Two top experts in topology， O.Ya. Viro and D.B. Fuchs， give an upto-date account of research in central areas of topology and the theory of Lie groups. They cover homotopy， homology and cohomology as well as the theory of manifolds， Lie groups， Grassmannians and lowdimensional manifolds.Their book will be used by graduate students and researchers in mathematics and mathematical physics.
Ⅰ. Introduction to Homotopy Theory Chapter 1.Basic Concepts 1.Terminology and Notations 1.1.Set Theory 1.2.Logical Equivalence 1.3.Topological Spaces 1.4.Operations on Topological Spaces 1.5.Operations on Pointed Spaces 2.Homotopy 2.1.Homotopies 2.2.Paths 2.3.Homotopy as a Path 2.4.Homotopy Equivalence 2.5.Retractions 2.6.Deformation Retractions 2.7.Relative Homotopies 2.8.k-connectedness 2.9.Borsuk Pairs 2.10.CNRS Spaces 2.11.Homotopy Properties of Topological Constructions 2.12.Natural Group Structures on Sets of Homotopy Classes 3.Homotopy Groups 3.1.Absolute Homotopy Groups 3.2.Digression: Local Systems 3.3.Local Systems of Homotopy Groups of a Topological Space 3.4.Relative Homotopy Groups 3.5.The Homotopy Sequence of a Pair 3.6.Splitting 3.7.The Homotopy Sequence of a Triple Chapter 2.Bundle Techniques 4.Bundles 4.1.General Definitions 4.2.Locally Trivial Bundles 4.3.Serre Bundles 4.4.Bundles of Spaces of Maps 5.Bundles and Homotopy Groups 5.1.The Local System of Homotopy Groups of the Fibres of a Serre Bundle 5.2.The Homotopy Sequence of a Serre Bundle 5.3.Important Special Cases 6.The Theory of Coverings 6.1.Coverings 6.2.The Group of a Covering 6.3.Hierarchies of Coverings 6.4.The Existence of Coverings 6.5.Automorphisms of a Coveting 6.6.Regular Coverings 6.7.Covering Maps Chapter 3 Cellular Techniques 7.Cellular Spaces 7.1.Basic Concepts 7.2.Gluing of Cellular Spaces from Balls 7.3.Examples of Cellular Decompositions 7.4.Topological Properties of Cellular Spaces 7.5.Cellular Constructions 8.Simplicial Spaces 8.1.Basic Concepts 8.2.Simplicial Schemes 8.3.Simplicial Constructions 8.4.Stars, Links, Regular Neighbourhoods 8.5.Simplicial Approximation of a Continuous Map 9.Cellular Approximation of Maps and Spaces 9.1.Cellular Approximation of a Continuous Map 9.2.Cellular k-connected Pairs 9.3.Simplicial Approximation of Cellular Spaces …… Capter4 The Simplest Calculations Ⅱ.Homology and Cohomology Ⅲ. Classical Manifolds Index
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