Combinatorial Foundation of Homology and Homotopy: Applications to Spaces, Diagrams, Transformation Groups, Compactifications, Differential Algebras, Algebraic Theories, Simplicial Objects, and Resolutions

378 Pages·1999·13.11 MB·English

by  Hans-Joachim Baues (auth.)

Springer Monographs in Mathematics
Springer-Verlag
Berlin Heidelberg GmbH
Hans-Joachim Baues
Combinatorial Foundation
of Homology and Homotopy
Applications to Spaces, Diagrams,
Transformation Groups, Compactifications,
Differential Algebras, Algebraic Theories,
Simplicial Objects, and Resolutions
, Springer
Hans-Joachim Baues
Max-Planck-Institut fiir Mathematik
Gottfried-Claren-Strasse 26
D-53225 Bonn, Germany
e-mail: [email protected]
Cataloging in Publication Data applied for
Die Deutsche Bibliothek -CIP-Einheitsaufnahme
Baues, Hans J.:
Combinatorial foundation of homology and homotopy: applications to spaces, diagrams,
transformation groups, compactifications, differential algebras, algebraic theories,
simplical objects,and resolutions I Hans Joachim Baues.
(Springer monographs in mathematics)
ISBN 978-3-642-08447-8 ISBN 978-3-662-11338-7 (eBook)
DOI 10.1007/978-3-662-11338-7
Mathematics Subject Classification (1991): 55-02
ISBN 978-3-642-08447-8
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Preface
The classical formulation of homology theory is based on the notion of ring and
module or more generally on abelian categories. The homology that one considers,
however, often comes from a group, or a Lie algebra, or a topological space, etc.
which are non-abelian objects. Therefore a general treatment of homology should
derive the abelian concept of homology from non-abelian data.
The notion of homology emerges in this book from a theory of cogroups or more
generally from a theory of coactions. Such theories arise frequently in algebra and
topology. For example, most algebraic objects like groups, algebras, Lie algebras,
etc. are models of theories of cogroups. Moreover, each homotopy theory contains
theories of coactions. A "theory of coact ions" is a very general concept related
to notions in the literature like near ring or Malcev variety. Nevertheless it has
exactly those properties which are needed to obtain a homology theory suitable
for obstruction theory.
Classical obstruction theory relies on the properties of CW-complexes. Here we
will show that fundamental results on CW-complexes have generalizations in the
realm of categorical algebra. For this we associate to a theory T of coactions the
notion of a T-complex in a cofibration category which is the categorical general
ization of a CW- complex.
We present a homology and cohomology theory for T-complexes which em
bodies numerous homology theories in various fields of algebra and topology. For
example, by suitable specialization one obtains the homology of groups, the homol
ogy in a variety of groups, the Hochschild homology of an algebra, the homology
of a Lie algebra, the homology of a topological space, the Bredon homology of a
G-space where G is a group, the homology theory for diagrams of spaces, the ho
mology theory for controlled spaces, or the homology theory for compactifications,
and many more examples. All these examples are homology theories associated to
theories T of coactions and T-complexes.
The book consists of two parts. The first part (Chapters A, B, C, D) furnishes
a long list of explicit examples and applications in various fields of topology and
algebra. The second part (Chapters I, ... , VIII) develops the axiomatic theory of
combinatorial homology and homotopy.
The unification in this book possesses all the usual advantages. One proof
replaces many different proofs in all such fields. In addition, an interplay takes
place among the various specializations, which thereby enrich one another. The
unified theory also applies to various new situations. Moreover, all definitions,
VI Preface
proofs and results in the second part use a categorical language, so that by a
duality which reverses the direction of arrows one obtains the corresponding dual
definitions, proofs and results, respectively.
May 1998
H.-J. Baues
Table of Contents
Preface ............................................................ V
Leitfaden ........................................................ " XI
Fields of Application ............................................. XIII
Part I. Examples and Applications
Chapter A: Examples and Applications in Topological Categories.. 3
1 Homotopy Theory of Spaces Under a Space ................... " 3
2 Homotopy Theory of Diagrams of Spaces ..................... " 18
3 Homotopy Theory of 'Transformation Groups. . . . . . . . . . . . . . . . . . .. 31
4 Homotopy Theory Controlled at Infinity ...................... " 40
Chapter B: Examples and Applications
in Algebraic Homotopy Theories. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 51
1 Homotopy Theory of Chain Algebras ......................... " 51
2 Homotopy Theory of Connected Simplicial Objects
in Algebraic Theories ........................................ 61
Chapter C: Applications and Examples
in Delicate Homotopy Theories of Simplicial Objects .......... 71
1 Homotopy Theory of Free Simplicial Objects
in Theories of Coact ions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 71
2 Examples of Theories of Coact ions Satisfying
the Delicate Blakers-Massey Property .......................... 87
3 Polynomial Theories of Cogroups ............................ " 89
4 Algebras over an Operad . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 95
Chapter D: Resolutions in Model Categories ..................... " 99
1 Quillen Model Categories ................................... " 99
2 Spiral Model Categories ...................................... 101
3 Spiral Homotopy Theory ..................................... 110
4 Spiral Homotopy Groups ..................................... 115
VIII Table of Contents
5 Examples of Spiral Model Categories ........................... 118
6 Homology and Cohomology in Spiral Homotopy Theory ....... : .. 120
7 Spiral Resolutions and Spiral Realizations ...................... 124
Part II. Combinatorial Homology and Homotopy
Chapter I: Theories of Coact ions and Homology .................. 129
1 Theories of Cogroups and Theories of Coactions ................. 129
