The Integral Cohomology Algebras of

Ordered Configuration Spaces of Spheres

Eva Maria Feichtner and Günter M. Ziegler

Received: August 12, 1999

Communicated by Ulf Rehmann

Abstract. We compute the cohomology algebras of spaces of ordered point configurations on spheres, F (S^{k}, n), with integer coefficients. For k = 2 we describe a product structure that splits F (S^{2}, n) into well-studied spaces. For k > 2 we analyze the spectral sequence associated to a classical fiber map on the configuration space. In both cases we obtain a complete and explicit description of the integer cohomology algebra of F (S^{k}, n) in terms of generators, relations and linear bases. There is 2-torsion occuring if and only if k is even. We explain this phenomenon by relating it to the Euler classes of spheres.

Our rather classical methods uncover combinatorial structures at the core of the problem.

2000 Mathematics Subject Classification: Primary 55M99; Secondary: 57N65, 55R20, 52C35 Keywords and Phrases: spheres, ordered configuration spaces, subspace arrangements, integral cohomology algebra, fibration, Serre spectral sequence

1 Introduction

The space of configurations of n pairwise distinct labelled points in a topological space X,

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

is called the n-th (ordered) configuration space of X. A systematic study of these spaces started with work by Fadell & Neuwirth [FaN] and Fadell [Fa] in the sixties. They introduced sequences of fibrations for configuration spaces and mainly concentrated on describing their homotopy groups for various instances of X. In 1969 Arnol′d [Ar] derived the integer cohomology algebra of F (ℂ, n) — the group cohomology of the colored braid group — and thereby initiated still ongoing research on the cohomology algebras of complements of linear subspace arrangements. Broader interest in the cohomology algebras of configuration spaces came up in the seventies: The cohomology of F (X, n) for a manifold X appeared as a basic ingredient in the E_{2}-terms of spectral sequences for the Gelfand-Fuks cohomology of the manifold [GF] and for the homology of certain function spaces [An]. Cohen [C1, C2] studied various aspects of the cohomology of configuration spaces of Euclidean spaces in view of its relation to homology operations for iterated loop spaces [C3]. Cohen & Taylor [CT1, CT2] described the cohomology algebras of configuration spaces of spheres with coefficients in a field of characteristic different from 2. Recently, compactifications of configuration spaces of algebraic varieties have been constructed by Fulton and MacPherson [FM]. As an application, they determine the rational homotopy type of configuration spaces of non-singular compact complex algebraic varieties F (X, n) in terms of invariants of X. Compare also work of Kriz [Kr] and Totaro [T], where alternative minimal models for F (X, n) are used. In contrast to these results on the rational homotopy type of configuration spaces, it seems that so far Arnol′d’s computation of the integer cohomology algebra of F (ℂ, n) remained the only instance where the integer cohomology algebra of an ordered configuration space was fully described.

Recently, Raoul Bott asked about the integer cohomology algebra of the ordered configuration space of the 2-sphere. We are able to answer his question by describing a product decomposition for F (S^{2}, n):

F (S^{2}, n) ≅ PSL(2, ℂ) × M_{0,n},

where M_{0,n}, the moduli space of n-punctured complex projective lines, is homotopy equivalent to the complement of an affine complex hyperplane arrangement. We deduce that H^{∗}(F (S^{2}, n), ℤ) has (only) 2-torsion that can be traced back to H^{2}(PSL(2, ℂ), ℤ) ≅ ℤ_{2} (Section 2). For spheres of higher dimension we use spectral sequences to obtain an analogous decomposition on the level of cohomology algebras:

H^{∗}(F (S^{k}, n), ℤ) ≅ (ℤ ⊕ ℤ) ⊗ H^{∗}(M(A_{n−2}^{(k)}), ℤ)  for odd k ,
H^{∗}(F (S^{k}, n), ℤ) ≅ (ℤ ⊕ ℤ_{2} ⊕ ℤ) ⊗ H^{∗}(M(A_{Π_{3}}), ℤ) for even k ,

where M(A_{n−2}^{(k)}) is the complement of a certain arrangement of real linear subspaces A_{n−2}^{(k)} and M(A_{Π_{3}}) is the complement of an arrangement of affine subspaces that is naturally related to the linear arrangement A_{n−2}^{(k)}. For both arrangement complements the integer cohomology algebra is torsion-free and we have explicit descriptions in terms of generators, relations and linear bases. In the following all (co)homology is taken with ℤ-coefficients.

The key for our approach is a family of locally trivial fiber maps on configuration spaces that appears already in the work by Fadell & Neuwirth [FaN] and Fadell [Fa]. The maps are given by “projection to the last r points” of a configuration. For configuration spaces of spheres F (S^{k}, n) and 1 ≤ r < n the projection Π_{r} reads as follows:

Π_{r} = Π_{r}(S^{k}, n) :  F (S^{k}, n) ⟶ F (S^{k}, r)
(x_{1}, …, x_{n}) ⟼ (x_{n−r+1}, …, x_{n}) .

We derive the integer cohomology algebra of F (S^{k}, n) for k > 2 by a complete discussion of the Leray-Serre spectral sequence associated to the fiber map Π_{1}(S^{k}, n). Our success with this rather classical approach depends on the fact that the fibers of Π_{1}(S^{k}, n) are complements of linear subspace arrangements. Their cohomology algebras are well-studied objects both from topological and combinatorial viewpoints [GM, BZ, Bj, DP]. The fibers of Π_{1}(S^{k}, n) are in fact the complements of codimension k versions of the classical braid arrangements, and thus they are particularly prominent examples of arrangement complements. This paves the way for a complete discussion of the associated spectral sequence (Section 3). A distinction between the configuration spaces of spheres of odd and even dimension emerges from the only possibly non-trivial differential of the spectral sequence. We present two methods to compute this differential (Section 4).

(1) It can be derived from one particular cohomology group of F (S^{k}, n). To obtain the latter we use an independent, rather elementary approach to the cohomology of configuration spaces, which may be of interest on its own right.

(2) We show that the differential can be interpreted as a map that is induced by “multiplication with the Euler class of S^{k}.” It is well-known that the Euler class depends on the parity of k.

To get the final tableau of the spectral sequence, and to derive the integer cohomology algebra of the configuration space F (S^{k}, n), we use combinatorially constructed ℤ-linear bases for the cohomology of the fiber (Section 5). In the last section of this paper we consider the bundle structures on F (S^{k}, n) given by the fiber maps Π_{r}(S^{k}, n), 1 < r < n. We show that the associated spectral sequences collapse in their second terms unless k is even and r equals 1 or 2. For some parameters we can decide the triviality of the bundle structure, which in general is a difficult question.

For configuration spaces of closed manifolds other than spheres, in principle one can attempt to follow the approach taken in this paper. However, with the cohomology of the manifold (i.e., of the base space of the considered fiber map) getting more complicated, the corresponding spectral sequence will be less sparse, and thus more non-trivial differentials will have to be considered. Even more importantly, if the manifold is not simply connected, then it is not straightforward, and not true in general, that the system of local coefficients on the manifold induced by the fiber map is simple. Already the entries of the second sequence tableau thus will be much harder to compute.

Acknowledgment: We are grateful for discussions with Ezra Getzler that influenced the course of these investigations. Also, we wish to thank Raoul Bott who asked us about connections to the Euler classes of spheres.

2 Configuration spaces of the 2-sphere

We first comment on some special cases for small values of n and on the configuration space of the 1-sphere. For n = 1, we see from the definition that F (X, 1) = X for all spaces X. For n = 2, we consider the projection Π_{1}, sending a configuration in F (S^{k}, 2) to its second point. We obtain a fiber bundle with contractible fiber Π_{1}^{−1}(x_{2}) = F (S^{k} \{x_{2}}, 1) ≅ ℝ^{k}, hence F (S^{k}, 2) ≃ S^{k}. In fact, F (S^{k}, 2) is equivalent to the tangent bundle over S^{k}. For the configuration space of the 1-sphere, F (S^{1}, n), we state an explicit trivialization of the fiber bundle given by Π_{1}, the projection to the last point of a configuration. Using the group structure on S^{1} we define a homeomorphism which shows that Π_{1}(S^{1}, n) is a trivial fiber map:

φ_{1} : F (S^{1} \ {e}, n − 1) × S^{1} ⟶ F (S^{1}, n)
((x_{1}, …, x_{n−1}) , y) ⟼ (yx_{1}, …, yx_{n−1}, y) .

