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Algebraic Topology 1st ed. 2021 [Pehme köide]

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  • Formaat: Paperback / softback, 209 pages, kõrgus x laius: 235x155 mm, kaal: 454 g, 28 Illustrations, color; 41 Illustrations, black and white; XIV, 209 p. 69 illus., 28 illus. in color., 1 Paperback / softback
  • Ilmumisaeg: 20-Jun-2021
  • Kirjastus: Springer Nature Switzerland AG
  • ISBN-10: 3030706079
  • ISBN-13: 9783030706074
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Algebraic Topology 1st ed. 2021Suurem pilt
  • Formaat: Paperback / softback, 209 pages, kõrgus x laius: 235x155 mm, kaal: 454 g, 28 Illustrations, color; 41 Illustrations, black and white; XIV, 209 p. 69 illus., 28 illus. in color., 1 Paperback / softback
  • Ilmumisaeg: 20-Jun-2021
  • Kirjastus: Springer Nature Switzerland AG
  • ISBN-10: 3030706079
  • ISBN-13: 9783030706074
Teised raamatud teemal:
Püsilink: https://www.kriso.ee/db/9783030706074.html

Algebraic Topology is an introductory textbook based on a class for advanced high-school students at the Stanford University Mathematics Camp (SUMaC) that the authors have taught for many years. Each chapter, or lecture, corresponds to one day of class at SUMaC. The book begins with the preliminaries needed for the formal definition of a surface. Other topics covered in the book include the classification of surfaces, group theory, the fundamental group, and homology.

This book assumes no background in abstract algebra or real analysis, and the material from those subjects is presented as needed in the text. This makes the book readable to undergraduates or high-school students who do not have the background typically assumed in an algebraic topology book or class. The book contains many examples and exercises, allowing it to be used for both self-study and for an introductory undergraduate topology course.

Arvustused

Algebraic topology provides a self-contained introduction to the field . the book thus provides a particularly well-organized, interesting, and smooth exposition of its subject. This particular book unique is that it provides a clear, elementary, but mathematically solid introduction to algebraic topology that keeps the subject interesting throughout. provides a clear, readable, and detailed treatment of the ideas and proofs in the subject . (Thomas Mack, Mathematical Reviews, July, 2022)

The book could easily be used in an undergraduate course or read by a bright high school student. It should certainly be in any high school library. (Jonathan Hodgson, zbMATH 1481.55001, 2022)

