Infinite homotopy theory / by Hans-Joachim Baues and Antonio Quintero.

Format:

Book

Author/Creator:

Baues, Hans J., 1943-

Contributor:

Quintero Toscano, Antonio.

Series:

K-monographs in mathematics ; v. 6.

K-monographs in mathematics ; v. 6

Language:

English

Subjects (All):

Homotopy theory.

Physical Description:

vii, 296 pages : illustrations ; 25 cm.

Place of Publication:

Dordrecht ; Boston : Kluwer Academic Publishers, [2001]

Contents:

Chapter I. Foundations of homotopy theory and proper homotopy theory 7

1 Compactifications and compact maps 8

2 Homotopy 18

3 Categories with a cylinder functor 22

4 Cofibration categories and homotopy theory in I-categories 29

5 Tracks and cylindrical homotopy groups 36

6 Homotopy groups 44

7 Cofibres 50

8 Appendix: Compact maps 53

9 Appendix: The Freudenthal compactification 57

Chapter II. Trees and spherical objects in the category Topp of compact maps 71

1 Locally finite trees and Freudenthal ends 71

Appendix Halin's tree lemma 78

2 Unions in Topp 80

Appendix The proper Hilton

Milnor theorem 87

3 Spherical objects and homotopy groups in Topp 89

4 The homotopy category of n-dimensional spherical objects in Topp 96

Appendix Classification of spherical objects under a tree 103

Chapter III. Tree-like spaces and spherical objects in the category End of ended spaces 107

1 Tree-like spaces in End 107

2 Unions in End 109

3 Spherical objects and homotopy groups in End 113

4 The homotopy category of n-dimensional spherical objects in End 117

Appendix Classification of spherical objects under a tree-like space 122

5 Z-sets and telescopes 124

6 ARZ-spaces 130

Chapter IV. CW-complexes 135

1 Relative CW-complexes in Top 135

2 Strongly locally finite CW-complexes 140

3 Relative CW-complexes in Topp 142

4 Relative CW-complexes in End 148

5 Normalization of CW-complexes 154

6 Push outs of CW-complexes 157

7 The Blakers-Massey theorem 159

8 The proper Whitehead theorem 163

Chapter V. Theories and models of theories 165

1 Theories of cogroups and Van Kampen theorem for proper fundamental groups 165

2 Additive categories and additivization 175

3 Rings associated to tree-like spaces 185

4 Inverse limits of gr(T)-models 192

5 Kernels in ab(T) 199

Chapter VI. T-controlled homology 203

1 R-modules and the reduced projective class group 203

2 Chain complexes in ringoids and homology 208

3 Cellular T-controlled homology 211

4 Coefficients for T-controlled homology and cohomology 215

5 The Hurewicz theorem in End 221

6 The proper homological Whitehead theorem (the 1-connected case) 224

7 Proper finiteness obstructions (the 1-connected case) 225

Chapter VII. Proper groupoids 229

1 Filtered discrete objects 229

2 The fundamental groupoid of ended spaces 232

3 The proper homotopy category of 1-dimensional reduced relative CW-complexes 236

4 Free D-groupoids under G 237

5 The proper fundamental groupoid of a 1-dimensional reduced relative CW-complex 242

6 Simplicial objects in proper homotopy theory 244

Chapter VIII. The enveloping ringoid of a proper groupoid 249

1 The homotopy category of 1-dimensional spherical objects under T 249

2 The ringoid S(X,T) associated to a pair (X,T) in End 250

3 The enveloping ringoid of the proper fundamental group 253

4 The enveloping ringoid of the proper fundamental groupoid 256

Chapter IX. T-controlled homology with coefficients 261

1 The T-controlled twisted chain complex of a relative CW-complex (X,T) 261

2 The T-controlled twisted chain complex of a CW-complex X 266

3 T-controlled cohomology and homology with local coefficients 268

4 Proper obstruction theory 269

5 The twisted Hurewicz homomorphism and the twisted T-sequence in [infinity]End 270

6 The proper homological Whitehead theorem (the 0-connected case) 273

7 Proper finiteness obstructions (the 0-connected case) 274

Chapter X. Simple homotopy types with ends 275

1 The torsion group K[subscript 1] 275

2 Simple equivalences and proper equivalences 277

3 The topological Whitehead group 279

4 The algebraic Whitehead group 280

5 The proper algebraic Whitehead group 282.

Notes:

Includes bibliographical references (pages [285]-290) and index.

ISBN:

0792369823

OCLC:

46808838