Description
Book SynopsisOffers a geometric introduction to the homology of topological spaces and the cohomology of smooth manifolds. This title introduces a class of stratified spaces, so-called stratifolds. It derives basic concepts from differential topology such as Sard's theorem, partitions of unity and transversality.
Trade ReviewDifferential Algebraic Topology: From Stratifolds to Exotic Spheres is a good book. It is clearly written, has many good examples and illustrations, and, as befits a graduate-level text, exercises. It is a wonderful addition to the literature." -
MAA Reviews"This book is a very nice addition to the existing books on algebraic topology. A careful effort has been made to give the intuitive background when a new concept is introduced. This and the choice of topics makes reading the book a real pleasure." - Marko Kranjc,
Mathematical ReviewsTable of Contents
- Introduction
- A quick introduction to stratifolds
- Smooth manifolds revisited
- Stratifolds
- Stratifolds with boundary: 𝑐-stratifolds
- ℤ/2-homology
- The Mayer-Vietoris sequence and homology groups of spheres
- Brouwer’s fixed point theorem, separation, invariance of dimension
- Homology of some important spaces and the Euler characteristic
- Integral homology and the mapping degree
- A comparison theorem for homology theories and 𝐶𝑊-complexes
- Künneth’s theorem
- Some lens spaces and quaternionic generalizations
- Cohomology and Poincaré duality
- Induced maps and the cohomology axioms
- Products in cohomology and the Kronecker pairing
- The signature
- The Euler class
- Chern classes and Stiefel-Whitney classes
- Pontrjagin classes and applications to bordism
- Exotic 7-spheres
- Relation to ordinary singular (co)homology
- Appendix A: Constructions of stratifolds
- Appendix B: The detailed proof of the Mayer-Vietoris sequence
- Appendix C: The tensor product
- Bibliography
- Index
