A rational model for the fiberwise THH transfer II: $A_\infty$-algebras

TL;DR

This paper provides an explicit $A_ abla$-algebraic description of the fiberwise THH transfer, extending previous models and applying to manifold topology and characteristic classes.

Contribution

It generalizes the Hochschild homology transfer to $A_ abla$-algebras and applies this to rational models in manifold topology, including characteristic classes and diffeomorphism spaces.

Findings

Explicit $A_ abla$-algebraic description of the Hochschild homology transfer.

Vanishing of certain characteristic classes for specific fibrations.

Rational models for fiberwise THH structures and classifying spaces.

Abstract

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 Poincar\'e 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.