Strictly commutative complex orientation theory

TL;DR

This paper develops an iterative process to lift complex orientations to strictly commutative orientations in structured ring spectra, revealing a tower of obstructions related to formal group laws and p-localization.

Contribution

It introduces a new tower-based method for lifting complex orientations to strictly commutative ones in highly structured spectra, extending previous work by Arone-Lesh.

Findings

The tower only changes at stages that are powers of p.

For E(n)-local spectra, the tower stabilizes after stage p^n.

Lifting from stage 1 to p relates to formal group law conditions.

Abstract

For a multiplicative cohomology theory E, complex orientations are in bijective correspondence with multiplicative natural transformations to E from complex bordism cohomology MU. If E is represented by a spectrum with a highly structured multiplication, we give an iterative process for lifting an orientation MU -> E to a map respecting this extra structure, based on work of Arone-Lesh. The space of strictly commutative orientations is the limit of an inverse tower of spaces parametrizing partial lifts; stage 1 corresponds to ordinary complex orientations, and lifting from stage (m-1) to stage m is governed by the existence of a orientation for a family of E-modules over a fixed base space F_m. When E is p-local, we can say more. We find that this tower only changes when m is a power of p, and if E is E(n)-local the tower is constant after stage p^n. Moreover, if the coefficient ring E^* is p-torsion free, the ability to lift from stage 1 to stage p is equivalent to a condition on the associated formal group law that was shown necessary by Ando.