On Samelson products in p-localized unitary groups

摘要

The unitary group U(n) has elements ε_i ∈ π_(2i+1)(U(n)) (0 ≤ i ≤ n - 1) of its homotopy groups in the stable range. In this paper we show that certain multi Samelson products of type < ε_i, < ε_j, ε_k > > are non-trivial. This leads us to the result that the nilpotency class of the group of the self homotopy set [SU(n), SU(n)] is no less than 3, if 4 ≤ n. Also by the power of generalized Samelson products, we can see the further result that, for a prime p and an integer n = pk, nil[SU(n),SU(n)]_((p)) ≥ 3, if (1) p ≥ 7 or (2) p = 5 and n = 0 or 1 mod 4.