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Infinite Homotopy Theory : K-Monographs in Mathematics - Hans-Joachim Baues

Infinite Homotopy Theory

K-Monographs in Mathematics

Hardcover Published: 30th June 2001
ISBN: 9780792369820
Number Of Pages: 296

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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.

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"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)

Introductionp. 1
Foundations of homotopy theory and proper homotopy theoryp. 7
Compactifications and compact mapsp. 8
Homotopyp. 18
Categories with a cylinder functorp. 22
Cofibration categories and homotopy theory in I-categoriesp. 29
Tracks and cylindrical homotopy groupsp. 36
Homotopy groupsp. 44
Cofibresp. 50
Appendicesp. 53
Appendix: Compact mapsp. 53
Appendix: The Freudenthal compactificationp. 57
Trees and spherical objects in the category Topp of compact mapsp. 71
Locally finite trees and Freudenthal endsp. 71
Halin's tree lemmap. 78
Unions in Toppp. 80
The proper Hilton--Milnor theoremp. 87
Spherical objects and homotopy groups in Toppp. 89
The homotopy category of n-dimensional spherical objects in Toppp. 96
Classification of spherical objects under a treep. 103
Tree-like spaces and spherical objects in the category End of ended spacesp. 107
Tree-like spaces in Endp. 107
Unions in Endp. 109
Spherical objects and homotopy groups in Endp. 113
The homotopy category of n-dimensional spherical objects in Endp. 117
Classification of spherical objects under a tree-like spacep. 122
Z-sets and telescopesp. 124
ARZ-spacesp. 130
CW-complexesp. 135
Relative CW-complexes in Topp. 135
Strongly locally finite CW-complexesp. 140
Relative CW-complexes in Toppp. 142
Relative CW-complexes in Endp. 148
Normalization of CW-complexesp. 154
Push outs of CW-complexesp. 157
The Blakers-Massey theoremp. 159
The proper Whitehead theoremp. 163
Theories and models of theoriesp. 165
Theories of cogroups and Van Kampen theorem for proper fundamental groupsp. 165
Additive categories and additivizationp. 175
Rings associated to tree-like spacesp. 185
Inverse limits of gr(T)-modelsp. 192
Kernels in ab(T)p. 199
T-controlled homologyp. 203
R-modules and the reduced projective class groupp. 203
Chain complexes in ringoids and homologyp. 208
Cellular T-controlled homologyp. 211
Coefficients for T-controlled homology and cohomologyp. 215
The Hurewicz theorem in Endp. 221
The proper homological Whitehead theorem (the 1-connected case)p. 224
Proper finiteness obstructions (the 1-connected case)p. 225
Proper groupoidsp. 229
Filtered discrete objectsp. 229
The fundamental groupoid of ended spacesp. 232
The proper homotopy category of 1-dimensional reduced relative CW-complexesp. 236
Free D-groupoids under Gp. 237
The proper fundamental groupoid of a 1-dimensional reduced relative CW-complexp. 242
Simplicial objects in proper homotopy theoryp. 244
The enveloping ringoid of a proper groupoidp. 249
The homotopy category of 1-dimensional spherical objects under Tp. 249
The ringoid S(X,T) associated to a pair (X,T) in Endp. 250
The enveloping ringoid of the proper fundamental groupp. 253
The enveloping ringoid of the proper fundamental groupoidp. 256
T-controlled homology with coefficientsp. 261
The T-controlled twisted chain complex of a relative CW-complex (X,T)p. 261
The T-controlled twisted chain complex of a CW-complex Xp. 266
T-controlled cohomology and homology with local coefficientsp. 268
Proper obstruction theoryp. 269
The twisted Hurewicz homomorphism and the twisted T-sequence in [infinity]Endp. 270
The proper homological Whitehead theorem (the 0-connected case)p. 273
Proper finiteness obstructions (the 0-connected case)p. 274
Simple homotopy types with endsp. 275
The torsion group K[subscript 1]p. 275
Simple equivalences and proper equivalencesp. 277
The topological Whitehead groupp. 279
The algebraic Whitehead groupp. 280
The proper algebraic Whitehead groupp. 282
Bibliographyp. 285
Subject Indexp. 291
List of symbolsp. 295
Table of Contents provided by Syndetics. All Rights Reserved.

ISBN: 9780792369820
ISBN-10: 0792369823
Series: K-Monographs in Mathematics
Audience: Professional
Format: Hardcover
Language: English
Number Of Pages: 296
Published: 30th June 2001
Publisher: Springer
Country of Publication: NL
Dimensions (cm): 23.4 x 15.6  x 1.9
Weight (kg): 1.35