Let G be a compact connected Lie group and p : E → ΣA be a principal G-bundle with a characteristic map α :up to homotopy for any i. Our main result is as follows: we have cat(X) ≤ m+1, if firstly the characteristic map α is compressible into F 1 , secondly the Berstein-Hilton Hopf invariant H 1 (α) vanishes in [A, ΩF 1 * ΩF 1 ] and thirdly K m is a sphere. We apply this to the principal bundle SO(9) → SO(10) → S 9 to determine L-S category of SO(10).Let R be a commutative ring and X a connected space. The cup-length of X with coefficients in R is the least non-negative integer k (or ∞) such that