Complex Topological K-Theory (Cambridge Studies in Advanced Mathematics Book 111) 1st Edition by Efton Park (PDF)

Description

Topological K-theory is a key tool in topology, differential geometry and index theory, yet this is the first contemporary introduction for graduate students new to the subject. No background in algebraic topology is assumed; the reader need only have taken the standard first courses in real analysis, abstract algebra, and point-set topology. The book begins with a detailed discussion of vector bundles and related algebraic notions, followed by the definition of K-theory and proofs of the most important theorems in the subject, such as the Bott periodicity theorem and the Thom isomorphism theorem. The multiplicative structure of K-theory and the Adams operations are also discussed and the final chapter details the construction and computation of characteristic classes. With every important aspect of the topic covered, and exercises at the end of each chapter, this is the definitive book for a first course in topological K-theory.

User’s Reviews

Reviews from Amazon users which were colected at the time this book was published on the website:

⭐This is a very user-friendly introduction to complex topological K-theory.It does not even assume the reader is familiar with inner product spaces in general.The first chapter provides all the required background for an introduction to the core material.This is very convenient for beginners such as me.The book provides a detailed proof for Bott periodicity, one of the main results of topological k-theory.Highly recommended.

⭐K-theory, whether it be algebraic or topological has many uses and applications in mathematics and high energy physics, and this entails that its understanding is crucial if one is to enter into those areas which it has found applicability. As in almost all areas of modern mathematics, the intuitive explanations of the ideas and concepts behind them are usually lacking, with emphasis placed on formal constructions and proofs. The latter of course is what makes mathematics what it is, but for those who thirst for a real understanding of a particular subject area, such as K-theory, the formal style of writing in modern mathematical texts and monographs will not quench this thirst.This book is different in that it offers such an understanding, but without sacrificing the rigor that is expected in mathematics. Students of K-theory, or those who want to understand its applications, will therefore benefit greatly from the study of this book, and definitely take away an appreciation of the context in which K-theory arose historically. This is especially the case in the manner in which the author discusses the needed mathematical tools in the first chapter of the book. Indeed, the notion of an idempotent is clearly understandable as being a generalization of an ordinary projection operator in Euclidean space. Readers will learn the enormously important role that idempotents play in K-theory, and good examples of them occur throughout the book.That one can treat vector bundles and idempotents as groups, even though they are not, is one of the unique features of K-theory, and being able to add and subtract vector bundles and idempotents comes from taking what is called the Grothendieck completion of these objects. The author shows in detail that when this is done vector bundles and idempotents become naturally isomorphic. This isomorphism between classes of idempotents and classes of ranges of these idempotents makes it crystal clear why idempotents and be viewed as generalizations of the projection operators in ordinary vector space theory. The Grothendieck completion of the class of vector bundles (or idempotents) of a compact Hausdorff space X is the zeroth K-group of of X.Just as in the ordinary theory of vector spaces, where one can study subspaces of the vector space at hand, K-theory can be done for closed subspaces of a compact Hausdorff space. This goes by the name of ‘relative K-theory’ and the author gives a good motivation from a geometric point of view in the book. Of particular importance in the study of relative K-theory is the construction of a ‘one-point compactification’, since in later developments and applications of K-theory to areas such as homotopy theory and the theory of spectra it is used quite extensively, along with its generalization called the ‘suspension.’ The one-point compactification is also used in the book to prove the famous Bott periodicity theorem, and in the proof of the latter the author is kind to the reader in discussing the general structure of the reader before jumping immediately into its details. Subtracting trivial vector bundles (which have a zeroth K-group isomorphic to the integers) from non-trivial vector bundles is the topic of ‘reduced K-theory’, where the intent is to concentrate the effort on the non-trivial part of the vector bundle. As the author shows, this is accomplished by using pointed spaces, which should be very familiar to the reader acquainted with homotopy theory.Readers familiar with differential topology will appreciate the discussion and the proof of the Thom isomorphism, due in part to the use of exterior calculus and the ball and sphere bundles. It is somewhat surprising to learn that the K-theory of a vector bundle V over a compact Hausdorff space X and the K-theory of X are in fact isomorphic, and readers who go through the proof of the Thom isomorphism and who are familiar with the suspension of a space will see it generalized to the case of vector bundles.

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