_{i=1}^{n} v_{i} = 0.

The differential d_{m} acts by

d_{m}(s_{i}) = v_{i}, d_{m}(v_{i}) = d_{m}(u) = 0.  (6)

Our goal is to compute the cohomology of (E_{m}, d_{m}).

Let A denote the quotient of Z_{2}[s_{1}, …, s_{n}, v_{1}, …, v_{n}, u] by the ideal generated by the relations (a), (c’), (d’), i.e., we simply ignore relations (b) and (e). We consider A and E_{m} with the total grading, where each s_{i} has degree m − 1 and the elements v_{i} and u have degree m. Let the differential d_{A} : A → A act by d_{A}(s_{i}) = v_{i} and d_{A}(v_{i}) = 0 = d_{A}(u). There is a canonical epimorphism of graded differential algebras f : A → E_{m}, so that fd_{A} = d_{m}f. The kernel of f is the ideal of A generated by ∑_{i=1}^{n} v_{i} and σ_{n−1}(s).

We claim that

H^{i}(A, d_{A}) = \{ \begin{matrix}0, if i ≠ m, \\ Z_{2}, if i = m, \end{matrix}  (7)

and the only nontrivial cohomology class is represented by u. In order to prove this, consider the filtration A_{0} ⊂ A_{1} ⊂ … ⊂ A_{n} = A, where A_{0} is generated by u, and each A_{k} is generated by u, s_{i}, v_{i} for i = 1, …, k. The differential d_{A} preserves this filtration. d_{A}, restricted to A_{0}, vanishes, and so (A_{0}, d_{A}) has a one-dimensional cohomology generated by the class of u. We will now show that all factors A_{k}/A_{k−1} are acyclic, where k = 1, …, n. Indeed, any element a ∈ A_{k}/A_{k−1} can be uniquely represented in the form a = s_{k}x + v_{k}y, where x, y ∈ A_{k−1}. If a is a cocycle then we have

d_{A}(a) = v_{k}x + s_{k}d_{A}(x) + v_{k}d_{A}(y) = s_{k}d_{A}(x) + v_{k}[x + d_{A}(y)] = 0

and hence x = d_{A}(y). Therefore a = d_{A}(s_{k}y). This proves that H^{∗}(A, d_{A}) = H^{∗}(A_{0}, d_{A}) and our claim (7) follows.

Introduce a new differential δ_{A} : A → A of degree m:

δ_{A}(x) = (∑_{i=1}^{n} v_{i})x.

Clearly, δ_{A}^{2} = 0 and δ_{A}d_{A} = d_{A}δ_{A}; however δ_{A} does not obey the Leibnitz rule. We claim that

H^{i}(A, δ_{A}) = \{ \begin{matrix}0, if i ≠ n(m − 1), i ≠ n(m − 1) + m, \\ Z_{2}, if i = n(m − 1) or i = n(m − 1) + m, \end{matrix}  (8)

and the only nontrivial cohomology classes are represented by s_{1}s_{2} … s_{n} and by s_{1}s_{2} … s_{n}u. Indeed, each element of A can be written as a sum of monomials in s_{i}, v_{i} and u. For I ⊂ {1, 2, … n}, denote by s_{I} the product of all s_{i} for i ∈ I. Similarly, label monomials in variables v_{i} as v_{J} for J ⊂ {1, 2, … n}. Note that the product s_{I}v_{J} ∈ A is nontrivial if and only if I and J are disjoint subsets of {1, …, n}. Note also that

δ_{A}(s_{I}v_{J}) = ∑_{i∉I∪J} s_{I}v_{J∪{i}}, δ_{A}(s_{I}v_{J}u) = ∑_{i∉I∪J} s_{I}v_{J∪{i}}u.

We see that application of δ_{A} does not change the multi-index I or the factor u. Hence, the complex (A, δ_{A}) splits into a direct sum over different multi-indices I. Fix a set I and denote by k the cardinality of the set {1, …, n} − I. Then the respective part of the complex (A, δ_{A}) is isomorphic to two copies (one with u and one without) of the standard cochain complex of the simplex with k vertices: the differential of an r-dimensional face (i.e., set J) is the sum of r + 1-dimensional faces that contain the given one (sets J ∪ {i}). Note that empty set J is also allowed. This complex has zero cohomology, unless k = 0 (empty simplex), in which case the cohomology is Z_{2}. This exceptional case corresponds to I = {1, …, n}, and the claim follows.

Now we are ready to compute the cohomology of (E_{m}, d_{m}), i.e., the term E_{m+1} of the spectral sequence. Denote by σ_{k} ∈ A the k-th elementary symmetric function in variables s_{1} …, s_{n}, i.e.,

σ_{0} = 1, and σ_{k} = ∑_{1≤i_{1}<...<i_{k}≤n} s_{i_{1}}s_{i_{2}} … s_{i_{k}} for k = 1, 2, …, n.

It is clear that

σ_{i}σ_{j} = \{ \begin{matrix}\binom{i + j}{i}σ_{i+1}, for i + j ≤ n, \\ 0, for i + j > n. \end{matrix}

Also note that

d_{A}(σ_{k+1}) = δ_{A}(σ_{k}), d_{A}(σ_{k+1}u) = δ_{A}(σ_{k}u).  (9)

Therefore the images of the classes σ_{i} and σ_{i}u under the projection f : A → E_{m} are cocycles. Note also that the classes f(σ_{n}), f(σ_{n−1}) vanish due to the relation (b). All other classes f(σ_{0}) = 1, f(σ_{1}), …, f(σ_{n−2}) are nonzero elements of E_{m}; this follows since the kernel of f is generated by the image of δ_{A} and by the element σ_{n−1}, cf. above. It is clear from dimension considerations that σ_{i} is not a coboundary; likewise, σ_{i}u is not a coboundary either. Hence we have nontrivial cohomology classes

f(σ_{i}) ∈ H^{i(m−1)}(E_{m}, d_{m}) and f(σ_{i}u) ∈ H^{i(m−1)+m}(E_{m}, d_{m}), i = 0, 1, …, n − 2.

Our aim now is to show that these classes constitute all the cohomology.

Let K ⊂ A denote the kernel of f. Using long exact sequences, we deduce from (7) that

H^{i}(E_{m}, d_{m}) ≃ H^{i+1}(K, d_{A}) for i ≠ m.  (10)

Note that K contains σ_{n}, and denote by \overline{K} the factor of K by the ideal, generated by σ_{n} and d_{A}(σ_{n}). This ideal is four-dimensional and is generated, as a vector space, by the classes σ_{n}, d_{A}(σ_{n}), σ_{n}u and d_{A}(σ_{n})u; it is closed under d_{A} and is acyclic. Hence we have

H^{∗}(K, d_{A}) ≃ H^{∗}(\overline{K}, d_{A}).

Define a chain map ϕ : (E_{m}, d_{m}) → (\overline{K}, d_{A}) increasing the degree by m. Given an element x ∈ E_{m}, consider a lift \overline{x} ∈ A and set: ϕ(x) = δ_{A}(\overline{x}) ∈ \overline{K}. To see that ϕ is well-defined, note that \overline{x} is determined up to summation with elements of the form δ_{A}(a) + σ_{n−1}b, where a, b ∈ A. Since δ_{A}(δ_{A}(a) + σ_{n−1}b) = d_{A}(σ_{n})b = 0 ∈ \overline{K}, we see that ϕ(x) does not depend on the choice of \overline{x}. It is clear that ϕ is a chain map, i.e., ϕd_{m} = d_{A}ϕ. Due to (8), ϕ is a monomorphism. The cokernel of ϕ is linearly generated by the classes σ_{n−1} and σ_{n−1}u, and we find from the short exact sequence 0 → E_{m} →^{ϕ} \overline{K} → Coker(ϕ) → 0 that

ϕ : H^{i}(E_{m}, d_{m}) → H^{i+m}(\overline{K}, d_{A})  (11)

is an isomorphism for all i except i = (n − 2)(m − 1) and i = (n − 1)(m − 1) + 1; for these two exceptional values of i the homomorphism (11) is an epimorphism with one dimensional kernel. To prove this claim about (11) notice that the induced differential on Coker(ϕ) is trivial, as well as the induced map H^{∗}(\overline{K}) → H^{∗}(Coker(ϕ)).

Combining isomorphisms (10) and (11), one finds that

H^{i}(E_{m}, d_{m}) →^{≃} H^{i+m−1}(E_{m}, d_{m}),  (12)

where i ≠ 1, i ≠ (n − 2)(m − 1) and i ≠ (n − 1)(m − 1) + 1. Note also that for i = (n − 2)(m − 1) and i = (n − 1)(m − 1) + 1 instead of isomorphism (12) we have an epimorphism with one dimensional kernel.

1  m^{u}^{σ_{1}u}^{σ_{2}u}^{σ_{1}}^{σ_{2}}^{σ_{n−2}}  ^{σ_{n−2}u} ⋮  ⋮  ⋮  ⋮  ⋮ ⋮  ⋮  ⋮  ⋮  ⋮ m − 1 2m − 2 (n − 2)(m − 1)

Figure 2: Term E_{m+1} of the spectral sequence.

