# Secondary power operations and the Brown-Peterson spectrum at the prime 2

## TL;DR

This paper constructs a secondary operation in mod-2 homology to demonstrate that the 2-primary Brown-Peterson spectrum cannot be given an E_n-algebra structure for any n ≥ 12, answering a longstanding question.

## Contribution

It introduces a new secondary operation in mod-2 homology and proves the non-existence of high-level E_n-structures on the Brown-Peterson spectrum at prime 2.

## Key findings

- The canonical subalgebra is not closed under the secondary operation.
- The Brown-Peterson spectrum does not admit an E_n-structure for n ≥ 12.
- Answers a question of May negatively.

## 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 greater than or equal to 12, answering a question of May in the negative.