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Equivariant algebraic models for relative self-equivalences and block diffeomorphisms

Alexander Berglund, Robin Stoll

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.

Equivariant algebraic models for relative self-equivalences and block diffeomorphisms

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.
Paper Structure (33 sections, 136 theorems, 292 equations)

This paper contains 33 sections, 136 theorems, 292 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Basic conventions and notation
  4. Simplicial model categories and relative mapping complexes
  5. Nilpotent and virtually nilpotent spaces
  6. Model structures on categories of spaces
  7. Classifying spaces of simplicial monoids
  8. Block diffeomorphisms and self-equivalences of bundles
  9. Local systems
  10. Polynomial forms with local coefficients
  11. The Federer spectral sequence
  12. Differential graded Lie algebras
  13. Lie models
  14. Algebraic groups
  15. Algebraicity of relative self-equivalences
  16. ...and 18 more sections

Key Result

Theorem A

Let $M$ be a simply connected compact smooth manifold of dimension $d \geq 6$ with simply connected (in particular non-empty) boundary. Assume that the rational Pontryagin classes of $\partial M \setminus *$ are trivial. Then there is a rational equivalence where $\langle\tilde{\mathfrak{g}}_{\partial}(M)\rangle$ denotes the geometric realization of a nilpotent dg Lie algebra $\tilde{\mathfrak{g}

Theorems & Definitions (357)

  • Theorem A: see \ref{['thm:block_diff']}
  • Corollary B: see \ref{['cor:coho_block_diff']}
  • Theorem C: see \ref{['cor:Baut_eq']}
  • Corollary D: see \ref{['cor:cohomology']}
  • Remark
  • Theorem E: see \ref{['thm:manifolds_block']}
  • Theorem F: see \ref{['cor:Baut_eq_natural']}
  • Corollary G: see \ref{['thm:manifolds_block_gluing']}
  • Lemma B
  • proof
  • ...and 347 more