We can finally show that the term E_{m+1} is linearly generated by the classes σ_{1}, …, σ_{n−2} and u, σ_{1}u, …, σ_{n−2}u, where, abusing notation, we identify f(σ_{i}) and f(u) with σ_{i} and u, respectively. Obviously, σ_{1} and u are the only nontrivial classes of E_{m+1} of total degree < 2(m − 1). The periodicity isomorphism (12) implies that in each degree i(m − 1) (where 1 ≤ i < n − 1) we have only one nontrivial class, which therefore must coincide with σ_{i}. Also, again from (12), one concludes that in each degree i(m − 1) + m, where 1 ≤ i < n − 1, there is a single nontrivial class, and so it must coincide with σ_{i}u. Hence the term E_{m+1} has the structure shown in Figure 2.

By dimension considerations, all further differentials d_{r} with r > m vanish and thus E_{m+1} = E_{∞}.

Now we want to reconstruct the cohomology algebra H^{∗}(G(S^{m}, n)) from E_{∞}. The m-th column is an ideal in H^{∗}(G(S^{m}, n)), and the factor of H^{∗}(G(S^{m}, n)) by this ideal is the 0-th column. Since m > 2, each diagonal x + y = c contains at most one nonzero term of E_{∞}. Hence the generators σ_{i} and u admit unique lifts to H^{∗}(G(S^{m}, n)), and we label the cohomology classes of H^{∗}(G(S^{m}, n)) as σ_{i} and σ_{i}u. Since the multiplication in E_{∞} is induced from the multiplication in H^{∗}(G(S^{m}, n)), the product of σ_{i}u and σ_{j}u is trivial, and the product of σ_{i} and σ_{j}u equals \binom{i+j}{i}σ_{i+j}u. The product of σ_{i} and σ_{j} may be equal to \binom{i+j}{i}σ_{i+j} plus a term from the m-th column, but, in fact, this additional term must vanish since the m-th column has no nonzero terms in dimensions, divisible by m − 1. We conclude that the cohomology algebra H^{∗}(G(S^{m}, n)) has the structure described in Theorem 4.

The dihedral group D_{n} acts trivially on H^{∗}(G(S^{m}, n)) since in each dimension we have at most one nonzero element.

The arguments in the two last paragraphs do not apply if m = 2. This explains the modifications to the statement of Theorem 4 in case m = 2, described in Remark 3.3 above. Let us show that Z_{n} ⊂ D_{n} acts trivially on H^{∗}(G(S^{2}, n); Z_{2}), assuming that n is odd. Indeed, it follows from the above calculation of the E_{∞}-term that H^{i}(G(S^{2}, n); Z_{2}) is at most two-dimensional for any i, and hence it has either 1 or 3 nonzero elements. Z_{n} acts by permutations of nonzero classes, and at least one of these classes is fixed, since D_{n} acts trivially on the E_{∞}-term. Therefore the action of Z_{n} is trivial provided n is odd. This completes the proof. □

Corollary 5 Consider the cohomology classes

σ_{1}, σ_{2}, σ_{4}, …, σ_{2^{k}} ∈ H^{∗}(G(S^{m}, n); Z_{2}), m > 2,

where k is the largest integer, not exceeding log_{2}(n − 2). Then, for any i ≤ n − 2, the class σ_{i} can be expressed as a product

σ_{i} = σ_{2^{k_{1}}}σ_{2^{k_{2}}} … σ_{2^{kr}} ,  (13)

where i = 2^{k_{1}} + 2^{k_{2}} + … + 2^{k_{r}} is the binary expansion of i; here k_{1} < k_{2} < … < k_{r}.

Proof. It follows from relation (ii) of Theorem 4 that

σ_{l_{1}} … σ_{l_{r}} = \binom{l_{1} + … + l_{r}}{l_{1}, …, l_{r}}σ_{l_{1}+...+l_{r}}, l_{1} + … + l_{r} ≤ n − 2.

Hence the statement of the Corollary will follow once we show that the multinomial coefficient

\binom{2^{k_{1}} + … + 2^{k_{r}}}{2^{k_{1}}, …, 2^{k_{r}}} = \frac{(2^{k_{1}} + … + 2^{k_{r}})!}{(2^{k_{1}})! … (2^{k_{r}})!}

is odd for k_{1} < … < k_{r}. This is a consequence of the well known fact that the maximal power of 2 that divides l! equals l − r, where r is the number of 1’s in the binary expansion of l. □

Corollary 6 For m > 1 the cup-length of H^{∗}(G(S^{m}, n); Z_{2}) equals

[log_{2}(n − 1)] + 1.

Proof. Assume first that m > 2. Then, by Corollary 5, the length of the longest nontrivial product of classes σ_{i} equals the maximal number of 1’s in the binary expansion, that integers, not exceeding n − 2, may have. It is easy to see that the latter number equals [log_{2}(n − 1)]. This implies Corollary 6, since we also have the class u at our disposal. For m = 2 the arguments are similar using relation (ii’) in Remark 3.3. □

4 Morse theory of closed billiard trajectories

In this section we describe Morse theory of closed billiard trajectories which is similar to the classical Morse theory of closed geodesics in Riemannian manifolds. Some of the results of this section are known to experts, but do not exist in the literature in an accessible form.

The proof of Theorem 1 will depend on the results of this section in two ways. First, Morse theory of closed billiard trajectories, in the simplest case when the billiard table is the round sphere (like in [1]), will provide a tool for computation of the D_{n}- equivariant cohomology ring of the cyclic configuration space G(S^{m}, n). Second, we will invoke equivariant Morse theory [7] and a version of equivariant Lusternik-Schnirelman theory developed in [17] to deduce Theorem 1 concerning arbitrary smooth strictly convex billiards.

Let X ⊂ R^{m+1} be a smooth closed strictly convex hypersurface, topologically the sphere, which is the boundary of the billiard table. Denote by

L_{X} : G(X, n) → R  (14)

the perimeter length function, taken with the minus sign,

L_{X}(x_{1}, …, x_{n}) = −|x_{1} − x_{2}| − |x_{2} − x_{3}| − … − |x_{n} − x_{1}|,

where (x_{1}, x_{2}, …, x_{n}) ∈ G(X, n) and the distance |x_{i}−x_{i+1}| is measured in the ambient Euclidean space R^{m+1}. The reason for the minus sign will become clear shortly.

It is well-known that n-periodic billiard orbits in X are precisely the critical points of the function L_{X}; this is the Maupertuis’ principle of the classical mechanics as applied to billiards (see, e.g., [16, 19]). Clearly, the function L_{X} is smooth and D_{n}- equivariant. Identifying G(X, n) with G(S^{m}, n), we see that the shape of the billiard domain X becomes encoded in the function L_{X} : G(S^{m}, n) → R, and the problem of finding the closed billiard trajectories inside X turns into a problem of Morse-Lusternik-Schnirelman theory.

We encounter the following difficulty: one cannot apply Morse-Lusternik-Schnirelman theory directly to G(X, n) since this manifold is not compact. The function L_{X} extends to a continuous function on the space of all n-tuples X^{×n} but this extension fails to be differentiable on the singular set Σ consisting of the points with x_{i} = x_{i+1} for some i. A way around this difficulty is in replacing G(X, n) by a compact manifold with boundary G_{ε}(X, n) ⊂ G(X, n), where ε > 0 is small enough and

G_{ε}(X, n) = {(x_{1}, …, x_{n}) ∈ X^{×n} : ∏_{i=1}^{n} |x_{i} − x_{i+1}| ≥ ε};  (15)

similar approach can be found in [1] and in [12], [16] for the two-dimensional case.

Proposition 4.1 If ε > 0 is sufficiently small then:

(a) G_{ε}(X, n) is a smooth manifold with boundary;

(b) the inclusion G_{ε}(X, n) ⊂ G(X, n) is a D_{n}-equivariant homotopy equivalence;

(c) all critical points of L_{X} : G(X, n) → R are contained in G_{ε}(X, n);

(d) at every point of ∂G_{ε}(X, n), the gradient of L_{X} has the outward direction.

Proof. The function ϕ(x_{1}, …, x_{n}) = ∏_{1}^{n} |x_{i} − x_{i+1}|^{2} is smooth on X^{×n}. Its zero level set ϕ^{−1}(0) is Σ = X^{×n} − G(X, n), which is a critical level. Statements (a) and (b) will follow once we show that there exists a constant δ > 0 such that the interval (0, δ) consists of regular values of ϕ.

To prove the claim in italic we need to understand geometry of n-tuples x̄ = (x_{1}, …, x_{n}) ∈ G(X, n) which are critical points of ϕ. An easy computation with Lagrange multipliers shows that x̄ is critical if and only if

\frac{x_{i−1} − x_{i}}{|x_{i−1} − x_{i}|^{2}} + \frac{x_{i+1} − x_{i}}{|x_{i+1} − x_{i}|^{2}} = t_{i}ν_{i}, i = 1, …, n  (16)

where ν_{i} = ν(x_{i}) is the unit normal vector to X at point x_{i} and t_{i} ∈ R is some constant. It will be convenient for us to assume that ν_{i} has the inward direction.

