The homotopy types of Sp(3)-gauge groups.

Let G k denote the gauge group of the principal S p ( 3 ) -bundle over S 4 with first symplectic Pontryagin class k ∈ H 4 S 4 = Z . We show that when localised at an odd prime there is a homotopy equivalence G k ≃ G l if and only if ( 21 , k ) = ( 21 , l ) . We also give bounds on the number of 2-local homotopy types amongst the G k . Our results follow from a thorough examination of the commutator map c : S p ( 1 ) ∧ S p ( 3 ) → S p ( 3 ) and we determine its order to be either 4 ⋅ 3 ⋅ 7 = 84 , 8 ⋅ 3 ⋅ 7 = 168 or 16 ⋅ 3 ⋅ 7 = 336 . In particular its order at odd primes is completely determined. [ABSTRACT FROM AUTHOR]