[Paper Review] A model for genuine equivariant commutative ring spectra away from the group order

Christian Wimmer|arXiv (Cornell University)|May 29, 2019

Homotopy and Cohomology in Algebraic Topology11 references6 citations

TL;DR

This paper establishes an algebraic model for rational genuine equivariant commutative ring spectra by leveraging geometric fixed points and the norm maps from Hill-Hopkins-Ravenel's construction. It shows that after inverting the group order, the homotopy theory of genuine $G$-equivariant commutative ring spectra is equivalent to a category of diagrams of rational $W_G H$-modules equipped with additional norm maps, providing a computationally tractable and symmetric monoidal description of the rational theory.

ABSTRACT

We use geometric fixed points to describe the homotopy theory of genuine equivariant commutative ring spectra after inverting the group order. The main innovation is the use of the extra structure provided by the Hill-Hopkins-Ravenel norms in the form of additional norm maps on geometric fixed point diagrams, which turns out to be computationally managable.

Motivation & Objective

• To provide a computable and symmetric monoidal algebraic model for rational genuine equivariant commutative ring spectra.
• To extend the classical rational stable homotopy equivalence to the genuine equivariant setting with full commutative ring structure.
• To incorporate the additional multiplicative structure from Hill-Hopkins-Ravenel norms into the geometric fixed point framework.
• To establish a Quillen equivalence between the model category of genuine $G$-equivariant commutative ring spectra and a diagram category of rational modules with norm maps.
• To show that inverting the group order simplifies the theory to a manageable form while preserving the essential multiplicative and equivariant structure.

Proposed method

• Use geometric fixed point spectra as a key invariant to decompose the homotopy theory of genuine $G$-spectra after inverting the group order.
• Leverage the norm maps from the Hill-Hopkins-Ravenel construction to enrich the geometric fixed point diagrams with additional multiplicative structure.
• Construct a comparison functor from genuine $G$-spectra to diagrams of rational $W_G H$-modules indexed by conjugacy classes of subgroups $H \leq G$.
• Show that the forgetful functor from orthogonal $G$-spectra to symmetric $G$-spectra of simplicial sets preserves homotopy colimits of simplicial objects, enabling the use of simplicial methods.
• Use the fact that geometric realization of simplicial objects in symmetric $G$-spectra preserves weak equivalences to compute homotopy colimits correctly.
• Establish a zig-zag of Quillen equivalences between commutative orthogonal $G$-ring spectra and commutative symmetric $G$-ring spectra in simplicial sets, enabling diagrammatic models.

Experimental results

Research questions

• RQ1 Can the rational homotopy theory of genuine equivariant commutative ring spectra be modeled algebraically using geometric fixed points?
• RQ2 How do the norm maps from the Hill-Hopkins-Ravenel construction refine the geometric fixed point diagram in the rational setting?
• RQ3 Is there a Quillen equivalence between the model category of genuine $G$-equivariant commutative ring spectra and a diagram category of rational $W_G H$-modules with norm data?
• RQ4 Does inverting the group order simplify the multiplicative structure of genuine $G$-spectra in a way that allows for a computable and symmetric monoidal model?
• RQ5 Can the homotopy colimits of simplicial objects in the category of commutative ring spectra be computed in the underlying category of spectra?

Key findings

• After inverting the group order, the homotopy theory of genuine $G$-equivariant commutative ring spectra is equivalent to a category of diagrams of rational $W_G H$-modules indexed by conjugacy classes of subgroups $H \leq G$.
• The geometric fixed point functor induces a Quillen equivalence between the model category of genuine $G$-spectra and the product of categories of rational $W_G H$-modules.
• The additional norm maps from the Hill-Hopkins-Ravenel construction endow the geometric fixed point diagram with a computationally manageable multiplicative structure.
• The forgetful functor from orthogonal $G$-spectra to symmetric $G$-spectra of simplicial sets preserves homotopy colimits of simplicial objects, enabling diagrammatic computations.
• There is a zig-zag of Quillen equivalences between commutative orthogonal $G$-ring spectra and commutative symmetric $G$-ring spectra in simplicial sets, allowing for a simplicial model.
• The realization of a simplicial object in commutative ring spectra computes the correct homotopy colimit, and the forgetful functor preserves this structure.