Beginner's course in topology : geometric chapters
著者
書誌事項
Beginner's course in topology : geometric chapters
(Springer series in Soviet mathematics)(Universitext)
Springer-Verlag, c1984
- : U.S.
- : Germany
- タイトル別名
-
Nachalʹnyĭ kurs topologii
大学図書館所蔵 全64件
  青森
  岩手
  宮城
  秋田
  山形
  福島
  茨城
  栃木
  群馬
  埼玉
  千葉
  東京
  神奈川
  新潟
  富山
  石川
  福井
  山梨
  長野
  岐阜
  静岡
  愛知
  三重
  滋賀
  京都
  大阪
  兵庫
  奈良
  和歌山
  鳥取
  島根
  岡山
  広島
  山口
  徳島
  香川
  愛媛
  高知
  福岡
  佐賀
  長崎
  熊本
  大分
  宮崎
  鹿児島
  沖縄
  韓国
  中国
  タイ
  イギリス
  ドイツ
  スイス
  フランス
  ベルギー
  オランダ
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  アメリカ
注記
Bibliography: p. [506]-507
Includes index
内容説明・目次
内容説明
This book is the result of reworking part of a rather lengthy course of lectures of which we delivered several versions at the Leningrad and Moscow Universities. In these lectures we presented an introduction to the fundamental topics of topology: homology theory, homotopy theory, theory of bundles, and topology of manifolds. The structure of the course was well determined by the guiding term elementary topology, whose main significance resides in the fact that it made us use a rather simple apparatus. tn this book we have retained {hose sections of the course where algebra plays a subordinate role. We plan to publish the more algebraic part of the lectures as a separate book. Reprocessing the lectures to produce the book resulted in the profits and losses inherent in such a situation: the rigour has increased to the detriment of the intuitiveness, the geometric descriptions have been replaced by formulas needing interpretations, etc. Nevertheless, it seems to us tha*t the book retains the main qualities of our lectures: their elementary, systematic, and pedagogical features. The preparation of the reader is assumed to be limi ted to the usual knowledge of set *theory, algebra, and calculus which mathematics students should master after the first year and a half of studies. The exposition is accompanied by examples and exercises. We hope that the book can be used as a topology textbook.
目次
Set-Theoretical Terms and Notations Used in This Book, but not Generally Adopted.- 1 Topological Spaces.- 1. Fundamental Concepts.- 1. Topologies.- 2. Metrics.- 3. Subspaces.- 4. Continuous Maps.- 5. Separation Axioms.- 6. Countability Axioms.- 7. Compactness.- 2. Constructions.- 1. Sums.- 2. Products.- 3. Quotients.- 4. Glueing.- 5. Projective Spaces.- 6. More Special Constructions.- 7. Spaces of Continuous Maps.- 8. The Case of Pointed Spaces.- 9. Exercises.- 3. Homotopies.- 1. General Definitions.- 2. Paths.- 3. Connectedness and k-Connectedness.- 4. Local Properties.- 5. Borsuk Pairs.- 6. CNRS-Spaces.- 7. Homotopy Properties of Topological Constructions.- 8. Exercises.- 2 Cellular Spaces.- 1. Cellular Spaces and Their Topological Properties.- 1. Fundamental Concepts.- 2. Glueing Cellular Spaces from Balls.- 3. The Canonical Cellular Decompositions of Spheres, Balls, and Projective Spaces.- 4. More Topological Properties of Cellular Spaces.- 5. Cellular Constructions.- 6. Exercises.- 2. Simplicial Spaces.- 1. Euclidean Simplices.- 2. Simplicial Spaces and Simplicial Maps.- 3. Simplicial Schemes.- 4. Polyhedra.- 5. Simplicial Constructions.- 6. Stars. Links. Regular Neighborhoods.- 7. Simplicial Approximation of Continuous Maps.- 8. Exercises.- 3. Homotopy Properties of Cellular Spaces.- 1. Cellular Pairs.- 2. Cellular Approximation of Continuous Maps.- 3. k-Connected Cellular Pairs.- 4. Simplicial Approximation of Cellular Spaces.- 5. Exercises.- 3 Smooth Manifolds.- 1. Fundamental Concepts.- 1. Topological Manifolds.- 2. Differentiable Structures.- 3. Orientations.- 4. The Manifold of Tangent Vectors.- 5. Embeddings, Immersions, and Submersions.- 6. Complex Structures.- 7. Exercises.- 2. Stiefel and Grassman Manifolds.- 1. Stiefel Manifolds.- 2. Grassman Manifolds.