Overview

This text is based on notes taken by the author at the DMV Seminar on Algebraic Models of Homotopy Types. The work aims to provide an overview of homotopy theory from the point of view of algebraic models of homotopy types, leading the reader from basic definitions in algebraic topology to specific fields of research. The exposition is addressed towards graduate students as well as to researchers from other fields. Due to the scope and size of the book, only a few complete proofs can be given; however, the fundamental concepts and methods of the subject are pointed out, and a number of recent results are included together with the essential bibliographic references. The book begins with a recollection of the basic notions of homotopy theory. A discussion of rational homotopy theory, emphasizing both the Sullivan and the Quillen models, follows. Next, the Adams-Hilton and the Anick models are shown to yield some integral information. The last chapter provides an introduction to the integral classification of homotopy types, along the lines initiated by J.H.C. Whitehead and considerably developed by H. Baues during the last decade.

Full Product Details

Author:   Marc Aubry
Publisher:   Birkhauser Verlag AG
Imprint:   Birkhauser Verlag AG
Edition:   1995 ed.
Volume:   24
Dimensions:   Width: 17.00cm , Height: 0.70cm , Length: 24.40cm
Weight:   0.252kg
ISBN:  

9783764351854

ISBN 10:   3764351853
Pages:   117
Publication Date:   27 March 1995
Audience:   College/higher education ,  Professional and scholarly ,  Postgraduate, Research & Scholarly ,  Professional & Vocational
Format:   Paperback
Publisher's Status:   Active
Availability:   Out of stock  
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Table of Contents

1: Basic Homotopy Theory.- 1. Homotopy.- 2. Cofibrations and fibrations.- 2: Homology and Homotopy Decomposition of Simply Connected Spaces.- 1. Eckmann-Hilton duality.- 2. Homology and homotopy decompositions.- 3. Application: Classification of 2-stage spaces.- 3: Cofibration Categories.- 1. Basic definitions.- 2. Homotopy in a cofibration category.- 3. Properties of cofibration categories.- 4. Properties of cofibrant models.- 5. The homotopy category as a localization.- 4: Algebraic Examples of Cofibration Categories.- 1. The category CDA.- 2. The category Chain+.- 3. The category DA.- 4. The category DL.- 5: The Rational Homotopy Category of Simply Connected Spaces.- 1. The category of rational spaces.- 2. Quillen's model category.- 3. Sullivan's model theory.- 4. Some easy applications.- Appendix: Relations between the Various Models of a Space.- A.1. A functor between DL and CDA.- A.2. Models over ?/p?.- A.3. Sullivan Models.- 6: Attaching Cells in Topology and Algebra.- 1. Algebraic models of spaces with a cell attached.- 2. Inertia.- 7: Elliptic Spaces.- 1. Finiteness of the formal dimension.- 2. Elliptic models.- 3. Some equalities and inequalities.- 4. Topological interpretation.- 8: Non Elliptic Finite C.W.-Complexes.- 1. Homotopy invariants of spaces.- 2. Sullivan models and the (algebraic) Lusternik-Schnirelmann category.- 3. Lie algebras of finite depth.- 4. The mapping theorem.- 5. Proof of Theorem 0.1.- 9: Towards Integral Algebraic Models of Homotopy Types.- 1. Introduction and general problem.- 2. Algebraic description of the integral homotopy types in dimension 4.- 3. Algebraic description of the integral homotopy types in dimension N.

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