Operations and quantum doubles in complex oriented cohomology theory

Abstract

We survey recent developments introducing quantum algebraic methods into the study of cohomology operations in complex oriented cohomology theory. In particular, we discuss geometrical and homotopy theoretical aspects of the quantum double of the Landweber-Novikov algebra, as represented by a subalgebra of operations in double complex cobordism. We work in the context of Boardman's eightfold way, which offers an important framework for clarifying the relationship between quantum doubles and the standard machinery of Hopf algebroids of homology cooperations. These considerations give rise to novel structures in double cohomology theory, and we explore the twist operation and extensions of the quantum antipode by way of example.