- 3. Some Low-Dimensional Stiefel and Grassman Manifolds.- 4. Exercises.- 3. A Digression: Three Theorems from Calculus.- 1. Polynomial Approximation of Functions.- 2. Singular Values.- 3. Nondegenerate Critical Points..- 4. Embeddings. Immersions. Smoothings. Approximations.- 1. Spaces of Smooth Maps.- 2. The Simplest Embedding Theorems.- 3. Transversalizations and Tubes.- 4. Smoothing Maps in the Case of Closed Manifolds.- 5. Glueing Manifolds Smoothly.- 6. Smoothing Maps in the Presence of a Boundary.- 7. General Position.- 8. Maps Transverse to a Submanifold.- 9. Raising the Smoothness Class of a Manifold.- 10. Approximation of Maps by Embeddings and Immersions.- 11. Exercises.- 5. The Simplest Structure Theorems.- 1. Morse Functions.- 2. Cobordisms and Surgery.- 3. Two-dimensional Manifolds.- 4. Exercises.- 4 Bundles.- 1. Bundles without Group Structure.- 1. General Definitions.- 2. Locally Trivial Bundles.- 3. Serre Bundles.- 4. Bundles With Map Spaces as Total Spaces.- 5. Exercises.- 2. A Digression: Topological Groups and Transformation Groups.- 1. Topological Groups.- 2. Groups of Homeomorphisms.- 3. Actions.- 4. Exercises.- 3. Bundles with a Group Structure.- 1. Spaces With F-Structure.- 2. Steenrod Bundles.- 3. Associated Bundles.- 4. Ehresmann-Feldbau Bundles.- 5. Exercises.- 4. The Classification of Steenrod Bundles.- 1. Steenrod Bundles and Homotopies.- 2. Universal Bundles.- 3. The Milnor Bundles.- 4. Reductions of the Structure Group.- 5. Exercises.- 5. Vector Bundles.- 1. General Definitions.- 2. Constructions.- 3. The Classical Universal Vector Bundles.- 4. The Most Important Reductions of the Structure Group.- 5. Exercises.- 6. Smooth Bundles.- 1. Fundamental Concepts.- 2. Smoothings and Approximations.- 3. Smooth Vector Bundles.- 4. Tangent and Normal Bundles.- 5. Degree.- 6. Exercises.- 5 Homotopy Groups.- 1. The General Theory.- 1. Absolute Homotopy Groups.- 2. A Digression: Local Systems.- 3. Local Systems of Homotopy Groups of a Topological Space.- 4. Relative Homotopy Groups.- 5. A Digression: Sequences of Groups and Homomorphisms, and ?-Sequences.- 6. The Homotopy Sequence of a Pair.- 7. The Local System of Homotopy Groups of the Fibers of a Serre Bundle.- 8. The Homotopy Sequence of a Serre Bundle.- 9. The Influence of Other Structures Upon Homotopy Groups.- 10. Alternative Descriptions of the Homotopy Groups.- 11. Additional Theorems.- 12. Exercises.- 2. The Homotopy Groups of Spheres and of Classical Manifolds.- 1. Suspension in the Homotopy Groups of Spheres.- 2. The Simplest Homotopy Groups of Spheres.- 3. The Composition Product.- 4. Information: Homotopy Groups of Spheres.- 5. The Homotopy Groups of Projective Spaces and Lenses.- 6. The Homotopy Groups of Classical Groups.- 7. The Homotopy Groups of Stiefel Manifolds and Spaces.- 8. The Homotopy Groups of Grassman Manifolds and Spaces.- 9. Exercises.- 3. Homotopy Groups of Cellular Spaces.- 1. The Homotopy Groups of One-dimensional Cellular Spaces.- 2. The Effect of Attaching Balls.- 3. The Fundamental Group of a Cellular Space.- 4. Homotopy Groups of Compact Surfaces.- 5. The Homotopy Groups of Bouquets.- 6. The Homotopy Groups of a k-Connected Cellular Pair.- 7. Spaces With Prescribed Homotopy Groups.- 8. Eight Instructive Examples.- 9. Exercises.- 4. Weak Homotopy Equivalence.- 1. Fundamental Concepts.- 2. Weak Homotopy Equivalence and Constructions.- 3. Cellular Approximations of Topological Spaces.- 4. Exercises.- 5. The Whitehead Product.- 1. The Class wd(m,n).- 2. Definition and the Simplest Properties of the Whitehead Product.- 3. Applications.- 4. Exercises.- 6. Continuation of the Theory of Bundles.- 1. Weak Homotopy Equivalence and Steenrod Bundles.- 2. Theory of Coverings.- 3. Orientations.- 4. Some Bundles Over Spheres.- 5. Exercises.- Glossary of Symbols.
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