**Graduate Studies in Mathematics**

Volume: 110;
2010;
218 pp;
Hardcover

MSC: Primary 55; 57;

Print ISBN: 978-0-8218-4898-2

Product Code: GSM/110

List Price: $58.00

Individual Member Price: $46.40

**Electronic ISBN: 978-1-4704-1592-1
Product Code: GSM/110.E**

List Price: $58.00

Individual Member Price: $46.40

#### You may also like

#### Supplemental Materials

# Differential Algebraic Topology: From Stratifolds to Exotic Spheres

Share this page
*Matthias Kreck*

This book presents a geometric introduction to the homology of topological spaces and the cohomology of smooth manifolds. The author introduces a new class of stratified spaces, so-called stratifolds. He derives basic concepts from differential topology such as Sard's theorem, partitions of unity and transversality. Based on this, homology groups are constructed in the framework of stratifolds and the homology axioms are proved. This implies that for nice spaces these homology groups agree with ordinary singular homology. Besides the standard computations of homology groups using the axioms, straightforward constructions of important homology classes are given. The author also defines stratifold cohomology groups following an idea of Quillen. Again, certain important cohomology classes occur very naturally in this description, for example, the characteristic classes which are constructed in the book and applied later on. One of the most fundamental results, Poincaré duality, is almost a triviality in this approach.

Some fundamental invariants, such as the Euler characteristic and the signature, are derived from (co)homology groups. These invariants play a significant role in some of the most spectacular results in differential topology. In particular, the author proves a special case of Hirzebruch's signature theorem and presents as a highlight Milnor's exotic 7-spheres.

This book is based on courses the author taught in Mainz and Heidelberg. Readers should be familiar with the basic notions of point-set topology and differential topology. The book can be used for a combined introduction to differential and algebraic topology, as well as for a quick presentation of (co)homology in a course about differential geometry.

#### Table of Contents

# Table of Contents

## Differential Algebraic Topology: From Stratifolds to Exotic Spheres

- Cover Cover11 free
- Title page iii4 free
- Contents v6 free
- Introduction ix10 free
- A quick introduction to stratifolds 114 free
- Smooth manifolds revisited 518
- Stratifolds 1528
- Stratifolds with boundary: 𝑐-stratifolds 3346
- ℤ/2-homology 3952
- The Mayer-Vietoris sequence and homology groups of spheres 5568
- Brouwer’s fixed point theorem, separation, invariance of dimension 6780
- Homology of some important spaces and the Euler characteristic 7184
- Integral homology and the mapping degree 7992
- A comparison theorem for homology theories and 𝐶𝑊-complexes 93106
- Künneth’s theorem 103116
- Some lens spaces and quaternionic generalizations 111124
- Cohomology and Poincaré duality 119132
- Induced maps and the cohomology axioms 127140
- Products in cohomology and the Kronecker pairing 135148
- The signature 147160
- The Euler class 153166
- Chern classes and Stiefel-Whitney classes 161174
- Pontrjagin classes and applications to bordism 167180
- Exotic 7-spheres 177190
- Relation to ordinary singular (co)homology 185198
- Appendix A: Constructions of stratifolds 191204
- Appendix B: The detailed proof of the Mayer-Vietoris sequence 197210
- Appendix C: The tensor product 209222
- Bibliography 215228
- Index 217230 free
- Back Cover Back Cover1234

#### Readership

Graduate students and research mathematicians interested in algebraic and differential topology.

#### Reviews

*Differential 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 Reviews