We denote the group of homotopy set [X, U(n)] by the unstable K1-group of X. In this paper, using the unstable K1-group of the multi-suspended CP2, we give a necessary condition for two principal SU(n)-bundles over §4 to have the associated gauge group of the same homotopy type, which is an improvement of the result of Sutherland and, particularly, show the complete classification of homotopy types of SU(3)-gauge groups over S4.

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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.