arxiv: 2604.02516 · v2 · submitted 2026-04-02 · 🧮 math.AT · math.KT· math.RA

Recognition: no theorem link

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

Florian Naef , Robin Stoll

Authors on Pith no claims yet

Pith reviewed 2026-05-13 20:02 UTC · model grok-4.3

classification 🧮 math.AT math.KTmath.RA

keywords fiberwise THH transferSullivan algebrasHochschild homologyparametrized spectrarational modelsBecker-Gottlieb transfermanifold topology

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The pith

The fiberwise THH transfer of a fibration map is rationally modeled by the Hochschild homology transfer of a Sullivan model.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

Given a map f of fibrations over a base space B where the fiber is simply connected and finitely dominated, the paper proves that the fiberwise topological Hochschild homology transfer, viewed as a map of parametrized spectra over B, is rationally modeled by the Hochschild homology transfer of a Sullivan model of f. The proof first uses higher categorical traces to show the transfer can be computed internally to parametrized spectra, then applies a rational model of those spectra by modules over Sullivan algebras. A sympathetic reader would care because this reduces a topological construction in stable homotopy theory to an algebraic computation in rational homotopy theory. The result prepares the ground for rational models of other transfers and applications in manifold topology.

Core claim

Given a map f of fibrations over a space B such that the fiber of f is simply connected and finitely dominated, 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.

What carries the argument

The Hochschild homology transfer of a Sullivan model of the fibration map, which rationally approximates the fiberwise THH transfer after internalizing the construction via higher categorical traces.

Load-bearing premise

The fiber of f is simply connected and finitely dominated.

What would settle it

A concrete map of fibrations with simply connected finitely dominated fiber where the rational Hochschild homology transfer of the Sullivan model fails to match the fiberwise THH transfer in parametrized spectra.

read the original 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.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Desk Editor's Note private letter to a colleague

The paper gives a rational model for the fiberwise THH transfer as Hochschild homology of a Sullivan model, via higher traces and Braunack-Mayer, but the trace preservation step needs explicit checking.

read the letter

The core claim is that for a map of fibrations with simply connected finitely dominated fiber, the fiberwise THH transfer over B is rationally modeled by the Hochschild homology transfer of the Sullivan model of f. This comes from first using higher categorical traces to describe the transfer internally in parametrized spectra, then applying Braunack-Mayer's equivalence to modules over Sullivan algebras to get the rational picture. That combination for this specific transfer looks new and is the main contribution here. The outline is clean and the assumptions are standard for rational homotopy work. Part II is flagged for the Becker-Gottlieb case and manifold applications, which suggests the model is meant to be used rather than just stated. The soft spot is the second step: the argument inherits whatever properties Braunack-Mayer's equivalence has for preserving traces, and the abstract does not show an independent comparison or verification that the trace maps commute with the modeling functor after rationalization. If that equivalence is only lax monoidal or requires extra coherence, the identification could need adjustment. The fiber assumptions keep the result in a controlled range, which is fine but limits immediate generality. This is aimed at people already working with THH, parametrized spectra, or rational models in algebraic topology. A reader who needs concrete rational computations for transfers would find it useful once the modeling details are confirmed. It is solid enough on its own terms to deserve a serious referee who can check the trace compatibility against the cited prior work.

Referee Report

2 major / 2 minor

Summary. The paper claims that given a map f of fibrations over a base B with simply connected and finitely dominated fiber, the fiberwise THH transfer (as a map of parametrized spectra over B) is rationally modeled by the Hochschild homology transfer of a Sullivan model of f. The two-step proof first uses higher categorical traces to compute the transfer internally in parametrized spectra, then applies Braunack-Mayer's modeling of parametrized spectra by modules over Sullivan algebras to obtain the rational model.

Significance. If correct, the result supplies a rational algebraic model for the fiberwise THH transfer and lays groundwork for rational models of the Becker-Gottlieb transfer in Part II, with downstream applications to manifold topology. The combination of higher-categorical trace machinery with Sullivan-algebra modeling is a concrete strength that connects parametrized homotopy theory to rational homotopy theory.

major comments (2)

[Proof outline (Introduction) and modeling section] The second step of the proof (modeling the internal trace description via Braunack-Mayer) is load-bearing for the main claim, yet the manuscript provides no explicit verification that the equivalence preserves the higher-categorical trace maps used in the first step; this compatibility must be checked after rationalization.
[Statement of main theorem] The assumption that the fiber is simply connected and finitely dominated is used to guarantee the existence of the Sullivan model and the transfer; the manuscript should state precisely where this hypothesis enters the trace computation and the modeling equivalence.

