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

Florian Naef, Robin Stoll

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.

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

Abstract

Given a map of fibrations over a space such that the fiber of is simply connected and finitely dominated, we prove that its fiberwise THH transfer, considered as a map of parametrized spectra over , is rationally modeled by the Hochschild homology transfer of a Sullivan model of . 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.

Paper Structure

This paper contains 28 sections, 62 theorems, 227 equations.

Table of Contents

  1. Introduction
  2. The Hochschild homology transfer.
  3. Equivariant models.
  4. The proof.
  5. Further directions.
  6. Applications.
  7. Preliminaries
  8. Model categories, infinity-categories, animas, and spectra
  9. Differential graded algebra
  10. Homology localization
  11. Sullivan's approach to rational homotopy theory
  12. Parametrized spectra
  13. Rational parametrized spectra
  14. The fiberwise THH transfer as a trace
  15. Dualizable objects and the trace
  16. ...and 13 more sections

Key Result

Theorem A

Let $f \colon X \to Y$ and $p \colon Y \to B$ be fibrations between nilpotent spaces of finite rational type that are respectively modeled by cofibrations $\varphi \colon R \to S$ and $\iota \colon \Bbbk \to R$ of cdgas. Assume that the fiber of $f$ is simply connected and finitely dominated, and th

Theorems & Definitions (167)

  • Theorem A: see \ref{['thm:transfer_model_cdga']}
  • Conjecture B
  • Definition F
  • Definition H
  • Proposition J: Shulman
  • proof
  • Lemma K
  • proof
  • Lemma P
  • proof
  • ...and 157 more