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 \(MU \rightarrow E\) 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 an 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.
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Acknowledgements
The authors would like to thank Greg Arone and Kathryn Lesh for discussions related to this material, to Andrew Baker, Eric Peterson, and Nathaniel Stapleton for their comments, and to Jeremy Hahn for locating an error in the previous version of this paper.
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Michael J. Hopkins partially supported by NSF Grant DMS-0906194.
Tyler Lawson partially supported by NSF Grant DMS-1206008.
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Hopkins, M.J., Lawson, T. Strictly commutative complex orientation theory. Math. Z. 290, 83–101 (2018). https://doi.org/10.1007/s00209-017-2009-6
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DOI: https://doi.org/10.1007/s00209-017-2009-6