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Lusternik-Schnirelmann category of

Authors:	Norio Iwase Akira Kono

Institution:	Faculty of Mathematics, Kyushu University, Fukuoka 810-8560, Japan ; Department of Mathematics, Faculty of Science, Kyoto University, Kyoto 607-8502, Japan

Abstract:	First we give an upper bound of $\mathrm{cat}{(E)}$ , the L-S category of a principal -bundle for a connected compact group with a characteristic map $\alpha: {\Sigma}V \to G$ . Assume that there is a cone-decomposition $\{F_{i}\,\vert\,0 \leq i\leq m\}$ of in the sense of Ganea that is compatible with multiplication. Then we have $\mathrm{cat}{(E)} \leq\mathrm{Max}(m{+}n,m{+}2)$ for $n \geq 1$ , if $\alpha$ is compressible into $F_{n} \subseteq F_{m}\simeq G$ with trivial higher Hopf invariant $H_n(\alpha)$ . Second, we introduce a new computable lower bound, $\mathrm{Mwgt} {(X; {\mathbb{F}_2}})$ for $\mathrm{cat}({X})$ . The two new estimates imply $\mathrm{cat}({\mathbf{Spin}{(9))}}=\mathrm{Mwgt} ({\mathbf{Spin}{(9)};{\mathbb{F}_2}}) = 8 > 6 =\mathrm{wgt}({\mathbf{Spin}{(9)};{\mathbb{F}_2}})$ , where $(\mathrm{wgt}{-;R})$ is a category weight due to Rudyak and Strom.

Keywords:	Lusternik-Schnirelmann category spinor groups partial products

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