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.