一般拓扑学引论
Syllabuses on Geometry and Topology
Space curves and surfaces
Curves and Parametrization, Regular Surfaces; Inverse Images of Regular Values. Gauss Map and Fundamental Properties; Isometries; Conformal Maps; Rigidity of the Sphere.
Topological space
Space, maps, compactness and connectedness, quotients; Paths and Homotopy. The Fundamental Group of the Circle. Induced Homomorphisms. Free Products of Groups. The van Kampen Theorem. Covering Spaces and Lifting Properties; Simplex and complexes. Triangulations. Surfaces and its classification.
Differential Manifolds
Differentiable Manifolds and Submanifolds, Differentiable Functions and Mappings; The Tangent Space, Vector Field and Covector Fields. Tensors and Tensor Fields and differential forms. The Riemannian Metrics as examples, Orientation and Volume Element; Exterior Differentiation and Frobenius’s Theorem; Integration on manifolds, Manifolds with Boundary and Stokes’ Theorem.
Homology and cohomology
Simplicial and Singular Homology. Homotopy Invariance. Exact Sequences and Excision. Degree. Cellular Homology. Mayer-Vietoris Sequences. Homology with Coefficients. The Universal Coefficient Theorem. Cohomology of Spaces. The Cohomology Ring. A Kunneth Formula. Spaces with Polynomial Cohomology. Orientations and Homology. Cup Product and Duality.
Riemannian Manifolds
Differentiation and connection, Constant Vector Fields and Parallel Displacement
Riemann Curvatures and the Equations of Structure Manifolds of Constant Curvature,
Spaces of Positive Curvature, Spaces of Zero Curvature, Spaces of Constant Negative Curvature
References:
- M. do Carmo , Differentia geometry of curves and surfaces.
- Prentice- Hall, 1976 (25th printing)
- Chen Qing and Chia Kuai Peng, Differential Geometry
- M. Armstrong, Basic Topology Undergraduate texts in mathematics
- W.M. Boothby, An Introduction to Differentiable Manifolds and Riemannian Geometry Academic Press, Inc., Orlando, FL, 1986
- M. Spivak, A comprehensive introduction to differential geometry
- N. Hicks, Notes on differential geometry, Van Nostrand.
- T. Frenkel, Geometry of Physics
- J. Milnor, Morse Theory
- 10.A Hatcher, Algebraic Topology (http://www.math.cornell.edu/~hatcher/AT/ATpage.html)
- 11.J. Milnor, Topology from the differentiable viewpoint
- 12.R. Bott and L. Tu, Differential forms in algebraic topology
- 13.V. Guillemin, A. Pollack, Differential topology
1，集合，映射。
2，拓扑空间。
3，拓扑学的基本概念（开集，闭集等）。
4，度量空间。
5，极限。
6，紧性与完备性。
7，连续映射。
8，度量的概念。
9，线性半连续函数与映射。
10，完备映射。
11，Tychonoff拓扑，Tychonoff定理。
12，乘积，同态，度量化。
13，度量与赋范一致收敛。
14，分离性兣，正则与赋范空间，Uryson定理。
15，连通性。
16，欧氏与赋范空间的性质。
17，选择性。
18，商拓扑。
19，Vietoris拓扑与Hausdorff度量。
Point-Set Topology
1，V.Fedorchuk、V.V.Filippov，一般拓扑学，莫斯科大学出版社，1988。
2，P.S.Alexandroff，集论与一般拓扑学初阶，科学出版社，1977。
3，R.Engelking，General Topology，Heldermann，1989。
4，V.V.Filippov，常微分方程的解空间，莫斯科大学出版社，1993。
《拓扑学奇趣》巴尔佳斯基叶弗来莫维契合著
巴兹列夫《几何与拓扑习题集》, 《几何学与拓扑学习题集》
Basic Topology by Armstrong
Munkries “Topology” 2nd ed. Prentice Hall
Hatcher “Algebraic Topology” Cambridge UP
Spaniers “Algebraic Topology”
J.L. Kelley “General Topology”(GTM 27) ，Springer-Verlag
Willard, General Topology：
I.M.Singer, J.A.Thorp “Lecture notes on elementary topology and geometry
Greenberg “Lectures on Algebraic Topology”
Glen Bredon ‘s ” Topology and Geometry”(GMT139) is praised as the successor of Spanier‘s great book.
Introduction to Topological Manifolds by John M. Lee：
From calculus to cohomology by Madsen：
Fomenko,Differential geometry and topology
Aleksandrov‘s ” Combinatorial Topology ” is very good for beginner.
J. Milnor, Morse Theory
J. Milnor, Topology from the differentiable viewpoint
尤承业，《基础拓扑学讲义》，北大版
熊金成，《点集拓扑讲义》，高教版
张筑生，《微分拓扑新讲》，北大版
李元熹,张国(木梁) “拓扑学”
儿玉之宏《拓扑空间论》科学出版社
陈肇姜《点集拓扑学》, 《点集拓扑学题解与反例》南京大学出版社
An introduction to General topology
Syllabuses on Geometry and Topology
Space curves and surfaces
Curves and Parametriz ation , Regular Surfaces; Inverse Images of Regular Values. Gauss Map and Fundamental Properties;Isometries; Conformal Maps; Rigidity of the Sphere.
