Operations on spectral partition Lie algebras and TAQ cohomology

Zhang, Adela YiYu

doi:10.2140/gt.2025.29.4477

Download this article
	For screen
	For printing

Recent Issues

The Journal

About the journal

Ethics and policies

Peer-review process

Submission guidelines

Submission form

Editorial board

Subscriptions

ISSN (electronic): 1364-0380

ISSN (print): 1465-3060

Author index

To appear

Other MSP journals

Operations on spectral partition Lie algebras and TAQ cohomology

Adela YiYu Zhang

Geometry & Topology 29 (2025) 4477–4529

DOI: 10.2140/gt.2025.29.4477

Abstract

We determine all natural operations and their relations on the homotopy groups of spectral partition Lie algebras, which coincide with $𝔽_{p}$ -linear topological André–Quillen cohomology operations at any prime. We construct unary operations and a shifted restricted Lie algebra structure on the homotopy groups of spectral partition Lie algebras. Then we prove a composition law for the unary operations, as well as a compatibility condition between unary operations and the shifted Lie bracket with restriction up to a unit for the restriction. Comparing with Brantner and Mathew’s result on the ranks of the homotopy groups of free spectral partition Lie algebras, we deduce that these generate all natural operations, thereby also recovering unpublished results of Kriz and of Basterra and Mandell on $𝔽_{p}$ -linear TAQ cohomology operations. As a corollary, we determine the structure of natural operations on mod $p$ $𝕊$ -linear TAQ cohomology.

Keywords

spectral partition Lie algebra, power operations, Koszul duality, topological Andre–Quillen cohomology, Quillen homology

Mathematical Subject Classification

Primary: 18C15, 55P43, 55S10, 55S12, 55S99

References

Publication

Received: 18 May 2022

Revised: 10 February 2025

Accepted: 24 April 2025

Published: 31 December 2025

Proposed: Mark Behrens

Seconded: Stefan Schwede, Ulrike Tillmann

Authors

Adela YiYu Zhang
Copenhagen Centre for Geometry and Topology University of Copenhagen Copenhagen Denmark

Open Access made possible by participating institutions via Subscribe to Open.

Volume 29, issue 9 (2025)