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Chromatic homotopy theory of spaces

Periodic Reporting for period 2 - ChromSpaces (Chromatic homotopy theory of spaces)

Reporting period: 2022-07-01 to 2023-12-31

This project sits at the interface of homotopy theory, which is the study of high-dimensional shapes (usually called 'spaces') and their continuous deformations, and the more rigid world of algebra. Specifically, this project studies the 'chromatic decomposition' of spaces; this is a procedure to break a space apart into pieces that only detect information corresponding to specific periodic families of elements in homotopy groups. The chromatic decomposition has been studied intensively in the setting of stable homotopy theory, but much less so in the unstable case on which this project focuses.

The objectives of this project can be summarized as follows:
1. Develop algebraic models for spaces 'of a single color', i.e. the individual pieces in the chromatic decomposition. Specifically, I have recently shown that a sophisticated version of the theory of Lie algebras can be used to obtain such models (generalizing Quillen's work on rational homotopy theory). This project aims to address a certain interplay ('Koszul duality') between such spectral Lie algebras and commutative algebras. Also, we will use spectral Lie algebras to study the existence of exponents in homotopy groups of certain spaces. This is a classical problem, on which the use of spectral Lie algebras sheds a new light.
2. Understand how to put the individual pieces back together. Specifically, we investigate how to combine the theories of spectral Lie algebras at different chromatic heights into a single algebraic entity that can be used to describe spaces. This 'assembly problem' is closely related to understanding the Frobenius map for commutative ring spectra (which has recently received much attention because of its role in topological Hochschild homology) in a chromatic setting.
We have made significant progress in part 1 of the project, partially in collaboration with Brantner (Oxford), Hahn (MIT), and Yuan (Columbia), and partially in collaboration with members of project team: Boyde, Taggart, and Blans. The main results achieved so far are the following:
1. I have resolved of a conjecture of Francis-Gaitsgory on Koszul duality in the negative, and proved a precise theorem with which to replace it. In particular, this clarifies the relation between spectral Lie algebras and commutative ring spectra.
2. With Brantner, Hahn, and Yuan we have generalized the classical Milnor-Moore theorem to a certain class of spectral Lie algebras (namely the abelian ones).
3. With Boyde, Taggart, and Blans we have elucidated the structure of free spectral Lie algebras on Moore spectra. We will apply this to exponent results for homotopy groups of spaces and to the theory of power operations for spectral Lie algebras.

We have also started to progress on part 2 of the project, partially in collaboration with Meier (Utrecht) and Nikolaus (Münster). We have identified the effect of the Tate diagonal for spectra chromatically localized at height 1 and have partial results at higher heights.
All three items listed above go beyond the state of the art. Specifically:

1. The resolution of the Francis-Gaitsgory conjecture has widespread implications in higher algebra. In this project we will use our results to understand the structure of spectral Lie algebras, but many other applications are expected.
2. The generalization of the Milnor-Moore theorem with Brantner-Hahn-Yuan is highly novel. We expect to generalize our results to all spectral Lie algebras, rather than only the abelian ones. This will have direct applications to the study of the Behrens-Rezk comparison map. Also, we expect our work on nilpotent spectral Lie algebras to give new results on exponents for homotopy groups of spaces.
3. With Boyde-Taggart-Blans we are developing the theory of power operations for spectral Lie algebras beyond the torsionfree cases that were previously known.

My work with Meier and Nikolaus sheds light on 'chromatic versions' of the Segal conjecture. I will now implement these results to construct a notion of transchromatic spectral Lie algebra, which should provide an answer to the problem of chromatic assembly for spaces.