Differential algebraic topology : from stratifolds to exotic spheres

著者

• Kreck, Matthias

書誌事項

Differential algebraic topology : from stratifolds to exotic spheres

Matthias Kreck

（Graduate studies in mathematics, v. 110）

American Mathematical Society, c2010

注記

Includes bibliographical references (p. 215-216) and index

内容説明・目次

内容説明

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, Poincare 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.

目次

Introduction A quick introduction to stratifolds Smooth manifolds revisited Stratifolds Stratifolds with boundary: 𝑐-stratifolds /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 Kunneth's theorem Some lens spaces and quaternionic generalizations Cohomology and Poincare 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

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詳細情報

• NII書誌ID(NCID): BB0193184X
• ISBN:
  • 9780821848982
• LCCN: 2009037982
• 出版国コード: us
• タイトル言語コード: eng
• 本文言語コード: eng
• 出版地: Providence, R.I.
• ページ数/冊数: xii, 218 p.
• 大きさ: 27 cm
• 分類:
  • LCC : QA612
  • DC22 : 514/.2
• 件名:
  • LCSH : Algebraic topology
  • LCSH : Differential topology
• 親書誌ID:
  • BA20542214