Infinite Homotopy Theory

H-J. Baues

ISBN 10: 9401064938 / ISBN 13: 9789401064934

Published by Springer, Springer, 2013

About this Item

Druck auf Anfrage Neuware - Printed after ordering - 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.

Seller Inventory # 9789401064934

Bibliographic Details

• Title: Infinite Homotopy Theory
• Publisher: Springer, Springer
• Publication Date: 2013
• Language: English
• ISBN 10: 9401064938
• ISBN 13: 9789401064934
• Binding: Taschenbuch
• Condition: Neu
• Dimensions: 240W x 160H x 17D millimeters 240 millimeters width by 160 millimeters height by 17 millimeters depth
• Weight: 495 grams

About this title

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.

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)