# Alexander Berglund: Algebraic models for classifying spaces of fibrations

**Time: **
Wed 2021-11-17 10.15 - 11.15

**Location: **
Kräftriket, House 6, Room 306 (also on Zoom)

**Lecturer: **
Alexander Berglund (Stockholm University)

**Abstract:**

For a simply connected finite CW-complex *X*, we construct a tractable model for the rational homotopy type of Baut(*X*), the classifying space for fibrations with fiber *X*. The space Baut(*X*) is in general far from nilpotent, so one should not expect to be able to model its rational homotopy type by a dg Lie algebra over \(\mathbb{Q}\) as in Quillen's theory. Instead, we work with dg Lie algebras in the category of algebraic representations of a certain reductive algebraic group associated to *X*.

A consequence of our results is that the computation of the rational cohomology of Baut(*X*) reduces to the computation of Chevalley–Eilenberg cohomology of dg Lie algebras and cohomology of arithmetic subgroups of reductive groups with coefficients in algebraic representations. Our results also simplify 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.