Central cohomology operations and K-theory

TL;DR

This paper proves that for all chromatic heights, the only central stable and additive unstable cohomology operations are those originating from K-theory, extending known results to BP<n> for any n.

Contribution

It generalizes the characterization of central cohomology operations from BP to BP<n> for all n in the unstable additive context.

Findings

Central operations for BP<n> are the same as those from K-theory.

The result extends known isomorphisms to all chromatic heights.

Centrality is preserved across different heights in the additive unstable setting.

Abstract

For stable degree zero operations, and also for additive unstable operations of bidegree (0,0), it is known that the centre of the ring of operations for complex cobordism is isomorphic to the corresponding ring of connective complex K-theory operations. Similarly, the centre of the ring of BP operations is the corresponding ring for the Adams summand of p-local connective complex K-theory. Here we show that, in the additive unstable context, this result holds with BP replaced by BP<n> for any n. Thus, for all chromatic heights, the only central operations are those coming from K-theory.