Secondary power operations and the Brown–Peterson spectrum at the prime $2$

Abstract

The dual Steenrod algebra has a canonical subalgebra isomorphic to the homology of the Brown–Peterson spectrum. We will construct a secondary operation in mod-2 homology and show that this canonical subalgebra is not closed under it. This allows us to conclude that the 2-primary Brown–Peterson spectrum does not admit the structure of an $E_n$-algebra for any $n \geq 12$, answering a question of May in the negative.