Infinite sums of additive unstable Adams operations and cobordism

TL;DR

This paper explicitly describes the structure of additive unstable cobordism operations as infinite sums of Adams operations, establishing the Adams subring as the center of the operation ring, including p-local cases.

Contribution

It extends the explicit description of additive unstable operations from K-theory to complex cobordism and identifies the Adams subring as the center of the operation ring.

Findings

The Adams subring is the center of the ring of additive unstable cobordism operations.

Infinite sums of Adams operations can be made sense of in complex cobordism.

The result holds in both global and p-local split settings.

Abstract

The elements of the ring of bidegree (0,0) additive unstable operations in complex K-theory can be described explicitly as certain infinite sums of Adams operations. Here we show how to make sense of the same expressions for complex cobordism MU, thus identifying the "Adams subring" of the corresponding ring of cobordism operations. We prove that the Adams subring is the centre of the ring of bidegree (0,0) additive unstable cobordism operations. For an odd prime p, the analogous result in the p-local split setting is also proved.