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First we give an upper bound of cat (E), the L-S category of a principal G-bundle E for a connected compact group G with a characteristic map a: ΣV → G. Assume that there is a cone-decomposition {F i | 0 1, if a is compressible into F n C F m ≃ G with trivial higher Hopf invariant H n (α). Second, we introduce a new computable lower bound, Mwgt(X; F 2 ) for cat(X). The two new estimates imply cat(Spin(9)) = Mwgt(Spin(9);F 2 ) = 8 > 6 = wgt(Spin(9);F 2 ), where (wgt-; R) is a category weight due to Rudyak and Strom.

Lusternik-Schnirelmann category of 𝐒𝐩𝐢𝐧(9)