For r > 1, the fiber of Π_{r}(S^{1}, n) is homeomorphic to the space of configurations of n − r points on r disjoint copies of the unit interval. We obtain a homeomorphism

φ_{r} : F ( ⨄_{r} (0, 1) , n − r) × F (S^{1}, r) ⟶ F (S^{1}, n)

that trivializes the bundle by “inserting” the points x_{1}, …, x_{n−r} from ⨄_{r} (0, 1) into the r open segments in which the points of the configuration (y_{1}, …, y_{r}) in F (S^{1}, r) separate S^{1}.

Compared to configuration spaces of higher dimensional spheres we gain the main structural advantage for the 2-dimensional case from the fact that the 2-sphere S^{2} is homeomorphic to the complex projective line ℂP^{1}. We will freely switch between the resulting two viewpoints on the configuration space in question. The group of projective automorphisms PSL(2, ℂ) of ℂP^{1} acts freely on the configuration space F (ℂP^{1}, n) by coordinatewise action, thus exhibiting F (ℂP^{1}, n) as the total space of a principal PSL(2, ℂ)-bundle for n ≥ 3 [Ge]. We identify the base space — the space of n-tuples of distinct points on the complex projective line modulo projective automorphisms — as the moduli space M_{0,n} of n-punctured complex projective lines. Compactifications of M_{0,n} and their cohomology algebras are the focus of recent research; for a brief account and further references see [FM, p.189].

Theorem 2.1 The configuration space F (ℂP^{1}, n) of the complex projective line is the total space of a trivial PSL(2, ℂ)-bundle over M_{0,n} for n ≥ 3; hence there is a homeomorphism

F (ℂP^{1}, n) ≅ PSL(2, ℂ) × M_{0,n} .

Proof. The automorphism group PSL(2, ℂ) acts sharply 3-transitive on ℂP^{1}. In particular, we obtain a homeomorphism between the configuration space of three distinct points on ℂP^{1} and the automorphism group PSL(2, ℂ):

ϕ : F (ℂP^{1}, 3) ⟶ PSL(2, ℂ) .

Here (x_{1}, x_{2}, x_{3}) ∈ F (ℂP^{1}, 3) is mapped to the unique automorphism that transforms x_{1} to \binom{1}{0}, x_{2} to \binom{0}{1}, and x_{3} to \binom{1}{1}, i.e., to the “standard projective basis” of ℂP^{1}. Given a configuration x = (x_{1}, …, x_{n}) of n distinct points on ℂP^{1}, the group element ϕ(x_{1}, x_{2}, x_{3}) transforms x to a configuration on ℂP^{1} that has the standard projective basis in its first three entries. We describe the resulting configuration by the columns of a (2 × n)-matrix:

ϕ(x_{1}, x_{2}, x_{3}) ∘ x = ( \begin{matrix}1 & 0 & 1 & z_{3} & … & z_{n−1} \\ 0 & 1 & 1 & 1 & … & 1 \end{matrix} ) ,

where z_{i} ∈ ℂ\{0, 1} for 3 ≤ i ≤ n − 1, z_{i} ≠ z_{j} for 3 ≤ i < j ≤ n − 1, and the columns are understood as vectors in ℂ^{2}\{0} that represent elements in ℂP^{1}. Lifting an element x̄ ∈ M_{0,n} to its “normal form” ϕ(x_{1}, x_{2}, x_{3}) ∘ x in the total space F (ℂP^{1}, n) defines a section for the PSL(2, ℂ)-bundle. Hence, the principal bundle is trivial [St, Part I, Thm. 8.3]. The resulting product decomposition on F (ℂP^{1}, n) can be described explicitly by the homeomorphism

Φ : F (ℂP^{1}, n) ⟶ PSL(2, ℂ) × M_{0,n}
(x_{1}, …, x_{n}) ⟼ ( ϕ(x_{1}, x_{2}, x_{3}) , x̄) . □

Remark 2.2 An analogous argument is not possible for S^{4}, since there are no sharply 3-transitive group actions in the case of a non-commutative field such as ℍ. The structural reason for this can be traced back to a theorem by von Staudt, see [P, Kap. 5.1.4].

In view of a description of the integer cohomology algebra of F (ℂP^{1}, n) we use the intimate relation of the base space M_{0,n} to a complex hyperplane arrangement — the complex braid arrangement A_{n−2}^{ℂ} of rank n − 2 in ℂ^{n−1} given by the hyperplanes

z_{j} − z_{i} = 0 for 1 ≤ i < j ≤ n − 1 .

This arrangement is a key example in the theory of hyperplane arrangements and initiated much of its development [Ar, OT]. Its complement, M(A_{n−2}^{ℂ}) := ℂ^{n−1}\ ⋃ A_{n−2}^{ℂ}, coincides with F (ℂ, n − 1), the configuration space of the complex plane.

The base space M_{0,n} is homotopy equivalent to the complement of the affine arrangement ^{aff}A_{n−2}^{ℂ}, which is obtained from A_{n−2}^{ℂ} by restriction to the affine hyperplane {z_{2} − z_{1} = 1} ≅ ℂ^{n−2}. A complete description of the integer cohomology algebra of the complement M(^{aff}A_{n−2}^{ℂ}) := ℂ^{n−2}\ ⋃^{aff}A_{n−2}^{ℂ} is provided by general theory on the topology of complex hyperplane arrangements [OS, BZ, OT]. The description depends only on combinatorial data of the arrangement, i.e., on the semi-lattice of intersections L(^{aff}A_{n−2}^{ℂ}) which is customarily ordered by reverse inclusion.

Proposition 2.3 The base space M_{0,n} is homotopy equivalent to the complement of the affine complex braid arrangement of rank n − 2, since

M_{0,n} × ℂ ≅ M(^{aff}A_{n−2}^{ℂ}) .

Its integer cohomology algebra is torsion-free. It is generated by one-dimen-sional classes e_{i,j} for 1 ≤ i < j ≤ n − 1, (i, j) ≠ (1, 2), and has a presentation as a quotient of the exterior algebra on these generators:

H^{∗}(M(^{aff}A_{n−2}^{ℂ})) ≅ Λ^{∗} ℤ^{\binom{n−1}{2}−1} / I ,

where I is the ideal generated by elements of the form

e_{i,l} ∧ e_{j,l} − e_{i,j} ∧ e_{j,l} + e_{i,j} ∧ e_{i,l} for 1 ≤ i < j < l ≤ n − 1, (i, j) ≠ (1, 2) ,
e_{1,i} ∧ e_{2,i} for 2 < i ≤ n − 1 .

Proof. We consider the homeomorphic image of M_{0,n} under the section defined in the proof of Proposition 2.1:

M_{0,n} ≅ \{ ( \begin{matrix}1 & 0 & 1 & z_{3} & … & z_{n−1} \\ 0 & 1 & 1 & 1 & … & 1 \end{matrix} ) | z_{i} ∈ ℂ\\{0, 1\}, z_{i} ≠ z_{j} for i ≠ j\}
≅ { (z_{1}, …, z_{n−1}) | z_{i} ∈ ℂ, z_{i} ≠ z_{j} for i ≠ j, z_{1} = 0, z_{2} − z_{1} = 1} .

