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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

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关于纤维化奇异纤维的一些新的组合条件

Northeastern Mathematical Journal 1994年第4期10卷 503-510页

作者：唐明元

For any relative minimal fibration of gender g, we have known some basic combinatorialproperties, i. e, the necessary conditions (1)-(5) in this paper. Xiao[1] showed that these conditionsare not sufficient for g>2 and gave another two nccry conditions. In this peper, we use themethed of the Stable Reduction and the roemann-Hurtvitz formula to obtain some new combinatorialconditions. These resultS are reduced to three propeitions.

关键词： fibration Singular Fibre Base Extension Principal component

Algebraic Surfaces of General Type with K^2 = 2p_g-1, p_g■5

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Acta Mathematica Sinica,English Series 1996年第3期12卷 234-243页

作者： Liu Xianfang, Department of Mathematics East China Normal University Shanghai, 200062 ChinaCurrent address: Institute of systems science Beijing, 100080 China Department of Mathematics East China Normal University Shanghai China

This paper mainly deals with minimal algebraic surfaces of general type with K^2= 2p_g-1. We prove that for p_g 7 all these surfaces are birational to a double cover of some rational surfaces, and all but a finite classes of them have a unique fibration of genus 2; then we study their structures by determining their branch loci and singular fibres. We study similarly for surfaces with p_g=5, 6. Lastly we show that when p_g 13 all these surfaces are simply-connected.

关键词： Double cover Canonical resolution fibration Singular fiber Ramification divisor Branch locus

A CLASS OF INCOMPRESSIBLE SURFACES IN 3-MANIFOLDS FIBERING OVER S^1

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Chinese Science Bulletin 1991年第19期36卷 1590-1594页

作者：陈仲鹤 Department of Mathematics Peking University Beijig 100871 PRC

An important class in the study of topology of 3-manifolds is the so-called 'sufficiently large 3-manifolds', i.e. those compact 3-manifolds which contain properly embedded, 2-sided incompressible surfaces. We know that every sufficiently large 3-manifold can bc cut into some 3-cells along incompressible surfaces in a finite process. For a 3-manifold, it is interesting to know whether there exists an incompressible surface embedded in it and how the incompressible surface embeds.

关键词： incompressible surface fibration Dehn twist.

On Monodromy of Semi-stable Fibres

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Chinese Science Bulletin 1994年第14期39卷 1145-1149页

作者：吴新文 Institute of Systems Science Academia Sinica Beijing 100080 PRC

The fibration is one of the fundamental methods in the study of algebraic surfaces. In the early years, fibration was studied by using the method of complete classification of singular fibres, which was obtained through studying the combinatorical properties of singular fibres. But with the rising of the genus of fibration, this method will not work now. In 1977, a more essential classification of singular fibres of genus two was given by Eiji Horikawa by using relative canonical maps. Prof. Xiao Gang has successfully improved the classification of Horikawa and effectively studied some *** surfaces by using his classification. We know that, ordinarily, through the classification of singular fibres, parameters of fibration can be obtained. Based on this, we study the properties of surfaces. But for some parameters, we can effectively describe them only by the study of combinatorical properties and topological

关键词： fibration of surfaces semi-stable fibres monodromy.

Bernstein theorem for harmonic morphisms from (N + 1)-sphere to N- manifold

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Chinese Science Bulletin 1997年第1期42卷 11-14页

作者： DONG YuxinDepartment of Mathematics, Hangzhou University, Hangzhou 310028, China Department of Mathematics Hangzhou University Hangzhou China

morphism. Harmonic morphisms may be characterized as harmonic mappings which are horizontally weakly conformal. In recent years a substantial progress has been made in the field of harmonic morphism, for example, refs. [ 3—7]. In ref. [4], Baird and Wood proved the following: (ⅰ) Any non-constant harmonic morphism from (S^3, g_(can)) to a Riemann

关键词： harmonic morphism Hopf fibration geodesic.