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An advanced monograph on the subject of topological transformation groups, this volume summarizes important research conducted during a period of lively activity in this area of mathematics. The book is of particular note because it represents the culmination of research by authors Deane Montgomery and Leo Zippin, undertaken in collaboration with Andrew Gleason of Harvard University, that led to their solution of a well-known mathematical conjecture, Hilbert's Fifth Problem. The treatment begins with an examination of topological spaces and groups and proceeds to locally compact groups and groups with no small subgroups. Subsequent chapters address approximation by Lie groups and transformation groups, concluding with an exploration of compact transformation groups.
“This book is a jewel – it explains important, useful and deep topics in Algebraic Topology that you won’t find elsewhere, carefully and in detail.” Prof. Günter M. Ziegler, TU Berlin
Contents: S. Bauer: The homotopy type of a 4-manifold with finite fundamental group.- C.-F. Bödigheimer, F.R. Cohen: Rational cohomology of configuration spaces of surfaces.- G. Dylawerski: An S1 -degree and S1 -maps between representation spheres.- R. Lee, S.H. Weintraub: On certain Siegel modular varieties of genus two and levels above two.- L.G. Lewis, Jr.: The RO(G)-graded equivariant ordinary cohomology of complex projective spaces with linear /p actions.- W. Lück: The equivariant degree.- W. Lück, A. Ranicki: Surgery transfer.- R.J. Milgram: Some remarks on the Kirby - Siebenmann class.- D. Notbohm: The fixed-point conjecture for p-toral groups.- V. Puppe: Simply connected manifolds without S1-symmetry.- P. Vogel: 2 x 2 - matrices and application to link theory.
Introduction to Compact Transformation Groups
Historically, applications of algebraic topology to the study of topological transformation groups were originated in the work of L. E. 1. Brouwer on periodic transformations and, a little later, in the beautiful fixed point theorem ofP. A. Smith for prime periodic maps on homology spheres. Upon comparing the fixed point theorem of Smith with its predecessors, the fixed point theorems of Brouwer and Lefschetz, one finds that it is possible, at least for the case of homology spheres, to upgrade the conclusion of mere existence (or non-existence) to the actual determination of the homology type of the fixed point set, if the map is assumed to be prime periodic. The pioneer result of P. A. Smith clearly suggests a fruitful general direction of studying topological transformation groups in the framework of algebraic topology. Naturally, the immediate problems following the Smith fixed point theorem are to generalize it both in the direction of replacing the homology spheres by spaces of more general topological types and in the direction of replacing the group tl by more general compact groups.
This book provides an overview of some of the most active topics in the theory of transformation groups over the past decades and stresses advances obtained in the last dozen years. The emphasis is on actions of Lie groups on manifolds and CW complexes. Manifolds and actions of Lie groups on them are studied in the linear, semialgebraic, definable, analytic, smooth, and topological categories. Equivalent vector bundles play an important role. The work is divided into fifteen articles and will be of interest to anyone researching or studying transformations groups. The references make it easy to find details and original accounts of the topics surveyed, including tools and theories used in these accounts.
This book presents an introduction to the theory of transformation groups with emphasis on the study of topological groups and the compact Lie groups acting on manifolds.
In recent years, there has been increasing interest and activity in the area of group actions on affine and projective algebraic varieties. Tech niques from various branches of mathematics have been important for this study, especially those coming from the well-developed theory of smooth compact transformation groups. It was timely to have an interdisciplinary meeting on these topics. We organized the conference "Topological Methods in Alg~braic Transformation Groups," which was held at Rutgers University, 4-8 April, 1988. Our aim was to facilitate an exchange of ideas and techniques among mathematicians studying compact smooth transformation groups, alge braic transformation groups and related issues in algebraic and analytic geometry. The meeting was well attended, and these Proceedings offer a larger audience the opportunity to benefit from the excellent survey and specialized talks presented. The main topics concerned various as pects of group actions, algebraic quotients, homogeneous spaces and their compactifications. The meeting was made possible by support from Rutgers University and the National Science Foundation. We express our deep appreciation for this support. We also thank Annette Neuen for her assistance with the technical preparation of these Proceedings.
Locally compact groups play an important role in many areas of mathematics as well as in physics. The class of locally compact groups admits a strong structure theory, which allows to reduce many problems to groups constructed in various ways from the additive group of real numbers, the classical linear groups and from finite groups. The book gives a systematic and detailed introduction to the highlights of that theory. In the beginning, a review of fundamental tools from topology and the elementary theory of topological groups and transformation groups is presented. Completions, Haar integral, applications to linear representations culminating in the Peter-Weyl Theorem are treated. Pontryagin duality for locally compact Abelian groups forms a central topic of the book. Applications are given, including results about the structure of locally compact Abelian groups, and a structure theory for locally compact rings leading to the classification of locally compact fields. Topological semigroups are discussed in a separate chapter, with special attention to their relations to groups. The last chapter reviews results related to Hilbert's Fifth Problem, with the focus on structural results for non-Abelian connected locally compact groups that can be derived using approximation by Lie groups. The book is self-contained and is addressed to advanced undergraduate or graduate students in mathematics or physics. It can be used for one-semester courses on topological groups, on locally compact Abelian groups, or on topological algebra. Suggestions on course design are given in the preface. Each chapter is accompanied by a set of exercises that have been tested in classes.
