Combinatorial foundation of homology and homotopy : applications to spaces, diagrams, transformation groups, compactifications, differential algebras, algebraic theories, simplicial objects, and resolutions

This book considers deep and classical results of homotopy theory like the homological Whitehead theorem, the Hurewicz theorem, the finiteness obstruction theorem of Wall, the theorems on Whitehead torsion and simple homotopy equivalences, and characterizes axiomatically the assumptions under which such results hold. This leads to a new combinatorial foundation of homology and homotopy. Numerous explicit examples and applications in various fields of topology and algebra are given. Front Matter....Pages I-XV Front Matter....Pages 1-2 Examples and Applications in Topological Categories....Pages 3-50 Examples and Applications in Algebraic Homotopy Theories....Pages 51-69 Applications and Examples in Delicate Homotopy Theories of Simplicial Objects....Pages 71-97 Resolutions in Model Categories....Pages 99-126 Front Matter....Pages 127-128 Theories of Coactions and Homology....Pages 129-168 Twisted Chain Complexes and Twisted Homology....Pages 169-202 Basic Concepts of Homotopy Theory....Pages 203-228 Complexes in Cofibration Categories....Pages 229-248 Homology of Complexes....Pages 249-265 Realization of Chain Maps....Pages 267-300 Finiteness Obstructions....Pages 301-313 Non-Reduced Complexes and Whitehead Torsion....Pages 315-354 Erratum....Pages 370-370 Back Matter....Pages 355-369

A new combinatorial foundation of the two concepts, based on a consideration of deep and classical results of homotopy theory, and an axiomatic characterization of the assumptions under which results in this field hold. Includes numerous explicit examples and applications in various fields of topology and algebra.