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Differential Algebraic Topology: From Stratifolds to Exotic Spheres
 
Matthias Kreck Hausdorff Research Institute for Mathematics, Bonn, Germany
Front Cover for Differential Algebraic Topology
HardcoverISBN:  978-0-8218-4898-2
Product Code:  GSM/110
List Price: $62.00
MAA Member Price: $55.80
AMS Member Price: $49.60
eBookISBN:  978-1-4704-1592-1
Product Code:  GSM/110.E
List Price: $58.00
MAA Member Price: $52.20
AMS Member Price: $46.40
HardcoverISBN:  978-0-8218-4898-2
eBookISBN:  978-1-4704-1592-1
Product Code:  GSM/110.B
List Price: $120.00$91.00
MAA Member Price: $108.00$81.90
AMS Member Price: $96.00$72.80
Front Cover for Differential Algebraic Topology
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  • Front Cover for Differential Algebraic Topology
  • Back Cover for Differential Algebraic Topology
Differential Algebraic Topology: From Stratifolds to Exotic Spheres
Matthias Kreck Hausdorff Research Institute for Mathematics, Bonn, Germany
Hardcover ISBN:  978-0-8218-4898-2
Product Code:  GSM/110
List Price: $62.00
MAA Member Price: $55.80
AMS Member Price: $49.60
eBook ISBN:  978-1-4704-1592-1
Product Code:  GSM/110.E
List Price: $58.00
MAA Member Price: $52.20
AMS Member Price: $46.40
Hardcover ISBN:  978-0-8218-4898-2
eBookISBN:  978-1-4704-1592-1
Product Code:  GSM/110.B
List Price: $120.00$91.00
MAA Member Price: $108.00$81.90
AMS Member Price: $96.00$72.80
  • Book Details
     
     
    Graduate Studies in Mathematics
    Volume: 1102010; 218 pp
    MSC: 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, 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.

    Readership

    Graduate 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. $\mathbb {Z}$/2-homology
    • Chapter 5. The Mayer-Vietoris 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 Stiefel-Whitney classes
    • Chapter 18. Pontrjagin classes and applications to bordism
    • Chapter 19. Exotic 7-spheres
    • Chapter 20. 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
  • 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
  • Requests
     
     
    Review Copy – for reviewers who would like to review an AMS book
    Permission – for use of book, eBook, or Journal content
    Accessibility – to request an alternate format of an AMS title
Volume: 1102010; 218 pp
MSC: 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, 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.

Readership

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. $\mathbb {Z}$/2-homology
  • Chapter 5. The Mayer-Vietoris 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 Stiefel-Whitney classes
  • Chapter 18. Pontrjagin classes and applications to bordism
  • Chapter 19. Exotic 7-spheres
  • Chapter 20. 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
  • 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
Review Copy – for reviewers who would like to review an AMS book
Permission – for use of book, eBook, or Journal content
Accessibility – to request an alternate format of an AMS title
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