Algebraandtopology,thetwofundamentaldomainsofmathematics,playcomplem- tary roles. Topology studies continuity and convergence and provides a general framework to study the concept of a limit. Much of topology is devoted to handling in?nite sets and in?nity itself; the methods developed are qualitative and, in a certain sense, irrational. - gebra studies all kinds of operations and provides a basis for algorithms and calculations. Very often, the methods here are ?nitistic in nature. Because of this difference in nature, algebra and topology have a strong tendency to develop independently, not in direct contact with each other. However, in applications, in higher level domains of mathematics, such as functional analysis, dynamical systems, representation theory, and others, topology and algebra come in contact most naturally. Many of the most important objects of mathematics represent a blend of algebraic and of topologicalstructures. Topologicalfunctionspacesandlineartopologicalspacesingeneral, topological groups and topological ?elds, transformation groups, topological lattices are objects of this kind. Very often an algebraic structure and a topology come naturally together; this is the case when they are both determined by the nature of the elements of the set considered (a group of transformations is a typical example). The rules that describe the relationship between a topology and an algebraic operation are almost always transparentandnatural—theoperationhastobecontinuous,jointlyorseparately.
This book is designed as an introduction into what I call 'abstract' Topological Dynamics (TO): the study of topological transformation groups with respect to problems that can be traced back to the qualitative theory of differential equa is in the tradition of the books [GH] and [EW. The title tions. So this book (,Elements . . . ' rather than 'Introduction . . . ') does not mean that this book should be compared, either in scope or in (intended) impact, with the 'Ele ments' of Euclid or Bourbaki. Instead, it reflects the choice and organisation of the material in this book: elementary and basic (but sufficient to understand recent research papers in this field). There are still many challenging prob lems waiting for a solution, and especially among general topologists there is a growing interest in this direction. However, the technical inaccessability of many research papers makes it almost impossible for an outsider to under stand what is going on. To a large extent, this inaccessability is caused by the lack of a good and systematic exposition of the fundamental methods and techniques of abstract TO. This book is an attempt to fill this gap. The guiding principle for the organization of the material in this book has been the exposition of methods and techniques rather than a discussion of the leading problems and their solutions. though the latter are certainly not neglected: they are used as a motivation wherever possible.
The theory of transformation groups studies symmetries of various mathematical objects such as topological spaces, manifolds, polyhedra and function spaces. It is thus a central concept in many branches of mathematics. This volume contains 25 of the papers submitted at the conference on transformation groups held at the University of Newcastle upon Tyne in August 1976.
The description for this book, Seminar on Transformation Groups. (AM-46), Volume 46, will be forthcoming.
This edition is an almost exact translation of the original Russian text. A few improvements have been made in the present- ation. The list of references has been enlarged to include some papers published more recently, and the latter are marked with an asterisk. THE AUTHOR vii LIST OF SYMBOLS M = M(X, T, rr. ) 1,3. 3 A(X, T) 2-7. 3 M(R) 2-9. 4 2 C [(Y, T, p), G, h] 3-16. 6 P = P(X, T, rr. ) 3,16. 12 1'3. 3 C9v [(Y, T, p), G, h] Px 2-8. 9 E = E(X, T, rr. ) 1,4. 7 Q = Q(X, T, rr. ) 1,3. 3 3,12. 8 Ey Q" = Q" (X, T, rr. ) = Q#(X, T, rr. ) Ext[(Y, T, p), G, h] 3,16. 4 Ext9v[(Y, T, p), G, h] 3,16. 12 2-8. 31 Q" (R) = Q#(R) 3-13. 5 3,12. 12 Gy 3,15. 4 Sx(A) 2,8. 18 G(X, Y) SeA) 2-8. 22 2 3,16. 8 H [cY, T, rr. ), G, h] HE, (X, T, rr. ) = (X, T) 3'12. 12 1'1. 1 Y (X, T, rr., G, a) 4-21. 4 3'16. 1 Hef) HK(f) 4-21. 9 H(X, T) 2,7. 3 1- 3,19. 1 L = L(X, T, rr. ) 1,3. 3 viii I NTRODUCTI ON 1. It is well known that an autonomous system of ordinary dif- ferential equations satisfying conditions that ensure uniqueness and extendibility of solutions determines a flow, i. e. a one- parameter transformation group. G. D.

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