Strictly commutative complex orientation theory

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 Fm. 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 pn. Moreover, if the coefficient ring E∗ is ptorsion 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. Characteristic classes play a fundamental role in algebraic topology, with the primary example being the family of Chern classes ci(ξ) ∈ H(X) associated to a complex vector bundle ξ → X. Not all generalized cohomology ∗Partially supported by NSF grant DMS–0906194. †Partially supported by NSF grant DMS–1206008.