From this description we see that M_{0,n} is homeomorphic to the complement of the affine braid arrangement ^{aff}A_{n−2}^{ℂ} intersected with the hyperplane {z_{1} = 0}. This intersection operation is equivalent to a projection parallel to the intersection of all the hyperplanes in A_{n−2}^{ℂ}, ⋂ A_{n−2}^{ℂ} = {z_{1} = … = z_{n−1}}. The fibers of this projection map are contractible: they are translates of ⋂ A_{n−2}^{ℂ}. Hence the projection does not alter the homotopy type, and we conclude that M_{0,n} is homotopy equivalent to M(^{aff}A_{n−2}^{ℂ}). The presentation of the integer cohomology algebra follows from general results on the topology of the complements of complex hyperplane arrangements (compare [OT]). □

We have seen that the fiber PSL(2, ℂ) is homeomorphic to F (ℂP^{1}, 3), resp. F (S^{2}, 3). By a result of Fadell [Fa, Thm. 2.4] there is a fiber homotopy equivalence between F (S^{k}, 3) and V_{k+1,2}, the Stiefel manifold of orthogonal 2-frames in ℝ^{k+1}. The cohomology of the latter is well-known, see [Bd, Ch. IV, Exp. 13.5]. Combining the product structure on F (ℂP^{1}, n) obtained in Theorem 2.1 with the information on the cohomology algebras of base space and fiber we conclude:

Theorem 2.4 The cohomology algebra of F (S^{2}, n) with integer coefficients is given by

H^{∗}(F (S^{2}, n)) ≅  H^{∗}(F (S^{2}, 3)) ⊗ H^{∗}(M(^{aff}A_{n−2}^{ℂ}))
≅ ( ℤ(0) ⊕ ℤ_{2}(2) ⊕ ℤ(3)) ⊗ Λ^{∗} ⨁_{\binom{n−1}{2}−1} ℤ(1) / I ,

where G(i) denotes a direct summand G in dimension i, and I is the ideal of relations described in Proposition 2.3.

3 A spectral sequence for H^{∗}(F (S^{k}, n))

Our approach for k > 2 uses the Leray-Serre spectral sequence associated with the projection Π_{1}:

Π_{1} :  F (S^{k}, n) ⟶ S^{k}
(x_{1}, …, x_{n}) ⟼ x_{n} .

For the construction and special features of Leray-Serre spectral sequences we refer to Borel [Bo2, Sect. 2]. Since the base space of the considered fiber bundle is a sphere we could equally work with the Wang sequence [Wh, Ch. VII, Sect. 3], a long exact sequence connecting the cohomology of the total space and of the fiber. However, the derivation of the multiplicative structure of the cohomology algebra gets more transparent with spectral sequence tableaux. Moreover, this approach extends to projections Π_{r} for r > 1 (see Section 6).

We meet especially favorable conditions in the second tableau of the Leray-Serre spectral sequence associated to the fiber map Π_{1}(S^{k}, n): The base space S^{k} is simply connected for k ≥ 2, hence the system of local coefficients on S^{k} induced by Π_{1} for k ≥ 2 is simple. As the fiber over x_{n} ∈ S^{k} we obtain:

Π_{1}^{−1}(x_{n}) = {(x_{1}, …, x_{n−1}) ∈ (S^{k})^{n−1} | x_{i} ≠ x_{j} for i ≠ j,
x_{i} ≠ x_{n} for i = 1, …, n−1}
≅ {(x_{1}, …, x_{n−1}) ∈ (ℝ^{k})^{n−1} | x_{i} ≠ x_{j} for i ≠ j} .

This is the complement of the real k-braid arrangement A_{n−2}^{(k)} of rank n−2 which is formed by linear subspaces U_{i,j} in (ℝ^{k})^{n−1}, 1 ≤ i < j ≤ n−1,

U_{i,j} = {(x_{1}, …, x_{n−1}) ∈ (ℝ^{k})^{n−1} | x_{i_{1}} = x_{j_{1}}, …, x_{i_{k}} = x_{j_{k}}} .

This arrangement, a direct generalization of the real and complex braid arrangements, is a k-arrangement in the sense of Goresky & MacPherson [GM, Part III, p. 239]: the subspaces have codimension k, and the codimensions of their intersections are multiples of k. Such arrangements have combinatorial properties analogous to those of complex hyperplane arrangements, which is reflected by strong similarities in their topological properties: The cohomology algebras of real k-arrangements are torsion-free [GM, Part III, Thm. B]; they are generated in dimension k − 1 by cohomology classes that naturally correspond to the subspaces of the arrangement [BZ, Sect. 9]. The complement of the real k-braid arrangement A_{n−2}^{(k)} is an ordered configuration space: the space F (ℝ^{k}, n − 1) of configurations of n − 1 pairwise distinct points in ℝ^{k}. The following thus complements work by Cohen [C1, C2], who discussed the cohomology of F (ℝ^{k}, n − 1) in connection with homology operations for iterated loop spaces.

Proposition 3.1 The integer cohomology algebra of M(A_{n−2}^{(k)}) is generated by (k − 1)-dimensional cohomology classes c_{i,j}, 1 ≤ i < j ≤ n − 1. It has a presentation as a quotient of the exterior algebra on these generators:

H^{∗}(M(A_{n−2}^{(k)})) ≅ Λ^{∗} ℤ^{\binom{n−1}{2}} / I ,

where I is the ideal generated by the elements

(c_{i,l}∧c_{j,l}) + (−1)^{k+1}(c_{i,j} ∧c_{j,l}) + (c_{i,j} ∧c_{i,l}) for 1 ≤ i < j < l ≤ n−1 .

Remark 3.2 The generating cohomology classes c_{i,j}, 1 ≤ i < j ≤ n−1, are defined by restricting cohomology generators ĉ_{i,j} for the subspace complements M({U_{i,j}}) ≃ S^{k−1} to the complement of the arrangement. A canonical choice of the generators ĉ_{i,j} results from fixing the natural “frame of hyperplanes” in the sense of [BZ, Sect. 9]. Proof. Björner & Ziegler [BZ, Sect. 9] derived a presentation for the cohomology algebras of real k-arrangements up to the signs in the relations. For the real k-braid arrangement their presentation specializes up to signs to the one stated above. Consider the relation for a triple 1 ≤ i < j < l ≤ n − 1:

ε_{1}(c_{i,l} ∧ c_{j,l}) + ε_{2}(c_{i,j} ∧ c_{j,l}) + ε_{3}(c_{i,j} ∧ c_{i,l}) = 0 , ε_{r} ∈ {±1} for r = 1, 2, 3 .

Transpositions of (i, j) and (i, l) and of (i, l) and (j, l) in the linear (lexicographic) order of the subspaces in A_{n−2}^{(k)} lead to similar relations among the cohomology classes c_{i,l} ∧ c_{j,l}, c_{i,j} ∧ c_{j,l}, and c_{i,j} ∧ c_{i,l}:

ε_{1}(c_{i,j} ∧ c_{j,l}) + ε_{2}(c_{i,l} ∧ c_{j,l}) + ε_{3}(c_{i,l} ∧ c_{i,j}) = 0
ε_{1}(c_{j,l} ∧ c_{i,l}) + ε_{2}(c_{i,j} ∧ c_{i,l}) + ε_{3}(c_{i,j} ∧ c_{j,l}) = 0 .

Anti-commutativity of the exterior product yields the signs in the relations.

□

We obtain the following tensor product decomposition on the E^{2}-tableau of the Leray-Serre spectral sequence associated with the fiber map Π_{1}(S^{k}, n):

k − 1^{0} 0  ^{k}  H^{∗}(M(A_{n−2}^{(k)})) E_{2}^{∗,∗}^{3k − 3}^{2k − 2}  ^{d_{k}}  \begin{matrix}E_{2}^{p,q} ≅ H^{p}(S^{k}) ⊗ H^{q}(M(A_{n−2}^{(k)})) , \\ p, q ≥ 0 . \end{matrix}

The location of non-zero entries shows that there is only one possibly non-trivial differential on stage k of the sequence.

4 The k-th differential

The tableaux of a cohomological spectral sequence are bigraded algebras. The differentials respect their multiplicative structure. In particular, the differentials are determined by their action on multiplicative generators of the sequence tableaux. Thus, it suffices in our case to describe d_{k} on the multiplicative generators c_{i,j}, 1 ≤ i < j ≤ n − 1, of E_{k}^{0,∗} ≅ H^{∗}(M(A_{n−2}^{(k)})) in dimension k − 1.

Actually, we can restrict our attention even further to the action of d_{k} on one single generator, say on c_{1,2}: The permutation of the first n − 1 points of a configuration in F (S^{k}, n) by 𝔖_{n−1} gives a group action on the considered fiber bundle and hence induces a 𝔖_{n−1}-action on the spectral sequence. The group 𝔖_{n−1} acts transitively on the generators c_{i,j} of E_{k}^{0,k−1}, whereas it keeps E_{k}^{k,0} fixed. We conclude that

d_{k}(c_{i,j}) = d_{k}(c_{1,2}) for 1 ≤ i < j ≤ n − 1 .