Since X is smooth and strictly convex one can find two positive numbers r < R such that for every x ∈ X

(1) there is a sphere s(x) of radius r, tangent to X at x and contained inside X; and

(2) there is a sphere S(x) of radius R, tangent to X at x and containing X.

Let x_{i−1}, x_{i}, x_{i+1} be three consecutive points on X from a critical n-tuple x̄ of the function ϕ. Note that the 2-plane through the points x_{i−1}, x_{i}, x_{i+1} contains the centers of both spheres s(x_{i}) and S(x_{i}). Let α_{i} be the angle between ν_{i} and x_{i−1} − x_{i}, and β_{i} the angle between ν_{i} and x_{i+1} − x_{i}. Set: l_{i} = |x_{i} − x_{i−1}|. Then (16) gives

sin α_{i}/l_{i} = sin β_{i}/l_{i+1}.  (17)

Since x_{i−1} and x_{i+1} lie outside the sphere s(x_{i}) and inside the sphere S(x_{i}), we have

2r cos α_{i} < l_{i} < 2R cos α_{i}, 2r cos β_{i} < l_{i+1} < 2R cos β_{i},  (18)

and hence

\frac{r cos α_{i}}{R cos β_{i}} ≤ \frac{l_{i}}{l_{i+1}} ≤ \frac{R cos α_{i}}{r cos β_{i}} .  (19)

We claim that

\frac{r}{\sqrt{r^{2} + R^{2}}} ≤ \frac{l_{i}}{l_{i+1}} ≤ \frac{\sqrt{r^{2} + R^{2}}}{r}  (20)

for any i = 1, 2, …, n. Indeed, if (20) fails and

\frac{l_{i}}{l_{i+1}} < \frac{r}{\sqrt{r^{2} + R^{2}}}  (21)

then, combining (21) with the left inequality in (19), we obtain

cos α_{i} < \frac{R cos β_{i}}{\sqrt{r^{2} + R^{2}}} ≤ \frac{R}{\sqrt{r^{2} + R^{2}}} .  (22)

On the other hand, combining (21) with (17), we obtain

sin α_{i} < \frac{r sin β_{i}}{\sqrt{r^{2} + R^{2}}} ≤ \frac{r}{\sqrt{r^{2} + R^{2}}} ,  (23)

which leads to a contradiction since (22) and (23) are incompatible. This argument shows that the left inequality in (20) holds. The right inequality in (20) follows similarly.

It follows from (20) that for any two edges l_{i} and l_{j} of a critical n-gon one has:

\frac{l_{i}}{l_{j}} < (1 + \frac{R^{2}}{r^{2}} )^{n/2}.

In other words, if one of the edges of a critical n-gon is ”short” then all others are also ”short”.

It remains to show that a critical n-gon cannot have all edges arbitrarily short. Indeed, assume that x̄ = (x_{1}, …, x_{n}) is a critical polygon such that all points x_{i} lie in a small neighborhood U ⊂ X of a point y ∈ U. We will assume that U is so small that for any x ∈ U the scalar product ⟨ν(x), ν(y)⟩ is positive (recall that ν(x) denotes the inward unit normal to X at point x). Taking scalar product of equation (16) with ν_{i} gives

\frac{cos α_{i}}{l_{i}} + \frac{cos β_{i}}{l_{i+1}} = t_{i},

and hence all numbers t_{i}, which appear in (16), are positive. Now, the scalar product of (16) with vector ν(y) gives

\frac{⟨(x_{i−1} − x_{i}), ν(y)⟩}{l_{i}^{2}} > \frac{⟨(x_{i} − x_{i+1}), ν(y)⟩}{l_{i+1}^{2}}, i = 1, 2, …, n.  (24)

Recall that in the last inequality the indices are understood cyclically, i.e. x_{n+j} = x_{j}. Hence (24) leads to a contradiction.

The above arguments prove statements (a) and (b).

Next we prove statement (c). The argument is similar. If x̄ = (x_{1}, …, x_{n}) ∈ G(X, n) is a billiard trajectory in X, then (instead of (16)) we have

\frac{x_{i−1} − x_{i}}{l_{i}} + \frac{x_{i+1} − x_{i}}{l_{i+1}} = t_{i}ν_{i}, i = 1, …, n,

and (17) becomes the usual reflection law

α_{i} = β_{i}.

We have the inequality

\frac{r}{R} ≤ \frac{l_{i}}{l_{i+1}} ≤ \frac{R}{r}

which is an analog of (20) in the present case. As above, this inequality implies that if one edge of an n-periodic billiard trajectory is ”short” then so are all its edges. The preceding argument shows that billiard n-gons cannot lie entirely in a small neighborhood of a point of the hypersurface X.

Finally we prove claim (d). Since the gradient ∇ϕ is orthogonal to the boundary ∂G_{ε}(X, n) and points inside G_{ε}(X, n), it suffices to show that the scalar product ⟨∇(ln ϕ), ∇L_{X}⟩ is negative along ∂G_{ε}(X, n) for every sufficiently small ε > 0.

Let us compute the gradients involved. Taking into account the decomposition Tx̄G(X, n) = T_{x_{1}}X × … × T_{x_{n}}X, where x̄ ∈ G(X, n), one finds that the i-th components of the gradients ∇L_{X} and ∇(ln ϕ) are given by

(∇L_{X})_{i} = −\frac{x_{i−1} − x_{i}}{l_{i}} − \frac{x_{i+1} − x_{i}}{l_{i+1}} + (cos α_{i} + cos β_{i})ν_{i},  (25)

\frac{1}{2} (∇ ln ϕ)_{i} = \frac{x_{i−1} − x_{i}}{l_{i}^{2}} + \frac{x_{i+1} − x_{i}}{l_{i+1}^{2}} − (\frac{cos α_{i}}{l_{i}} + \frac{cos β_{i}}{l_{i+1}} )ν_{i}.  (26)

Denote by θ_{i} the angle between the vectors x_{i−1} − x_{i} and x_{i+1} − x_{i}. Due to strict convexity of X, we have θ_{i} ∈ [0, π). A direct computation using (25) and (26) shows that

⟨∇L, ∇ ln ϕ⟩ = 2(S_{1} + S_{2}),  (27)

where

S_{1} = − ∑_{i=1}^{n}(\frac{1}{l_{i}} + \frac{1}{l_{i+1}} )(1 + cos θ_{i}), S_{2} = ∑_{i=1}^{n}(cos α_{i} + cos β_{i})(\frac{cos α_{i}}{l_{i}} + \frac{cos β_{i}}{l_{i+1}} ). (28)

We want to show that the right-hand side of (27) is negative for all x̄ ∈ ∂G_{ε}(X, n) with ε > 0 small enough. It follows from inequalities (18) that

\frac{cos α_{i}}{l_{i}} + \frac{cos β_{i}}{l_{i+1}} < \frac{1}{r},

therefore S_{2} < 2n/r. We will be done once we show that S_{1} tends to −∞ as ε goes to zero.

Assume, to the contrary, that there exists a constant C and an infinite sequence x̄_{k} ∈ G(X, n), where k = 1, 2, …, such that ϕ(x̄_{k}) = ε_{k} tends to 0 and S_{1}(x̄_{k}) > −C. One has

(\frac{1}{l_{i}} + \frac{1}{l_{i+1}} )(1 + cos θ_{i}) < C, i = 1, …, n,  (29)

and l_{1} … l_{n} = ε_{k}^{1/2}.

Suppose that for an n-gon x̄_{k} ∈ ∂G_{ε_{k}}(X, n) and some index j one has an inequality l_{i} ≤ bϵ_{k}^{a}, where k is large enough, a > 0 and b > 0 (for example, for the smallest link l_{i}, one has: l_{i} ≤ (ε_{k})^{1/2n}. Then it follows from (29) that

1 + cos θ_{j} < Cbε_{k}^{a}, and hence π − θ_{j} < \frac{2}{\sqrt{3}}\sqrt{Cb}ε_{k}^{a/2}.

Since θ_{j} ≤ α_{j} + β_{j}, one concludes that

π/2 − β_{j} < \frac{2}{\sqrt{3}}\sqrt{Cb}ε_{k}^{a/2}, and cos β_{j} = sin(\frac{π}{2} − β_{i}) < \frac{2}{\sqrt{3}}\sqrt{Cb}ε_{k}^{a/2}.

It follows then from inequalities (18) that

l_{j+1} < \frac{4R}{\sqrt{3}}\sqrt{Cb}ε_{k}^{a/2}.  (30)

The argument that derives (30) from the initial assumption l_{i} ≤ bϵ_{k}^{a} can be repeated n times to conclude that there exist positive constants b, b′ and a_{0}, so that for any j = 1, 2, …, n and for any sufficiently large k one has:

l_{j} < bε_{k}^{a_{0}} and π − θ_{j} < b′ε_{k}^{a_{0}/2}

(we may take a_{0} = 2^{−n}n^{−1}).

