A rational model for the fiberwise THH transfer I: Sullivan algebras

TL;DR

This paper provides a rational model for the fiberwise THH transfer using Sullivan algebras, connecting higher categorical traces with algebraic models in parametrized spectra.

Contribution

It introduces a novel rational modeling approach for the fiberwise THH transfer via Sullivan models, bridging higher categorical traces and algebraic topology.

Findings

Fiberwise THH transfer is modeled by Hochschild homology transfer of Sullivan models.

Uses higher categorical traces to compute the transfer internally.

Models parametrized spectra as modules over Sullivan algebras.

Abstract

Given a map $f$ of fibrations over a space $B$ such that the fiber of $f$ is simply connected and finitely dominated, we prove that its fiberwise THH transfer, considered as a map of parametrized spectra over $B$, is rationally modeled by the Hochschild homology transfer of a Sullivan model of $f$. The proof goes in two steps. Firstly, we use the machinery of higher categorical traces to show that the fiberwise THH transfer can be computed internally to parametrized spectra. Secondly, we model the resulting description rationally using work of Braunack-Mayer, who proved that parametrized spectra can be modeled by modules over Sullivan algebras. In Part II, we will use our result to obtain a rational model of the Becker-Gottlieb transfer, and for applications to manifold topology.