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Rational homotopy theory studies the free parts of homotopy and homology groups. By ignoring torsion, the remaining structure can be represented by algebraic models. This approach works particularly well for simply connected spaces with cohomology of finite type. The Sullivan model, a special type of differentially graded algebra, is one of the central algebraic models in rational homotopy theory.
In this lecture series, I will introduce the construction of the Sullivan model, explain how it encodes topological information, and describe how to reconstruct a space from such an algebra. My current research focuses on understanding when the realization of a non-simply-connected Sullivan algebra preserves cohomology.
If time permits, I will also discuss another related topic: a space is called formal if its rational cohomology ring itself can serve as its algebraic model. Although a sphere bundle over a compact formal manifold may not be formal, its formality can be determined arithmetically using the Bianchi-Massey tensor introduced by Crowley and Nordström. |