2 Examples ................................................... 134
3 The Category of Twisted Maps ................................ 139
4 The Category of Coefficients .................................. 145
5 Enveloping Functors and the Categories
of Premodules and Modules ................................... 149
6 Chain Complexes and Homology .............................. 156
7 Augmented Theories of Coact ions ............................. 161
Chapter II: Twisted Chain Complexes and Twisted Homology ..... 169
1 Twisted Chain Complexes .................................... 170
2 The Module r1 . ............................................. 174
3 The Obstruction for the Twisted Realization of a Chain Map ..... 177
4 Twisted Homotopies ......................................... 182
5 Twisted Homotopy Equivalences .............................. 187
6 The Augmentation Functor ................................... 192
7 Appendix: Homology of Coefficient Objects ..................... 193
8 Appendix: Twisted Homology of Coefficient Objects ............. 197
Chapter III: Basic Concepts of Homotopy Theory ................. 203
1 Cofibration Categories ....................................... 203
2 Homotopy Groups ........................................... 207
3 Principal Cofibrations ........................................ 209
4 The Cylinder of Pairs ........................................ 214
5 Homotopy Cogroups and Homotopy Coactions .................. 215
6 The Theories susp( *) and cone( *) ............................ 217
7 Appendix: Categories with a Cylinder Functor .................. 221
8 Appendix: Natural Cylinder Categories
and Homotopy Theory of Diagrams ............................ 223
9 Appendix: Homotopy Theory of Chain Complexes ............... 225
Chapter IV: Complexes in Cofibration Categories ................. 229
1 Filtered Objects ............................................. 229
2 Complexes Associated to Theories of Coactions ................. 231
3 The Whitehead Theorem ..................................... 234
4 Cellular Approximation ...................................... 240
5 The Blakers-Massey Property ................................. 245
Table of Contents IX
Chapter V: Homology of Complexes ............................... 249
1 Homological Cofibration Categories ............................ 249
2 The Chains of a Complex ..................................... 254
3 The Homology of a Complex .................................. 257
4 The Obstruction Cocycle ..................................... 260
5 The Hurewicz Homomorphism and Whitehead's Exact Sequence ... 262
Chapter V: Homology of Complexes ............................... 267
1 Twisted Homotopy Systems of Order n ......................... 267
2 Obstructions for the Realizability of Chain Maps ................ 271
3 The Homotopy Lifting Property of the Chain Functor ............ 276
4 Counting Realization of Chain Maps ........................... 277
5 Linear Extensions and Towers of Categories ..................... 279
6 The Homological Tower of Categories .......................... 283
7 The Homological Whitehead Theorem .......................... 286
8 The Model Lifting Property of the Twisted Chain Functor ........ 287
9 Obstructions for the Realizability of Twisted Chain Complexes .... 291
10 The Hurewicz Theorem ...................................... 294
11 Appendix: Eilenberg-Mac Lane Complexes and (C, T)-Homology
of Coefficient Objects ........................................ 296
Chapter VII: Finiteness Obstructions .............................. 301
1 The Reduced Projective Class Group .......................... 301
2 The Finiteness Obstruction Theorem ........................... 303
3 Finiteness Obstructions for Twisted Chain Complexes ............ 304
4 Proof of the Finiteness Obstruction Theorem ................... 312
Chapter VIII: Non-Reduced Complexes and Whitehead Torsion ... 315
1 Classes of Discrete Objects ................................... 315
2 Cells in a Cofibration Category ................................ 317
3 Non-Reduced Complexes ..................................... 319
4 The Ball Pair Axiom ......................................... 323
5 Cellular I-Categories ......................................... 327
6 Elementary Expansions ...................................... 328
7 Formal Deformations and Simple Homotopy Equivalences ........ 330
8 The Whitehead Group and Whitehead Torsion .................. 334
9 Simplified Form of Elements in the Whitehead Group ............ 339
10 The Torsion Group K 1 ....................................... 342
11 The Algebraic Whitehead Group .............................. 344
12 The Isomorphism Between the Geometric
and Algebraic Whitehead Group .............................. 345
Index .............................................................. 355
List of Notations .................................................. 361
Leitfaden
The main concepts studied in the axiomatic theory of Part 2 are given by the
following list. We start with a
theory of cogmups T, or a (1.l. 9)
theory of coactions T. (1.1.11)
All the results in Chapter I, II and in VII, § 3 deal with properties of T. This is
pure categorical algebra. We derive from T the
enveloping functor U: Coef ---+ Ringoids (1.5.11)
which is needed in all chapters. In order to introduce homotopy theory we recall
from Baues [AH] some properties of a
cofibmtion category C, or an (III.Ll )
I -category C. (IIl.7.1)
A T -complex can be defined in a
cofibmtion category under T (IV.2.1)
and homology of a T-complex can be obtained in a
homological cofibmtion category under T. (V. 1.1)
Chapter IV deals with cofibration categories under T; in particular, we discuss the
Whitehead theorem, cellular approximation, and the Blakers-Massey property in
such categories. If the Blakers-Massey property holds then one obtains a homo
logical cofibration category under T and all the results of Chapters V, VI, VII are
available.
In particular, we prove the following results in a homological cofibration cate
gory:
- definition of homology and cohomology in terms of a chain functor
- obstruction theory for the extension of maps
- Whitehead's exact sequence for the Hurewicz homomorphism
- homotopy lifting property of the chain functor