Calculating obstruction groups for E-infinity ring spectra

TL;DR

This paper applies a specialized obstruction theory to compute the moduli of $E_$ ring spectra, focusing on the 2-primary Brown-Peterson spectrum, and interprets obstructions via secondary operations.

Contribution

It adapts Senger's version of the Goerss-Hopkins obstruction theory to explicitly calculate obstruction groups for specific $E_$ ring spectra.

Findings

Chain complex for first obstruction groups of the 2-primary Brown-Peterson spectrum

Identification of potential genuine obstructions

Interpretation of obstruction classes through secondary operations

Abstract

We describe a special instance of the Goerss-Hopkins obstruction theory, due to Senger, for calculating the moduli of $E_\infty$ ring spectra with given mod-$p$ homology. In particular, for the $2$-primary Brown-Peterson spectrum we give a chain complex that calculates the first obstruction groups, locate the first potential genuine obstructions, and discuss how some of the obstruction classes can be interpreted in terms of secondary operations.