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Search: MSC category 55P45 ( $H$-spaces and duals )

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1. CJM 2009 (vol 62 pp. 284)

Grbić, Jelena; Theriault, Stephen
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)

Boardman, J. Michael; Wilson, W. Stephen
$k(n)$-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.

Categories:55N22, 55P45

3. CJM 2003 (vol 55 pp. 181)

Theriault, Stephen D.
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

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