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Rational Homotopy Theory
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========================
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Master thesis on Rational Homotopy Theory, the study of homotopy without
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torsion. The thesis mainly focuses on the Sullivan equivalence, which models
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rational spaces by commutative differential graded algebras (contrary to
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Quillen's dual approach which considers coalgebras). These objects are nice
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since they are very similar to polynomial rings and hence allow for easy
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calculations. The construction which allows us to go back and forth between
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spaces and algebras resembles ideas from differential geometry (namely De Rham's
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cochain complex), but can be applied to any topological space. Applications
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include the calculation of all rational homotopy groups of all spheres (in
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contrast to the integral case where many groups remain unknown as of today) and
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a result towards Adams' theorem (stating that only the spheres S0, S1, S3 and S7
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are H-spaces).
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## Contents:
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1. Basics of Rational Homotopy theory
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- Rational Homotopy Theory
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- Serre Theorems mod C
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- Rationalizations
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2. CDGAs as Algebraic Models
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- Homotopy theory for CDGAs
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- Polynomial Forms
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- Minimal Models
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- The Main Equivalence
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3. Applications and Further Topics
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- Rational Homotopy Groups of the Spheres and Other Calculations
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- Further Topics
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4. Appendices
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- Differential Graded Algebra
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- Model Categories
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## Building the pdf
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cd thesis
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make
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open thesis.pdf
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