1 Surface Preliminaries
1(18)
1.1 Surfaces
1(1)
1.2 Euclidean Space
2(1)
1.3 Open Sets
3(6)
1.4 Functions and Their Properties
9(2)
1.5 Continuity
11(5)
1.6 Problems
16(3)
2 Surfaces
19(12)
2.1 The Definition of a Surface
19(1)
2.2 Examples of Surfaces
19(3)
2.3 Spheres as Surfaces
22(1)
2.4 Surfaces with Boundary
23(1)
2.5 Closed, Bounded, and Compact Surfaces
24(1)
2.6 Equivalence Relations and Topological Equivalence
24(2)
2.7 Homeomorphic Spaces
26(1)
2.8 Invariants
27(1)
2.9 Problems
28(3)
3 The Euler Characteristic And Identification Spaces
31(20)
3.1 Triangulations and the Euler Characteristic
31(4)
3.2 Invariance of the Euler Characteristic
35(2)
3.3 Identification Spaces
37(2)
3.4 ID Spaces as Surfaces
39(1)
3.5 Abstract Topological Spaces
40(2)
3.6 The Quotient Topology
42(1)
3.7 Further Examples of ID Spaces
43(2)
3.8 Triangulations of ID Spaces
45(1)
3.9 The Connected Sum
46(1)
3.10 The Euler Characteristic of a Compact Surface with Boundary
47(1)
3.11 Problems
48(3)
4 Classification Theorem Of Compact Surfaces
51(12)
4.1 The Geometry of the Projective Plane and the Klein Bottle
51(3)
4.2 Orientable and Nonorientable Surfaces
54(2)
4.3 The Classification Theorem for Compact Surfaces
56(1)
4.4 Compact Surfaces Have Finite Triangulations
57(1)
4.5 Proof of the Classification Theorem
58(3)
4.6 Problems
61(2)
5 Introduction To Group Theory
63(14)
5.1 Why Use Groups?
63(1)
5.2 A Motivating Example
64(1)
5.3 Definition of a Group
64(1)
5.4 Examples of Groups
65(5)
5.5 Free Groups, Generators, and Relations
70(3)
5.6 Free Products
73(1)
5.7 Problems
74(3)
6 Structure Of Groups
77(14)
6.1 Subgroups
77(1)
6.2 Direct Products of Groups
78(2)
6.3 Homomorphisms
80(3)
6.4 Isomorphisms
83(1)
6.5 Existence of Homomorphisms
84(3)
6.6 Finitely Generated Abelian Groups
87(2)
6.7 Problems
89(2)
7 Cosets, Normal Subgroups, And Quotient Groups
91(14)
7.1 Cosets
91(3)
7.2 Lagrange's Theorem and Its Consequences
94(1)
7.3 Coset Spaces and Quotient Groups
95(1)
7.4 Properties and Examples of Normal Subgroups
96(2)
7.5 Coset Representatives
98(1)
7.6 A Quotient of a Dihedral Group
98(1)
7.7 Building up Finite Groups
99(2)
7.8 An Isomorphism Theorem
101(1)
7.9 Problems
101(4)
8 The Fundamental Group
105(10)
8.1 Paths and Loops on a Surface
105(1)
8.2 Equivalence of Paths and Loops
106(1)
8.3 Equivalence Classes of Paths and Loops
107(1)
8.4 Multiplication of Path and Loop Classes
108(2)
8.5 Definition of the Fundamental Group
110(3)
8.6 Problems
113(2)
9 Computing The Fundamental Group
115(12)
9.1 Homotopies of Maps and Spaces
115(8)
9.2 Computing the Fundamental Group of a Circle
123(2)
9.3 Problems
125(2)
10 Tools For Fundamental Groups
127(14)
10.1 More Fundamental Groups
127(2)
10.2 The Degree of a Loop
129(3)
10.3 Fundamental Group of a Circle--Redux
132(2)
10.4 The Induced Homomorphism on Fundamental Groups
134(3)
10.5 Retracts
137(2)
10.6 Problems
139(2)
11 Applications Of Fundamental Groups
141(10)
11.1 The Fundamental Theorem of Algebra
141(4)
11.2 Further Applications of the Fundamental Group
145(4)
11.3 Problems
149(2)
12 The Seifert-Van Kampen Theorem
151(14)
12.1 The Fundamental Group of a Wedge of Circles
151(2)
12.2 The Seifert-Van Kampen Theorem: First Version
153(2)
12.3 More Fundamental Groups
155(1)
12.4 The Seifert-Van Kampen Theorem: Second Version
156(1)
12.5 The Fundamental Group of a Compact Surface
157(2)
12.6 Even More Fundamental Groups
159(1)
12.7 Proof of the Second Version of the Seifert-Van Kampen Theorem
160(1)
12.8 General Seifert-Van Kampen Theorem
161(1)
12.9 Groups as Fundamental Groups
161(2)
12.10 Problems
163(2)
13 Introduction To Homology
165(16)
13.1 The Idea of Homology
165(1)
13.2 Chains
166(2)
13.3 The Boundary Map
168(1)
13.4 Homology
169(2)
13.5 The Zeroth Homology Group
171(1)
13.6 Homology of the Klein Bottle
172(1)
13.7 Homology and Euler Characteristic
173(1)
13.8 Homology and Orientability
174(1)
13.9 Smith Normal Form
175(3)
13.10 The Induced Map on Homology
178(2)
13.11 Problems
180(1)
14 The Mayer-Vietoris Sequence
181(12)
14.1 Exact Sequences
181(2)
14.2 The Mayer-Vietoris Sequence
183(3)
14.3 Homology of Orientable Surfaces
186(2)
14.4 The Jordan Curve Theorem
188(1)
14.5 The Hurewicz Map
189(2)
14.6 Problems
191(2)
Appendix A Topological Notions 193(4)
Appendix B A Brief Look at Singular Homology 197(4)
Appendix C Hints for Selected Problems 201(2)
References 203(4)
Index 207
Clark Bray is an Associate Professor of the Practice in the Department of Mathematics at Duke University. He graduated from Rice University in 1993, cum laude, with majors in Mathematics and Physics, and finished his Ph.D. in Mathematics at Stanford University in 1999, studying algebraic topology with Professor Gunnar Carlsson. While subsequently a Lecturer at Stanford, he created and taught a program in algebraic topology for talented high school math students at the Stanford University Mathematics Camp (SUMaC) in 2001-2003. 

Adrian Butscher received his B.Sc. in mathematics and physics at the University of Toronto and his Ph.D. in mathematics at Stanford University.  After a fifteen-year period in academia, conducting research in pure and applied differential geometry and teaching at all levels of the undergraduate and graduate curricula, Adrian now works as a research scientist with Autodesk, Inc., a global software company that creates digital design tools for the manufacturing, architecture, building, construction, media, and entertainment industries.  While at Stanford, Adrian served as a counsellor and instructor for SuMAC, the Stanford University Mathematics Camp, a month-long, immersive educational experience for mathematically gifted high school students.



Simon Rubinstein-Salzedo received his PhD in mathematics from Stanford University in 2012. Afterwards, he taught at Dartmouth College and Stanford University. In 2015, he founded Euler Circle, a mathematics institute in the San Francisco Bay Area, dedicated to teaching college-level mathematics classes to advanced high-school students, as well as mentoring them on mathematics research. His research interests include number theory, algebraic geometry, combinatorics, probability, and game theory.