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Victor Buchstaber, The Chern–Dold character in cobordisms. I,
Russian original: Mat. Sb. (N.S.), 1970 Volume 83(125), Number 4(12), Pages 575–595 (mathnet:3530)
English translation: Mathematics of the USSR-Sbornik, Volume 12, Number 4, AMS 1970 (doi:10.1070/SM1970v012n04ABEH000939)
Victor Buchstaber, A. P. Veselov, Chern-Dold character in complex cobordisms and abelian varieties (arXiv:2007.05782)
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As noted in the article on complex orientable cohomology theories, complex orientations do not correspond to maps of E-infinity rings, just homotopy-commutative rings. Hopkins and Lawson (https://arxiv.org/abs/1603.00047) have given conditions under which this map can be made E-infinity, but that’s a more complicated story.
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