Title: k(n)-torsion-free H-spaces and P(n)-cohomology
Authors: J. Michael Boardman, W. Stephen Wilson
E-mail: boardman@math.jhu.edu, wsw@math.jhu.edu
Address: Dept. of Mathematics, Johns Hopkins University,
3400 N. Charles St., Baltimore MD 21218-2686
AMS Classifications: Primary 55N22, 55P45
Abstract: In his thesis, the second author split the H-space that
represents Brown-Peterson cohomology BP^k(-) into indecomposable factors,
which 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 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(n) using only v_n-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.