Combinatorial Homotopy and 4-Dimensional Complexes

✍ Scribed by Hans-Joachim Baues; Ronald Brown

• Publisher: De Gruyter
• Year: 1991
• Tongue: English
• Leaves: 408
• Series: De Gruyter Expositions in Mathematics; 2
• Edition: Reprint 2011
• Category: Library

✦ Synopsis

"This is an impressive and substantial book containing significant new ideas in a rather unfashionable area." Mathematical Reviews

✦ Table of Contents

Preface by Ronald Brown
Introduction
Chapter I. Homotopy, homology, and Whitehead’s classification of simply connected 4-dimensional CW-complexes
§ 0 Sufficiency, realizability and detecting functors
§ 1 Homotopy groups
§ 2 CW-complexes and homology with local coefficients
§ 3 Whitehead’s certain exact sequence
§ 4 The quadratic functor Γ and central extensions
§ 5 Cup products and Pontrjagin squares
§ 6 Cohomological invariants
§ 7 An2-polyhedra and An2-forms
§ 8 Whitehead’s classification of An2-polyhedra, n ≥ 2
Chapter II. The CW-tower of categories
§ 1 Linear extensions of categories and exact sequences for functors
§ 2 Homotopy systems of order (n + 1)
§ 3 The CW-tower of categories
§4 The Postnikov chain functor
§ 5 Three formulas for the obstruction
§ 6 Trees of homotopy types
§ 7 On the homotopy classification of manifolds with finite fundamental group
Chapter III. Crossed modules and homotopy systems of order 3
§ 1 Nilpotent groups and Peiffer nilpotent pre-crossed modules
§ 2 Crossed chain complexes and homotopy systems of order 3
§ 3 Cylinders of CW-complexes and of crossed chain complexes
§4 Cofibrations in the category of crossed chain complexes
§ 5 The homotopy addition lemma
§ 6 A model functor from spaces to crossed chain complexes
§ 7 The homotopy category of 2-dimensional CW-complexes
§ 8 The homotopy category of 2-types
§ 9 The crossed chain complex of a product
§ 10 The action of the fundamental group and free homotopy classes
Appendix A. Obstructions for the realizability of chain complexes
Appendix B. The homotopy category of pseudo projective planes
Appendix C. On the suspension and the James construction
Appendix D. The homotopy category of suspended pseudo projective planes
Chapter IV. Quadratic modules and homotopy systems of order 4
§ 1 Quadratic modules
§ 2 Free quadratic modules
§ 3 Quadratic chain complexes
§ 4 Homotopies for quadratic chain maps
§ 5 Cofibrations in the category of quadratic chain complexes
§ 6 The secondary homotopy addition lemma and a model functor from spaces to quadratic chain complexes
§ 7 Homotopy systems of order 4
§ 8 The homotopy category of 3-dimensional CW-complexes
§ 9 The CW-tower in degree ≤ 4
§ 10 The homotopy category of 3-types
§ 11 The action of the fundamental group for quadratic chain complexes
§ 12 The quadratic chain complex of a product
Appendix A. Some diverse examples and applications of quadratic chain complexes
Appendix B. Quadratic chain complexes and simplicial groups
Appendix C. Reduced and stable quadratic modules
Appendix D. On the homotopy classification of semi free group actions
Chapter V. Cohomological invariants
§ 1 The classification of 4-dimensional homotopy types
§ 2 A new cohomological invariant and the cup product
§ 3 Obstructions for the existence of certain chain maps and the primary obstruction for the realizability of a chain complex
§ 4 The classification of special 4-dimensional homotopy types by Pontrjagin squares
§ 5 Natural quotients of the Postnikov chain functor
§ 6 Pontrjagin squares with local coefficients
Appendix A. The stable equivalence classes of finite 4-dimensional complexes
Chapter VI. The cohomology of categories and the calculus of tracks
§ 1 The cohomology of categories
§ 2 The category of free nil(2)-groups and the existence of Pontrjagin squares
§ 3 Linear track extensions of categories
§ 4 Free nil(2)-groups and tracks for one point unions of n-spheres
§ 5 Tracks for one point unions of n-spheres with operators in groups
§ 6 Free nil(2)-modules and tracks for 2-dimensional CW-complexes
§ 7 Track-models for 4-dimensional CW-complexes
Bibliography
List of Symbols
Index