1. CJM 2009 (vol 62 pp. 284)
||Self-Maps of Low Rank Lie Groups at Odd Primes|
Let G be a simple, compact, simply-connected Lie group localized at an odd prime~p. We study the group of homotopy classes of self-maps $[G,G]$ when the rank of G is low and in certain cases describe the set of homotopy classes of multiplicative self-maps $H[G,G]$. The low rank condition gives G certain structural properties which make calculations accessible. Several examples and applications are given.
Keywords:Lie group, self-map, H-map
Categories:55P45, 55Q05, 57T20
2. CJM 2007 (vol 59 pp. 1154)
||$k(n)$-Torsion-Free $H$-Spaces and $P(n)$-Cohomology |
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 $Y_i$ 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 \linebeak$P(n)_*$-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.
3. CJM 2003 (vol 55 pp. 181)
||Homotopy Decompositions Involving the Loops of Coassociative Co-$H$ Spaces |
James gave an integral homotopy decomposition of $\Sigma\Omega\Sigma X$,
Hilton-Milnor one for $\Omega (\Sigma X\vee\Sigma Y)$, and Cohen-Wu gave
$p$-local decompositions of $\Omega\Sigma X$ if $X$ is a suspension. All
are natural. Using idempotents and telescopes we show that the James and
Hilton-Milnor decompositions have analogues when the suspensions are
replaced by coassociative co-$H$ spaces, and the Cohen-Wu decomposition
has an analogue when the (double) suspension is replaced by a coassociative,
cocommutative co-$H$ space.