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

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Chinese Annals of Mathematics,Series B 2017年第6期38卷 1223-1246页

作者： daciberg Lima goncalves John GUASCHI Departamento de Matematica IME-USPRua do Matao 1010CEP:05508-090Sao PauloSPBrazil Normandie Universite UNICAENLaboratoire de Mathematiques Nicolas Oresme UMR CNRS 6139CS 1403214032 Caen Cedex 5France

Let X be a topological *** 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_nwith respect to a free action of a group G on X,and the graph configuration spaces F_n~Γ(X)and F_n~Γ(X)/H,whereΓis a graph and H is a suitable subgroup of the symmetric group S_*** ordered configuration spaces F_n(X),F_n^G(X),F_n~Γ(X)are all subsets of the n-fold Cartesian product ∏_1~nX of X with itself,and satisfy F_n^G(X)?F_n(X)?F_n~Γ(X)?∏_1~*** A denotes one of these configuration spaces,the authors analyse the difference between A and ∏_1~nXfrom a topological and homotopical point of *** principal results known in the literature concern the usual configuration *** authors are particularly interested in the homomorphism on the level of the homotopy groups of the spaces induced by the inclusionι:A-→∏_1~nX,the homotopy type of the homotopy fibre I_ιof the mapιvia certain constructions on various spaces that depend on X,and the long exact sequence in homotopy of the fibration involving I_ιand arising from the inclusionι.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/Gof 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 Golasi′nski for the second and third *** authors also briefly indicate some older results relative to the homotopy of these spaces that are related to the problems of *** 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.

关键词： Configuration space Equivariant configuration space Fibration Homotopy fibre K(π,1)space Braid groups

Obstruction Theory and Coincidences in Positive Codimension

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Acta Mathematica Sinica,English Series 2006年第5期22卷 1591-1602页

作者： daciberg goncalves Jerzy JEZIERSKI Peter WONG Departamento de Matemática-Instituto de Matemdtica e Estatística-Universidade de Sǎo Paulo Caixa Postal 66.281-CEP 05311-970 Sǎo Paulo-SP Brasil Department of Mathematics University of Agriculture Nowoursynowska 166 02766 Warszawa Poland Department of Mathematics Bates College Lewiston ME 04240 USA

Let f, g ： X→Y be two maps between closed manifolds with dim X ≥ dim Y = n ≥ 3. We study the primary obstruction on（f, g） to deforming f and g to be coincidence free on the n-th skeleton of X. We give examples f... 详细信息

关键词： Nielsen number Reidemeister number coincidence theory obstruction theory

Coincidence Properties for Maps from the Torus to the Klein Bottle

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Chinese Annals of Mathematics,Series B 2008年第4期29卷 425-440页

作者： daciberg L.goncalves Michael R.KELLY Departamento de Matemática IME USP Department of Mathematics Loyola University

The authors study the coincidence theory for pairs of maps from the Torus to the Klein bottle. Reidemeister classes and the Nielsen number are computed, and it is shown that any given pair of maps satisfies the Wecken property. The 1-parameter Wecken property is studied and a partial negative answer is derived. That is for all pairs of coincidence free maps a countable family of pairs of maps in the homotopy class is constructed such that no two members may be joined by a coincidence free homotopy.

关键词： Coincidence point Nielsen number Wecken property

Free Cyclic Actions on Surfaces and the Borsuk-Ulam Theorem

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Acta Mathematica Sinica,English Series 2022年第10期38卷 1803-1822页

作者： daciberg Lima goncalves John GUASCHI Vinicius Casteluber LAASS Departamento de Matematica IMEUniversidade de Sao PauloRua do Matao 1010 CEP:05508-090Sao Paulo-SPBrazil Normandie Univ. UNICAENCNRSLMNO14000 CaenFrance Universidade Federal da Bahia IMEDepartamento de MatematicaAv.Milton SantosS/N Ondina CEP:40170-110Salvador-BABrazil

Let M and N be topological spaces,let G be a group,and letτ:G×M→M be a proper free action of *** this paper,we define a Borsuk-Ulam-type property for homotopy classes of maps from M to N with respect to the pair(G,τ)that generalises the classical antipodal Borsuk-Ulam theorem of maps from the n-sphere S^(n) to R^(n).In the cases where M is a finite pathwise-connected CWcomplex,G is a finite,non-trivial Abelian group,τis a proper free cellular action,and N is either R^(2) or a compact surface without boundary different from S^(2) and RP^(2),we give an algebraic criterion involving braid groups to decide whether a free homotopy class β∈[M,N]has the Borsuk-Ulam *** an application of this criterion,we consider the case where M is a compact surface without boundary equipped with a free actionτof the finite cyclic group *** terms of the orient ability of the orbit space Mof M by the actionτ,the value of n modulo 4 and a certain algebraic condition involving the first homology group of M,we are able to determine if the single homotopy class of maps from M to R^(2) possesses the Borsuk-Ulam property with respect to(Z_(n),τ).Finally,we give some examples of surfaces on which the symmetric group acts,and for these cases,we obtain some partial results regarding the Borsuk-Ulam property for maps whose target is R^(2).

关键词： Borsuk-Ulam theorem cyclic groups braid groups surfaces