For a simply connected finite CW-complex X, we construct a tractable model for the rational homotopy type of the classifying space Baut(X) of the topological monoid of self-homotopy equivalences of X, aka 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 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 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.