## Introduction to Differential and Algebraic TopologySpringer 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. |

### Avis des internautes - Rédiger un commentaire

Aucun commentaire n'a été trouvé aux emplacements habituels.

### Table des matières

3 | |

11 | |

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 |

477 | |

481 | |

About the authors and the book | 491 |

### Autres éditions - Tout afficher

Introduction to Differential and Algebraic Topology Yu.G. Borisovich,N.M. Bliznyakov,T.N. Fomenko,Y.A. Izrailevich Affichage d'extraits - 1995 |

Introduction to Differential and Algebraic Topology Yu.G. Borisovich,N.M. Bliznyakov,T.N. Fomenko,Y.A. Izrailevich Aucun aperçu disponible - 2010 |

Introduction to Differential and Algebraic Topology Yu. G. Borisovich,N. M. Bliznyakov,T. N. Fomenko Aucun aperçu disponible - 2014 |

### Expressions et termes fréquents

algebraic arbitrary atlas axiom barycentric bijective boundary C*-atlas C*-diffeomorphism C*-manifold C*-mapping called cell complex chain complexes chart commutative compact space concept connected consequently Consider consisting construct continuous mapping coordinates corresponding countable covering critical points defined definition denoted diffeomorphism dimension disc edge element equal example Exercise exists fibre bundle finite number fixed point formula function f functor fundamental group given gluing Hausdorff homeomorphism homology groups homology theory homotopy class homotopy equivalent induced intersection isomorphism lemma Let f linear loop mapping f matrix metric space Möbius strip n-dimensional obtain open neighbourhood open set orientation path path—connected plane point x e point x0 polygon polyhedron preimage Proof properties Prove quotient space sequence Show simplex simplices simplicial complex simplicial mapping singular points smooth manifold sphere submanifold subset subspace tangent bundle tangent vector theorem topological space torus triangulation vector field vector space Verify vertices zero