Let G be a compact connected Lie group and p : E \to { \Sigma}^2V a principal G-bundle with a characteristic map \alpha : A={ \Sigma}V \to G. By combining cone decomposition arguments in Iwase-Mimura-Nishimoto [3,5] with computations of higher Hopf invariants introduced in Iwase [8], we generalize the result in Iwase-Mimura [12]: Let { F_{ i}|0 \leq i \leq m} be a cone-decomposition of G with a canonical structure map \sigma_{ i} of cat(F_{ i}) \leq i for i \leq m. We have cat(E) \leq \Max(m+n,m+2) for n \geq 1, if \alpha is compressible into F_{ n} \subseteq F_{ m} \simeq G and H^{ \sigma_n}_n(\alpha) = 0, under a suitable compatibility condition. On the other hand, calculations of Hamanaka-Kono [3] and Ishitoya-Kono-Toda [5] on spinor groups yields a lower estimate for the L-S category of spinor groups by means of a new computable invariant Mwgt(-;{ mathbb{ F}_2}) which is stronger than wgt(-;{ \mathbb{ F}_2}) introduced in Rudyak [16] and Strom [18]. As a result, we obtain cat(Spin(9)) = Mwgt(Spin(9);\mathbb{ F}_2) = 8 > 6 = wgt(Spin(9);\mathbb{ F}_2).
Categories: math.AT