Thus, for k large enough, ∑_{i=1}^{n}(π − θ_{i}) is close to zero. On the other hand, given a closed polygonal line in Euclidean space, the sum of its exterior angles, that is, the angles π − θ_{i}, is at least 2π (a smooth version of this statement holds too: the total curvature of a closed curve is at least 2π). This is a contradiction. □

Remark 4.2 L_{X} and ϕ are particular cases of the following more general function on inscribed polygons:

F (x_{1}, …, x_{n}) = ∑_{1}^{n} f(|x_{i+1} − x_{i}|)

where f(t) is a function of one variable (we obtain L_{X} when f(t) = t and ln ϕ when f(t) = 2 ln t). For some of these functions it is not true that all the critical n-tuples lie off a neighborhood of the singular set Σ. The simplest example is provided by f(t) = t^{2}. An analog of the condition (16) reads

\overrightarrow{x_{i}x_{i−1}} + \overrightarrow{x_{i}x_{i+1}} = t_{i}ν(x_{i}).

If X is a circle then this equation holds whenever either x_{i−1}x_{i}x_{i+1} is a right angle or |x_{i−1} − x_{i}| = |x_{i} − x_{i+1}|. In this ”billiard” we have closed trajectories in the form of arbitrary rectangles inscribed into the circle. Thus, we may have one pair of sides arbitrarily small.

Definition 4.3 Let X ⊂ R^{m+1} be a smooth hypersurface. Then X is n-generic if L_{X} : G(X, n) → R is a Morse function.

A justification for this definition is provided by the next lemma.

Lemma 4.4 There is a massive subset E in the space of embeddings S^{m} → R^{m+1} such that for every f ∈ E the hypersurface Im f is n-generic for all n.

Recall that the space of smooth maps from one manifold to another is considered in the Whitney C^{∞} topology; a massive set is a countable intersection of open dense sets. Due to the Baire property, a massive set is dense – see, [14].

Proof. Consider n germs of immersions

ϕ_{i} : (S^{m}, s_{i}) → (R^{m+1}, x_{i}), i = 1, …, n,

and assume that the targets x_{1}, …, x_{n} satisfy x_{i} ≠ x_{i+1} for i = 1, …, n. Then a germ of the respective perimeter length function (S^{m}, s_{1}) × … × (S^{m}, s_{n}) → R is defined:

L(t_{1}, …, t_{n}) = ∑_{1}^{n} |ϕ_{i+1}(t_{i+1}) − ϕ_{i}(t_{i})|.

Clearly, the first partial derivatives of L depend on the first derivatives of ϕ_{i}. Therefore we may consider the situation on the level of 1-jets. Namely, let

U ⊂ J^{1}(S^{m}, R^{m+1}) × … × J^{1}(S^{m}, R^{m+1})

consist of multi-jets of immersions satisfying the following three conditions:

(i) the targets x_{1}, …, x_{n} satisfy x_{i} ≠ x_{i+1} for i = 1, …, n;

(ii) the vector

ν_{i} = \frac{x_{i} − x_{i−1}}{|x_{i} − x_{i−1}|} + \frac{x_{i} − x_{i+1}}{|x_{i} − x_{i+1}|}  (31)

does not vanish for i = 1, …, n;

(iii) the sources t_{1}, …, t_{n} satisfy t_{i} ≠ t_{j} for all i ≠ j.

The first requirement is needed for L to be smooth and the third for the multi-jet transversality theorem to be applicable; the role of the second one will become clear shortly. Note that U is an open subset of the multi-jet space. Also consider the space of 1-jets of functions J^{1}(S^{m} × … × S^{m}), and let D be its submanifold of codimension mn consisting of the 1-jets with trivial differential. Assigning the 1-jet of the perimeter length function L to a multi-jet in U provides a map

π : U → J^{1}((S^{m})^{×n}).

Claim: the map π is transversal to the submanifold D.

Assuming the claim, the proof proceeds as follows. The 1-jet extension of the perimeter length function L determines the section j^{1}(L) of the 1-jet bundle

J^{1}((S^{m})^{×n}) → (S^{m})^{×n},

and the critical points of L are non-degenerate if and only if this section is transversal to D. The claim implies that Δ = π^{−1}(D) is a submanifold in U. According to the multi-jet transversality theorem (see [14]), there exists a massive set E_{n} of embeddings S^{m} → R^{m+1} such that for f ∈ E_{n} the multi-jet extension j^{1}(f) : G(S^{m}, n) → U is transversal to Δ. Since j^{1}(L) = π ∘ j^{1}(f), it follows from the claim that j^{1}(L) is transversal to D for f ∈ E_{n}, that is, the critical points of the perimeter length function are non-degenerate. Setting E = ∩_{n=2}^{∞}E_{n} completes the proof.

It remains to prove the italized claim above. It is convenient to choose local coordinates in the 1-jet spaces involved. Consider a multi-jet ϕ̄ = (ϕ_{1}, …, ϕ_{n}) ∈ Δ, and let s_{i} ∈ S^{m} be the source of ϕ_{i}. For each i identify a neighborhood of s_{i} with an open disk U ⊂ R^{m}. Then a neighborhood of ϕ̄ in J^{1}(S^{m}, R^{m+1})^{×n} is identified with J^{1}(U, R^{m+1})^{×n} and π becomes a map

π : J^{1}(U, R^{m+1})^{×n} → J^{1}(U^{×n}).

The 1-jet space J^{1}(U, R^{m+1}) consists of triples (u, x, A) where u ∈ U is the source, x ∈ R^{m+1} is the target and A : R^{m} → R^{m+1} is a linear map (derivative of a map U → R^{m+1}); the multi-jet space J^{1}(U, R^{m+1})^{×n} consists of n-tuples (u_{i}, x_{i}, A_{i}), i = 1, …, n, of such triples. The space J^{1}(U^{×n}) consists of 1-jets of functions ψ : U^{×n} → R, that is, of 2n + 1-tuples (u_{i}, p_{i}, z), i = 1, …, n, where u_{i} ∈ U, p_{i} = ∂ψ/∂u_{i} ∈ (R^{m+1})^{∗} is a covector and z = ψ(u_{1}, …, u_{n}) ∈ R. In these coordinates we explicitly describe the map π : (u_{i}, x_{i}, A_{i}) → (u_{i}, p_{i}, z):

z = ∑_{i=1}^{n} |x_{i+1} − x_{i}|, p_{i}(v_{i}) = ⟨ν_{i}, A_{i}(v_{i})⟩

where v_{i} ∈ R^{m} is a test vector and the vector ν_{i} is as in (31). The first formula is obvious and the second was established in the proof of Proposition 4.1. Identifying vectors and covectors by the Euclidean structure, one has:

p_{i} = A_{i}^{∗}(ν_{i}).  (32)

Consider a multi-jet ϕ̄ ∈ Δ ⊂ J^{1}(U, R^{m+1})^{×n}; as before, ϕ_{i} = (u_{i}, x_{i}, A_{i}). We want to show that

dπ(Tϕ̄J^{1}(U, R^{m+1})^{×n}) + T_{π(ϕ̄)}D = T_{π(ϕ̄)}J^{1}(U^{×n}).  (33)

The space T_{π(ϕ̄)}D consists of vectors whose p_{i} components vanish, while the u_{i} and z components are arbitrary. Thus the equality (33) will follow once we show that every vector in T_{π(ϕ̄)}J^{1}(U^{×n}) with trivial u_{i} and z components is in the image of dπ.

Consider an infinitesimal deformation (u_{i}, x_{i}, A_{i} + εB_{i}) of ϕ̄ ∈ J^{1}(U, R^{m+1})^{×n} where B_{i} : R^{m} → R^{m+1} is a linear map; this deformation determines a tangent vector ξ ∈ Tϕ̄J^{1}(U, R^{m+1})^{×n}. Since π(ϕ̄) ∈ D, it follows from (32) that A_{i}^{∗}(ν_{i}) = 0. Therefore dπ(ξ) is a vector in T_{π(ϕ̄)}J^{1}(U^{×n}) whose p_{i} component is B_{i}^{∗}(ν_{i}), while the u_{i} and z components vanish. Since ν_{i} ≠ 0, the vector B_{i}^{∗}(ν_{i}) ∈ R^{m} can be made arbitrary by varying B_{i}, and the result follows. □

We will use Proposition 4.1 to tackle the problem of finding topological lower bounds on the number of closed billiard trajectories by applying the methods of Morse theory. The function L_{X} being D_{n}-invariant, we will use equivariant Morse and Lusternik-Schnirelman theories. Namely, one has the next result.

Proposition 4.5 Let X ⊂ R^{m+1} be a smooth strictly convex hypersurface. Then for any odd n ≥ 3 the number of D_{n}-orbits of n-periodic billiard trajectories in X is greater than the cup-length of

H^{∗}(G(S^{m}, n)/D_{n}; Z_{2}).  (34)

If X is n-generic then the number of D_{n}-orbits of n-periodic billiard trajectories in X is not less than the sum of Betti numbers

∑_{i} dim_{Z_{2}} H^{i}(G(S^{m}, n)/D_{n}; Z_{2}).  (35)

Proof. Start with the following claim: for odd n the cohomology group of the quotient G(S^{m}, n)/D_{n} coincides with the equivariant cohomology of G(S^{m}, n):

H^{j}(G(S^{m}, n)/D_{n}; Z_{2}) ≃ H^{j}(ED_{n} ×_{D_{n}} G(S^{m}, n), Z_{2})  (36)

where ED_{n} is a contractible space with a free D_{n}-action.

