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Algebraic models for
classifying spaces of fibrations
Alexander Berglund and Tomáš Zeman |
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Geometry & Topology 29 (2025) 3567–3634
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Abstract |
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We prove new structural results for the rational homotopy type of the classifying space of fibrations with fiber a simply connected finite CW-complex . We first study nilpotent covers of and show that their rational cohomology groups are algebraic representations of the associated transformation groups. For the universal cover, this yields an extension of the Sullivan–Wilkerson theorem to higher homotopy and cohomology groups. For the cover corresponding to the kernel of the homology representation, this proves algebraicity of the cohomology of the homotopy Torelli space. For the cover that classifies what we call normal unipotent fibrations, we then prove the stronger result that there exists a nilpotent dg Lie algebra in algebraic representations that models its equivariant rational homotopy type. This leads to an algebraic model for the space and to a description of its rational cohomology ring as the cohomology of a certain arithmetic group with coefficients in the Chevalley–Eilenberg cohomology of . This has strong structural consequences for the cohomology ring and, in certain cases, allows it to be completely determined using invariant theory and calculations with modular forms. We illustrate these points with concrete examples. As another application, we significantly improve on certain results on self-homotopy equivalences of highly connected even-dimensional manifolds due to Berglund and Madsen, and we prove parallel new results in odd dimensions. |
Keywords
self-homotopy equivalences, rational homotopy theory,
arithmetic groups
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Mathematical Subject Classification
Primary: 55P62, 55R40
Secondary: 11F75, 55R15
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References |
Publication
Received: 23 August 2022
Revised: 17 September 2024
Accepted: 25 January 2025
Published: 10 October 2025
Proposed: Nathalie Wahl
Seconded: Haynes R Miller, Ulrike Tillmann
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Authors |
| Alexander Berglund | |
| Department of Mathematics Stockholm University Stockholm Sweden |
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| Tomáš Zeman | |
| Department of Mathematics Stockholm University Stockholm Sweden |
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| © 2025 The Author(s), under exclusive license to MSP (Mathematical Sciences Publishers). Distributed under the Creative Commons Attribution License 4.0 (CC BY). |
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