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Simplicial Homotopy Theory

2012-12-06 Mathematics
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Release : 2012-12-06
Genre : Mathematics
Kind : eBook
Book Rating : 078/5 ( reviews)

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Book Synopsis Simplicial Homotopy Theory by : Paul G. Goerss

Download or read book Simplicial Homotopy Theory written by Paul G. Goerss. This book was released on 2012-12-06. Available in PDF, EPUB and Kindle. Book excerpt: Since the beginning of the modern era of algebraic topology, simplicial methods have been used systematically and effectively for both computation and basic theory. With the development of Quillen's concept of a closed model category and, in particular, a simplicial model category, this collection of methods has become the primary way to describe non-abelian homological algebra and to address homotopy-theoretical issues in a variety of fields, including algebraic K-theory. This book supplies a modern exposition of these ideas, emphasizing model category theoretical techniques. Discussed here are the homotopy theory of simplicial sets, and other basic topics such as simplicial groups, Postnikov towers, and bisimplicial sets. The more advanced material includes homotopy limits and colimits, localization with respect to a map and with respect to a homology theory, cosimplicial spaces, and homotopy coherence. Interspersed throughout are many results and ideas well-known to experts, but uncollected in the literature. Intended for second-year graduate students and beyond, this book introduces many of the basic tools of modern homotopy theory. An extensive background in topology is not assumed.

Lectures on Homotopy Theory

1992-01-21 Mathematics
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Release : 1992-01-21
Genre : Mathematics
Kind : eBook
Book Rating : 824/5 ( reviews)

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Book Synopsis Lectures on Homotopy Theory by : R.A. Piccinini

Download or read book Lectures on Homotopy Theory written by R.A. Piccinini. This book was released on 1992-01-21. Available in PDF, EPUB and Kindle. Book excerpt: The central idea of the lecture course which gave birth to this book was to define the homotopy groups of a space and then give all the machinery needed to prove in detail that the nth homotopy group of the sphere Sn, for n greater than or equal to 1 is isomorphic to the group of the integers, that the lower homotopy groups of Sn are trivial and that the third homotopy group of S2 is also isomorphic to the group of the integers. All this was achieved by discussing H-spaces and CoH-spaces, fibrations and cofibrations (rather thoroughly), simplicial structures and the homotopy groups of maps. Later, the book was expanded to introduce CW-complexes and their homotopy groups, to construct a special class of CW-complexes (the Eilenberg-Mac Lane spaces) and to include a chapter devoted to the study of the action of the fundamental group on the higher homotopy groups and the study of fibrations in the context of a category in which the fibres are forced to live; the final material of that chapter is a comparison of various kinds of universal fibrations. Completing the book are two appendices on compactly generated spaces and the theory of colimits. The book does not require any prior knowledge of Algebraic Topology and only rudimentary concepts of Category Theory are necessary; however, the student is supposed to be well at ease with the main general theorems of Topology and have a reasonable mathematical maturity.

Motivic Homotopy Theory

2007-07-11 Mathematics
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Release : 2007-07-11
Genre : Mathematics
Kind : eBook
Book Rating : 972/5 ( reviews)

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Book Synopsis Motivic Homotopy Theory by : Bjorn Ian Dundas

Download or read book Motivic Homotopy Theory written by Bjorn Ian Dundas. This book was released on 2007-07-11. Available in PDF, EPUB and Kindle. Book excerpt: This book is based on lectures given at a summer school on motivic homotopy theory at the Sophus Lie Centre in Nordfjordeid, Norway, in August 2002. Aimed at graduate students in algebraic topology and algebraic geometry, it contains background material from both of these fields, as well as the foundations of motivic homotopy theory. It will serve as a good introduction as well as a convenient reference for a broad group of mathematicians to this important and fascinating new subject. Vladimir Voevodsky is one of the founders of the theory and received the Fields medal for his work, and the other authors have all done important work in the subject.

Stable Homotopy and Generalised Homology

1974 Mathematics
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Release : 1974
Genre : Mathematics
Kind : eBook
Book Rating : 240/5 ( reviews)

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Book Synopsis Stable Homotopy and Generalised Homology by : John Frank Adams

Download or read book Stable Homotopy and Generalised Homology written by John Frank Adams. This book was released on 1974. Available in PDF, EPUB and Kindle. Book excerpt: J. Frank Adams, the founder of stable homotopy theory, gave a lecture series at the University of Chicago in 1967, 1970, and 1971, the well-written notes of which are published in this classic in algebraic topology. The three series focused on Novikov's work on operations in complex cobordism, Quillen's work on formal groups and complex cobordism, and stable homotopy and generalized homology. Adams's exposition of the first two topics played a vital role in setting the stage for modern work on periodicity phenomena in stable homotopy theory. His exposition on the third topic occupies the bulk of the book and gives his definitive treatment of the Adams spectral sequence along with many detailed examples and calculations in KU-theory that help give a feel for the subject.

Lectures on Homotopy Theory

1992-01-01 Mathematics
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Release : 1992-01-01
Genre : Mathematics
Kind : eBook
Book Rating : 386/5 ( reviews)

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Book Synopsis Lectures on Homotopy Theory by : Renzo A. Piccinini

Download or read book Lectures on Homotopy Theory written by Renzo A. Piccinini. This book was released on 1992-01-01. Available in PDF, EPUB and Kindle. Book excerpt: The central idea of the lecture course which gave birth to this book was to define the homotopy groups of a space and then give all the machinery needed to prove in detail that the n th homotopy group of the sphere S n, for n greater than or equal to 1 is isomorphic to the group of the integers, that the lower homotopy groups of S n are trivial and that the third homotopy group of S 2 is also isomorphic to the group of the integers. All this was achieved by discussing H-spaces and CoH-spaces, fibrations and cofibrations (rather thoroughly), simplicial structures and the homotopy groups of maps. groups, to construct a special class of CW-complexes (the Eilenberg-Mac Lane spaces) and to include a chapter devoted to the study of the action of the fundamental group on the higher homotopy groups and the study of fibrations in the context of a category in which the fibres are forced to live; the final material of that chapter is a comparison of various kinds of universal fibrations. Completing the book are two appendices on compactly generated spaces and the theory of colimits. The book does not require any prior knowledge of Algebraic Topology and only rudimentary concepts of Category Theory are necessary; however, the student is supposed to be well at ease with the main general theorems of Topology and have a reasonable mathematical maturity.

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