Rational Homotopy Theory and Differential Forms Progress in Mathematics

This completely revised and corrected version of the well-known Florence notes circulated by the authors together with E. Friedlander examines basic topology, emphasizing homotopy theory. Included is a discussion of Postnikov towers and rational homotopy theory. This is then followed by an in-depth look at differential forms and de Tham’s theorem on simplicial complexes. In addition, Sullivan’s results on computing the rational homotopy type from forms is presented. New to the Second Edition: \*Fully-revised appendices including an expanded discussion of the Hirsch lemma \*Presentation of a natural proof of a Serre spectral sequence result \*Updated content throughout the book, reflecting advances in the area of homotopy theory With its modern approach and timely revisions, this second edition of __Rational Homotopy Theory and Differential Forms__ will be a valuable resource for graduate students and researchers in algebraic topology, differential forms, and homotopy theory. “rational Homotopy Theory Is Today One Of The Major Trends In Algebraic Topology. Despite The Great Progress Made In Only A Few Years, A Textbook Properly Devoted To This Subject Still Was Lacking Until Now... The Appearance Of The Text In Book Form Is Highly Welcome, Since It Will Satisfy The Need Of Many Interested People. Moreover, It Contains An Approach And Point Of View That Do Not Appear Explicitly In The Current Literature.” —zentralblatt Math (review Of First Edition) “the Monograph Is Intended As An Introduction To The Theory Of Minimal Models. Anyone Who Wishes To Learn About The Theory Will Find This Book A Very Helpful And Enlightening One. There Are Plenty Of Examples, Illustrations, Diagrams And Exercises. The Material Is Developed With Patience And Clarity. Efforts Are Made To Avoid Generalities And Technicalities That May Distract The Reader Or Obscure The Main Theme. The Theory And Its Power Are Elegantly Presented.^ This Is An Excellent Monograph.” —bulletin Of The American Mathematical Society (review Of First Edition) This Completely Revised And Corrected Version Of The Well-known Florence Notes Circulated By The Authors Together With E. Friedlander Examines Basic Topology, Emphasizing Homotopy Theory. Included Is A Discussion Of Postnikov Towers And Rational Homotopy Theory. This Is Then Followed By An In-depth Look At Differential Forms And De Tham’s Theorem On Simplical Complexes. In Addition, Sullivan’s Results On Computing The Rational Homotopy Type From Forms Is Presented.^ New To The Second Edition: *fully-revised Appendices Including An Expanded Discussion Of The Hirsch Lemma *presentation Of A Natural Proof Of A Serre Spectral Sequence Result *updated Content Throughout The Book, Reflecting Advances In The Area Of Homotopy Theory With Its Modern Approach And Timely Revisions, This Second Edition Of Rational Homotopy Theory And Differential Forms Will Be A Valuable Resource For Graduate Students And Researchers In Algebraic Topology, Differential Forms, And Homotopy Theory. 1 Introduction -- 2 Basic Concepts -- 3 Cw Homology Theorem -- 4 The Whitehead Theorem And The Hurewicz Theorem.- 5 Spectral Sequence Of A Fibration -- 6 Obstruction Theory -- 7 Eilenberg-maclane Spaces, Cohomology, And Principal Fibrations -- 8 Postnikov Towers And Rational Homotopy Theory -- 9 Derham's Theorem For Simplicial Complexes -- 10 Differential Graded Algebras -- 11 Homotopy Theory Of Dgas -- 12 Dgas And Rational Homotopy Theory -- 13 The Fundamental Group -- 14 Examples And Computations -- 15 Functorality -- 16 The Hirsch Lemma -- 17 Quillen's Work On Rational Homotopy Theory -- 18 A1-structures And C1-structures -- 19 Exercises. Phillip Griffiths, John Morgan. Includes Bibliographical References (pages 223-224). Front Matter....Pages i-xi Introduction....Pages 1-3 Basic Concepts....Pages 5-20 CW Homology Theorem....Pages 21-25 The Whitehead Theorem and the Hurewicz Theorem....Pages 27-40 Spectral Sequence of a Fibration....Pages 41-52 Obstruction Theory....Pages 53-61 Eilenberg–MacLane Spaces, Cohomology, and Principal Fibrations....Pages 63-67 Postnikov Towers and Rational Homotopy Theory....Pages 69-81 deRham’s Theorem for Simplicial Complexes....Pages 83-93 Differential Graded Algebras....Pages 95-102 Homotopy Theory of DGAs....Pages 103-111 DGAs and Rational Homotopy Theory....Pages 113-118 The Fundamental Group....Pages 119-126 Examples and Computations....Pages 127-140 Functorality....Pages 141-149 The Hirsch Lemma....Pages 151-163 Quillen’s Work on Rational Homotopy Theory....Pages 165-176 A ∞ -Structures and C ∞ -Structures....Pages 177-185 Exercises....Pages 187-221 Back Matter....Pages 223-227 Phillip A. Griffiths And John W. Morgan. Originated As A Set Of Informal Notes From A Summer Course Taught By The Present Authors, Together With Eric Friedlander, At The Istituto Matematico 'ulisse Dini' In Florence During The Summer Of 1972--introd. Includes Bibliographical References And Index.