Indeed, consider the Leray spectral sequence of the projection ED_{n} ×_{D_{n}} G(S^{m}, n) → G(S^{m}, n)/D_{n}, see [6]. The E_{2}-term has the form E_{2}^{p,q} = H^{p}(G(S^{m}, n)/D_{n}; L^{q}), where L^{q} is the Leray sheaf. Let x̄ ∈ G(S^{m}, n) be an orbit. Since n is odd, no reflection in D_{n} belongs to the stabilizer Hx̄. Thus the stabilizer of x̄ is a cyclic subgroup Hx̄ ⊂ D_{n} of odd order. The stalk of L^{q} over the orbit of x̄ is

H^{q}(ED_{n} ×_{D_{n}} D_{n}/Hx̄; Z_{2}) = H^{q}(ED_{n}/Hx̄; Z_{2}).

It follows that L^{q} = 0 for q > 0. Therefore the Leray spectral sequence is nonzero only along the p-axis. This implies (36).

Now we use Proposition 4.1. Fix a sufficiently small ε > 0 and consider the function L_{X} : G_{ε}(X, n) → R, determined by the billiard hypersurface X ⊂ R^{m+1}. Since L_{X} is D_{n}-invariant, the set of critical points of L_{X} is also D_{n}-invariant. Our task is to estimate the number of critical D_{n}-orbits. We have to take into account the presence of the boundary. However, due to statement (d) of Proposition 4.1, the boundary points make no contribution to the topology of G_{ε}(X, n) – cf. [2] and also [13], chapter 1. In other words, the principles of the critical point theory apply to L_{X} : G_{ε}(X, n) → R the same way as if G_{ε}(X, n) were a manifold without boundary.

Assume that X is n-generic. Then, using the negative gradient flow of L_{X}, we obtain a D_{n}-equivariant cell decomposition of G_{ε}(X, n). The number of cells in the resulting cell decomposition of the quotient space G_{ε}(X, n)/D_{n} equals the number of critical D_{n}-orbits of L_{X}, i.e., the number of D_{n}-orbits of n-periodic billiard trajectories in X. This proves the second statement of Proposition 4.5.

To prove the first statement, apply equivariant Lusternik - Schnirelman theory developed in [17], see also [9]. We use Theorems 3.2 and 1.13 from [17]. The first example in section 1.14 of [17] of singular multiplicative D_{n}-cohomology theory is given by Y ↦ H^{∗}(Y/D_{n}; Z_{2}); in our case H^{∗}(G(X, n)/D_{n}; Z_{2}) coincides with the equivariant cohomology H_{D_{n}}^{∗}(G(X, n); Z_{2}) due to (36) and statement (b) of Proposition 4.1. □

5 Equivariant cohomology of the cyclic configuration space of the sphere

In view of Proposition 4.5, in order to find lower bounds on the number of periodic trajectories of billiards we need to compute the equivariant cohomology ring of the cyclic configuration space G(S^{m}, n). This is the main goal of this section.

Our computation of the equivariant cohomology will be based on Theorem 4, giving the structure of the usual cohomology ring. We will also use the method which was first suggested by I. Babenko [1]. It consists in applying Morse theory in the opposite direction, that is, studying the topology of the cyclic configuration space G(S^{m}, n) by examining the billiard inside the round ball in R^{m+1}; in the latter case the closed trajectories are readily described. Note that a similar idea of studying the closed geodesics on the round sphere leads to a computation of the homology of the loop space of the sphere – see [7] and the references therein.

From now on we will assume that the number n is odd and that m > 1.

Let X = S^{m} ⊂ R^{m+1} be the unit sphere. Consider the corresponding length function L_{X} : G(X, n) → R. The critical points of L_{X} are precisely the closed billiard trajectories inside X having n reflections. Each such n-periodic trajectory lies in a two-dimensional plane P through the center of the sphere X. The intersection P ∩ X is a unit circle and the reflections go the same way as in the plane circular billiard P ∩ X. Hence any billiard trajectory (x_{1}, x_{2}, …, x_{n}) is a plane regular n-gon, possibly star-shaped, inscribed into the circle P ∩ X. If n is not prime then such a polygon may be multiple, i.e., it may traverse itself several times. The angle α between x_{1} and x_{2} is of the form α = 2πr/n, where 1 ≤ r ≤ (n − 1)/2. This number is clearly related to the rotation number (cf. Example 2.1). The following picture shows the 7-periodic trajectories:

[figure: r = 3; r = 2; r = 1]

Figure 3: Critical submanifolds for G(S^{n}, 7)

Note that any two n-periodic trajectories in X with the same rotation number r can be continuously deformed one to another. We conclude that the critical points of the function L_{X} : G(X, n) → R form a disjoint union of connected submanifolds

V_{0}, V_{1}, …, V_{(n−3)/2},  (37)

where V_{p} denotes the set of all closed trajectories with the rotation number (n − 1 − 2p)/2. Each V_{p} is diffeomorphic to the Stiefel manifold V_{2,m+1} and hence is a closed manifold of dimension 2m − 1. The next result is due to Babenko [1].

Proposition 5.1 If X = S^{m} ⊂ R^{m+1} is a round sphere then

(a) The function L_{X} : G(X, n) → R is nondegenerate in the sense of Bott.

(b) The index of the critical manifold V_{p} equals 2p(m − 1).

(c) The critical values of the function L_{X} on the critical manifolds V_{p} increase: L_{X}(V_{p}) < L_{X}(V_{p′}) for p < p′.

As an important addition to Proposition 5.1 we make the following observation.

Proposition 5.2 If X is a round sphere then L_{X} : G(X, n) → R is a perfect Bott function with respect to the field Z_{2}.

Proof. Let us first explain the meaning of our statement. Choose ε > 0 as in Proposition 4.1 holds. We may find constants c_{0}, …, c_{(n−3)/2} with

L_{X}(V_{p}) < c_{p} < L_{X}(V_{p+1}).

Set: F_{p} = L_{X}^{−1}((−∞, c_{p}]). We obtain a filtration

F_{0} ⊂ F_{1} ⊂ … ⊂ F_{(n−3)/2} = G_{ε}(X, n),  (38)

and our statement means that the sum of the Poincaré polynomials of the pairs (F_{p}, F_{p−1}) with Z_{2} coefficients equals the Poincaré polynomial of G_{ε}(X, n).

Indeed, the mod 2 Poincaré polynomial of (F_{p}, F_{p−1}) equals

∑_{j} t^{j} dim_{Z_{2}} H^{j}(F_{p}, F_{p−1}; Z_{2}) = t^{2p(m−1)}[t^{2m−1} + t^{m} + t^{m−1} + 1].  (39)

Here we used the fact that (F_{p}, F_{p−1}) is homotopy equivalent to the Thom space of the negative normal bundle of V_{p}, the Thom isomorphism, the index computation, given by Proposition 5.1, and the fact that the Poincaré polynomial with coefficients in Z_{2} of the Stiefel manifold V_{2,m+1} is t^{2m−1} + t^{m} + t^{m−1} + 1, cf. [5]. Summing formulae (39) for all p = 0, …, (n − 3)/2 we obtain

∑_{p=0}^{(n−3)/2} ∑_{j} t^{j} dim_{Z_{2}} H^{j}(F_{p}, F_{p−1}; Z_{2}) =

= (t^{2m−1} + t^{m} + t^{m−1} + 1) · \frac{t^{(n−1)(m−1)} − 1}{t^{2(m−1)} − 1} =

= \frac{(t^{m} + 1)(t^{(n−1)(m−1)} − 1)}{t^{(m−1)} − 1},

which, according to Theorem 4, coincides with the Poincaré polynomial of G(X, n). □

Propositions 5.1 and 5.2 hold for even n as well, but then their statements are slightly different. We will not need these results in this paper.

Remark 5.3 For m > 3 there exists a different proof of Proposition 5.2, which does not use Theorem 4 and provides an independent computation of the Poincaré polynomial of the cyclic configuration space of the sphere. It is quite straightforward. One considers the spectral sequence of the filtration (38), where

E_{1}^{p,q} = H^{p+q}(F_{p}, F_{p−1}; Z_{2}).

The calculation of the relative homology H^{p+q}(F_{p}, F_{p−1}; Z_{2}) as in the above proof and elementary geometric considerations show that all differentials must be zero provided m > 3.

This argument fails for m = 2 and for m = 3. The nonzero terms and the differential d_{1} of the spectral sequence for m = 2 and m = 3 are shown in Figure 4. Unlike the case m > 3, this picture does not imply that the differentials vanish.

Note also that for n = 3 the proof of Proposition 5.2 trivializes: the filtration (38) consists of a single term only.

