Algebraic models for classifying spaces of fibrations

Alexander Berglund, Stockholm University

Online Talk 

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For a simply connected finite CW-complex X, we construct an algebraic model for the rational homotopy type of Baut(X), the classifying space of fibrations with fiber X. This space is in general far from nilpotent, so its rational homotopy type cannot be modeled by a dg Lie algebra over Q as in Quillen's theory. Instead, we work with dg Lie algebras in algebraic representations of a certain reductive algebraic group.

An important consequence of our results is that the computation of the rational cohomology ring of Baut(X) reduces to the computation of cohomology of arithmetic groups and dg Lie algebras. In special cases, this reduces further to calculations with modular forms and invariant theory. We also prove that the representations of the homotopy mapping class group of X in the higher rational homotopy groups of Baut(X) are algebraic in a suitable sense, extending a classical result of Sullivan and Wilkerson. Our results moreover improve and generalize certain earlier results of Ib Madsen and myself on Baut(M) for highly connected manifolds M. This is joint work with Tomas Zeman.