Algebraic Topology

Description

Title page Preface I. INTRODUCTION TO GENERAL TOPOLOGY 1. Primitive concepts 2. Topological spaces 3. Aggregates of sets. Coverings. Dimension 4. Connectedness 5. Compact spaces 6. Separation axioms 7. Inverse mapping systems 8. Metrization 9. Homotopy. Deformation. Retraction II. ADDITIVE GROUPS 1. General properties 2. Generators of a group 3. Limit-groups 4. Group multiplication 5. Characters. Duality 6. Vector spaces III. COMPLEXES 1. Complexes. Definitions and examples 2. Homology theory of finite complexes. (a) Generalities 3. Homology theory of finite complexes. (b) Integral groups 4. Homology theory of finite complexes. (c) Arbitrary groups of coefficients 5. Application to some special complexes 6. Duality theory for finite complexes 7. Linking coefficients. Duality in the sense of Alexander 8. Homology theory of infinite complexes 9. Augmentable and simple complexes IV. COMPLEXES: PRODUCTS. TRANSFORMATIONS. SUBDIVISIONS 1. Products of complexes 2. Products of chains and cycles 3. Set-transformations 4. Chain-mappings 5. Chain-homotopy 6. Complements 7. Subdivision. Derivation. Partition V. COMPLEXES: MULTIPLICATIONS AND INTERSECTIONS. FIXED ELEMENTS. MANIFOLDS 1. Multiplications 2. Intersections 3. Coincidences and fixed elements 4. Combinatorial manifolds VI. NETS OF COMPLEXES 1. Definition of nets and their groups 2. Duality and intersections 3. Further properties of nets 4. Spectra 5. Application to infinite complexes 6. Webs 7. Metric complexes VII. HOMOLOGY THEORY OF TOPOLOGICAL SPACES 1. Homology theory: foundations and general properties 2. Relations between connectedness and homology 3. Groups related to webs 4. Groups related to the union and intersection of two sets 5. The Vietoris homology theory for compacta 6. Reduction of the Vietoris theory to the Cech theory 7. Homology theories of Kurosch and Alexander-Kolmogorov VIII. TOPOLOGY OF POLYHEDRA AND RELATED QUESTIONS 1. Geometrie complements 2. Homology theory 3. Geometrie manifolds 4. Continuous and singular complexes 5. Coincidences and fixed points 6. Quasi-complexes and the fixed point theorem 7. Topological complexes 8. Differentiable complexes and manifolds 9. Group manifolds 10. Nomenclature of complexes and manifolds APPENDIX A. On homology groups of infinite complexes and compacta. By Samuel Eilenberg and Saunders MacLane B. Fixed points of periodic transformations. By P. A. Smith BIBLIOGRAPHY INDEX OF SPECIAL SYMBOLS AND NOTATION INDEX