A Survey of the Homotopy Properties of Inclusion of Certain Types of Configuration Spaces into the Cartesian Product

Autor(es)

Goncalves, Daciberg Lima ^[1] ; Guaschi, John ^[2]

Afiliação do(s) autor(es)

^[1] Univ Sao Paulo, IME, Dept Matemat, Rua Matao 1010, BR-05508090 Sao Paulo, SP - Brazil

^[2] Normandie Univ, CNRS, UNICAEN, Lab Math Nicolas Oresme, UMR 6139, CS 14032, F-14032 Caen 5 - France

• Tipo de documento: Artigo Científico
• Fonte: CHINESE ANNALS OF MATHEMATICS SERIES B; v. 38, n. 6, p. 1223-1246, NOV 2017.
• Citações Web of Science: 0

Resumo

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-n(G)(X) and F-n(G)(X)/S-n, with respect to a free action of a group G oil X, and the graph configuration spaces F-n(Gamma)(X)(X) and F-n(Gamma)(X)(X)/H, where F is a graph and 11 is a suitable subgroup of the symmetric group Sn. The ordered configuration spaces F-n(X), F-n(G)(X), F-n(Gamma)(X), are all subsets of the n-fold (artesian product Pi(n)(1) X of X with itself, and satisfy F-n(G)(X) subset of F-n(G)(X) subset of F-n(Gamma)(X) subset of Pi(n)(1) If A denotes one of these configuration spaces, the authors analyse the difference between A and Pi(n)(1) from a topological and boinotopical 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 n. inclusion t: A -> Pi(n)(1) X, the homotopy type of the homotopy fibre I-i, of the map i via certain constructions on various spaces that depend on X, and the long exact sequence in homotopy of the fibration involving I-i, and arising from the inclusion i. 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 spaceS(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 Colasinski 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 configurat ion spaces, a few of their basic properties are described and proved. A fist of open questionti and problems is given at the end of the paper. (AU)

• Processo FAPESP: 14/50131-7 - Algebraic and topological properties of the braid groups of the real projective plane, sphere, disk, orbit configuration spaces, and relations with crystallographic groups
• Beneficiário: Daciberg Lima Gonçalves
• Modalidade de apoio: Auxílio à Pesquisa - Regular
• Processo FAPESP: 12/24454-8 - Topologia algébrica, geométrica e diferencial
• Beneficiário: Daciberg Lima Gonçalves
• Modalidade de apoio: Auxílio à Pesquisa - Temático