Basic Homology
Authors
Description
The book deals with homology groups. A homology theory is a method of associating to each topological space of a certain category a series of groups (or, more generally, modules) called the homology groups of that space, in such a way that homeomorphic spaces have isomorphic homology groups. Unlike the fundamental group, homology groups are abelian. The book is intended as an introductory text to Algebraic Topology, as part of the beginning of postgraduate studies.
Target audience
Higher education
Name: Basic Homology
Author(s): e Elon Lages Lima
Pages: 191
Publication: IMPA, 2012
ISBN: 978-85-244-0286-9
Edition: 2
Foreword
Preface to the second edition
Chapter I. Formal homology
1. Chain complexes
2. Algebraic homotopy
3. Exact sequences
4. Cohomology
5. Inductive limits
Chapter II. De Rham’s Cohomology
1. The de Rham complex
2. Homotopic invariance
3. The Mayer-Vietoris sequence
4. Cohomology with compact supports
5. Coverings vs cohomology
6. The topological Jordan-Brouwer theorem
7. Poincaré’s Duality Theorem
8. The degree of an application
9. Cohomology of a compact
10. The exact Cech-Alexander-Spanier sequence
Chapter III. Simplicial Homology
1. Polyhedra
2. The simplicial complex
3. First examples of simplicial homology
4. Barycentric subdivision
5. Simplicial approximation
6. Pseudo-varieties
7. The Lefschetz Fixed Point Theorem
8. Ordered homology
9. Simplicial cohomology
10. The cohomology ring
Chapter IV. Singular Homology
1. First definitions
2. Homotopic invariance
3. Barycentric subdivision in singular homology
4. Singular cohomology
5. De Rham’s theorem
6. Cohomology in terms of homology
Chapter V. Exercises
Bibliographical References
Index