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Obstruction theory and the strict associativity of Morava K-theories. (English) Zbl 0688.55008
Advances in homotopy theory, Proc. Conf. in Honour of I.M. James, Cortona/Italy 1988, Lond. Math. Soc. Lect. Note Ser. 139, 143-152 (1989).
[For the entire collection see Zbl 0682.00015.]
In this paper, the author studies the obstructions to refining a ring structure on a spectrum E to an \(A_{\infty}\) ring structure (i.e.: obstructions to finding a strictly associative ring spectrum which is homotopy equivalent to E). The main result is Theorem 1.11, where he shows that the obstructions lie in the Hochschild cohomology of the dual Steenrod algebra \(E_*E\), when the E-cohomology of each smash power \(E^ r\) is given by perfect K√ľnneth and duality theorems: \[ E^*(E^ r)\approx Hom_{E_*}((E_*E)^{\otimes r},E_*),\quad k\geq 1. \] This result is an analogue of Stasheff’s theorems about H- spaces.
As an application, the author considers the case in which E is Morava’s nth K-theory K(n) and he proves that K(n) has an \(A_{\infty}\) structure for every n, but it is not unique; there are uncountably many different structures in each case.
Reviewer: C.Costinescu

55N99 Homology and cohomology theories in algebraic topology