Rational homotopy theory: a brief introduction

By Kathryn Hess. 2006

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Abstract. These notes contain a brief introduction to rational homotopytheory: its model category foundations, the Sullivan model and interactionswith the theory of local commutative rings. ContentsIntroduction1. Foundations2. Sullivan models3. Commutative algebra and rational homotopy theoryReferencesIntroductionThis overview of rational homotopy theory consists of an extended version oflecture notes from a minicourse based primarily on the encyclopedic text [18] ofF´elix, Halperin and Thomas. With only three hours to devote to such a broadand rich subject, it was difficult to choose among the numerous possible topics topresent. Based on the subjects covered in the first week of this summer school,I decided that the goal of this course should be to establish carefully the founda-tions of rational homotopy theory, then to treat more superficially one of its mostimportant tools, the Sullivan model. Finally, I provided a brief summary of the ex-tremely fruitful interactions between rational homotopy theory and local algebra,in the spirit of the summer school theme “Interactions between Homotopy Theoryand Algebra.” I hoped to motivate the students to delve more deeply into thesubject themselves, while providing them with a solid enough background to do sowith relative ease.As these lecture notes do not constitute a history of rational homotopy theory,I have chosen to refer the reader to [18], instead of to the original papers, for the

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