Geometry and topology of configuration spaces

Abstract

The space of configurations of k distinct points in a manifold M, Fk(M), appears and plays an important role in various contexts. The book under review is mainly devoted to a detailed study of the topology of Fk(M) and its loop spaces (based and free) for M=Rn+1 and Sn+1. We are going to describe the main results for Euclidean spaces (the case of spheres being similar, but slightly more complicated). The basic geometric idea is the existence of a natural fibration, Fk+1(Rn+1)→Fk(Rn+1), with fiber a wedge of k spheres, ⋁kSn. As a consequence, Fk(Rn+1) [resp. the based loop space, ΩFk(Rn+1)] is topologically an iterated (nontrivial) twisted product of simple spaces, namely wedges, W, of n-spheres [resp. ΩW]. The twisting is encoded by the so-called (infinitesimal) Yang-Baxter relations, (I)YB, which also appear in quantum group theory [see V. G. Drinfelʹd, Algebra i Analiz 2 (1990), no. 4, 149–181; MR1080203 (92f:16047)] and in the theory of finite-type invariants of braids and links [see T. Kohno, in Symplectic geometry and quantization (Sanda and Yokohama, 1993), 123–138, Contemp. Math., 179, Amer. Math. Soc., Providence, RI, 1994; MR1319605 (96g:57010)]. The authors use the above approach to derive a complete description of the cohomology algebras, H∗Fk(Rn+1). The defining algebra relations are obtained, by duality, from the IYB relations, and the result is independent of n (modulo mild rescaling). Explicit (minimal) cell decompositions of Fk(Rn+1) are also obtained. At the level of homotopy groups, the discussion naturally breaks into two cases, n=1 and n>1. The spaces Fk(R2) are aspherical, and YB relations are used to exhibit a group presentation for π1Fk(R2). In the 1-connected case (n>1), the authors obtain a complete description of the Hopf algebra H∗ΩFk(Rn+1) (and consequently, of the rational homotopy Lie algebra, π∗ΩFk(Rn+1)⊗Q). Modulo rescaling, they find that H∗ΩFk(Rn+1) is isomorphic to the universal enveloping algebra of Lk, ULk, where Lk is "the universal IYB Lie algebra on k strings''. That is, Lk is the associated graded Lie algebra (with respect to the lower central series) of π1Fk(R2), the pure braid group of k strings; see Kohno [Invent. Math. 82 (1985), no. 1, 57–75; MR0808109 (87c:32015a)], for the computation of Lk with Q-coefficients (subsequently extended over Z by a number of authors). This intriguing rescaling formula, H∗ΩFk(Rn+1)=Ugrπ1Fk(R2), has been found independently by F. R. Cohen and S. Gitler [Trans. Amer. Math. Soc. 354 (2002), no. 5, 1705–1748 (electronic) MR1881013 ]. An interesting direction of generalization has been proposed by D. C. Cohen, F. R. Cohen, and M. Xicotencatl ["Lie algebras associated to fiber-type arrangements'', preprint, arXiv.org/abs/math.AT/0005091]. They replace Fk(R2) by the complement of an arrangement, A, of complex hyperplanes, which is assumed to be fiber-type (that is, to possess an iterated fibered structure, like the one of Fk(R2)). They go onto replace Fk(Rn+1) above by the complement of an associated arrangement, Aq, of complex subspaces of codimension q, and prove that the rescaling formula still holds. Denote by X the complement of A, and let Y be a homological rescaling of X. (The complement of Aq is such a rescaling.) The reviewer and A. Suciu ["Homotopy Lie algebras, lower central series, and the Koszul property'', preprint, arXiv.org/abs/math.AT/0110303] showed that the rescaling formula for X and Y is actually equivalent to the Koszul property for the cohomology algebra H∗X. As expected, the computation of free loop space homology, H∗ΛFk(Rn+1), n>1, turns out to be a more difficult problem. In the book under review, the authors attack it by using a variety of additional tools (models of James reduced product type and Adams-Hilton type, Serre and Eilenberg-Moore spectral sequences). Their main results here are: the computation of the mod 2 Poincaré series of ΛF3(Rn+1); the equality cup-length(H∗(ΛFk(Rn+1);Z2))=∞, and the related result that ΛFk(Rn+1) contains compact subsets of arbitrarily high Lyusternik-Shnirelʹman category. The result on LS category is finally used to study certain Hamiltonian systems of k-body type.