Calculating obstruction groups for 𝐸_{∞} ring spectra

Abstract

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

E n -spectra and Dyer-Lashof operations

“…In unpublished work [88], Senger has given a development of this theory for E ∞ -algebras where the obstructions occur in nonabelian Ext-groups calculated in the category of graded-commutative rings with Dyer-Lashof operations and Steenrod operations satisfying the Nishida relations, and provided tools for calculating with them. This played a critical role in [48,47].…”

“…If the generators of π * R have nontrivial image in H * R, then the spectral sequence Finally, we must determine a candidate secondary operation in H * H to which we can apply this procedure-there are many candidate operations and many dead ends. The secondary operation is rather large: it was found using a calculation in Goerss-Hopkins obstruction theory that is detailed at length in [47].…”