In the following we provide two independent ways to evaluate d_{k}.

4.1 … via a homology group of the discriminant.

Here the key observation is that knowing H^{k}(F (S^{k}, n)) is sufficient to determine d_{k}. To obtain this specific group, we use a “Vassiliev type” argument that allows one to compute, in favorable situations, some cohomology groups of configuration spaces. Using a smooth compactification, in our case given by F (S^{k}, n) ⊆ (S^{k})^{n}, we set

F (S^{k}, n) = (S^{k})^{n} \ Γ_{n} = (S^{k})^{n} \ ⋃_{1≤i<j≤n}(Γ_{n})_{i,j ,}

where

(Γ_{n})_{i,j} = {(x_{1}, …, x_{n}) ∈ (S^{k})^{n} | x_{i} = x_{j}} for 1 ≤ i < j ≤ n .

The idea is to use duality theorems in (S^{k})^{n} for transferring homology information about the discriminant Γ_{n} to the cohomology of F (S^{k}, n). For this, we proceed in three steps. Step 1. Determine H_{∗}(Γ_{n}) in dimensions (n − 1)k and (n − 1)k − 1. The spaces (Γ_{n})_{i,j} are homeomorphic to (S^{k})^{n−1}; they intersect in spaces homeomorphic to (S^{k})^{n−2}, hence in dimension k(n − 2). By a Mayer-Vietoris argument we obtain the top two homology groups of the discriminant:

H_{(n−1)k} (Γ_{n}) ≅ ⨁_{1≤i<j≤n} H_{(n−1)k}((Γ_{n})_{i,j}) ≅ ℤ^{\binom{n}{2}}
H_{(n−1)k−1} (Γ_{n}) =  0 .

Step 2. Determine the relative homology H_{∗}((S^{k})^{n}, Γ_{n}) in dimension (n−1)k. The relevant part of the long exact sequence in homology for the pair ((S^{k})^{n}, Γ_{n}) is the following:

→ H_{(n−1)k} (Γ_{n}) →^{i_{∗}} H_{(n−1)k} ((S^{k})^{n}) → H_{(n−1)k} ((S^{k})^{n}, Γ_{n}) → H_{(n−1)k−1} (Γ_{n}) →

We had computed that the last group is zero, and thus

H_{(n−1)k} ((S^{k})^{n}, Γ_{n}) ≅ coker i_{∗} ,

where i_{∗} is induced by the inclusion i : Γ_{n} ↪ (S^{k})^{n}. We intend to write i_{∗} as a (n × \binom{n}{2})-matrix over ℤ and to read the cokernel from its Smith normal form [Mu, § 11]. For this we choose ℤ-bases for the homology groups that are involved, and determine i_{∗} in terms of these bases. According to the Künneth Theorem, H_{(n−1)k}((S^{k})^{n}) has a basis that consists of tensor products of k-dimensional classes ω_{j}, j = 1, …, n, of the form

ν_{i} = ω_{1} ⊗ … ⊗ ω̂_{i} ⊗ … ⊗ ω_{n} , i = 1, …, n ,

where ω_{j} is an orientation class for the j-th factor in (S^{k})^{n}, and ω̂_{i} denotes that we omit the i-th class. Generating homology classes of Γ_{n} in dimension (n − 1)k are given by the \binom{n}{2} generating homology classes for the spaces (Γ_{n})_{i,j}, 1 ≤ i < j ≤ n. These spaces are products of k-spheres,

(Γ_{n})_{i,j} ≅ S_{i,j} × S_{1} × … × Ŝ_{i} × … × Ŝ_{j} × … × S_{n} ,

with S_{l} denoting the l-th k-sphere appearing as a factor in (S^{k})^{n}, whereas S_{i,j} denotes the k-sphere diagonally embedded in the i-th and j-th k-sphere. A generating homology class for (Γ_{n})_{i,j} in dimension (n − 1)k can be described

as ν_{ij} = ω_{ij} ⊗ ω_{1} ⊗ … ⊗ ω̂_{i} ⊗ … ⊗ ω̂_{j} ⊗ … ⊗ ω_{n} , 1 ≤ i < j ≤ n ,

where ω_{ij} is a homology generator for S_{i,j} in dimension k.

To understand how i_{∗} maps such generators ν_{ij} we use the following lemma. It tells how to describe the homology generator of the diagonal in S_{i} × S_{j} in terms of homology classes of the product.

Lemma 4.1 Let ω denote a generating homology class in dimension k for the k-sphere. Under the diagonal map Δ : S^{k} → S^{k}×S^{k}, Δ(x) = (x, x) for x ∈ S^{k}, the homology class ω is mapped to

Δ_{∗}(w) = ω ⊗ 1 + 1 ⊗ ω .

Proof. By the Künneth Theorem the two summands form a basis of H_{k}(S^{k} × S^{k}), so Δ_{∗}(ω) is a ℤ-linear combination of those. Moreover, the diagonal map combined with one of the projections pr_{i} to the respective factor is the identity map on S^{k}. Hence the result follows from (pr_{i})_{∗} ∘ Δ_{∗}(ω) = ω for i = 1, 2. □ We conclude that

i_{∗}(ν_{ij}) = ((ω_{i} ⊗ 1) + (1 ⊗ ω_{j})) ⊗ ⨂_{{l=1 \atop l≠i,j}}^{n} ω_{l} , 1 ≤ i < j ≤ n .

To write this in terms of the generators ν_{i} for H_{(n−1)k} ((S^{k})^{n}) we have to permute the factors of the underlying product space to the order used above in the definition of the classes ν_{i}. The tensor product of homology classes is anti-commutative [FFG, Ch. II, §16]; i.e., under the transposition map τ : X × X ⟶ X × X, (x_{1}, x_{2}) ↦ (x_{2}, x_{1}), a product of homology classes α ⊗ β, α, β ∈ H_{∗}(X), is mapped to

τ_{∗}(α ⊗ β) = (−1)^{deg(α) deg(β)}β ⊗ α.

This is the point where the distinction between odd and even dimensions comes up:

i_{∗}(ν_{ij}) = \{ \begin{matrix}(−1)^{i−1}ν_{j} + (−1)^{j−2}ν_{i} & for odd k , \\ ν_{j} + ν_{i} & for even k \end{matrix} (1 ≤ i < j ≤ n).

Writing i_{∗} as a (n × \binom{n}{2})-matrix M(n) we obtain the (unsigned) incidence matrix of 2-element subsets of an n-set for even k, whereas for odd k a certain sign pattern occurs on the matrix entries. For example,

M(3) = ( \begin{matrix}& 12 & 13 & 23 \\ 1 & 1 & (−1)^{k} & 0 \\ 2 & 1 & 0 & (−1)^{k} \\ 3 & 0 & 1 & (−1)^{k} \end{matrix} ) ,

M(4) = ( \begin{matrix}& 12 & 13 & 14 & 23 & 24 & 34 \\ 1 & 1 & (−1)^{k} & 1 & 0 & 0 & 0 \\ 2 & 1 & 0 & 0 & (−1)^{k} & 1 & 0 \\ 3 & 0 & 1 & 0 & (−1)^{k} & 0 & 1 \\ 4 & 0 & 0 & 1 & 0 & (−1)^{k} & 1 \end{matrix} ) .

We now derive the Smith normal forms of the matrices M(n) by describing elementary row and column operations. Ordering the columns of M(n) – corresponding to the 2-element subsets of {1, …, n} – lexicographically, we see that

M(n) = ( \begin{matrix}1 & ⋯ & · · · · · · & 1 & 0 & ⋯ & · · · · · · · · · · · · & ⋯ & 0 \\ 1 \\ . & . & . \\ & & ... & & & & M(n−1) \\ & & & 1 \end{matrix} ) for even k, and

M(n) = ( \begin{matrix}1 & −1 & ⋯ & (−1)^{n} & 0 & ⋯ & · · · · · · · · · · · · & ⋯ & 0 \\ 1 \\ . & . & . \\ & & ... & & & & −M(n−1) \\ & & & 1 \end{matrix} ) for odd k .

