Intended for use both as a text and a reference, this book is an exposition of the fundamental ideas of algebraic topology. The first third of the book covers the fundamental group, its definition and its application in the study of covering spaces. The focus then turns to homology theory, including cohomology, cup products, cohomology operations, and topological manifolds. The remaining third of the book is devoted to Homotropy theory, covering basic facts about homotropy groups, applications to obstruction theory, and computations of homotropy groups of spheres. In the later parts, the main emphasis is on the application to geometry of the algebraic tools developed earlier.
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ISBN-13
978-1-4684-9322-1 (9781468493221)
DOI
10.1007/978-1-4684-9322-1
Schweitzer Klassifikation
1 Set theory.- 2 General topology.- 3 Group theory.- 4 Modules.- 5 Euclidean spaces.- 1 Homotopy and The Fundamental Group.- 1 Categories.- 2 Functors.- 3 Homotopy.- 4 Retraction and deformation.- 5 H spaces.- 6 Suspension.- 7 The fundamental groupoid.- 8 The fundamental group.- Exercises.- 2 Covering Spaces and Fibrations.- 1 Covering projections.- 2 The homotopy lifting property.- 3 Relations with the fundamental group.- 4 The lifting problem.- 5 The classification of covering projections.- 6 Covering transformations.- 7 Fiber bundles.- 8 Fibrations.- Exercises.- 3 Polyhedra.- 1 Simplicial complexes.- 2 Linearity in simplicial complexes.- 3 Subdivision.- 4 Simplicial approximation.- 5 Contiguity classes.- 6 The edge-path groupoid.- 7 Graphs.- 8 Examples and applications.- Exercises.- 4 Homology.- 1 Chain complexes.- 2 Chain homotopy.- 3 The homology of simplicial complexes.- 4 Singular homology.- 5 Exactness.- 6 Mayer-Vietoris sequences.- 7 Some applications of homology.- 8 Axiomaticcharacterization of homology.- Exercises.- 5 Products.- 1 Homology with coefficients.- 2 The universal-coefficient theorem for homology.- 3 The Künneth formula.- 4 Cohomology.- 5 The universal-coefficient theorem for cohomology.- 6 Cup and cap products.- 7 Homology of fiber bundles.- 8 The cohomology algebra.- 9 The Steenrod squaring operations.- Exercises.- 6 General Cohomology Theory and Duality.- 1 The slant product.- 2 Duality in topological manifolds.- 3 The fundamental class of a manifold.- 4 The Alexander cohomology theory.- 5 The homotopy axiom for the Alexander theory.- 6 Tautness and continuity.- 7 Presheaves.- 8 Fine presheaves.- 9 Applications of the cohomology of presheaves.- 10 Characteristic classes.- Exercises.- 7 Homotopy Theory.- 1 Exact sequences of sets of homotopy classes.- 2 Higher homotopy groups.- 3 Change of base points.- 4 The Hurewicz homomorphism.- 5 The Hurewicz isomorphism theorem.- 6 CW complexes.- 7 Homotopy functors.- 8 Weak homotopy type.- Exercises.-8 Obstruction Theory.- 1 Eilenberg-MacLane spaces.- 2 Principal fibrations.- 3 Moore-Postnikov factorizations.- 4 Obstruction theory.- 5 The suspension map.- Exercises.- 9 Spectral Sequences and Homotopy Groups of Spheres.- 1 Spectral sequences.- 2 The spectral sequence of a fibration.- 3 Applications of the homology spectral sequence.- 4 Multiplicative properties of spectral sequences.- 5 Applications of the cohomology spectral sequence.- 6 Serre classes of abelian groups.- 7 Homotopy groups of spheres.- Exercises.
