A survey of the homotopy properties of inclusion of certain types of configuration spaces into the Cartesian product

Abstract

Let X be a topological space. In this survey the authors consider several types of configuration spaces, namely, the classical (usual) configuration spaces F _n(X) and D _n(X), the orbit configuration spaces F ^G _n (X) and F ^G _n (X)/S _n with respect to a free action of a group G on X, and the graph configuration spaces F ^Г _n (X) and F ^Г _n (X)/H, where Г is a graph and H is a suitable subgroup of the symmetric group S _n. The ordered configuration spaces F _n(X), F ^G _n (X), F ^Г _n (X) are all subsets of the n-fold Cartesian product n Π 1 X of X with itself, and satisfy F ^G _n (X) → F _n(X) → F ^Г _n (X) → n Π 1 X. If A denotes one of these configuration spaces, the authors analyse the difference between A and n Π 1 X from a topological and homotopical point of view. The principal results known in the literature concern the usual configuration spaces. The authors are particularly interested in the homomorphism on the level of the homotopy groups of the spaces induced by the inclusion ι: A → n Π 1 X, the homotopy type of the homotopy fibre I _ι of the map ι via certain constructions on various spaces that depend on X, and the long exact sequence in homotopy of the fibration involving I _ι and arising from the inclusion ι. In this respect, if X is either a surface without boundary, in particular if X is the 2-sphere or the real projective plane, or a space whose universal covering is contractible, or an orbit space S ^k/G of the k-dimensional sphere by a free action of a Lie group G, the authors present recent results obtained by themselves for the first case, and in collaboration with Golasiński for the second and third cases. The authors also briefly indicate some older results relative to the homotopy of these spaces that are related to the problems of interest. In order to motivate various questions, for the remaining types of configuration spaces, a few of their basic properties are described and proved. A list of open questions and problems is given at the end of the paper.

Keywords

Configuration space / Equivariant configuration space / Fibration / Homotopy fibre / K(π,1) space / Braid groups

References

[1] Arkowitz M.. Introduction to homotopy theory, 2011, New York: Springer-Verlag

[2] Baranovsky V., Sazdanovic R.. Graph homology and graph configuration spaces. J. Homotopy Relat. Struct., 2012, 7: 223-235

[3] Birman J. S.. On braid groups. Comm. Pure and Appl. Math., 1969, 22: 41-72

[4] Birman, J. S., Braids, Links and Mapping Class Groups, Ann. Math. Stud., 82, Princeton University Press, Princeton, 1974.

[5] Bödigheimer C.-F., Cohen F. R., Peim M. D.. Mapping class groups and function spaces. Homotopy methods in Algebraic Topology (Boulder, CO, 1999), Amer. Math. Soc., Providence, RI, Contemp. Math., 2001, 271: 17-39

[6] Cohen F. R.. Introduction to configuration spaces and their applications, in Braids, Vol. 19. Lect. Notes Ser. Inst. Math. Sci. Natl. Univ. Singap., 2010, Hackensack, NJ: World Sci. Publ. 183-261

[7] Cohen F. R., Xicoténcatl M. A.. On orbit configuration spaces associated to the Gaussian integers: Homotopy and homology groups. Arrangements in Boston: A Conference on Hyperplane Arrangements, 1999, Topol. Appl., 2002, 118: 17-29

[8] Edmonds A. L.. Taming free circle actions. Proc. Amer. Math. Soc., 1977, 62: 337-343

[9] Fadell E., Husseini S. Y.. Fixed point theory for non-simply-connected manifolds. Topology, 1981, 20: 53-92

[10] Fadell E., Husseini S. Y.. Geometry and topology of configuration spaces, 2001, Berlin: Springer-Verlag

[11] Fadell E., Neuwirth L.. Configuration spaces. Math. Scand., 1962, 10: 111-118

[12] Fadell E., Van Buskirk J.. The braid groups of E² and S². Duke Math. J., 1962, 29: 243-257

[13] Feichtner E. M., Ziegler G. M.. The integral cohomology algebras of ordered configuration spaces of spheres. Doc. Math., 2000, 5: 115-139

[14] Fox R. H., Neuwirth L.. The braid groups. Math. Scandinavica, 1962, 10: 119-126

[15] Ganea T.. A generalization of the homology and homotopy suspension. Comment. Math. Helv., 1965, 39: 295-322

[16] Ganea T.. Induced fibrations and cofibrations. Trans. Amer. Math. Soc., 1967, 127: 442-459

[17] Gillette R., Van Buskirk J.. The word problem and consequences for the braid groups and mapping class groups of the 2-sphere. Trans. Amer. Math. Soc., 1968, 131: 277-296

[18] Golasiński M., Gonçalves D. L., Guaschi J.. On the homotopy fibre of the inclusion map F _n(X) →IIⁿ ₁ (X) for some orbit spaces X. Bol. Soc. Mat. Mexicana, 2017, 23: 457-485

[19] Goldberg C. H.. An exact sequence of braid groups. Math. Scand., 1973, 33: 69-82

[20] Gonçalves D. L., Guaschi J.. The roots of the full twist for surface braid groups. Math. Proc. Camb. Phil. Soc., 2004, 137: 307-320

[21] Gonçalves D. L., Guaschi J.. The braid groups of the projective plane. Algebr. Geom. Topol., 2004, 4: 757-780

[22] Gonçalves D. L., Guaschi J.. The braid group B _n,_m(S²) and the generalised Fadell-Neuwirth short exact sequence. J. Knot Theory Ramif., 2005, 14: 375-403

[23] Gonçalves D. L., Guaschi J.. Inclusion of configuration spaces in Cartesian products, and the virtual cohomological dimension of the braid groups of S² and RP ². Pac. J. Math., 2017, 287: 71-99

[24] Gonçalves, D. L. and Guaschi, J., The homotopy fibre of the inclusion F _n(X) →II ⁿ ₁ M for M either S² or RP ² and orbit configuration spaces, arXiv:1710.11544.

[25] González-Meneses J., Paris L.. Vassiliev invariants for braids on surfaces. Trans. Amer. Math. Soc., 2004, 356: 219-243

[26] Milnor J.. On spaces having the homotopy type of a CW-complex. Trans. Amer. Math. Soc., 1959, 90: 272-280

[27] Murasugi K.. Seifert fibre spaces and braid groups. Proc. London Math. Soc., 1982, 44: 71-84

[28] Tochimani A.. Grupos de trenzas de superficies compactas, Master’s Thesis. Centro de Investigación y de Estudios Avanzados (CINVESTAV), 2011

[29] Van Buskirk J.. Braid groups of compact 2-manifolds with elements of finite order. Trans. Amer. Math. Soc., 1966, 122: 81-97

[30] Xicoténcatl M. A.. Orbit configuration spaces, infinitesimal braid relations in homology and equivariant loop spaces, 1997, Rochester, NY: niversity of Rochester

[31] Xicoténcatl M. A.. Orbit configuration spaces. Contemp. Math., 2014, 621: 113-132