Synopses & Reviews
This book deals with algebraic topology, homotopy theory and simple homotopy theory of infinite CW-complexes with ends. Contrary to most other works on these subjects, the current volume does not use inverse systems to treat these topics. Here, the homotopy theory is approached without the rather sophisticated notion of pro-category. Spaces with ends are studied only by using appropriate constructions such as spherical objects of CW-complexes in the category of spaces with ends, and all arguments refer directly to this category. In this way, infinite homotopy theory is presented as a natural extension of classical homotopy theory. In particular, this book introduces the construction of the proper groupoid of a space with ends and then the cohomology with local coefficients is defined by the enveloping ringoid of the proper fundamental groupoid. This volume will be of interest to researchers whose work involves algebraic topology, category theory, homological algebra, general topology, manifolds, and cell complexes.
Review
From the reviews: "In this book the authors try to deal with more general spaces in a fundamental way by setting up algebraic topology in an abstract categorical context which encompasses not only the usual category of topological spaces and continuous maps, but also several categories related to proper maps. ... all concepts are carefully explained and detailed references for the proofs are given. ... a good understanding of the basics of ordinary homotopy theory is all that is needed to enjoy reading this book." (F. Clauwens, Nieuw Archief voor Wiskunde, Vol. 7 (2), 2006)
Review
From the reviews:
"In this book the authors try to deal with more general spaces in a fundamental way by setting up algebraic topology in an abstract categorical context which encompasses not only the usual category of topological spaces and continuous maps, but also several categories related to proper maps. ... all concepts are carefully explained and detailed references for the proofs are given. ... a good understanding of the basics of ordinary homotopy theory is all that is needed to enjoy reading this book." (F. Clauwens, Nieuw Archief voor Wiskunde, Vol. 7 (2), 2006)
Synopsis
Compactness in topology and finite generation in algebra are nice properties to start with. However, the study of compact spaces leads naturally to non-compact spaces and infinitely generated chain complexes; a classical example is the theory of covering spaces. In handling non-compact spaces we must take into account the infinity behaviour of such spaces. This necessitates modifying the usual topological and algebraic cate gories to obtain "proper" categories in which objects are equipped with a "topologized infinity" and in which morphisms are compatible with the topology at infinity. The origins of proper (topological) category theory go back to 1923, when Kere kjart6 VT] established the classification of non-compact surfaces by adding to orien tability and genus a new invariant, consisting of a set of "ideal points" at infinity. Later, Freudenthal ETR] gave a rigorous treatment of the topology of "ideal points" by introducing the space of "ends" of a non-compact space. In spite of its early ap pearance, proper category theory was not recognized as a distinct area of topology until the late 1960's with the work of Siebenmann OFB], IS], DES] on non-compact manifolds."
Description
Includes bibliographical references (p. [285]-290) and index.
Table of Contents
Introduction.
I. Foundations of homotopy theory and proper homotopy theory.
II. Trees and Spherical objects in the category Topp of compact maps.
III. Three-like spaces and spherical objects in the category End of ended spaces.
IV. CW-complexes.
V. Theories and models of theories.
VI. T-controlled homology.
VII. Proper groupoids.
VIII. The enveloping ringoid of a proper groupoid.
IX. T-controlled homology with coefficients.
X. Simple homotopy types with ends. Bibliography. Subject Index. List of symbols.