For even k, we subtract the i-th row from the first row for i = 2, …, n, and thus create 0-entries in the left part of the first row and entries −2 on top of the submatrix M(n−1). Note that the column sum in M(n−1) is 2. Adding multiples of the first n−1 columns to the rest of the matrix, we transform M(n−1) to 0. The remaining entries in the first row can be reduced to one single entry 2, and after switching rows and columns we obtain the following Smith normal form:

SNF( M(n) ) = ( \begin{matrix}1 \\ & ... & & & 0 \\ & & 1 \\ & & & 2 \end{matrix} ) for even k.

For odd k, we add the t-th row multiplied with (−1)^{t−1} to the first row for i = 2, …, n. This creates 0-entries in the first row. This is obvious for the first n−1 columns. For an entry on top of a column of the submatrix −M(n−1) which contains entries in its i-th and j-th rows, we obtain

(−1)^{i} · (−(−1)^{j−2}) + (−1)^{j} · (−(−1)^{i−1}) = 0 .

As before, we transform the submatrix −M(n−1) to 0 by adding multiples of the first n−1 columns. Thus, after switching rows, we obtain:

SNF( M(n) ) = ( \begin{matrix}1 \\ & ... & & & 0 \\ & & 1 \\ & & & 0 \end{matrix} ) for odd k.

We read off the cokernel of i_{∗} as

H_{(n−1)k} ((S^{k})^{n}, Γ_{n}) ≅ \{ \begin{matrix}ℤ & for odd k , \\ ℤ_{2} & for even k . \end{matrix}

Step 3. Apply Poincaré-Lefschetz duality between relative homology of the pair ((S^{k})^{n}, Γ_{n}) and cohomology of F (S^{k}, n).

Proposition 4.2 The k-th cohomology group of F (S^{k}, n), k > 2, n > 2, is given by

H^{k}(F (S^{k}, n)) ≅ \{ \begin{matrix}ℤ & for odd k , \\ ℤ_{2} & for even k . \end{matrix}

Remark 4.3 In principle, the discriminant approach can be used to determine the cohomology of F (S^{k}, n) as a graded group. However, to compute H_{∗}(Γ_{n}) is difficult and requires extra tools (interpretation of Γ_{n} as a homotopy limit of a diagram of spaces, study of a spectral sequence converging to the homology of a homotopy limit [ZŽ, Sect. 3(e)]). Also, the study of the pair sequence gets considerably more involved. Moreover, because of the use of Poincaré-Lefschetz duality the multiplicative structure of H^{∗}(F (S^{k}, n)) seems out of reach for this approach.

The partial result of Proposition 4.2 allows us to determine the differential in the spectral sequence associated to Π_{1}(S^{k}, n). Taking cohomology of E_{k}^{∗,∗} with respect to the differential d_{k} leads to the final sequence tableau E_{k+1}^{∗,∗}:

0  ^{0}  ^{0}^{k}ν  H^{k}(F (S^{k}, n)) E_{k+1}^{∗,∗}^{k − 1}  d_{k}  E_{k}^{∗,∗}  0^{0}  ^{k} c_{i,j}  ker d_{k}^{k − 1}  coker d_{k}

Since there is only one non-zero entry on the k-th diagonal for k > 2, H^{k}(F (S^{k}, n)) can be read from E_{k+1}^{∗,∗}:

H^{k}(F (S^{k}, n)) ≅ coker d_{k} .

Our result on H^{k}(F (S^{k}, n)) in Proposition 4.2 implies that

d_{k}(c_{1,2}) = d_{k}(c_{i,j}) = \{ \begin{matrix}0 & for odd k \\ 2ν & for even k , \end{matrix}

where ν is a generator of H^{k}(S^{k}).

4.2 … via an interpretation in terms of the Euler class.

Our second approach to the differential d_{k} stays within the setting of fiber bundles. We study an inclusion of fiber bundles and transfer information on the differentials via the induced homomorphism of spectral sequences. We will find that the differential is determined by the Euler class of the base space S^{k}, which depends on the parity of k.

Consider, for n ≥ 3, the following space of point configurations on S^{k}, k > 2:

F̂ := {(x_{1}, …, x_{n}) ∈ (S^{k})^{n} | x_{1} ≠ x_{2}, x_{j} ≠ x_{n} for j = 1, …, n − 1} .

Projection of a configuration to its last point, Π̂ : F̂ → S^{k}, makes it the total space of a fiber bundle with spherical fiber: the complement of the codimension k subspace U_{1,2} in (ℝ^{k})^{n−1},

Π̂^{−1}(x_{n}) = {(x_{1}, …, x_{n−1}) ∈ (S^{k})^{n−1} | x_{1} ≠ x_{2}, x_{j} ≠ x_{n} for 1 ≤ j ≤ n−1}
≅ {(x_{1}, …, x_{n−1}) ∈ (ℝ^{k})^{n−1} | x_{1} ≠ x_{2}}
= M({U_{1,2}}) .

The spectral sequence Ê_{∗} associated to Π̂ has an Ê_{2}-tableau of the form

\begin{matrix}Ê_{2}^{p,q} ≅ H^{p}(S^{k}) ⊗ H^{q}(M(\{U_{1,2}\})) , \\ p, q ≥ 0 . \end{matrix} ^{k − 1}^{0}  ^{k}^{0}  Ê_{2}^{∗,∗} d̂_{k}

From the location of non-zero entries in Ê_{2}^{∗,∗} we easily see that there is only one possibly non-trivial differential d̂_{k} on stage k of the sequence. The inclusion of F (S^{k}, n) into F̂ is a map of fiber bundles.

F̂ S^{k} F (S^{k}, n) S^{k} M(A_{n−2}^{(k)}) M( {U_{1,2}})

The homomorphism of spectral sequences induced by the inclusion of the fiber bundles factors on the Ê_{k}-tableau into the induced map between the cohomology of the fibers and the identity on the cohomology of the base space [Bo1, Exp. VIII, Thm. 4]. The map i^{∗} between the cohomology of the fibers maps the generator ĉ_{1,2} of H^{k−1}(M({U_{1,2}})) to c_{1,2} in H^{k−1}(M(A_{n−2}^{(k)})) (compare Remark 3.2). Hence, we are left to determine the action of the k-th differential on Ê_{k}^{0,k−1}:

d_{k}  i_{k}^{∗} Ê_{k}^{∗,∗} E_{k}^{∗,∗} c_{1,2}  ^{id^{∗}} ĉ_{1,2}  d̂_{k} d_{k}(c_{1,2}) = d_{k}(i^{∗}(ĉ_{1,2})) = d̂_{k}(ĉ_{1,2}) .

Proposition 4.4 The fiber bundle F̂ over S^{k} is fiber homotopy equivalent to V_{k+1,2}, the Stiefel manifold of orthogonal 2-frames in ℝ^{k+1}, considered as fiber bundle over S^{k}.

Proof. F̂ is fiber homotopy equivalent to F (S^{k}, 3), both spaces considered as fiber bundles over S^{k}. The fiber homotopy equivalence is realized by the projection of configurations in F̂ to their first, second and last points. In turn, F (S^{k}, 3) is fiber homotopy equivalent to the Stiefel manifold V_{k+1,2} [Fa, Thm. 2.4]. □

For a simply connected, k-dimensional, orientable manifold M the only possibly non-trivial differential in the spectral sequence associated to the unit tangent bundle can be described as a cup product multiplication with the Euler class of the manifold:

d_{k}(x ⊗ μ) = d_{k}(μ) ⌣ x = χ_{M} ⌣ x ,

where μ is a generator of H^{k−1}(S^{k−1}), x ∈ H^{∗}(M), and χ_{M} denotes the Euler class of the manifold (compare [MS, Thm. 12.2]). The Stiefel manifold V_{k+1,2} coincides with the unit tangent bundle on S^{k}. Given an orientation on S^{k} and a generator ν of H^{k}(S^{k}) that evaluates to 1 on the orientation class, the Euler class of S^{k} is given by

χ_{S_{k}} = \{ \begin{matrix}0 & for odd k , \\ 2ν & for even k . \end{matrix}

We conclude that in the spectral sequence for F̂ the differential d̂_{k} maps the generator ĉ_{1,2} of H^{k−1}(M({U_{1,2}})) to the Euler class χ of the base space, once an orientation for the base S^{k} and with it the Euler class have been chosen appropriately. In particular, d̂_{k} is the zero-map for odd k. For our initial fiber bundle we thus derive

d_{k}(c_{i,j}) = d_{k}(c_{1,2}) = \{ \begin{matrix}0 & for odd k , \\ 2ν & for even k , \end{matrix}

where 2ν is the Euler class of the k-sphere under appropriate orientation.

