Algebraic K-Theory and Equivariant Homotopy Theory (12w5116)

Objectives

Our goal is to bring together homotopy theorists and motivic homotopy theorists to attempt to unify the approaches and strengthen the connections. Tools like the slice spectral sequence in both the equivariant and motivic contexts already bridge the gap, and it would be helpful for each side to see the approaches and techniques of the other. Additionally, a broader goal of the workshop is to render the relatively impenetrable fields of equivariant homotopy theory and algebraic $K$-theory more approachable to young researchers. Many of the foundational results are buried in the literature or simply known to experts, and there is a paucity of elementary treatments. The organizers themselves are within $10$ years of their PhDs, and we have first hand experienced the wide-spread difficulty surrounding these topics. Bringing together researchers of all levels should ground the conversation in the specifics of the aforementioned questions while helping clarify many mysterious points in the literature. To ground the workshop proposal, we now present several themes to be discussed.

Even the basic constructions are mysterious from the point of view of equivariant homotopy, and several foundational questions will be addressed:

Why does the standard smash product in structured spectra fail to produce a model for $THH$ with the right properties? Is there a way to fix this without relying on the B"okstedt model?

How does the interplay between finite subgroups of $S^1$ and $S^1$ affect our choice of model and universe? Can we produce a conceptually cleaner result by ignoring the $S^1$ part and focusing on the pieces seen by finite subgroups?

Though they seem quite narrow in scope, these questions touch on several deep and broad-reaching points of the equivariant homotopy around algebraic $K$-theory. There are several rather plausible constructions for various ``fake'' $THH$s, and it is not at all clear why these fail to give the right construction and why no such construction would be possible. Moreover, the failure of the na"ive models centers on a fundamental difference in behavior between equivariant homotopy for finite subgroups and for compact Lie groups.

People studying general equivariant algebraic topology encounter many of these problems and discrepancies, and there are several machines in practice to separate out the contribution of finite subgroups versus the full $S^1$. It is, however, unclear to what extent the $S^1$ part is extraneous from the point of view of algebraic $K$-theory, and one of the over-arching themes of the workshop would be to determine exactly how the $S^1$-information is present.

From a computational perspective, there are also several natural questions.

What sorts of additional, computational restrictions does the cyclotomic structure place on the $RO(G)$-graded homotopy groups? Are there coherence relations between the obvious restrictions that can be exploited?
item What is the $RO(S^1)$-graded analogue of the de Rham-Witt complex? Since this is in some sense initial, how can we identify the $TR$ and $TC$ groups out of these?

What happens in two families of important examples: Thom spectra and free commutative algebras? How does the organizers' model of $THH$ of a Thom spectrum play out computationally?

A careful treatment of the last part can provide a springboard for the other questions. The Thom spectrum case is a natural generalization of the ``group ring'' known example, and though it will undoubtedly be less simple, understanding even a few cases would greatly help in teasing apart the general relationships mentioned in the first two computational questions.

Finally, building Hill-Hopkins-Ravenel's solution to the Kervaire problem, we can ask how to apply their techniques to algebraic $K$-theory:

What are the slices of $THH$? Can we use slice-theoretic machinery to simplify classical computations?

Here we note that the current approaches to $TC$ and $TR$ computations are largely inductive. The slice machinery could provide another avenue for describing quite naturally the inductive phenomena, tying in more basically the cyclotomic structure.

A final, broader goal of the workshop is to render the relatively impenetrable fields of equivariant homotopy theory and algebraic $K$-theory more approachable to young researchers. The organizers themselves are within $10$ years of their PhDs, and we have first hand experienced the wide-spread confusion and difficulty surrounding these topics. Bringing together researchers of all levels should ground the conversation in the specifics of the aforementioned questions while helping clarify many mysterious points in the literature.