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On Topological André-Quillen homology of Eilenberg-MacLane spectra

Cyril Barlasov

Abstract

Based on the work of Dundas, Lindenstrauss and Richter we compute the topological André-Quillen homology with reduced coefficients for Eilenberg-MacLane spectra such as $H\mathbb{Z}$ and $H\mathbb{Z}/p^n$. The case of $H\mathbb{F}_p$ was settled in an unpublished work of Basterra and Mandell, which was refined later by Brantner and Mathew in the context of spectral partition Lie algebras. Our approach is similar to Cartan's calculation of the Steenrod algebra, and eventually shorter. We also present some computations relative to the $H\mathbb{Z}$ base.

On Topological André-Quillen homology of Eilenberg-MacLane spectra

Abstract

Based on the work of Dundas, Lindenstrauss and Richter we compute the topological André-Quillen homology with reduced coefficients for Eilenberg-MacLane spectra such as and . The case of was settled in an unpublished work of Basterra and Mandell, which was refined later by Brantner and Mathew in the context of spectral partition Lie algebras. Our approach is similar to Cartan's calculation of the Steenrod algebra, and eventually shorter. We also present some computations relative to the base.
Paper Structure (6 sections, 24 theorems, 70 equations, 1 figure)

This paper contains 6 sections, 24 theorems, 70 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Topological Hochschild homology
  3. Topological André-Quillen homology
  4. Iterated bar construction
  5. Richter spectral sequence
  6. Spectral Lie algebras

Key Result

Proposition \ref{TAQ(Z/p^n,Z)}

There is an isomorphism of graded $\mathbb{F}_p$-modules where $\mathcal{A}_p$ is the mod $p$ Steenrod algebra. The quotient is taken wrt the two-sided ideal generated by the Bockstein operation $\beta$. There is a copy of $\mathbb{F}_p$ in degree $1$ and not $2$.

Figures (1)

  • Figure 1: The algebras $B^n$ for $1\leq n\leq 4$

Theorems & Definitions (43)

  • Proposition \ref{TAQ(Z/p^n,Z)}
  • Definition \ref{TAQ(Z/p^n,Z)}
  • Theorem \ref{TAQ(Z/p^n,Z)}: Veen
  • Definition \ref{TAQ(Z/p^n,Z)}
  • Definition \ref{TAQ(Z/p^n,Z)}: Suspension map $\sigma_*$
  • Proposition \ref{TAQ(Z/p^n,Z)}
  • proof
  • Definition \ref{TAQ(Z/p^n,Z)}
  • Definition \ref{TAQ(Z/p^n,Z)}
  • Theorem \ref{TAQ(Z/p^n,Z)}: see 3.5 in BM
  • ...and 33 more