5 Recovering H^{∗}(F (S^{k}, n)) from the spectral sequence

For configuration spaces of odd-dimensional spheres we now have enough information to derive a complete description of the integer cohomology algebra. In the previous section we showed that the k-th differential is trivial on multiplicative generators of the sequence tableau E_{k}^{∗,∗}, therefore it is trivial on all of E_{k}^{∗,∗}. The spectral sequence collapses in its second term; a favorable location of non-zero tableau entries allows us to get both the linear and the multiplicative structure of H^{∗}(F (S^{k}, n)) directly from the second tableau:

Theorem 5.1 For a sphere S^{k} of odd dimension k ≥ 3, and n ≥ 3, the integer cohomology algebra of F (S^{k}, n) is given by

H^{∗}(F (S^{k}, n)) ≅ H^{∗}(S^{k}) ⊗ H^{∗}(M(A_{n−2}^{(k)}))
≅ ( ℤ(0) ⊕ ℤ(k) ) ⊗ Λ^{∗} \begin{matrix}⨁ \\ ( & {n−1 \atop 2} & ) \end{matrix} ℤ(k − 1) / I ,

where I is the ideal described in Proposition 3.1. In particular, the cohomology is free.

For the case of even-dimensional spheres the considerations in the previous section show that the k-th differential is non-zero. We have to describe the kernel and cokernel of that differential and with it the final sequence tableau E_{k+1}^{∗,∗} in a manageable form. The cohomology algebra of the fiber, hence of the left-most column of the second, resp. k-th tableau, is given by Proposition 3.1. A linear basis for this algebra is given by the products of (k − 1)-dimensional classes c_{i,j} associated with the faces of the broken circuit complex BC(L) of the intersection lattice L = L(A_{n−2}^{(k)}) [BZ, Sect. 9]:

B_{BC} = {c_{α_{1}} ∧ … ∧ c_{α_{t}} | {α_{1}, …, α_{t}} ∈ BC(L)} .

Here is a different basis which enables us to describe the kernel of d_{k} both as a direct summand and as a subalgebra of H^{∗}(M(A_{n−2}^{(k)})):

Proposition 5.2 The following set is a ℤ-linear basis for H^{∗}(M(A_{n−2}^{(k)})) :

\begin{aligned}B′ & = & \{c_{1,2} ∧ (c_{α_{1}} − c_{1,2}) ∧ … ∧ (c_{α_{t}} − c_{1,2}) | \{α_{1}, …, α_{t}\} ∈ BC(L), α_{i} ≠ (1, 2)\} \\ & & ∪ \{(c_{α_{1}} − c_{1,2}) ∧ … ∧ (c_{α_{t}} − c_{1,2}) | \{α_{1}, …, α_{t}\} ∈ BC(L), α_{i} ≠ (1, 2)\} . \end{aligned}

Proof. Each element in B_{BC} can be written as a linear combination of elements in B′. This is true for each element having c_{1,2} as a factor because those are themselves elements in B′. For c_{α_{1}} ∧ … ∧ c_{α_{t}}, {α_{1}, …, α_{t}} ∈ BC(L), α_{i} ≠ (1, 2),

(c_{α_{1}} − c_{1,2}) ∧ … ∧ (c_{α_{t}} − c_{1,2}) = c_{α_{1}} ∧ … ∧ c_{α_{t}} + β ,

where β is a linear combination of products containing c_{1,2}, hence of elements in B′. Thus c_{α_{1}} ∧ … ∧ c_{α_{t}} can be written as a linear combination of those. □

Let T^{•} denote the submodule of H^{∗}(M(A_{n−2}^{(k)})) generated by those elements of B′ that contain c_{1,2} as a factor, whereas T^{∘} denotes the submodule generated by all other elements of B′:

H^{∗}(M(A_{n−2}^{(k)})) ≅ T^{∘} ⊕ T^{•} .

Obviously, multiplication within T^{•} is trivial, whereas for T^{∘} we can state the following:

Proposition 5.3 The submodule T^{∘} is a subalgebra of H^{∗}(M(A_{n−2}^{(k)})) generated by the elements c̄_{i,j} := (c_{i,j} − c_{1,2}) in dimension k − 1, 1 ≤ i < j ≤ n − 1, (i, j) ≠ (1, 2). It has a presentation as a quotient of the exterior algebra on these generators:

T^{∘} ≅ Λ^{∗}ℤ^{\binom{n−1}{2}−1} / J ,

where J is the ideal generated by elements of the form

(c̄_{i,l} ∧ c̄_{j,l}) + (−1)^{k+1}(c̄_{i,j} ∧ c̄_{j,l}) + (c̄_{i,j} ∧ c̄_{i,l}) , 1 ≤ i < j < l ≤ n−1 ,
(i, j) ≠ (1, 2) ,
(c̄_{1,i} ∧ c̄_{2,i}) , 2 < i ≤ n−1 .

Proof. It is clear that T^{∘} has a presentation as a quotient of the exterior algebra on the generators c̄_{i,j} = (c_{i,j} − c_{1,2}), 1 ≤ i < j ≤ n − 1, (i, j) ≠ (1, 2). Moreover, it is easy to check that the proposed relations hold in H^{∗}(M(A_{n−2}^{(k)})). To see that they generate the ideal for a presentation of T^{∘} note that they allow one to write each product in the generators c̄_{i,j} as a linear combination of elements from the linear basis for T^{∘}: Assume that for a product of generators

c̄_{α_{1}} ∧ … ∧ c̄_{α_{t}}

all products with lexicographically smaller index set can be written as a linear combination of basis elements from T^{∘}. If this product is not itself a basis element then {α_{1}, …, α_{t}} contains a broken circuit of L(A_{n−2}^{(k)}). In case (1, 2) extends it to a circuit the product is zero by a relation of the second type. Otherwise, a relation of the first type allows to write it as a linear combination of products with lexicographically smaller index set, and hence as a linear combination of basis elements. □

Our results on d_{k} now read as follows:

d_{k}(c_{1,2}) = 2 ν
d_{k}(c_{i,j} − c_{1,2}) = 0  for 1 ≤ i < j ≤ n − 1 ,

where ν is a generator of H^{k}(S^{k}). Evaluating d_{k} by the Leibniz rule on the basis elements of B′ we exhibit T^{∘} as the kernel of d_{k}, whereas im d_{k} = 2 T^{∘}, and hence coker d_{k} ≅ T^{∘}/2T^{∘} ⊕ T^{•}. We thus obtain the final sequence tableau E_{k+1}^{∗,∗} with entries E_{k+1}^{0,∗} = T^{∘} and E_{k+1}^{k,∗} = T^{∘}/2T^{∘} ⊕ T^{•}. From the sequence tableau E_{k+1}^{∗,∗} we can read the cohomology algebra of F (S^{k}, n): Free generators for T^{∘} = E_{k+1}^{0,∗} are located in E_{k+1}^{0,0} and E_{k+1}^{0,k−1}. Together with the free generator in E_{k+1}^{k,k−1} and the generator of order two in E_{k+1}^{k,0} they generate T^{∘}/2T^{∘} ⊕ T^{•} = E_{k+1}^{k,∗}.

T^{∘}  E_{k+1}^{∗,∗} T^{∘}/2T^{∘} ⊕ T^{•} 2k − 2 0 k − 1 3k − 3 0  k  ν/2ν 10  c_{1,2} 0 k − 1 c_{i,j} − c_{1,2}  k

Linearly, the cohomology of F (S^{k}, n) is isomorphic to a tensor product of two free generators in dimension 0 and 2k − 1 and a generator of order 2 in dimension k − 1 with the algebra T^{∘}:

H^{∗}(F (S^{k}, n)) ≅ (ℤ(0) ⊕ ℤ_{2}(k) ⊕ ℤ(2k − 1)) ⊗ T^{∘} .

