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Foundational Essays on Topological Manifolds, Smoothings, and Triangulations. (AM-88), Volume 88
Since Poincaré's time, topologists have been most concerned with three species of manifold. The most primitive of these--the TOP manifolds--remained rather mysterious until 1968, when Kirby discovered his now famous torus unfurling device. A period of rapid progress with TOP manifolds ensued, including, in 1969, Siebenmann's refutation of the Hauptvermutung and the Triangulation Conjecture. Here is the first connected account of Kirby's and Siebenmann's basic research in this area. The five sections of this book are introduced by three articles by the authors that initially appeared between 1968 and 1970. Appendices provide a full discussion of the classification of homotopy tori, including Casson's unpublished work and a consideration of periodicity in topological surgery.
Induction Theorems for Groups of Homotopy Manifold Structures
Classifying spaces in surgery theory were first used by Sullivan and Casson in their (independent) unpublished work on the Hauptvermutung for PL manifolds. In his 1968 Ph.D. thesis, F. Quinn developed a general theory of surgery classifying spaces, realizing the Wall surgery groups as the homotopy groups [italic]L[subscript]*([italic]G) = [lowercase Greek]Pi[subscript]*([italic]L([italic]G)) of a spectrum of manifold n-ad surgery problems with fundamental group G. This work presents a detailed account of Quinn's theory. Geometric methods are used to view the Sullivan-Wall manifold structure sequence as an exact sequence of abelian groups (as suggested by Quinn). The intersection of the known induction theorems for generalized cohomology groups and [italic]L-groups then gives an induction theorem for the structure sequence with finite [italic]G.
New Scientific Applications of Geometry and Topology
Geometry and topology are subjects generally considered to be "pure" mathematics. Recently, however, some of the methods and results in these two areas have found new utility in both wet-lab science (biology and chemistry) and theoretical physics. Conversely, science is influencing mathematics, from posing questions that call for the construction of mathematical models to exporting theoretical methods of attack on long-standing problems of mathematical interest. Based on an AMS Short Course held in January 1992, this book contains six introductory articles on these intriguing new connections. There are articles by a chemist and a biologist about mathematics, and four articles by mathematicians writing about science and mathematics involved. Because this book communicates the excitement and utility of mathematics research at an elementary level, it is an excellent textbook in an advanced undergraduate mathematics course.
Discrete Geometry for Computer Imagery
These proceedings contain papers presented at the 8th Discrete Geometry for Computer Imagery conference, held 17-19, March 1999 at ESIEE, Marne-la- Vall ee. The domains of discrete geometry and computer imagery are closely related. Discrete geometry provides both theoretical and algorithmic models for the p- cessing, analysis and synthesis of images; in return computer imagery, in its variety of applications, constitutes a remarkable experimentational eld and is a source of challenging problems. The number of returning participants, the arrival each year of contributions from new laboratories and new researchers, as well as the quality and originality of the results have contributed to the s...
Algebraic and Geometric Topology, Part 2
Contains sections on Structure of topological manifolds, Low dimensional manifolds, Geometry of differential manifolds and algebraic varieties, $H$-spaces, loop spaces and $CW$ complexes, Problems.
Classgroups of Group Rings
This book is a self-contained account of the theory of classgroups of group rings. The guiding philosophy has been to describe all the basic properties of such classgroups in terms of character functions. This point of view is due to A. Frohlich and it achieves a considerable simplification and clarity over previous techniques. A main feature of the book is the introduction of the author's group logarithm, with numerous examples of its application. The main results dealt with are: Ullom's conjecture for Swan modules of p-groups; the self-duality theorem for rings of integers of tame extensions; the fixed-point theorem for determinants of group rings; the existence of Adams operations on classgroups. In addition, the author includes a number of calculations of classgroups of specific families of groups such as generalized dihedral groups, and quaternion and dihedral 2-groups. The work contained in this book should be readily accessible to any graduate student in pure mathematics who has taken a course in the representation theory of finite groups. It will also be of interest to number theorists and algebraic topologists.
Geometric Aspects of General Topology
This book is designed for graduate students to acquire knowledge of dimension theory, ANR theory (theory of retracts), and related topics. These two theories are connected with various fields in geometric topology and in general topology as well. Hence, for students who wish to research subjects in general and geometric topology, understanding these theories will be valuable. Many proofs are illustrated by figures or diagrams, making it easier to understand the ideas of those proofs. Although exercises as such are not included, some results are given with only a sketch of their proofs. Completing the proofs in detail provides good exercise and training for graduate students and will be usefu...
Diagram Cohomology and Isovariant Homotopy Theory
Obstruction theoretic methods are introduced into isovariant homotopy theory for a class of spaces with group actions; the latter includes all smooth actions of cyclic groups of prime power order. The central technical result is an equivalence between isovariant homotopy and specific equivariant homotopy theories for diagrams under suitable conditions. This leads to isovariant Whitehead theorems, an obstruction-theoretic approach to isovariant homotopy theory with obstructions in cohomology groups of ordinary and equivalent diagrams, and qualitative computations for rational homotopy groups of certain spaces of isovariant self maps of linear spheres. The computations show that these homotopy groups are often far more complicated than the rational homotopy groups for the corresponding spaces of equivariant self maps. Subsequent work will use these computations to construct new families of smooth actions on spheres that are topologically linear but differentiably nonlinear.
Representations of General Linear Groups
This book examines the representation theory of the general linear groups, and reveals that there is a close analogy with that of the symmetric groups.
Proceedings of the Conference on Transformation Groups
These Proceedings contain articles based on the lectures and in formal discussions at the Conference on Transformation Groups held at Tulane University, May 8 to June 2, 1967 under the sponsorship of the Advanced Science Seminar Projects of the National Science Foun dation (Contract No. GZ 400). They differ, however, from many such Conference proceedings in that particular emphasis has been given to the review and exposition of the state of the theory in its various mani festations, and the suggestion of direction to further research, rather than purely on the publication of research papers. That is not to say that there is no new material contained herein. On the contrary, there is an abund...