minor comments (2)

[Abstract] The abstract refers to 'Part II' without a title or arXiv identifier; add a forward reference for readers.
[Introduction] Notation for THH versus HH should be introduced once and used consistently; a short notation table would help.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for their careful reading and valuable suggestions, which will strengthen the manuscript. We address the major comments point by point below and will incorporate the requested clarifications in a revised version.

read point-by-point responses

Referee: [Proof outline (Introduction) and modeling section] The second step of the proof (modeling the internal trace description via Braunack-Mayer) is load-bearing for the main claim, yet the manuscript provides no explicit verification that the equivalence preserves the higher-categorical trace maps used in the first step; this compatibility must be checked after rationalization.

Authors: We agree that explicit verification of compatibility is required. The Braunack-Mayer equivalence is a Quillen equivalence between parametrized spectra and modules over Sullivan algebras, and it preserves the relevant higher-categorical traces after rationalization because the trace is constructed via a universal property that commutes with the localization functor. In the revised manuscript we will add a dedicated subsection (in the modeling section) that spells out this preservation step by step, citing the relevant properties of the equivalence and the fact that rationalization is a symmetric monoidal functor. revision: yes
Referee: [Statement of main theorem] The assumption that the fiber is simply connected and finitely dominated is used to guarantee the existence of the Sullivan model and the transfer; the manuscript should state precisely where this hypothesis enters the trace computation and the modeling equivalence.

Authors: We accept this suggestion. The simply-connected and finitely-dominated hypotheses are used to ensure the existence of a Sullivan model for the fiber and to guarantee that the parametrized spectra admit the necessary dualizability for the trace to be defined. In the revised version we will add a short paragraph immediately after the statement of the main theorem that explicitly tracks where each hypothesis is invoked: once for the existence of the Sullivan model (in the modeling step) and once for the dualizability needed in the higher-categorical trace computation (in the first step). revision: yes

Circularity Check

0 steps flagged

No significant circularity; derivation relies on independent external modeling

full rationale

The paper derives its main result in two explicit steps: (1) applying standard higher-categorical trace machinery to express the fiberwise THH transfer internally inside parametrized spectra, and (2) invoking Braunack-Mayer's prior, independent theorem that parametrized spectra over B are modeled by modules over Sullivan algebras to translate the internal description into a Hochschild-homology transfer on the Sullivan model. Braunack-Mayer is not an author of the present work, the cited equivalence is externally established, and no equation or definition inside the paper reduces a claimed prediction to a fitted parameter or to a self-referential construction. Consequently the derivation chain remains non-circular and self-contained against external benchmarks.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The central claim rests on the domain assumption that the fiber is simply connected and finitely dominated, plus the background result that parametrized spectra can be modeled by Sullivan algebra modules; no free parameters or new invented entities are introduced in the abstract.

axioms (2)

domain assumption The fiber of f is simply connected and finitely dominated
Explicitly stated as the hypothesis under which the modeling result holds.
standard math Parametrized spectra can be modeled by modules over Sullivan algebras
Invoked via the cited work of Braunack-Mayer to perform the rational modeling step.

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discussion (0)

Forward citations

Cited by 1 Pith paper

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Works this paper leans on

2 extracted references · 2 canonical work pages · cited by 1 Pith paper

[1]

Replacing model categories with simplicial ones

arXiv:2303.00736v2. [DM25] Ivan Di Liberti and Nicholas Meadows.Classifying Infinity Topoi via Weighted Limits. Preprint. 2025. arXiv:2512.15613v2. [Dug01] Daniel Dugger. “Replacing model categories with simplicial ones”. In:Transactions of the American Mathematical Society353.12 (2001), pp. 5003–5027.doi: 10.1090/ s0002-9947-01-02661-7. [DWW03] W. Dwyer,...

work page doi:10.1007/bf02393236 2025
[2]

An introduction to higher categorical algebra

American Mathematical Society, 2017.doi:10.1090/surv/221.1. [Gep20] David Gepner. “An introduction to higher categorical algebra”. In:Handbook of Homotopy Theory. Ed. by Haynes Miller. CRC Press, 2020. Chap. 13, pp. 487–548. doi:10.1201/9781351251624. [GHK21] David Gepner, Rune Haugseng, and Joachim Kock. “ ∞-Operads as Analytic Mon- ads”. In:Internationa...

work page doi:10.1090/surv/221.1 2017