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Topological CoHochschild Homology and Thom Spectra

Jiaxi Zha

TL;DR

The paper develops a framework for topological coHochschild homology with Thom-spectrum coefficients, focusing on Thom spectra Th(f) over simply connected spaces X and input data f: X → Pic(R). It establishes a duality-like result coTHH^R(Th f; R[X]) ≃ Th f ⊗ Σ^{∞}_+ ΩX and constructs a skeleton filtration whose associated graded pieces are controlled by the loop space ΩX, reducing computations to coTHH on group-like spaces. A geometric description expresses coTHH^R(Th f;R[X]) as a colimit over Δ^{op} of coTHH^R(R[G^n]) with G = ΩX, and orientation-compatible base-change results yield spectral sequences for A-homology in favorable cases. Together, these results provide concrete tools for calculating coTHH in Thom-spectra contexts and illuminate the interplay between Thom constructions, comodule structures, and loop-space data.

Abstract

For an $\E$-ring spectrum $R$ and a map $f:X\to Pic(R)$ of spaces, the Thom spectrum $\T f$ is a comodule over $R\otimes\Si X$. In this parper we study the topological coHochschild homology of $R\otimes\Si X$ with coefficient $\T f$. More concretely, for a simply connected space $X$, we will give a filtration on $\mathrm{coTHH}^R(\T f;R\otimes\Si X)$ via the cellular structure of $X$. Furthermore, we will reduce the computation of $\mathrm{coTHH}^R(\T f;R\otimes\Si X)$ to that of $\mathrm{coTHH}^R(R\otimes\Si G)$ for some group-like $\mathbb{E}_1$-spaces. Finally, we will use these results to study properties of $\mathrm{coTHH}(\T f;R\otimes\Si X)$.

Topological CoHochschild Homology and Thom Spectra

TL;DR

The paper develops a framework for topological coHochschild homology with Thom-spectrum coefficients, focusing on Thom spectra Th(f) over simply connected spaces X and input data f: X → Pic(R). It establishes a duality-like result coTHH^R(Th f; R[X]) ≃ Th f ⊗ Σ^{∞}_+ ΩX and constructs a skeleton filtration whose associated graded pieces are controlled by the loop space ΩX, reducing computations to coTHH on group-like spaces. A geometric description expresses coTHH^R(Th f;R[X]) as a colimit over Δ^{op} of coTHH^R(R[G^n]) with G = ΩX, and orientation-compatible base-change results yield spectral sequences for A-homology in favorable cases. Together, these results provide concrete tools for calculating coTHH in Thom-spectra contexts and illuminate the interplay between Thom constructions, comodule structures, and loop-space data.

Abstract

For an -ring spectrum and a map of spaces, the Thom spectrum is a comodule over . In this parper we study the topological coHochschild homology of with coefficient . More concretely, for a simply connected space , we will give a filtration on via the cellular structure of . Furthermore, we will reduce the computation of to that of for some group-like -spaces. Finally, we will use these results to study properties of .
Paper Structure (6 sections, 25 theorems, 94 equations)

This paper contains 6 sections, 25 theorems, 94 equations.

Table of Contents

  1. Introduction
  2. Orgnization
  3. Conventions
  4. Thom spectra and comodules
  5. Topological coHochschild homology of Thom spectra
  6. The proof of Theorem \ref{['thm:geo']}

Key Result

Theorem 1.1

Let $R$ be an $\mathbb{E}_{\infty}$-ring spectrum and $X$ be a connected space. Given a map of spaces $f\colon X\to BGL_1(R)$, the Thom spectrum $\mathrm{Th}(R_X)\simeq R\otimes\Sigma^{\infty}_+ X$ is an $\mathbb{E}_{\infty}$-R-coalgebra, and $\mathrm{Th} f$ is a $\mathrm{Th}(R_X)$-comodule, where $

Theorems & Definitions (57)

  • Theorem 1.1
  • Theorem 1.2
  • Theorem 1.3
  • Theorem 1.4
  • Remark 2.1
  • Definition 2.2
  • Remark 2.3
  • Lemma 2.4
  • proof
  • Remark 2.5
  • ...and 47 more