[figure: p; q; p; q; m = 2; m = 3]

Figure 4: Nonzero terms of the spectral sequence for m = 2 and m = 3. The main result of this section is the next theorem.

Theorem 7 Let m ≥ 3 and let n be an odd integer. Then the cohomology ring H^{∗}(G(S^{m}; n)/D_{n}; Z_{2}) is multiplicatively generated by cohomology classes

σ_{2i} ∈ H^{2i(m−1)}(G(S^{m}; n)/D_{n}; Z_{2}), where i = 1, 2, …

and by classes

e ∈ H^{1}(G(S^{m}; n)/D_{n}; Z_{2}) and u ∈ H^{m}(G(S^{m}; n)/D_{n}; Z_{2}).

These classes satisfy the following relations (i) σ_{i} = 0 for i ≥ n − 1;

(ii) σ_{i}σ_{j} = \binom{i + j}{i}σ_{i+j} + ϵ_{ij}σ_{i+j−2}ue^{m−2}, where i and j are even and ϵ_{ij} ∈ Z_{2};

(iii) e^{m} = 0; (iv) u^{2} = 0.

The Poincaré polynomial of the quotient space G(S^{m}, n)/D_{n} with coefficients in Z_{2} equals

\frac{(t^{(n−1)(m−1)} − 1)}{(t^{2(m−1)} − 1)} · \frac{t^{m} − 1}{t − 1} · (t^{m} + 1),

and the sum of Betti numbers is m(n − 1).

The rest of this section consists of the proof of Theorem 7.

Start with the following lemma.

Lemma 5.4 Let G be a finite group acting simplicially on a finite polyhedron Y such that the action of G on the cohomology H^{∗}(Y; Z_{2}) is trivial. Suppose that G′ ⊂ G is a subgroup of odd index such that G acts trivially on H^{∗}(G′; Z_{2}). Then the induced homomorphism of the equivariant cohomology

H_{G}^{∗}(Y; Z_{2}) → H_{G′}^{∗}(Y; Z_{2})  (40)

is an isomorphism.

The notation H_{G}^{∗}(Y; Z_{2}) stands for the equivariant cohomology H^{∗}(EG ×_{G} Y; Z_{2}).

Proof. We will use the comparison theorem for spectral sequences. Since G acts trivially on H^{∗}(Y; Z_{2}), the Serre spectral sequence of the fibration EG ×_{G} Y → BG with fiber Y, converging to H_{G}^{∗}(Y; Z_{2}), has the initial term

E_{2}^{p,q} = H^{p}(G; Z_{2}) ⊗ H^{q}(Y; Z_{2}).

Similarly, we have a spectral sequence with the initial term

E′_{2}^{p,q} = H^{p}(G′; Z_{2}) ⊗ H^{q}(Y; Z_{2})

converging to H_{G′}^{∗}(Y; Z_{2}). The inclusion G′ → G induces a homomorphism of the spectral sequences E → E′ which is an isomorphism of the E_{2}-terms (cf. [8], Proposition 10.4, chapter 3). Hence, by the comparison theorem for spectral sequences, (40) is an isomorphism. □

Next we compute the D_{n}-equivariant cohomology of the critical manifolds of the function L_{X}.

Proposition 5.5 Suppose that m ≥ 3 and n is odd. Let V_{p} be any of the critical submanifolds (37) with the induced action of the dihedral group D_{n}. Then

(a) the equivariant cohomology ring H_{D_{n}}^{∗}(V_{p}; Z_{2}) has two multiplicative generators

e ∈ H_{D_{n}}^{1}(V_{p}; Z_{2}) and u ∈ H_{D_{n}}^{m}(V_{p}; Z_{2});

(b) they satisfy the relations e^{m} = 0 and u^{2} = 0; (c) the classes e^{i}u^{j}, with i = 0, 1, …, m − 1 and j = 0, 1 form an additive basis of

H_{D_{n}}^{∗}(V_{p}; Z_{2});

(d) the canonical homomorphism H^{∗}(V_{p}/D_{n}; Z_{2}) → H_{D_{n}}^{∗}(V_{p}; Z_{2}) is an isomorphism;

(e) the kernel of the canonical homomorphism Φ : H^{∗}(V_{p}/D_{n}; Z_{2}) → H^{∗}(V_{p}; Z_{2}) coincides with the ideal generated by e, i.e., is the linear span of the classes e^{i}u^{j} where i ≥ 1.

Proof. First note that (d) follows from the isomorphism (36) in the proof of Proposition 4.5. Indeed, the stabilizer of any orbit in V_{p} is a cyclic subgroup of odd order.

To compute the equivariant cohomology ring H_{D_{n}}^{∗}(V_{p}; Z_{2}) we will apply Lemma 5.4 with G = D_{n}, Y = V_{p} and G′ ≃ Z_{2} being a subgroup of the dihedral group D_{n} generated by a reflection. Since V_{p} is homeomorphic to the Stiefel manifold V_{2,m+1}, the cohomology of V_{p} with Z_{2} coefficients is isomorphic to Z_{2} in dimensions 0, m − 1, m, 2m − 1 and is trivial in all other dimensions. It follows that D_{n} acts trivially on H^{∗}(V_{p}; Z_{2}); likewise, D_{n} acts trivially on H^{∗}(G′; Z_{2}). Applying Lemma 5.4, we conclude that there exists a ring isomorphism

H_{D_{n}}^{∗}(V_{p}; Z_{2}) ≃ H_{G′}^{∗}(V_{p}; Z_{2}) ≃ H^{∗}(V_{p}/G′; Z_{2}).  (41)

The second isomorphism follows since the reflection acts freely on V_{p} for odd n. Formulae (41) show that, computing the equivariant cohomology H_{D_{n}}^{∗}(V_{p}; Z_{2}), we may ignore a large part of the D_{n}-action and keep track only of the G′-action.

Note that V_{p} can be identified with the variety of ordered pairs (v_{1}, v_{2}) of unit vectors in R^{m+1} making the angle of α_{p} = π(n − 1 − 2p)/n. The reflection (i.e. the generator of G′) acts on such pairs by sending (v_{1}, v_{2}) to (v_{1}, v′_{2}), where v′_{2} is the reflection of v_{2} in the line spanned by v_{1}.

Applying the Gram - Schmidt orthogonalization, construct a diffeomorphism between the quotient space V_{p}/G′ and the set of pairs (v_{1}, v_{2}) of mutually orthogonal unit vectors in R^{m+1} with identification (v_{1}, v_{2}) ≃ (v_{1}, −v_{2}). In other words, V_{p}/G′ is diffeomorphic to space of pairs (v, ℓ), where e ∈ R^{m+1} is a unit vector, and ℓ ⊂ R^{m+1} is a onedimensional linear subspace orthogonal to v. The projection (v, ℓ) ↦ v identifies V_{p}/G′ with the projective tangent bundle of S^{m}.

Alternatively, projecting (v, ℓ) ↦ ℓ, we view V_{p}/G′ as the space of a unit sphere bundle of a rank m vector bundle ξ over the projective space RP^{m}. The fiber of ξ over a line ℓ ∈ RP^{m} is the orthogonal complement ℓ^{⊥} of ℓ. The spectral sequence of this unit sphere bundle

E_{2}^{p q} = H^{p}(RP^{m}; Z_{2}) ⊗ H^{q}(S^{m−1}; Z_{2}) ⇒ H^{p+q}(V_{p}/G′; Z_{2})

has only two rows and the only possibly nontrivial differential is the transgression

d_{m} : E_{m}^{0,m−1} = H^{m−1}(S^{m−1}; Z_{2}) → E_{m}^{m,0} = H^{m}(RP^{m}; Z_{2}).

We claim that the differential d_{m} : E_{m}^{0,m−1} → E_{m}^{m,0} is an isomorphism. The image of the generator of H^{m−1}(S^{m−1}; Z_{2}) under d_{m} is the top Stiefel-Whitney class w_{m}(ξ) ∈ H^{m}(RP^{m}; Z_{2}). Let η be the tautological line bundle over RP^{m} whose fiber over a line ℓ is ℓ itself. Then the Whitney sum ξ ⊕ η is the trivial bundle of rank m + 1. Since the total Stiefel-Whitney class of η is 1 + e, where e ∈ H^{1}(RP^{m}; Z_{2}) is the generator, the Cartan’s formula

(1 + e)(1 + w_{1}(ξ) + … + w_{m}(ξ)) = 1  (42)

gives w_{j}(ξ) = e^{j}, for all j = 1, …, m. In particular, w_{m}(ξ) = e^{m}.

These arguments completely describe the ring structure of H^{∗}(V_{p}/G′; Z_{2}) ≃ H_{D_{n}}^{∗}(V_{p}; Z_{2}). Namely, in the above spectral sequence, the classes 1, e, …, e^{m−1} survive in the bottom row; also there is a class u ∈ E_{∞}^{1,m−1} such that the nonzero classes in row q = m − 1, surviving in E_{∞}, are u, ue, …, ue^{m−1}. The cohomology classes e and u lift uniquely from E_{∞} to H^{∗}(V_{p}/G′; Z_{2}) and satisfy the same relations therein. This proves statements (a), (b), (c).

