Operations and quantum doubles in complex oriented cohomology theory (Q1296339)

The authors present a survey of recent developments concerning the relation between two types of doubling. The first is the quantum double construction \({\mathcal D}(H)\) for a Hopf algebra \(H\) introduced by \textit{V. G. Drinfel'd} [Proc. Int. Congr. Math., Berkeley/Calif. 1986, Vol. 1, 789-820 (1987; Zbl 0667.16003)] and the second is the double complex cobordism theory of the second author's thesis. One formal connection between the two was given by the first author [Transl., Ser. 2, Am. Math. Soc. 170(27), 9-31 (1995; Zbl 0846.57023)]: if \(S^*\) is the Landweber-Novikov algebra then \({\mathcal D}(S^*)\) can be considered as a subalgebra of the algebra of operations in double complex cobordism theory. The structures present in \({\mathcal D}(S^*)\) are rather different from those traditionally considered by topologists and the authors investigate the implications of this, working in the context of Boardman's eightfold way. Along the way the authors give an exposition of the foundations of double complex cobordism theory which will make this paper a particularly valuable addition to the literature.