Differential Algebraic Topology: From Stratifolds to Exotic Spheres
- ISBN-10
- 0821848984
- ISBN-13
- 9780821848982
- Category
- Algebraic topology
- Pages
- 218
- Language
- English
- Published
- 2010
- Publisher
- American Mathematical Soc.
- Author
- Matthias Kreck
Description
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.
Get the book
Similar books
-
Differential Forms in Algebraic Topology
By Loring W. Tu, Raoul BottDeveloped from a first-year graduate course in algebraic topology, this text is an informal introduction to some of the main ideas of contemporary homotopy and cohomology theory.
-
Introduction to Differential and Algebraic Topology
By Yu.G. Borisovich, N.M. Bliznyakov, T.N. FomenkoTopology as a subject, in our opinion, plays a central role in university education.
-
Introduction to Differential and Algebraic Topology
By Yu.G. Borisovich, N.M. Bliznyakov, T.N. FomenkoTopology as a subject, in our opinion, plays a central role in university education. It is not really possible to design courses in differential geometry, mathematical analysis, differential equations, mechanics,...
-
Algebraic Topology Via Differential Geometry
By M. Karoubi, C. LerusteIn this volume the authors seek to illustrate how methods of differential geometry find application in the study of the topology of differential manifolds.
-
A History of Algebraic and Differential Topology, 1900 - 1960
By Jean DieudonnéThis book demonstrates in the case of topology how these obstacles can be overcome, with enlightening results.... Within its chosen boundaries the coverage of this book is superb.
-
Algebraic and Differential Topology
By R. V. GamkrelidzeThe third volume in L.S. Pontryagin's Selected Works, this book provides as many insights into algebraic topology for today's mathematician as it did when the author was making his initial endeavors into this field.
-
Differential Topology: First Steps
By Andrew H. WallaceKeeping mathematical prerequisites to a minimum, this undergraduate-level text stimulates students' intuitive understanding of topology while avoiding the more difficult subtleties and technicalities.
-
Differential Topology
By Morris W. HirschThere is an abundance of exercises, which supply many beautiful examples and much interesting additional information, and help the reader to become thoroughly familiar with the material of the main text." —MATHEMATICAL REVIEWS
-
Elements of Differential Topology
By Anant R. ShastriDerived from the author's course on the subject, Elements of Differential Topology explores the vast and elegant theories in topology developed by Morse, Thom, Smale, Whitney, Milnor, and others.
-
Differential Topology
By Alan Pollack, Victor GuilleminAn exercise section in Chapter 4 leads the student through a construction of de Rham cohomology and a proof of its homotopy invariance. The book is suitable for either an introductory graduate course or an advanced undergraduate course.