Abstract
The purpose of this paper is to give new stable splittings of the quotients spaces BO(2n)/BO(2n-2) and BU(np)/BU(np-p). That is, we will prove the stable splittings, localized at the prime 2, BO(2n)/BO(2n-2) ≃ F 2n ∨ M 2n , n ≥ 1, and for each prime p and localized at the prime p, BU(np)/BU(np-p) ≃ F̄ np ∨ M̄ np , n ≥ 1, where F 2n and M 2n are certain 2-local spectra and F̄ np and M̄ np are certain p-local spectra. Then as corollaries, we will stably split BSO(2n+1) and BSU(np+1), that is, localized at the prime 2, BSO(2n+1) ≅ M 2 ∨ M 4 ∨ M 6 ∨ ⋯ ∨ M 2n , and localized at the prime p (for each prime p), BSU(np+1) ≅ M̄ p ∨ M̄ 2p ∨ M̄ 3p ∨ ⋯ ∨ M̄ np .