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k(n)-Torsion-Free H-Spaces and P(n)-Cohomology
Published online by Cambridge University Press: 20 November 2018
Abstract
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The 
$H$-space that represents Brown-Peterson cohomology 
$\text{B}{{\text{P}}^{k}}\left( - \right)$ 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 
$P\left( n \right)$, by constructing idempotent operations in $P\left( n \right)$-cohomology 
$P{{(n)}^{k}}\left( - \right)$ in the style of Boardman-Johnson-Wilson; this relies heavily on the Ravenel-Wilson determination of the relevant Hopf ring. The resulting 
$\left( i\,-\,1 \right)$-connected $H$-spaces 
${{Y}_{i}}$ have free connective Morava 
$K$-homology 
$k{{(n)}_{*}}({{Y}_{i}})$, and may be built from the spaces in the 
$\Omega$-spectrum for 
$k\left( n \right)$ using only 
${{v}_{n}}$-torsion invariants.
We also extend Quillen's theorem on complex cobordism to show that for any space 
$X$, the 
$P{{\left( n \right)}_{*}}$-module 
$P{{(n)}^{*}}\,(X)$ is generated by elements of 
$P{{(n)}^{i}}(X)$ for 
$i\,\ge \,0$. 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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- Research Article
- Information
- Copyright
- Copyright © Canadian Mathematical Society 2007
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