[Paper Review] Equivariant Structure on Smash Powers

Morten Brun, Bjørn Ian Dundas+1 more|arXiv (Cornell University)|Apr 20, 2016

Homotopy and Cohomology in Algebraic TopologyMathematics47 references19 citations

TL;DR

This paper establishes a foundational framework for understanding the equivariant structure of smash powers in stable homotopy theory, particularly focusing on the geometric diagonal and its role in connecting commutative ring spectra, fixed point spectra, and Witt vector constructions. It proves that the fixed points of the $C_2$-action on $E^{ ext{∧}2}$ admit a natural restriction map to $E$, which generalizes to higher smash powers and underlies the construction of Witt vectors and the chromatic tower in stable homotopy theory.

ABSTRACT

We provide foundations for dealing with the equivariant structure of "smash powers" of commutative orthogonal ring spectra. The category of commutative orthogonal ring spectra $A$ is tensored over spaces $X$, so that $A \otimes X$ is a commutative orthogonal ring spectrum. If $X$ is a discrete space, this is literally the smash power of $A$ with itself indexed over $X$, and we keep this language also in the nondiscrete case. In particular $A \otimes S^1$ is a model for topological Hochschild homology. We provide a framework where a generalization of the cyclotomic structure of topological Hochschild homology is visible in a categorical framework, also for more general $G$ and $X$. Similar situations have been studied by others, e.g., in Hill, Hopkins and Ravenel's treatment of the norm construction and Brun, Carlsson, Dundas' covering homology. In the case of non-commutative $A$ and $X=S^1$, the situation is somewhat easier and has already been covered by Kro. We are motivated by applications to $G$ being a torus in order to study the iterated algebraic $K$-theory, and have to develop a categorical theory that in some ways goes beyond what has been done before. Most of the material appeared in the last author's thesis which was defended in 2011. We apologize for the delay.

Motivation & Objective

• To develop a systematic understanding of the equivariant structure of smash powers of $G$-spectra for compact Lie groups $G$.
• To clarify the role of the geometric diagonal in relating fixed point spectra to algebraic structures such as Witt vectors.
• To establish model categorical foundations for equivariant orthogonal spectra and their smash products.
• To demonstrate how the restriction map from $(E^{ ext{∧}2})^{C_2}$ to $E$ encodes deep algebraic and topological information, especially in the context of $H\mathbb{Z}/2$ and the $2$-adic integers.
• To unify algebraic and topological perspectives by showing that the fixed points of higher smash powers capture increasing chromatic complexity.

Proposed method

• Utilizes equivariant orthogonal spectra and level model structures to define and analyze the smash product in the stable category.
• Applies Illman’s triangulation theorem and mixed model structures to handle equivariant spaces and spectra.
• Employs cofibrant and fibrant replacements in model categories to define the geometric diagonal as a natural transformation from $E$ to $(E^{ ext{∧}n})^{G}$ for finite groups $G$.
• Applies Kan extension and coend calculus to assemble model structures on diagram categories of spectra.
• Uses Beck’s monadicity theorem to show that the category of $T$-algebras is isomorphic to the category of diagrams $[\mathscr{D}, \mathscr{C}]_0$, under suitable smallness and cellularity conditions.
• Establishes a product model structure on $[\mathscr{E}_{\mathscr{D}}, \mathscr{C}]_0$ via generating sets $\mathcal{G}^\mathscr{E}I$ and $\mathcal{G}^\mathscr{E}J$, ensuring cofibrantly generated model structures on diagram categories.

Experimental results

Research questions

• RQ1 How can the geometric diagonal be functorially defined for equivariant $G$-ring spectra, and what structure does it preserve?
• RQ2 What is the precise relationship between the fixed points of smash powers and the construction of Witt vectors in stable homotopy theory?
• RQ3 How do the fixed points of the $C_{2^n}$-action on $E^{ ext{∧}2^n}$ for $E = H\mathbb{Z}/2$ relate to the $2$-adic integers and the chromatic tower?
• RQ4 In what sense does the restriction map $(E^{ ext{∧}2})^{C_2} \to E$ generalize multiplication in commutative rings and capture algebraic data?
• RQ5 How can model structures on diagram categories of spectra be assembled to preserve fibrations, cofibrations, and weak equivalences pointwise?

Key findings

• The restriction map $(E^{ ext{∧}2})^{C_2} \to E$ is a natural transformation that generalizes multiplication and captures algebraic data such as Witt vector structures.
• For $E = H\mathbb{Z}/2$, the fixed points $(E^{ ext{∧}2^n})^{C_{2^n}}$ have path components isomorphic to $\mathbb{Z}/2^n$, and the inverse limit over $n$ yields the $2$-adic integers $\mathbb{Z}_2$, demonstrating a transition from finite to infinite characteristic.
• The $C_2 \times C_2$-fixed points of $E^{ ext{∧}4}$ capture additional topological complexity, including shadows of Bott periodicity.
• The geometric diagonal arises as a natural transformation from $E$ to the fixed points of its smash power, and this construction is functorial in the input spectrum and group action.
• A cofibrantly generated model structure exists on the diagram category $[\mathscr{E}_{\mathscr{D}}, \mathscr{C}]_0$, with fibrations, cofibrations, and weak equivalences defined pointwise, under suitable smallness and cellularity assumptions.
• The category of $T$-algebras for the monad $T$ induced by left adjoints is isomorphic to $[\mathscr{D}, \mathscr{C}]_0$, confirming the monadicity of the diagram category under the given conditions.