Algebraic Topology: A First CourseTo the Teacher. This book is designed to introduce a student to some of the important ideas of algebraic topology by emphasizing the re lations of these ideas with other areas of mathematics. Rather than choosing one point of view of modem topology (homotopy theory, simplicial complexes, singular theory, axiomatic homology, differ ential topology, etc.), we concentrate our attention on concrete prob lems in low dimensions, introducing only as much algebraic machin ery as necessary for the problems we meet. This makes it possible to see a wider variety of important features of the subject than is usual in a beginning text. The book is designed for students of mathematics or science who are not aiming to become practicing algebraic topol ogists-without, we hope, discouraging budding topologists. We also feel that this approach is in better harmony with the historical devel opment of the subject. What would we like a student to know after a first course in to pology (assuming we reject the answer: half of what one would like the student to know after a second course in topology)? Our answers to this have guided the choice of material, which includes: under standing the relation between homology and integration, first on plane domains, later on Riemann surfaces and in higher dimensions; wind ing numbers and degrees of mappings, fixed-point theorems; appli cations such as the Jordan curve theorem, invariance of domain; in dices of vector fields and Euler characteristics; fundamental groups |
Contents
4 | |
CHAPTER 2 | 17 |
PART II | 34 |
Applications of Winding Numbers | 48 |
PART III | 59 |
CHAPTER 6 | 78 |
PART IV | 94 |
CHAPTER 8 | 106 |
COHOMOLOGY AND HOMOLOGY III | 205 |
CHAPTER 16 | 219 |
17b Classification of Compact Oriented Surfaces | 236 |
18b Integrals of 2Forms | 251 |
19b Branched Coverings | 268 |
CHAPTER 20 | 277 |
20e Elliptic and Hyperelliptic Curves | 291 |
21d The AbelJacobi Theorem | 306 |
CHAPTER 9 | 115 |
PART VIII | 122 |
Holes and Integrals | 123 |
CHAPTER 10 | 137 |
PART VI | 151 |
CHAPTER 12 | 165 |
The Fundamental Group and Covering Spaces | 179 |
The Van Kampen Theorem | 193 |
22c Higher Homotopy Groups | 324 |
23c Spheres and Degree | 339 |
24c Cohomology and Cohomology with Compact Supports | 355 |
A2 Connected Components | 369 |
A3 Patching | 370 |
C3 Polynomials Gausss Lemma | 385 |
References | 419 |
Other editions - View all
Common terms and phrases
1-cycle a₁ Algebraic analytic Aut(Y/X base point boundary Chapter circle closed 1-chain closed 1-form closed paths cohomology compact complement complex connected components constant path construct contained continuous mapping coordinate Corollary covering map define denoted determines diffeomorphic differential disjoint union disk element endpoints equivalence Exercise fact follows formula free abelian group function f fundamental group G-covering H₁U H₁X homology group homomorphism homotopy integral isomorphism Jordan curve theorem k-form Lemma lifting linear mapping locally constant locally constant function locally path-connected loop manifold mapping f Mayer-Vietoris meromorphic function neighborhood nonzero open set P₁ polynomial Problem Proof Proposition prove quotient rectangle Rham Riemann surface Show simply connected sphere subgroup Suppose surjective T₁ takes theorem topological space topology trivial U₁ unique universal covering vector field vector space winding number y₁ zero