A rational model for the fiberwise THH transfer II: $A_\infty$-algebras
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arXiv
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| Format: | Preprint |
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2026
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| _version_ | 1866914511247638528 |
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| author | Naef, Florian Stoll, Robin |
| author_facet | Naef, Florian Stoll, Robin |
| contents | In Part I, we proved that a rational model for the fiberwise THH transfer of a map $f$ of fibrations over a base space is given by the Hochschild homology transfer of a cdga model of $f$. In this paper, we provide an explicit description of this Hochschild homology transfer in terms of $A_\infty$-algebras, generalizing work of Bouc. Using a result of Lind-Malkiewich, we deduce a rational model for the Becker-Gottlieb transfer. We furthermore use our results for the following applications to manifold topology.
Firstly, we consider the rational characteristic classes constructed by Berglund for fibrations with fiber a Poincaré complex (which generalize classes found by Berglund-Madsen); they are defined via the Lie graph complex, and we prove that the classes corresponding to non-trivalent graphs with exactly one loop vanish when evaluated on fiber bundles with fiber a compact simply connected topological manifold.
Secondly, we provide a rational model for the space of fiberwise THH-simple structures, which is a step towards obtaining rational models for the classifying spaces of diffeomorphisms and homeomorphisms of a compact simply connected manifold in the rational concordance stable range. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2604_24709 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | A rational model for the fiberwise THH transfer II: $A_\infty$-algebras Naef, Florian Stoll, Robin Algebraic Topology Geometric Topology K-Theory and Homology Rings and Algebras 55P62 (Primary) 55R12, 16E40, 55R70, 19D55, 19D10, 18G85, 55R40, 55P50 (Secondary) In Part I, we proved that a rational model for the fiberwise THH transfer of a map $f$ of fibrations over a base space is given by the Hochschild homology transfer of a cdga model of $f$. In this paper, we provide an explicit description of this Hochschild homology transfer in terms of $A_\infty$-algebras, generalizing work of Bouc. Using a result of Lind-Malkiewich, we deduce a rational model for the Becker-Gottlieb transfer. We furthermore use our results for the following applications to manifold topology. Firstly, we consider the rational characteristic classes constructed by Berglund for fibrations with fiber a Poincaré complex (which generalize classes found by Berglund-Madsen); they are defined via the Lie graph complex, and we prove that the classes corresponding to non-trivalent graphs with exactly one loop vanish when evaluated on fiber bundles with fiber a compact simply connected topological manifold. Secondly, we provide a rational model for the space of fiberwise THH-simple structures, which is a step towards obtaining rational models for the classifying spaces of diffeomorphisms and homeomorphisms of a compact simply connected manifold in the rational concordance stable range. |
| title | A rational model for the fiberwise THH transfer II: $A_\infty$-algebras |
| topic | Algebraic Topology Geometric Topology K-Theory and Homology Rings and Algebras 55P62 (Primary) 55R12, 16E40, 55R70, 19D55, 19D10, 18G85, 55R40, 55P50 (Secondary) |
| url | https://arxiv.org/abs/2604.24709 |