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Brown-Peterson cohomology from Morava \(K\)-theory. (English) Zbl 0912.55002

This paper gives the structure of the Brown-Peterson cohomology of a wide class of spaces – those with Morava \(K\)-theory concentrated in even dimensions. These spaces include: Eilenberg-Maclane spaces, loop spaces of finite Postnikov systems, and classifying spaces of most finite groups whose Morava \(K\)-theory is known. The main result is that for such spaces the Brown-Peterson cohomology is also concentrated in even dimensions and is flat as a \(BP^{*}\)-module for the category of finitely presented \(BP^{*}(BP)\)-modules. The proofs rely heavily on computations with the Atiyah-Hirzebruch spectral sequence.

MSC:

55N15 Topological \(K\)-theory
19D10 Algebraic \(K\)-theory of spaces
55N22 Bordism and cobordism theories and formal group laws in algebraic topology
55P20 Eilenberg-Mac Lane spaces
19L20 \(J\)-homomorphism, Adams operations
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