Geometry and Topology of Configuration Spaces

Our aim in Part One is to develop basic methods for Parts Two and Three, where we study the associated spaces of the based and free loops of, respectively, \(\mathbb{F}_k (M)\), with M being the Euclidean space ℝ ^m , or the sphere S ^m . Part One is devoted to the homotopy theory of configuration spaces.

Consider now the natural projection proj _k,r : \(\mathbb{F}_k (M) \to \mathbb{F}_r (M)\), r < k that sends (x ₁, …, x _k ) to (x ₁, …, x _r ), where M is a connected manifold of dimension m. With the configuration spaces being regarded as the space of imbeddings of the sets k and r, respectively, one sees that these projections are just the restriction maps induced by the injection r = {1, …, r} ⊂ k = {1, …, k}. Hence, according to [107, Thom] (see also [87, Palais]), these projections are locally trivial fibrations. As the fibration of \(\mathbb{F}_k (M)\) over \(\mathbb{F}_r (M)\) plays a central role in the study of the geometry and topology of these configuration spaces ([43, Fadell-Neuwirth], [17, Cohen]), a simple and direct proof of the fact that proj _k,r : \(\mathbb{F}_k (M) \to \mathbb{F}_r (M)\) is locally trivial, independent of [107, Thom] (or [87, Palais]), is in order. In §1 below, we give such a proof. The structure group will be a subgroup of Top(M), and its fiber the configuration space \(\mathbb{F}_{k - r} (M - Q_r )\), where Q _r = {q ₁, …, q _r } and q = (q ₁, …, q _r ), is the basepoint of \(\mathbb{F}_r (M)\).

In this chapter we shall consider the configuration space \(\mathbb{F}_k (\mathbb{R}^{n + 1} )\) and n < 1. The space is simply connected. The case when n = 1 will be taken up in Chapter IV.

The configuration spares of spheres, as to be expected, are intimately related to those of the Euclidean spaces of the same dimension. Nevertheless, they present important novel features. The primary difference is due to the fact that the tangent bundle of the sphere is nontrivial except for the cases when the sphere S ^m is S ¹, S ³, or S ⁷. In this chapter we consider the case m > 2 only, so the relevant configuration spaces are simply connected. The case S ² presents a new kind of difficulty, as the corresponding configuration spaces are no longer simply connected. It will be taken up in Chapter IV.

As the spaces \(\mathbb{F}_k (\mathbb{R}^2 )\) and \(\mathbb{F}_k (S^2 )\) are not simply connected, the methods in the previous chapters need to be adapted accordingly. In particular, the choice of the basepoint q = (q ₁ …, q _k ) must always be considered.

In Part Two our aim is to define minimal CW-complexes X _k , Y _k+1 homotopy equivalent to \({\mathbb{F}_k}({\mathbb{R}^{n + 1}})\) and \({\mathbb{F}_{k + 1}}({S^{n + 1}})\), respectively. In Chapter V we determine the structure of \({H^*}({\mathbb{F}_k}(M);\mathbb{Z})\), as an algebra, when M is ℝ ⁿ⁺¹ or S ⁿ⁺¹. We view the generators α _rs of the group \({\pi _n}({\mathbb{F}_k}({\mathbb{R}^{n + 1}}),q)\), defined in Chapter II, §2, as spherical homology classes and introduce the elements \(\left\{ {\alpha _{rs}^* \in {H^n}({\mathbb{F}_k}({\mathbb{R}^{n + 1}};\mathbb{Z})|1 \leqslant s < r \leqslant r} \right\}\) dual to the α _rs . These elements generate the group \({H^n}({\mathbb{F}_k}({\mathbb{R}^{n + 1}}),\mathbb{Z})\) and are invariant, set-wise, up to sign, under the action of the symmetric group ∑ _k . Moreover, they satisfy the cohomological version of the Y-B relations of Chapter II, §3. We show that \({H^*}({\mathbb{F}_k}({\mathbb{R}^{n + 1}}),\mathbb{Z})\) is the universal, commutative, graded algebra generated by the set of all α _rs ^* modulo the ideal generated by the Y-B relations. The proof is by induction on the natural filtration in diagram F _k (ℝ ⁿ⁺¹) of Chapter II. The rest of Chapter V is devoted to determining the cohomology algebra of \(\mathbb{F}_{k + 1} (S^{n + 1} )\). These results lead to cohomology bases consisting of multifold products of the elements of \(\left\{ {\alpha _{rs}^*|1 \leqslant s < r \leqslant k} \right\}\).

Our aim in this chapter is to determine the structure of \(H^* (\mathbb{F}_k (M);\mathbb{Z})\), as an algebra, when M is ℝ ⁿ⁺¹ or S ⁿ⁺¹ (cf. [17, Cohen], [19, Cohen-Taylor]). Note that we are including the case n = 1. In this case some notational change is necessary.

