Fundamental Group and Covering Spaces
Authors
Description
This is a simple and practical introduction to algebraic topology. The author starts with the fundamental group and covering spaces. This topic was chosen because of its elementary nature, because it illustrates the use of algebraic invariants in topological problems and because it has applications to other areas of mathematics such as real analysis, complex variables and differential geometry.
The book is divided into two parts. The first covers the fundamental group, with examples and applications, and introduces notions of homotopy, the most important idea in algebraic topology. Its notions are used throughout the book. In the second, the focus is on covering spaces and topics such as the fundamental theorem of lifting, homomorphisms between coverings, orientation in a vector space and orientable variables.
Target audience
Higher education
Name: Fundamental Group and Covering Spaces
Author(s): e Elon Lages Lima
Pages: 192
Publication: IMPA, 2018
ISBN: 978-85-244-0429-0
Edition: 5
Part One
1 Homotopy
1 Homotopic applications
2 Type of homotopy
3 Contractible spaces
4 Homotopy and extension of applications
5 Trees
6 Homotopy of pairs and relative homotopy
2 The Fundamental Group
1 Homotopy of paths
2 The fundamental group
3 Induced homomorphism
4 Other descriptions of the fundamental group
5 Simply connected spaces
6 Some properties of the fundamental group
3 Examples and Applications of the Fundamental Group
1 The fundamental group of the circle
2 Some consequences of the isomorphism π 1 (S1) ≈ Z
3 The number of turns of a closed plane curve
4 The number of turns expressed as a curvilinear integral
5 Real projective spaces
6 Fibrations and complex projective spaces
7 Rotations in Euclidean space
8 The fundamental group of some classical groups
Part Two
4 Covering Spaces
1 Local homeomorphisms
2 Covering applications
3 Properly discontinuous groups
4 Lifting paths and homotopies
5 Differentiable coverings
5 Recoveries and the Fundamental Group
1 The conjugacy class associated with a recovery
2 The Fundamental Lifting Theorem
3 Homomorphisms between recoveries
4 Automorphisms of recoveries
5 Properly discontinuous groups vs. regular recoveries
6 Existence of recoveries
7 Fundamental group of a compact surface
6 Orientable varieties and double oriented covering
1 Orientation in a vector space
2 Orientable varieties
3 Properly discontinuous groups of diffeomorphisms
4 Double oriented covering
5 Relations with the fundamental group
A Appendix: Own applications
Bibliography
Index