There is a product decomposition of a compact connected Lie group G at the prime p, called the mod p decomposition, when G has no p-torsion in homology. Then in studying the multiplicative structure of the p-localization of G, the Samelson products of the factor space inclusions of the mod p decomposition are fundamental. This paper determines (non-)triviality of these fundamental Samelson products in the p-localized exceptional Lie groups when the factor spaces are of rank ≤ 2, that is, G is quasi-p-regular.

Abstract. Let P n be the real n-dimensional projective space. We determine the group structure of the self-homotopy set of the double suspension of P n where n is 3, 4, 5 and 6 using the ideas and methods of the second author (The suspension order of the real even dimensional projective space, J. Math. Kyoto Univ. 43(4) (2003), 755-769).

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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

For certain pairs of Lie groups (G, H) and primes p, Harris showed a relation of the p-localized homotopy groups of G and H. This is reinterpreted as a p-local homotopy equivalence G ≃ (p)H × G/H, and so there is a projection G(p) → H(p). We show how much this projection preserves the higher homotopy associativity.