Our objective here is to describe cellular structures of \(\mathbb{F}_k (M)\) naturally associated with the bases of \(H^* (\mathbb{F}_k (M))\) for M = ℝ ⁿ⁺¹, S ⁿ⁺¹ of Theorem 4.2 and §6 of Chapter V. The basic ideas are the following: first, that the twisted product representation \(\mathbb{F}_k (\mathbb{R}^{n + 1} ) \simeq \,\mathbb{R}_1^{n + 1} \propto \cdots \propto \mathbb{R}_{k - 1}^{n + 1} \) introduced in Chapter II, §4 leads to a twisted product on homology, which we write as for all 1 ≤ p ≤ (k - 1); and, second, that each p-fold twisted product ω leads to an imbedding of a certain kind. These maps provide us with the cells and attaching maps of the desired complex.

Our aim in this chapter is to present cellular chain models for the singular chain coalgebras \(C_* S(\mathbb{F}_k (M))\), where M is ℝ ⁿ⁺¹ or S ⁿ⁺¹. The models are based on the cellular models \(X \simeq \mathbb{F}_k (\mathbb{R}^{n + 1} )\) and \(Y \simeq \mathbb{F}_{k + 1} (S^{n + 1} )\) of Chapter VI, with n < 1. The results here are to be used in Part Three, where cellular models for the loop spaces of the configuration spaces are studied.

In Part Three we turn to the loop spaces, first based and then free, of the configuration spaces.

The object of this chapter is to determine the structure of the Pontryagin algebra \(H_* (\Omega \mathbb{F}_k (M);\mathbb{K})\) when M is ℝ ⁿ⁺¹, or S ⁿ⁺¹, and \(\mathbb{K} = \mathbb{Z}\) or ℤ₂. As we have seen in the previous chapters, the two cases of ℝ ⁿ⁺¹, or S ⁿ⁺¹ are best treated separately.

Our aim in this chapter is to represent the spaces of based and free loops Ω(X) and Λ(X) as well as other associated spaces of a simply connected, countable CW-complex X with a single 0-vertex, e, as CW-complexes of a certain kind. In the case of Ω(X), the complex will have a nondegenerate product corresponding to the usual path multiplication. The first example of such constructions is the reduced-product complex X _∞ of [63, James]. The complexes we construct generalize his construction and will be called constructions of the reduced-product type, or RPT-complexes for short.

Our aim in this chapter is to present cellular chain algebra models for \(C_* D_N (\Omega (\mathbb{F}_k (M)))\), the normalized chain algebra of singular cubes, where M is either ℝ ⁿ⁺¹ or S ⁿ⁺¹. The models are based on the RPT-models for \(\Omega (\mathbb{F}_k (M))\) given in Chapter IX.

In this chapter we study the Serre spectral sequence \(\{ E_{*,*}^r (p),d^r \} \) over \(\mathbb{K}\, = \mathbb{Z}\), and ℤ₂, of the path space fibration with \(M = \,\mathbb{F}_k (\mathbb{R}^{n + 1} )\) or \(\mathbb{F}_{k + 1} (S^{n + 1} )\). Here the paths are based at an appropriate basepoint.

We have seen in the previous chapters that the space \(\mathbb{F}_k (M)\) can be described as a twisted product of simpler spaces when M is ℝ ⁿ⁺¹ or S ⁿ⁺¹. The simpler spaces are bouquets of n-dimensional spheres when M = ℝ ⁿ⁺¹; when M = S ⁿ⁺¹, they include the Stiefel manifold O _n+2,2 of orthonormal 2-frames in ℝ ⁿ⁺², as well. We have also seen that the space \(\Omega \mathbb{F}_k (M)\) of based loops splits as a product of the loop spaces of the split factors as spaces, but not as loop spaces. A natural question to ask is whether the space of free loops \(\Lambda \mathbb{F}_k (M)\) splits, at the homology level, as a tensor product of the homology of the split factors of \(\mathbb{F}_k (M)\). We shall see in this chapter that this is the exception: it is true for k = 3, but not in general.

In this chapter we prepare the ground for our applications to problems of the k-body type, to be presented in Chapter XIV. We are chiefly motivated by the notion of neighborhoods of infinity, introduced in [8, Bahri-Rabinowitz] in their study of 3-body problems. Intuitively speaking, the neighborhoods of infinity consist of configurations of three bodies that separate into simpler clusters moving away from each other. Another approach that deals with this is that of admissible and non-admissible sets, introduced in [75, 76, 77, Majer-Terracini]. In an attempt to bring the two points of view together, we offer the concept of ends. This concept makes it possible to adopt a variational approach that depends on the use of a category theory of the relative type.

Our basic objective here is to illustrate how the results of Chapter XIII can be used in the study of problems of the k-body type. More specifically, we discuss two examples to show how our results relate to the studies of the 3- body type in [8, Bahri-Rabinowitz] and of the k-body type in [75, 76, 77, 78, Majer-Terracicni].