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Product Code:  GSM/110 
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Hardcover ISBN:  9780821848982 
eBook: ISBN:  9781470415921 
Product Code:  GSM/110.B 
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Hardcover ISBN:  9780821848982 
Product Code:  GSM/110 
List Price:  $99.00 
MAA Member Price:  $89.10 
AMS Member Price:  $79.20 
eBook ISBN:  9781470415921 
Product Code:  GSM/110.E 
List Price:  $85.00 
MAA Member Price:  $76.50 
AMS Member Price:  $68.00 
Hardcover ISBN:  9780821848982 
eBook ISBN:  9781470415921 
Product Code:  GSM/110.B 
List Price:  $184.00 $141.50 
MAA Member Price:  $165.60 $127.35 
AMS Member Price:  $147.20 $113.20 

Book DetailsGraduate Studies in MathematicsVolume: 110; 2010; 218 ppMSC: Primary 55; 57
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, socalled 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 7spheres.
This book is based on courses the author taught in Mainz and Heidelberg. Readers should be familiar with the basic notions of pointset 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.
ReadershipGraduate students and research mathematicians interested in algebraic and differential topology.

Table of Contents

Chapters

Chapter 0. A quick introduction to stratifolds

Chapter 1. Smooth manifolds revisited

Chapter 2. Stratifolds

Chapter 3. Stratifolds with boundary: $c$stratifolds

Chapter 4. $\mathbb {Z}$/2homology

Chapter 5. The MayerVietoris sequence and homology groups of spheres

Chapter 6. Brouwer’s fixed point theorem, separation, invariance of dimension

Chapter 7. Homology of some important spaces and the Euler characteristic

Chapter 8. Integral homology and the mapping degree

Chapter 9. A comparison theorem for homology theories and $CW$complexes

Chapter 10. Künneth’s theorem

Chapter 11. Some lens spaces and quaternionic generalizations

Chapter 12. Cohomology and Poincaré duality

Chapter 13. Induced maps and the cohomology axioms

Chapter 14. Products in cohomology and the Kronecker pairing

Chapter 15. The signature

Chapter 16. The Euler class

Chapter 17. Chern classes and StiefelWhitney classes

Chapter 18. Pontrjagin classes and applications to bordism

Chapter 19. Exotic 7spheres

Chapter 20. Relation to ordinary singular (co)homology

Appendix A. Constructions of stratifolds

Appendix B. The detailed proof of the MayerVietoris sequence

Appendix C. The tensor product


Additional Material

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 graduatelevel 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


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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, socalled 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 7spheres.
This book is based on courses the author taught in Mainz and Heidelberg. Readers should be familiar with the basic notions of pointset 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.
Graduate students and research mathematicians interested in algebraic and differential topology.

Chapters

Chapter 0. A quick introduction to stratifolds

Chapter 1. Smooth manifolds revisited

Chapter 2. Stratifolds

Chapter 3. Stratifolds with boundary: $c$stratifolds

Chapter 4. $\mathbb {Z}$/2homology

Chapter 5. The MayerVietoris sequence and homology groups of spheres

Chapter 6. Brouwer’s fixed point theorem, separation, invariance of dimension

Chapter 7. Homology of some important spaces and the Euler characteristic

Chapter 8. Integral homology and the mapping degree

Chapter 9. A comparison theorem for homology theories and $CW$complexes

Chapter 10. Künneth’s theorem

Chapter 11. Some lens spaces and quaternionic generalizations

Chapter 12. Cohomology and Poincaré duality

Chapter 13. Induced maps and the cohomology axioms

Chapter 14. Products in cohomology and the Kronecker pairing

Chapter 15. The signature

Chapter 16. The Euler class

Chapter 17. Chern classes and StiefelWhitney classes

Chapter 18. Pontrjagin classes and applications to bordism

Chapter 19. Exotic 7spheres

Chapter 20. Relation to ordinary singular (co)homology

Appendix A. Constructions of stratifolds

Appendix B. The detailed proof of the MayerVietoris sequence

Appendix C. The tensor product

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 graduatelevel 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