It remains to prove statement (e). Clearly, Φ is a ring homomorphism and Φ(e) = 0. Therefore (e) will follow once we show that Φ : H^{m}(V_{p}/G′; Z_{2}) → H^{m}(V_{p}; Z_{2}) is an isomorphism in degree m.

Consider the product V_{p}×[0, 1] and identify the points (x, 1) and (T x, 1) for all x ∈ V_{p}, where T ∈ G′ denotes the reflection. After the identification, we obtain a compact 2mdimensional manifold with boundary Y, so that Y is homotopy equivalent to V_{p}/G′ and ∂Y is diffeomorphic to V_{p}. The restriction homomorphism H^{j}(Y; Z_{2}) → H^{j}(∂Y; Z_{2}) coincides with Φ. Using the Poincaré duality we obtain

H^{j}(Y, ∂Y; Z_{2}) ≃ H^{2m−j}(Y; Z_{2}) ≃ Z_{2} for all j.  (43)

Consider the exact cohomological sequence of (Y, ∂Y). The homomorphisms H^{j}(Y; Z_{2}) → H^{j}(∂Y; Z_{2}) are zero for j = m − 1 and j = m + 1, therefore the homomorphisms Z_{2} = H^{m−1}(∂Y; Z_{2}) → H^{m}(Y, ∂Y; Z_{2}) = Z_{2} and Z_{2} = H^{m+1}(Y, ∂Y; Z_{2}) → H^{m+1}(Y; Z_{2}) = Z_{2} are isomorphisms. It follows that Φ is an isomorphism in the following exact sequence (we suppress the coefficients Z_{2} from notation):

H^{m−1}(∂Y) →^{≃} H^{m}(Y, ∂Y) → H^{m}(Y) →^{Φ} H^{m}(∂Y) → H^{m+1}(Y, ∂Y) →^{≃} H^{m+1}(Y).

The proposition is proved. □

Proof of Theorem 7. Let X ⊂ R^{m+1} be a round sphere. Consider filtration (38) of the space G_{ε}(X, n), where ε > 0 is small enough, so that the claims of Proposition 4.1 hold. Denote the space F_{i}/D_{n} by F′_{i}. Hence we obtain a filtration

F′_{0} ⊂ F′_{1} ⊂ … ⊂ F′_{(n−3)/2} = G_{ε}(X, n)/D_{n}  (44)

of the space of D_{n}-orbits.

Formulate the following inductive hypothesis F_{p}, depending on a number p = 0, 1, …, (n − 3)/2:

The cohomology ring H^{∗}(F′_{p}; Z_{2}) is multiplicatively generated by cohomology classes

σ_{2i} ∈ H^{2i(m−1)}(F′_{p}; Z_{2}), where i = 1, 2, …

and by classes

e ∈ H^{1}(F′_{p}; Z_{2}) and u ∈ H^{m}(F′_{p}; Z_{2}).

These classes satisfy the following relations

(i) σ_{i} = 0 for i > 2p;

(ii) σ_{i}σ_{j} = \binom{i + j}{i}σ_{i+j} + ϵ_{ij}σ_{i+j−2}ue^{m−2}, where i and j are even and ϵ_{ij} ∈ Z_{2};

(iii) e^{m} = 0;

(iv) u^{2} = 0.

The Poincaré polynomial of the quotient space F′_{p} with coefficients in Z_{2} equals

\frac{(t^{2(p+1)(m−1)} − 1)}{(t^{2(m−1)} − 1)} · \frac{t^{m} − 1}{t − 1} · (t^{m} + 1),

and the sum of Betti numbers is 2m(p + 1).

The kernel of the canonical homomorphism Φ : H^{∗}(F′_{p}; Z_{2}) → H^{∗}(F_{p}; Z_{2}) coincides with the ideal generated by e.

Our aim is to show that statement F_{p} holds for p = (n − 3)/2. This would imply Theorem 7. Argue by induction. Proposition 5.5 implies that F_{0} holds. Hence we need to show that F_{p} implies F_{p+1}.

Assuming that F_{p} is satisfied, consider the boundary homomorphism

δ : H^{i}(F′_{p}; Z_{2}) → H^{i+1}(F′_{p+1}, F′_{p}; Z_{2}).  (45)

We claim that this homomorphism is trivial for all i. Indeed, the group H^{i}(F′_{p}; Z_{2}) is nonzero only for i ≤ 2(p + 1)(m − 1) + 1 (by the assumption F_{p}) and the group H^{i+1}(F′_{p+1}, F′_{p}; Z_{2}) is nonzero only for i + 1 ≥ 2(p + 1)(m − 1). The latter follows since F′_{p+1}/F′_{p} is homotopy equivalent to the Thom space of a vector bundle of rank 2(p + 1)(m − 1) over V_{p+1}/D_{n}, the space of D_{n}-orbits of the critical manifold V_{p+1}. Therefore homomorphism (45) can be nonzero only for three values of i, namely for

i = 2(p + 1)(m − 1) − 1, i = 2(p + 1)(m − 1) and i = 2(p + 1)(m − 1) + 1.

Let us first show that δ vanishes for i = 2(p + 1)(m − 1) − 1, i.e., in the lowest possible dimension. Consider the critical manifold V_{p+1} ⊂ F_{p+1} and the decomposition of its normal bundle into the negative and positive parts with respect to the Hessian of function L_{X}. This decomposition is D_{n}-equivariant, and after factorization by the action of the dihedral group D_{n} it produces two bundles (the negative and the positive) over V_{p+1}/D_{n}. The space F′_{p+1}/F′_{p} is homotopy equivalent to the Thom space of the negative bundle. Consider the commutative diagram

H^{i}(F′_{p}; Z_{2}) ⟶^{δ} H^{i+1}(F′_{p+1}, F′_{p}; Z_{2})
Φ ↓  ↓ Φ_{1}
H^{i}(F_{p}; Z_{2}) ⟶^{δ_{1}} H^{i+1}(F_{p+1}, F_{p}; Z_{2}).

Here Φ_{1} is defined similarly to Φ, i.e. it is induced by the canonical projection (F_{p+1}, F_{p}) → (F′_{p+1}, F′_{p}). We know that δ_{1} vanishes (by Proposition 5.2). Hence, to show that δ vanishes for i = 2(p + 1)(m − 1) − 1, it suffices to show that Φ_{1} is a monomorphism for this value of i. We have the following commutative diagram with the vertical maps being the Thom isomomorphisms

H^{i+1}(F′_{p+1}, F′_{p}; Z_{2}) ⟶^{Φ_{1}} H^{i+1}(F_{p+1}, F_{p}; Z_{2})
≃ ↓  ↓ ≃
H^{0}(V′_{p+1}; Z_{2}) ⟶^{Φ_{2}} H^{0}(V_{p+1}; Z_{2}).

It is clear that the homomorphism Φ_{2} is an isomorphism (compare statement (e) of Proposition 5.5), and therefore Φ_{1} is also an isomorphism for i = 2(p + 1)(m − 1) − 1.

Hence δ vanishes for i = 2(p + 1)(m − 1) − 1, in other words, δ(σ_{2p}ue^{m−3}) = 0. Recall that we assume that m ≥ 3. Now we will show that δ vanishes in the two other dimensions as well:

δ(σ_{2p}ue^{m−2}) = 0, δ(σ_{2p}ue^{m−1}) = 0.

It is clear that there exists a class ẽ ∈ H^{1}(F′_{p+1}; Z_{2}) such that ẽ|_{F′_{p}} = e. Using statement 12 in §6, chapter 5 of [18], we obtain

δ(σ_{2p}ue^{m−2}) = δ(σ_{2p}ue^{m−3}) · ẽ = 0.

Similarly, δ(σ_{2p}ue^{m−1}) = 0.

The vanishing of the boundary homomorphism (45) means that filtration (44) is also perfect, that is, we have an isomorphism

H^{∗}(F′_{p+1}; Z_{2}) ≃ H^{∗}(F′_{p}; Z_{2}) ⊕ H^{∗}(F′_{p+1}/F′_{p}; Z_{2})  (46)

The additive structure of the relative homology H^{∗}(F′_{p+1}/F′_{p}; Z_{2}) is given by Proposition 5.5 with a shift of all degrees by 2(p + 1)(m − 1). Here we use the Thom isomorphism and the equality between the equivariant cohomology and the cohomology of the factor space V_{p+1}/D_{n}, which holds by statement (d) of Proposition 5.5. Hence (46) fully describes the additive structure of the cohomology H^{∗}(F′_{p+1}; Z_{2}), which coincides with the statement of the hypothesis F_{p+1}.

Now we want to show that the multiplicative structure of H^{∗}(F′_{p+1}; Z_{2}) is as stated in the hypothesis F_{p+1}. Notice first that the cohomology classes e, u, σ_{2}, …, σ_{2p} in H^{∗}(F′_{p}; Z_{2}) extend uniquely to cohomology classes in H^{∗}(F′_{p+1}; Z_{2}) and denote the extensions by the same symbols. Our next problem is to identify the class σ_{2p+2} ∈ H^{2(p+1)(m−1)}(F′_{p+1}; Z_{2}); caution must be exercised since the cohomology of F′_{p+1} in this degree is two-dimensional.

