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k(n)-Torsion-Free H-Spaces and P(n)-Cohomology

  • J. Michael Boardman (a1) and W. Stephen Wilson (a1)
Abstract

The H-space that represents Brown–Peterson cohomology BP k (–) was split by the second author into indecomposable factors, which all have torsion-free homotopy and homology. Here, we do the same for the related spectrum P(n), by constructing idempotent operations in P(n)–cohomology P(n) k (–) in the style of Boardman–Johnson–Wilson; this relies heavily on the Ravenel–Wilson determination of the relevant Hopf ring. The resulting (i – 1)-connected H-spaces Yi have free connective Morava K-homology k(n)*(Yi ), and may be built from the spaces in the Ω-spectrum for k(n) using only vn -torsion invariants.

We also extend Quillen's theorem on complex cobordism to show that for any space X, the P(n)*-module P(n)*(X) is generated by elements of P(n) i (X) for i ≥ 0. This result is essential for the work of Ravenel–Wilson–Yagita, which in many cases allows one to compute BP–cohomology from Morava K-theory.

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References
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Canadian Journal of Mathematics
  • ISSN: 0008-414X
  • EISSN: 1496-4279
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