This isomorphism is an algebra isomorphism: This is obvious for multiplication among elements represented by entries in the left-most column E_{k+1}^{0,∗}. Also, multiplication between entries of E_{k+1}^{0,∗} and E_{k+1}^{k,∗} is correctly described in the proposed tensor product. Moreover, the trivial multiplication among entries in E_{k+1}^{k,∗} has its correspondence in the tensor algebra since multiplication within the left-hand factor is trivial. We conclude:

Theorem 5.4 For a sphere S^{k} of even dimension, k ≥ 4, the integer cohomology algebra of F (S^{k}, n), n ≥ 3, is given by

H^{∗}(F (S^{k}, n)) ≅ (ℤ(0) ⊕ ℤ_{2}(k) ⊕ ℤ(2k − 1)) ⊗ T^{∘}
≅ (ℤ(0) ⊕ ℤ_{2}(k) ⊕ ℤ(2k − 1)) ⊗ Λ^{∗} ⨁_{\binom{n−1}{2}−1} ℤ(k − 1) / J ,

where J is the ideal described in Proposition 5.3.

In the next section we will give a topological interpretation for this product decomposition of the cohomology algebra (see Remark 6.1).

6 A family of fiber bundles

The bundle structure on F (S^{k}, n) given by the projection Π_{1} was the key to determine the integer cohomology algebra of F (S^{k}, n). This projection Π_{1} is one instance from a family of fiber maps Π_{r} = Π_{r}(S^{k}, n), 1 ≤ r < n, that are given by projection of a configuration in F (S^{k}, n) to its last r points. In this section we will have a closer look at these fiber maps, at their spectral sequences, and at the question whether the induced bundle structures are trivial.

For the fiber map Π_{r}(S^{k}, n), 1 ≤ r < n, we obtain the following space as the fiber over a point configuration q = (q_{1}, …, q_{r}) on S^{k}:

Π_{r}^{−1}(q) = {(x_{1}, …, x_{n−r}) ∈ (S^{k})^{n−r} | x_{i} ≠ x_{j} for i ≠ j, x_{i} ≠ q_{t}
for i = 1, …, n − r, t = 1, …, r} .

This space is again a configuration space:

Π_{r}^{−1}(q) = F ( S^{k} \ {q_{1}, …, q_{r}}, n − r) .

Configurations on S^{k} that avoid r ≥ 1 (fixed) points q_{1}, …, q_{r} are equivalent to configurations in ℝ^{k} that avoid r−1 points q_{1}, …, q_{r−1}. Thus the fiber of Π_{r} is homeomorphic to the complement of the arrangement A_{Π_{r}(S^{k},n)} of (affine) subspaces in ℝ^{k(n−r)} given by

U_{i,j} = {(x_{1}, …, x_{n−r}) ∈ (ℝ^{k})^{n−r} | x_{i} = x_{j} },  1 ≤ i < j ≤ n − r,
U_{i}^{t} = {(x_{1}, …, x_{n−r}) ∈ (ℝ^{k})^{n−r} | x_{i} = t · (1, 0, …, 0)^{T} },
1 ≤ i ≤ n − r, 0 ≤ t ≤ r − 2 .

For r = 1, the arrangement A_{Π_{1}(S^{k},n)} coincides with the k-braid arrangement A_{n−2}^{(k)} — a fact that we used extensively in the previous sections. For r > 2, A_{Π_{r}(S^{k},n)} contains affine subspaces, the subspaces U_{i}^{t} for 0 < t ≤ r − 2. In the complex case, for k = 2, these arrangements were extensively studied by Welker [We].

6.1 The spectral sequences

We proved in the previous sections that the spectral sequence E_{∗}(Π_{1}) associated to the fiber map Π_{1}(S^{k}, n) collapses in E_{2} for odd k, and in E_{k+1} for even k. We obtain a similar picture for the spectral sequence E_{∗}(Π_{2}) associated to the fiber map Π_{2}(S^{k}, n): The base space F (S^{k}, 2) is homotopy equivalent to S^{k}. Hence, it is simply connected for k ≥ 2, and the system of local coefficients on S^{k} induced by Π_{2} is simple. The fiber M(A_{Π_{2}(S^{k},n)}) is homotopy equivalent to the complement of the k-braid arrangement A_{n−2}^{(k)}. In fact, the homotopy equivalence is realized by projection of M(A_{n−2}^{(k)}) along ⋂ A_{n−2}^{(k)} on the linear subspace

U_{n−1}^{0} = { (x_{1}, …, x_{n−1}) ∈ (ℝ^{k})^{n−1} | x_{n−1} = 0 } .

Thus, the E_{2}-tableaux of the spectral sequences induced by Π_{1} and Π_{2} coincide. For dimensional reasons, the collapsing results on E_{∗}(Π_{1}) translate to analogous collapsing results on E_{∗}(Π_{2}).

The picture changes for the spectral sequences E_{∗}(Π_{3}) associated to Π_{3}(S^{k}, n). In fact, we have all arguments at hand to discuss them briefly: The base space F (S^{k}, 3) of the fiber map Π_{3}(S^{k}, n) is homotopy equivalent to the Stiefel manifold V_{k+1,2} of orthogonal 2-frames in ℝ^{k+1} [Fa, Thm. 2.4], hence it is simply connected for k ≥ 2. We conclude that the system of local coefficients on F (S^{k}, 3) induced by Π_{3} is simple. We have seen above that the fiber of Π_{3} is homeomorphic to the complement of the (affine) subspace arrangement A_{Π_{3}(S^{k},n)}. Comparison to the complement of the k-braid arrangement A_{n−2}^{(k)} yields a homotopy equivalence,

M(A_{Π_{3}(S^{k},n)}) ≃ M(A_{n−2}^{(k)}_{⌈U}) ,

where A_{n−2}^{(k)}_{⌈U} denotes the restriction of the k-braid arrangement to the affine subspace

U = {(x_{1}, …, x_{n−1}) ∈ (ℝ^{k})^{n−1} | x_{n−2} − x_{n−1} = (1, 0, …, 0)^{T} } .

The homotopy equivalence is realized by projection of M(A_{n−2}^{(k)}_{⌈U}) along the

intersection ⋂ A_{n−2}^{(k)} to the linear subspace

U_{n−1}^{0} = {(x_{1}, …, x_{n−1}) ∈ (ℝ^{k})^{n−1} | x_{n−1} = 0 } .

The affine arrangement A_{n−2}^{(k)}_{⌈U} is associated to the k-braid arrangement in the same way as we associated before an affine complex hyperplane arrangement to the complex braid arrangement (compare Section 2). This analogy allows one to state a presentation for its cohomology algebra in terms of generators and relations. In fact, one obtains an algebra presentation that coincides with the one that we stated for T^{∘} in Proposition 5.3:

H^{∗}(M(A_{n−2}^{(k)}_{⌈U})) ≅ T^{∘} .

In particular, H^{∗}(M(A_{n−2}^{(k)}_{⌈U})) is torsion-free and it is generated in dimension k − 1 by cohomology classes that are in one-to-one correspondence with the inclusion maximal subspaces of the arrangement. For both odd and even k the E_{2}-tableaux of the spectral sequences associated to Π_{3}(S^{k}, n) carry the structure of tensor products. We content ourselves with discussing the spectral sequences for k ≥ 3; for k = 2, we already showed in Section 2 that the bundle structure induced by Π_{3} is trivial.

0 T^{∘}  T^{∘} T^{∘}  T^{∘} 0  d_{k−1} d_{k}  E_{2}^{∗,∗}(Π_{3}) k odd 3k − 3 2k − 2 k − 1  k k−1  2k−1  0 T^{∘}  T^{∘} T^{∘} ⊗ ℤ_{2} E_{2}^{∗,∗}(Π_{3}) k even d_{k}  d_{k−1} 3k − 3 2k − 2 k − 1 0  2k−1 k
E_{2}^{p,q}(Π_{3}) ≅ H^{p}(V_{k+1,2}) ⊗ H^{q}(M(A_{n−2}^{(k)}_{⌈U})) , p, q ≥ 0 .

