In this book we consider 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 we characterize 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.

Includes supplementary material: sn.pub/extras

Inhalt
I. Examples and Applications.- A: Examples and Applications in Topological Categories.- B: Examples and Applications in Algebraic Homotopy Theories.- C: Applications and Examples in Delicate Homotopy Theories of Simplicial Objects.- D: Resolutions in Model Categories.- II. Combinatorial Homology and Homotopy.- I: Theories of Coactions and Homology.- II: Twisted Chain Complexes and Twisted Homology.- III: Basic Concepts of Homotopy Theory.- IV: Complexes in Cofibration Categories.- V: Homology of Complexes.- V: Homology of Complexes.- VII: Finiteness Obstructions.- VIII: Non-Reduced Complexes and Whitehead Torsion.- List of Notations.