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The odd primary $$H$$-structure of low rank Lie groups and its application to exponents. (English) Zbl 1117.55005
If $$A$$ is a space, a space $$B$$ is said universal for $$A$$ if $$B$$ is a homotopy commutative, homotopy associative $$H$$-space and there is a map $$A\to B$$ which is universal in a natural way. Now localize at an odd prime and take homology with mod-$$p$$ coefficients. If $$G$$ is a compact, connected, simple Lie group which is torsion free, the author proves the existence of a co-$$H$$ space $$A(G)$$ such that $$H_*(G)\cong S(\tilde{H}_*A(G))$$ with a decomposition $$A(G)\simeq \vee_{i=1}^{p-1} A_i(G)$$. One of the main results of the paper is the following one.
Theorem: Suppose $$G$$ is one of the following: $$SU(n)$$ if $$n\leq (p - 1)(p - 3)$$, $$Sp(n)$$ if $$2n \leq (p - 1)(p - 3)$$, $$Spin(n)$$ if $$2n + 1 \leq (p - 1)(p - 3)$$; $$G_2$$, $$F_4$$, or $$E_6$$ if $$p \geq 5$$; $$E_7$$ or $$E_8$$ if $$p \geq 7$$. Then there is a homotopy decomposition $$G \simeq \prod_{i=1}^{p-1} B_i(G)$$ where each $$B_i(G)$$ is homotopy associative, homotopy commutative, and universal for $$A_i(G)$$.
In some particular cases (for instance, $$SU(n)$$ with $$2n<p$$), the previous decomposition is an equivalence of $$H$$-spaces. Applications of these results to the torsion in the homotopy groups of $$G$$, including an upper bound on its exponent, are given.

##### MSC:
 55P45 $$H$$-spaces and duals 55Q52 Homotopy groups of special spaces 57T20 Homotopy groups of topological groups and homogeneous spaces
##### Keywords:
Lie group; exponent; $$H$$-space, Whitehead product
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