It is easy to see that E_{∗}(Π_{3}) collapses in its second term for both odd and even k: The location of non-zero entries in the respective tableaux suffices to see the triviality of differentials d_{r} with r ≠ k. The k-th cohomology group of F (S^{k}, n) can be read already from the k-th diagonal in E_{k+1}(Π_{3}). Our results on H^{k}(F (S^{k}, n)) (Proposition 4.2) allow to deduce triviality of the differential d_{k} as we did in Section 4.1. Remark 6.1 There is a topological explanation for the product decomposition of the integer cohomology algebra of F (S^{k}, n) for even k that we obtained in Theorem 5.4: The factors are the cohomology algebras of base space and fiber for the fiber bundle structure on F (S^{k}, n) given by Π_{3}. We showed above that the associated spectral sequence E_{∗}(Π_{3}) collapses in its second term, which explains the product structure in cohomology. The collapsing result on E_{∗}(Π_{3}) extends to the spectral sequences associated to the fiber maps Π_{r} for r > 3, and we can summarize as follows:

Proposition 6.2 The spectral sequence E_{∗}(Π_{r}) of the fiber map Π_{r}(S^{k}, n) on the configuration space F (S^{k}, n) collapses in its second term unless k is even and r equals 1 or 2. For those parameters the spectral sequence collapses in E_{k+1}.

Proof. We are left to show the triviality of the spectral sequence E_{∗}(Π_{r}) for r > 3. This we will derive from the triviality of E_{∗}(Π_{3}), thereby involving several applications of the following Lemma.

Lemma 6.3 [Bo2, Ch. II, Thm. 14.1] Let F ↪^{i} E →^{Π} B be a fiber bundle with path-connected base B and assume that the cohomology of the base or the cohomology of the fiber is torsion-free. Then the following assertions are equivalent:

(1) The system of local coefficients on B induced by Π is simple and the associated spectral sequence with integer coefficients collapses in its second term.

(2) The induced map i^{∗} : H^{∗}(E) → H^{∗}(F) is surjective. Consider the map of fiber bundles between Π_{r}(S^{k}, n) and Π_{3}(S^{k}, n) given by (id, Π_{3}(S^{k}, r)). For simplicity of notation we denote with Q_{t} a fixed set of pairwise distinct points {q_{1}, …, q_{t}} in S^{k} and thus write F (S^{k}\Q_{t}, n − t) for the respective fibers. The fibers are complements of affine k-arrangements, thus their cohomology algebras are torsion-free.

F (S^{k}, n) Π_{r} F (S^{k} \ Q_{r}, n − r)  id i  F (S^{k}, r) F (S^{k} \ Q_{3}, n − 3)  i_{Π_{3}} i_{Π_{r}}  F (S^{k}, n) Π_{3} F (S^{k}, 3) Π_{3}

The configuration space F (S^{k}, 3) is simply connected for k ≥ 2, due to the homotopy equivalence with the Stiefel manifold V_{k+1,2}. With the collapsing result on E_{∗}(Π_{3}) we deduce that i_{Π_{3}}^{∗} is surjective by the equivalence stated above. We are left to show that the inclusion i between the fibers induces a surjective homomorphism in cohomology. Then i_{Π_{r}}^{∗} = i^{∗} ∘ i_{Π_{3}}^{∗} is surjective, and another application of Lemma 6.3 yields the collapsing result on E^{∗}(Π_{r}). To see that i^{∗} is surjective we interpret i as a concatenation of inclusions in a sequence of fiber maps. Namely, we consider the sequence of fiber maps obtained by successively projecting F (S^{k} \Q_{1}, n−1) to its last coordinate. We picture the part of this sequence which is relevant to our investigation:

F (S^{k}\Q_{r}, n − r) ⟶^{j_{r−1}} … ⟶^{j_{4}} F (S^{k}\Q_{4}, n − 4) ⟶^{j_{3}} F (S^{k}\Q_{3}, n − 3)
↓ p_{4}  ↓ p_{3}
S^{k} \ Q_{4}  S^{k} \ Q_{3}

The base spaces of the fiber bundles given by p_{t}, 1 ≤ t ≤ n − 2, are simply connected for k > 2, thus the systems of local coefficients are simple. The same holds for k = 2, which is a result of Cohen [C2, Lemma 6.3]. The fibers are complements of affine k-arrangements, thus their cohomology groups are nontrivial only in dimensions that are multiples of k − 1 [GM, Part III, Thm. B]. For dimensional reasons, the associated spectral sequences E_{∗}(p_{t}) collapse in their second terms and we conclude by Lemma 6.3 that the j_{t}^{∗} are surjective for 1 ≤ t ≤ n−2. Thus, i^{∗} = j_{r−1}^{∗} ∘…∘j_{3}^{∗} is a surjective map, which concludes our proof. □

6.2 Triviality of the fiber bundles

The fiber bundle structure induced by Π_{3} on F (S^{2}, n) for n ≥ 3 is trivial (Theorem 2.1). One is led to ask: For which parameters do the fiber maps Π_{r} induce a trivial fiber bundle structure on F (S^{k}, n)?

We observed in Section 2 that the bundle structure on F (S^{k}, 2) given by Π_{1} is equivalent to the tangent bundle over S^{k}. Thus, Π_{1}(S^{k}, 2) is a trivial fiber map if and only if S^{k} is parallelizable (see Hirzebruch [H]). This indicates that the triviality question for the fiber maps Π_{r} is difficult in general. Our results on the cohomology algebra of F (S^{k}, n) for even k, k ≥ 2, exclude a trivial bundle structure on F (S^{k}, n) induced by Π_{1}: There is 2-torsion in H^{∗}(F (S^{k}, n)) while the cohomology algebra of the cartesian product of base space and fiber is torsion-free. However, the cohomology algebra of F (S^{k}, n) for odd k coincides with the cohomology algebra of the cartesian product of base space and fiber. Such product decomposition might as well hold beyond the level of cohomology. Recall from previous arguments that F (S^{k}, 3) is fiber homotopy equivalent to the Stiefel manifold V_{k+1,2} of orthogonal 2-frames in ℝ^{k+1}, both considered as fiber bundles over S^{k}. Fiber bundles are trivial if and only if their associated principal bundles are trivial [St, Part I, Cor. 8.4]. Hence, V_{k+1,2} is a trivial fiber bundle if and only if O(k+1), considered as a fiber bundle over S^{k}, admits a section — which again is the case iff k = 1, 3 or 7. Moreover, V_{k+1,2} is fiber homotopy equivalent to a trivial bundle if and only if it is trivial itself, hence iff k = 1, 3 or 7 [Ja, Thm. 1.11]. We conclude that F (S^{k}, 3) is a non-trivial fiber bundle over S^{k} for k ≠ 1, 3 or 7. For the 1-sphere we have shown triviality of F (S^{1}, n) as a fiber bundle over S^{1} in Section 2. Analogously, we obtain a trivialization of the fiber bundle structure on F (S^{3}, n) given by Π_{1}, using the group structure of S^{3}. The 7-sphere does not carry the structure of a topological group [Bd, VI, Cor. 15.21]. However, one can establish an explicit equivalence of fiber bundles between F (S^{7}, 3) and V_{8,2} × ℝ^{7} × ℝ, both considered as fiber bundles over S^{7} in the natural way. As mentioned above, V_{8,2} is a trivial fiber bundle over S^{7}, and we can thus conclude triviality of F (S^{7}, 3) over S^{7}. Thus it remains to decide whether the bundle structure on F (S^{k}, n) induced by Π_{1} is trivial for n > 3 and odd k ≥ 5.

We have seen in Section 2 that Π_{3} induces a trivial bundle structure on F (S^{2}, n). Our collapsing results on the spectral sequences E_{∗}(Π_{3}(S^{k}, n)) for both odd and even k would be consistent with triviality of the fiber bundle structure given by Π_{3}. However, except for k = 2 this leaves us with an open question. Remark 6.4 After completion of this paper, we learned about recent work by Fadell & Husseini [FaH] which addresses the question of configuration space bundles being (fiber-homotopically) trivial. The paper is mostly concerned with configuration spaces of Euclidean spaces; a complete discussion for configuration spaces of spheres is announced, but the results are stated only for spheres of odd dimension.

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Eva Maria Feichtner

Dept. Mathematics, MA 7-1 TU Berlin 10623 Berlin, Germany Current address: Dept. Mathematics ETH Zürich 8092 Zürich, Switzerland feichtne@math.ethz.ch Günter M. Ziegler Dept. Mathematics, MA 7-1 TU Berlin 10623 Berlin, Germany ziegler@math.tu-berlin.de 140

Documenta Mathematica 5 (2000)