Circle actions on simply connected $4$-manifolds

Locally smooth ${S^1}$-actions on simply connected 4-manifolds are studied in terms of their weighted orbit spaces. An equivariant classification theorem is proved, and the weighted orbit space is used to compute the quadratic form of a given simply connected 4-manifold with ${S^1}$-action. This is used to show that a simply connected 4-manifold which admits a locally smooth ${S^1}$-action must be homotopy equivalent to a connected sum of copies of ${S^4},C{P^2}, - C{P^2}$, and ${S^2} \times {S^2}$.