A rational model for the fiberwise THH transfer I: 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.

• The paper establishes that the fiberwise THH transfer for fibrations is captured, after rationalization, by the Hochschild homology transfer via Sullivan models.
• It employs advanced techniques including higher-categorical trace analysis and explicit Quillen adjunctions to bridge parametrized stable homotopy and differential graded algebra computations.
• The results pave the way for extending these methods to cyclic homology and A-theory transfers, enhancing computational approaches in manifold topology and characteristic classes.

Rational Models for Fiberwise Topological Hochschild Homology Transfer: The Case of Sullivan Algebras

Introduction

This paper provides a rigorous treatment of rational models for the fiberwise topological Hochschild homology (THH) transfer in the setting of parametrized spectra over a base anima, specifically when the underlying spaces are nilpotent and of finite rational type. The main result establishes that for a fibration of such spaces, the fiberwise THH transfer is captured, after rationalization, by the Hochschild homology transfer at the level of Sullivan models. The results are situated within the higher-categorical framework of parametrized stable homotopy theory and provide a bridge to explicit computations in rational homotopy theory using differential graded algebra (DGA) models.

Context and Motivation

The classical Becker–Gottlieb transfer associates to a fibration $f: X \to Y$ a wrong-way map in stable homotopy, and, as shown by Lind–Malkiewich, the THH transfer $f^*:\mathrm{THH}(Y)\to\mathrm{THH}(X)$ refines this construction at the level of free loop spaces and traces in higher category theory. More generally, given $f$ as a map of fibrations over a base $B$, one obtains a map of spectra over $B$, called the fiberwise THH transfer. Understanding the rational structure of such transfers is of particular interest, especially given their connections to algebraic K-theory and the broader landscape of characteristic classes in geometric topology.

Main Results

The central result states that given a fibration $f: X \to Y$ over a base $B$, under suitable nilpotency and finiteness hypotheses, the fiberwise THH transfer is rationally modeled by the Hochschild homology transfer at the level of Sullivan models. Explicitly, for cdga models $\phi: R \to S$ of the underlying spaces, the map $f^*: \THH_B(Y)\to \THH_B(X)$ corresponds under realization to $\phi^*: \HH_\k(S)\to \HH_\k(R)$, where $f^*:\mathrm{THH}(Y)\to\mathrm{THH}(X)$0 denotes the (relative) Hochschild complex.

The methodology unfolds in two principal steps:

1. Higher-Categorical Trace Analysis: The fiberwise THH transfer is recast as a trace in the $f^*:\mathrm{THH}(Y)\to\mathrm{THH}(X)$1-category of parametrized spectra, employing the full machinery of higher categorical traces developed by Hoyois--Scherotzke--Sibilla and Carmeli--Cnossen--Ramzi--Yanovski.
2. Rational Modeling via Sullivan Algebras: Following the paradigm of Sullivan and subsequent extensions (notably by Braunack–Mayer), parametrized spectra are modeled rationally by modules over cdgas, and explicit functorial constructions for the associated six-functor formalism are provided. This includes a careful treatment of dualizing spectra and transfer maps at the level of modules.

The paper also rigorously addresses the subtleties arising in the equivariant context when the nilpotence assumption is only satisfied on universal covers, by employing equivariant cdga models and careful descent arguments.

Technical Innovations

A key technical contribution is the systematic use of model-categorical and enriched-categorical techniques to construct and control rational models for functors and transfers in parametrized stable homotopy. The approach provides fully coherent, functorial comparisons between parametrized stable homotopy categories and corresponding categories of cdga modules (and their equivariant analogues) through a succession of explicit Quillen adjunctions and left/right derived functors. The explicit bar constructions and derived Hom/tensor adjunctions for modeling Hochschild complexes and transfers are essential for computational accessibility.

Furthermore, the paper gives an explicit and verifiable dictionary between homotopical operations in the parametrized spectra setting (such as pushforwards, pullbacks, and dualizing objects) and their algebraic counterparts in the setting of Sullivan models. For example, the dualizing spectrum associated to a map modeled by $f^*:\mathrm{THH}(Y)\to\mathrm{THH}(X)$2 is, up to rational equivalence, represented by the derived $f^*:\mathrm{THH}(Y)\to\mathrm{THH}(X)$3-dual of $f^*:\mathrm{THH}(Y)\to\mathrm{THH}(X)$4 as an $f^*:\mathrm{THH}(Y)\to\mathrm{THH}(X)$5-module, and the various compatibility isomorphisms in the six-functor calculus are likewise modeled by explicit quasi-isomorphisms.

Numerical and Conceptual Consequences

The results enable strong explicitness for computational purposes. For instance, in the example of the canonical map $f^*:\mathrm{THH}(Y)\to\mathrm{THH}(X)$6, the THH transfer is computed via the map of polynomial algebras induced by $f^*:\mathrm{THH}(Y)\to\mathrm{THH}(X)$7 (rationally), and the resulting transfer on Hochschild complexes yields verifiable formulas that agree with classical calculations.

The model provides a method for proving vanishing results for graph characteristic classes associated to fiber bundles with structure group block diffeomorphisms, by straightforward reduction to calculations in Hochschild homology and the use of Lie graph complexes.

A salient and potentially contradictory claim in the paper, formulated as a conjecture, is the expectation that the transfer for negative topological cyclic homology (TC$f^*:\mathrm{THH}(Y)\to\mathrm{THH}(X)$8) is likewise modeled by the transfer in cyclic homology, as constructed by Keller. This shapes the expectation that the rational models for A-theory transfer (and thus for assembly-type maps in manifold topology) can be approached via cyclic homology, which, if validated, could significantly change the practical computation of secondary characteristic invariants of bundles.

Theoretical and Practical Implications

The results reinforce the principle that the intricate categorical and topological operations intrinsic to parametrized stable homotopy theory admit explicit and computable algebraic models in the rational context. This makes available the full computational apparatus of homological algebra and opens the way to new theorems using explicit models, particularly in the rational study of classifying spaces of diffeomorphism and homeomorphism groups and the associated characteristic classes.

On a theoretical level, the paper clarifies the deep relationship between traces in the parametrized ($f^*:\mathrm{THH}(Y)\to\mathrm{THH}(X)$9)-categorical setting and classical algebraic homological constructions via Sullivan models. The techniques are robust and poised to be adapted to generalizations such as equivariant settings, twisted THH and Reidemeister traces, and eventually to the study of assembly, transfer, and trace maps in the algebraic K-theory and cyclic homology of ring spectra.

Conclusions

This work provides a comprehensive and explicit account of rational models for the fiberwise THH transfer in terms of Sullivan algebras, including all necessary functorial and higher-categorical structures. The algebraic nature of the models provides both computational power and conceptual clarity, situating THH transfer (and potentially more general trace operations) firmly within (equivariant) rational homotopy theory. The results facilitate the explicit computation of localization, transfer, and characteristic maps that arise in manifold topology and homotopy theory, thereby constructing a solid foundation for future advances in the chromatic and arithmetic approaches to parametrized stable homotopy and its algebraic models.