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The image of the $BP$ Thom map for Eilenberg-MacLane spaces


Author: Hirotaka Tamanoi
Journal: Trans. Amer. Math. Soc. 349 (1997), 1209-1237
MSC (1991): Primary 55N22, 55P20, 55S25
DOI: https://doi.org/10.1090/S0002-9947-97-01826-6
MathSciNet review: 1401530
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Abstract: Fundamental classes in $BP$ cohomology of Eilenberg-MacLane spaces are defined. The image of the Thom map from $BP$ cohomology to mod-$p$ cohomology is determined for arbitrary Eilenberg-MacLane spaces. This image is a polynomial subalgebra generated by infinitely many elements obtained by applying a maximum number of Milnor primitives to the fundamental class in mod-$p$ cohomology. This subalgebra in mod $p$ cohomology is invariant under the action of the Steenrod algebra, and it is annihilated by all Milnor primitives. We also show that $BP$ cohomology determines Morava $K$ cohomology for Eilenberg-MacLane spaces.


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Additional Information

Hirotaka Tamanoi
Affiliation: Institut des Hautes Études Scientifiques, 35 Route de Chartres, 91440 Bures-sur-Yvette, France
Address at time of publication: Department of Mathematics, University of California at Santa Cruz, Santa Cruz, California 95064
Email: tamanoi@cats.ucsc.edu

DOI: https://doi.org/10.1090/S0002-9947-97-01826-6
Keywords: $BP$ cohomology theory, $BP$ fundamental class, Eilenberg--Mac Lane spaces, Milnor primitives, Morava $K$ theory, Steenrod algebra, Thom map
Received by editor(s): October 5, 1995
Article copyright: © Copyright 1997 American Mathematical Society

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