Introduction to Differential and Algebraic Topology

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Springer Science & Business Media, 9 mars 2013 - 493 pages
Topology as a subject, in our opinion, plays a central role in university education. It is not really possible to design courses in differential geometry, mathematical analysis, differential equations, mechanics, functional analysis that correspond to the temporary state of these disciplines without involving topological concepts. Therefore, it is essential to acquaint students with topo logical research methods already in the first university courses. This textbook is one possible version of an introductory course in topo logy and elements of differential geometry, and it absolutely reflects both the authors' personal preferences and experience as lecturers and researchers. It deals with those areas of topology and geometry that are most closely related to fundamental courses in general mathematics. The educational material leaves a lecturer a free choice in designing his own course or his own seminar. We draw attention to a number of particularities in our book. The first chap ter, according to the authors' intention, should acquaint readers with topolo gical problems and concepts which arise from problems in geometry, analysis, and physics. Here, general topology (Ch. 2) is presented by introducing con structions, for example, related to the concept of quotient spaces, much earlier than various other notions of general topology thus making it possible for students to study important examples of manifolds (two-dimensional surfaces, projective spaces, orbit spaces, etc.) as topological spaces, immediately.
 

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Table des matières

First notions of topology
3
Generalization of the concepts of space and function
11
From metric space to topological space visual material
16
4 The concept of Riemann surface
30
Something about knots
38
On some topological applications in physics
41
Review of the recommended literature
58
General Topology
63
Manifolds and fiberings
219
Smooth submanifolds in Euclidean space
229
Smooth manifolds
235
4 Smooth functions in manifolds and smooth partition of unity
254
Mappings of manifolds
263
Tangent bundle and tangent map
301
Tangent vector as differential operator Differential of a function and cotangent bundle
317
Vector fields on smooth manifolds
329

Topology and continuous mappings of metric spaces The spaces R S and D
71
Quotient space and quotient topology
80
4 Classification of surfaces
87
Orbit spaces Projective and lens spaces
100
Operations on sets in a topological space
103
Operations on sets in a metric space Sphere and ball Completeness
108
Properties of continuous mappings
113
Product of topological spaces
118
Connectedness of topological spaces
123
Countability and separability axioms
130
Normal spaces and functional separability
138
Compact locally compact and paracompact spaces and their mappings
144
Compact extensions of topological spaces Metrization
155
Review of the recommended literature
160
Homotopy theory
163
Category functor and algebraization of topological problems
173
Homotopy group functors
178
4 Computing the fundamental and homotopy groups of some spaces
193
Review of the recommended literature
215
Fibre bundles and coverings
336
Smooth functions on a manifold and the cellular structure of a manifold example
368
Nondegenerate critical point and their indices
374
Critical points and homotopy type of manifold
379
Review of the recommended literature
386
Homology theory
389
Homology of chain complexes
392
Homology groups of simplicial complexes
396
4 Singular homology theory
413
Axioms of homology theory Cohomology
426
Homology of spheres Degree of a mapping
429
Homology of a cell complex
447
Euler characteristic and Lefschetz number
453
Review of the recommended literature
476
References
477
Subject index
481
About the authors and the book
491
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