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Products on $MU$-modules


Author: N. P. Strickland
Journal: Trans. Amer. Math. Soc. 351 (1999), 2569-2606
MSC (1991): Primary 55T25
DOI: https://doi.org/10.1090/S0002-9947-99-02436-8
Published electronically: March 1, 1999
MathSciNet review: 1641115
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Abstract: Elmendorf, Kriz, Mandell and May have used their technology of modules over highly structured ring spectra to give new constructions of $MU$-modules such as $BP$, $K(n)$ and so on, which makes it much easier to analyse product structures on these spectra. Unfortunately, their construction only works in its simplest form for modules over $MU[{\textstyle\frac{1}{2}}]_*$ that are concentrated in degrees divisible by $4$; this guarantees that various obstruction groups are trivial. We extend these results to the cases where $2=0$ or the homotopy groups are allowed to be nonzero in all even degrees; in this context the obstruction groups are nontrivial. We shall show that there are never any obstructions to associativity, and that the obstructions to commutativity are given by a certain power operation; this was inspired by parallel results of Mironov in Baas-Sullivan theory. We use formal group theory to derive various formulae for this power operation, and deduce a number of results about realising $2$-local $MU_*$-modules as $MU$-modules.


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Additional Information

N. P. Strickland
Affiliation: Trinity College, Cambridge CB2 1TQ, England
Address at time of publication: Department of Pure Mathematics, University of Sheffield, Sheffield S3 7RH, United Kingdom
Email: n.p.strickland@sheffield.ac.uk

DOI: https://doi.org/10.1090/S0002-9947-99-02436-8
Received by editor(s): January 9, 1997
Published electronically: March 1, 1999
Article copyright: © Copyright 1999 American Mathematical Society

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