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The homotopy types of U(n)-gauge groups over the two most fundamental 4-manifolds S and CP 2 are studied. We give homotopy decompositions of the U(n)-gauge groups over S in terms of certain SU(n)and PU(n)-gauge groups and use these decompositions to enumerate the homotopy types of the U(2)-, U(3)and U(5)-gauge groups. Over CP 2 we provide bounding results on the number of homotopy types of U(n)-gauge groups, provide p-local decompositions and give homotopy decompositions of certain U(n)-gauge groups in terms of certain SU(n)gauge groups. Applications are then given to count the number of homotopy types of U(2)-gauge groups over CP .

The homotopy types of $U(n)$-gauge groups over $S^4$ and $\mathbb{C}P^2$