Equivariant algebraic models for relative self-equivalences and block diffeomorphisms

TL;DR

This paper develops rational algebraic models for classifying spaces of self-equivalences and block diffeomorphisms of simply connected manifolds, linking their cohomology to arithmetic groups and dg Lie algebras.

Contribution

It introduces equivariant algebraic models for these classifying spaces, compatible with gluing and boundary operations, advancing understanding of their rational homotopy types.

Findings

Provides a formula for the rational cohomology of classifying spaces

Models are compatible with boundary connected sums

Connects cohomology to arithmetic groups and dg Lie algebras

Abstract

We construct rational models for classifying spaces of self-equivalences of bundles over simply connected finite CW-complexes relative to a given simply connected subcomplex. Via work of Berglund-Madsen and Krannich this specializes to rational models for classifying spaces of block diffeomorphism groups of simply connected smooth manifolds of dimension at least 6 with simply connected boundary. The main application is a formula for the rational cohomology of these classifying spaces in terms of the cohomology of arithmetic groups and dg Lie algebras. We furthermore prove that our models are compatible with gluing constructions, and deduce that the model for block diffeomorphisms is compatible with boundary connected sums of manifolds whose boundary is a sphere. As in preceding work of Berglund-Zeman on spaces of self-homotopy equivalences, a key idea is to study equivariant algebraic models for nilpotent coverings of the classifying spaces.