Recent and Forthcoming Papers

by J. Michael Boardman

Last revised 12 DEC 2008 by JMB

The ring spectrum P(n) for the prime 2

has now been published, as Homology, Homotopy Appl. 10 (2008), 101-135. It is available on-line, direct from the publisher, at http://intlpress.com/HHA/v10/n3/a6

Abstract This paper is a companion to the Boardman-Wilson paper [below] on the ring spectrum P(n). When the prime is 2, this spectrum is not commutative, which introduces several complications. Here, we supply the necessary details of the relevant Hopf algebroids and Hopf ring for this case.

k(n)-torsion-free H-spaces and P(n)-cohomology

(with W. Stephen Wilson) The electronic reprint of this paper on P(n)-cohomology is now available here. Note that the item numbering is vastly different from previous editions (not our choice). The Abstract is below.

Abstract: In his thesis, the second author split the $H$-spaces that represent Brown-Peterson cohomology $BP^k(-)$ for all $k$ 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 $i \ge 0$. 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.

An earlier version of the paper (47 pages) is available in your choice of

both set in 12-point, as well as in a compact 10-point hard copy edition (52 logical pages on 13 sheets) by mail. WARNING: The numbering of theorems and equations in the final published edition will be totally different from these editions, though the text will be much the same.

Unstable Splittings related to Brown-Peterson Cohomology

(with W. Stephen Wilson) This is in the collection Cohomological Methods in Homotopy Theory, the proceedings of the Barcelona Conference on Algebraic Topology in Bellaterra, Spain in 1998, printed as Progress in Mathematics 196, Birkhauser Verlag (Basel 2001), 35-45.
Abstract We give a new and relatively easy proof of the splitting theorem of the second author for the spaces in the omega spectrum for the Brown-Peterson spectrum BP. We then give the first published proofs of our similar theorems for the Johnson-Wilson spectrum P(n).

The first item above gives far more detailed information on the splitting of P(n).

Spheres and Hopf Rings

This short (3-page) note explains in detail why the sign in a standard Hopf ring formula had to be changed. It is available as Spheres and Hopf Rings, in DVI format, in a 12-point edition only. There are no plans for formal publication.

Conditionally Convergent Spectral Sequences

based on two old preprints, appeared in Contemp. Math. volume 239 (1999), 49-84. There are two versions here, both in DVI format:

The standard 12-point version, with 39 pages.
The compact 10-point version, which puts two logical pages on each physical letter-size page, in landscape mode, and is intended for two-sided printing. This has 42 logical pages, on 21 physical pages. It is also available in a dead tree edition by mail.

These and the published version all have slight differences in wording, just to make everything fit, but the theorem and equation numbers are identical in all three editions. Neither of these versions uses the lamsarrow fonts.

Abstract Convergence criteria for spectral sequences are developed that apply more widely than the traditional concepts. In the presence of additional conditions that depend on data internal to the spectral sequence, they lead to satisfactory convergence and comparison theorems. The techniques apply to whole-plane as well as half-plane spectral sequences.