Topological space
Space, maps, compactness and connectedness, quotients; Paths and Homotopy. The Fundamental Group of the Circle. Induced Homomorphisms. Free Products of Groups. The van Kampen Theorem. Covering Spaces and Lifting Properties; Simplex and complexes. Triangulations. Surfaces and its classification.
Differential Manifolds
Differentiable Manifolds and Submanifolds, Differentiable Functions and Mappings; The Tangent Space, Vector Field and Covector Fields. Tensors and Tensor Fields and differential forms. The Riemannian Metrics as examples, Orientation and Volume Element; Exterior Differentiation and Frobenius’s Theorem; Integration on manifolds, Manifolds with Boundary and Stokes’ Theorem.
Homology and cohomology
Simplicial and Singular Homology. Homotopy Invariance. Exact Sequences and Excision. Degree. Cellular Homology. Mayer-Vietoris Sequences. Homology with Coefficients. The Universal Coefficient Theorem. Cohomology of Spaces. The Cohomology Ring. A Kunneth Formula. Spaces with Polynomial Cohomology. Orientations and Homology. Cup Product and Duality.
Riemannian Manifolds
Differentiation and connection , Constant Vector Fields and Parallel Displacement
Riemann Curvature s and the Equations of Structure Manifolds of Constant Curvature,
Spaces of Positive Curvature, Spaces of Zero Curvature, Spaces of Constant Negative Curvature
References:
1. M. do Carmo , Differentia geometry of curves and surfaces.
2. Prentice- Hall, 1976 (25th printing)
3. Chen Qing and Chia Kuai Peng, Differential Geometry
4. M. Armstrong, Basic Topology Undergraduate texts in mathematics
5. W.M. Boothby, An Introduction to Differentiable Manifolds and Riemannian Geometry Academic Press, Inc., Orlando, FL, 1986
6. M. Spivak, A comprehensive introduction to differential geometry
7. N. Hicks, Notes on differential geometry, Van Nostrand.
8. T. Frenkel, Geometry of Physics
9. J. Milnor, Morse Theory
10. A Hatcher, Algebraic Top o logy (http://www.math.cornell.edu/~hatcher/AT/ATpage.html)
11. J. Milnor, Topology from the differentiable viewpoint
12. R. Bott and L. Tu, Differential forms in algebraic topology
13. V. Guillemin, A. Pollack, Differential topology
1 , Collection, maps.
2 Topological spaces.
3 And the basic concepts of topology (open sets, closed sets, and so on).
4 And metric spaces.
5 Ultimate.
6 , Compactness and completeness.
7 And continuous maps.
8 And measurement concepts.
9 , Semi-continuous linear functions and mappings.
10 , Complete maps.
11 , Tychonoff Topology, Tychonoff Theorem.
12 Product, homomorphisms, metric.
13 , Measurement and uniform convergence in normed.
14 , Separation of Gong, and normed spaces, Uryson Theorem.
15 And connectedness.
16 , Euclidean and property of normed spaces.
17 And selectivity.
18 , Hypertopology.
19 , Vietoris Topology and Hausdorff Metric.
Point-Set Topology
1 , V.Fedorchuk 、 V.V.Filippov , General topology, Moscow University Press, 1988 。
2 , P.S.Alexandroff Set theory and General topology preliminaries, science press, 1977 。
3 , R.Engelking , General Topology , Heldermann , 1989 。
4 , V.V.Filippov , Solution of ordinary differential equations, Moscow University Press, 1993 。
The topology Trolltech Ba Erjia Polanski Ye Fulai co-author Mo Weiqi
Bazlev of the geometric and topological problem set , The geometry and topology of problem sets
Basic Topology by Armstrong
Munkries “Topology” 2nd ed. Prentice Hall
Hatcher “Algebraic Topology” Cambridge UP
Spaniers “Algebraic Topology”
J.L. Kelley “General Topology”(GTM 27) , Springer-Verlag
Willard, General Topology :
I.M.Singer, J.A.Thorp “Lecture notes on elementary topology and geometry
Greenberg “Lectures on Algebraic Topology”
Glen Bredon ‘s ” Topology and Geometry”(GMT139) is praised as the successor of Spanier’s great book.
Introduction to Topological Manifolds by John M. Lee :
From calculus to cohomology by Madsen :
Fomenko,Differential geometry and topology
Aleksandrov’s ” Combinatorial Topology ” is very good for beginner.
J. Milnor, Morse Theory
J. Milnor, Topology from the differentiable viewpoint
You Chengye, lectures on the basic topology, Peking University
Xiong Jincheng, lectures on the point set topology, higher education
Zhang Zhu, the new talk of differential topology, Peking University
Li Yuanxi , Zhang Guoli ( Wood ) ” Topology”
Kodama macro theory of topological space science press
Chen Zhaojiang of the point set topology , Solution to the point set topology and anti-Nanjing University Press
Serendipities 