The homotopy types of $$mathrm{SO}(4)$$-gauge groups

The homotopy types of gauge groups of principal SO(4)-bundles over S⁴ are classified p-locally for every prime p, and partial results are obtained integrally. The method generalizes to deal with any quotient of the form (S³)n/Z where Z is a subgroup generated by (−1, …, −1).