Consider again the critical manifold V_{p+1} ⊂ F_{p+1} and the D_{n}-equivariant decomposition of its normal bundle into the negative and positive parts with respect to the Hessian of the function L_{X}. After factorization by the action of the dihedral group, these bundles give two bundles (the negative and the positive) over V_{p+1}/D_{n}; the space F′_{p+1}/F′_{p} is homotopy equivalent to the Thom space of the negative bundle. The Thom class of the negative bundle lies in H^{2(p+1)(m−1)}(F′_{p+1}/F′_{p}; Z_{2}), but the last group is canonically embedded into H^{2(p+1)(m−1)}(F′_{p+1}; Z_{2}) via (46). Hence we define the class σ_{2p+2} as representing this Thom class.

All the generators having been defined, we want to check that the hypothesis F_{p+1} is satisfied. First we note that σ_{2p+2}u and σ_{2p+2}e^{m−1} are nonzero cohomology classes in H^{∗}(F′_{p+1}; Z_{2}). This would follow from the Thom Isomorphism Theorem once we show that e^{m−1} and u restrict to nontrivial cohomology classes on V_{p+1}/D_{n}. Nontriviality of the restriction of e^{m−1} is almost obvious; indeed, for any (m−2)-connected D_{n}-invariant subset A ⊂ F_{p+1} we have e^{m−1}|_{A/D_{n}} ≠ 0, as follows by considering the Serre spectral sequence. In order to show that u|_{V_{p+1}/D_{n}} ≠ 0, it is enough to show that u|_{V_{p+1}} ≠ 0 (by statement (e) of Proposition 5.5). Let W = V_{3,m+1} be the Stiefel manifold of triples (e_{1}, e_{2}, e_{3}) of mutually orthogonal unit vectors in R^{m+1}. For any p = 0, 1, …, (n−3)/2 denote by I_{p} : W → G(S^{m}, n) the following map

I_{p}(e_{1}, e_{2}, e_{3}) = (x_{1}, x_{2}, …, x_{n}), x_{j+1} = cos(jα_{p})e_{1} + sin(jα_{p})e_{2},

where j = 0, …, n − 1 and α_{p} = π(n − 1 − 2p)/n. The image of I_{p} is the critical submanifold V_{p}. Hence, it is enough to show that I_{p}^{∗}(u) ≠ 0 for any p. Construct a homotopy between I_{p} and I_{0}. Let

H_{τ} : W → G(S^{m}, n), τ ∈ [0, 1],

be defined by H_{τ}(e_{1}, e_{2}, e_{3}) = (x_{1}, x_{2}, …, x_{n}), where

x_{j+1} = cos(jα_{τ})e_{1} + sin(jα_{τ})e_{2}, and α_{τ} = π(n − 1 − 2p(1 − τ))/n,

for j = 0, 1, 2, …, n − 2, while

x_{n} = cos((n − 1)α_{τ})e_{1} + sin((n − 1)α_{τ})e_{2} + sin(πτ)e_{3}.

It is clear that H_{0} = I_{p} and H_{1} = I_{0}. Since u|_{V_{0}} ≠ 0 we obtain I_{0}^{∗}(u) ≠ 0 as a consequence of Proposition 10.3 in [5]. This Proposition describes the cohomology of Stiefel manifolds with Z_{2} coefficients; it implies that I_{0}^{∗} : H^{∗}(V_{p}; Z_{2}) → H^{∗}(W; Z_{2}) is a monomorphism. Hence, it follows I_{p}^{∗}(u) = I_{0}^{∗}(u) ≠ 0 and thus u|_{V_{p}} ≠ 0.

We want to show that for even i and j with i + j = 2p + 2 the following relation holds in the ring H^{∗}(F′_{p+1}; Z_{2}):

σ_{i}σ_{j} = \binom{2p + 2}{i}σ_{2p+2} + ϵ_{ij}σ_{2p}ue^{m−2},  (47)

where ϵ_{ij} ∈ Z_{2}. Note that for r = 2(p + 1)(m − 1) the homomorphism

Φ : H^{r}(F′_{p+1}; Z_{2}) → H^{r}(F_{p+1}; Z_{2})

has one-dimensional image and one-dimensional kernel. For any even r = 2, 4, …, 2p+2 the image of the Thom class σ_{r} under Φ is nonzero and so it equals the class Φ(σ_{r}) ∈ H^{∗}(F_{p+1}; Z_{2}), which was denoted in Theorem 4 by σ_{r}. From Theorem 4 we know that

Φ(σ_{i})Φ(σ_{j}) = \binom{2p + 2}{i}Φ(σ_{2p+2}).

This proves (47) since σ_{2p}ue^{m−2} belongs to the kernel of Φ.

The rest of properties in hypothesis F_{p+1} are now obvious. Thus F_{p} implies F_{p+1}, and the proof of Theorem 7 is complete. □

Remark 5.6 The above proof shows that the function L_{X} : G(X, n) → R is perfect with respect to the field Z_{2} in the D_{n}-equivariant sense as well, compare Proposition 5.2.

We conclude the paper with a proof of Theorem 1 formulated in Introduction.

Proof of Theorem 1. Statement (B) follows from Proposition 4.5 and Theorem 7 that states that the sum of Betti numbers of the space G(S^{m}; n)/D_{n} is m(n − 1).

To prove statement (A) of Theorem 1 one needs to find the cup-length of H^{∗}(G(S^{m}, n)/D_{n}; Z_{2}). The argument is similar to the one in Corollaries 5 and 6. Namely, the length of the longest nontrivial product of classes σ_{2i} from Theorem 7 equals the maximal number of 1’s in the binary expansion, that even integers, not exceeding n − 3, may have. This number is equal to the maximal number of 1’s in the binary expansion, that integers, not exceeding (n − 3)/2, may have, that is, to

[log_{2}((n − 3)/2 + 1)] = [log_{2}(n − 1)] − 1.

We also have the classes u and e^{m−1} at our disposal; therefore the cup-length equals [log_{2}(n − 1)] + m − 1. Statement (A) follows now from Proposition 4.5. □

References

[1] I. Babenko, Periodic trajectories in three-dimensional Birkhoff billiards, Math. USSR Sbornik, 71 (1992), 1–13.

[2] E. Baiada, M. Morse, Homotopy and homology related to the Schoenflies problem, Annals of Math., 58 (1953), 142–165.

[3] V. Bangert, Mather sets for twist maps and geodesics on tori, Dynamics Reported, 1 (1988), 1–56.

[4] G. Birkhoff. Dynamical systems. Amer. Math. Soc. Coll. Publ., 9, 1927.

[5] A. Borel, Sur la cohomologie des espaces principaux et des homogenes de groupes de Lie compactes, Ann. Math., 57(1953), 115 - 207

[6] A. Borel. Seminar on transformation groups. Ann. of Math. Stud., 46, Princeton Univ. Press, 1960.

[7] R. Bott, Lectures on Morse theory, old and new, Bull. Amer. Math. Soc., 7 (1982), 331–358.

[8] K. Brown. Cohomology of groups. Springer-Verlag, 1982.

[9] M. Clapp, D. Puppe, Critical point theory with symmetries, J. reiner angew. Math., 418 (1991), 1–29.

[10] F. Cohen, Artin’s braid group, classical homotopy theory, and sundry other curiosities, Amer. Math. Soc. Contemp. Math., 78 (1988), 167–206.

[11] F. Cohen, On configuration spaces, their homology, and Lie algebras, J. Pure Appl. Alg., 100 (1995), 19–42.

[12] H. Croft, H. Swinnerton-Dyer, On the Steinhaus billiard table problem, Proc. Camb. Phil. Soc., 59 (1963), 37–41.

[13] M. Farber. Critical point theory for closed 1-forms. Monograph (in preparation).

[14] M. Golubitsky, V. Guillemin. Stable mappings and their singularities. Springer-Verlag, 1974.

[15] M. Goresky, R. MacPherson. Stratified Morse theory. Springer-Verlag, 1988.

[16] V. Kozlov, D. Treshchev. Billiards, a generic introduction to the dynamics of systems with impacts. Amer. Math. Soc. Transl. Math. Monogr., 89, 1991.

[17] W. Marzantowicz, A G-Lusternik-Schnirelman category of a space with an action of a compact Lie group, Topology, 28 (1989), 403–412.

[18] E. Spanier. Algebraic topology. McGraw-Hill Book Co, 1966.

[19] S. Tabachnikov. Billiards. SMF ”Panoramas et Syntheses”, 1, 1995.

[20] B. Totaro, Configuration spaces of algebraic varieties, Topology, 35 (1996), 1057– 1067.

[21] V. Vassiliev. Complements of discriminants of smooth maps: topology and applications. Amer. Math. Soc. Transl. of Math. Monogr., 98, 1992.