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1. CJM 2012 (vol 65 pp. 82)
The Ranks of the Homotopy Groups of a Finite Dimensional Complex Let $X$ be an
$n$-dimensional, finite, simply connected CW complex and set
$\alpha_X =\limsup_i \frac{\log\mbox{ rank}\, \pi_i(X)}{i}$. When
$0\lt \alpha_X\lt \infty$, we give upper and lower bound for $
\sum_{i=k+2}^{k+n} \textrm{rank}\, \pi_i(X) $ for $k$ sufficiently
large. We show also for any $r$ that $\alpha_X$ can be estimated
from the integers rk$\,\pi_i(X)$, $i\leq nr$ with an error bound
depending explicitly on $r$.
Keywords:homotopy groups, graded Lie algebra, exponential growth, LS category Categories:55P35, 55P62, , , , 17B70 |
2. 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.
Categories:55P35, 55P45 |