- 6635 South Hall
About 25 years ago Hsiang and Kleiner showed that a simply connected positively curved 4-manifold with circle symmetry has euler characteristic at most 3 (at most 4 in non-negative curvature [Kleiner and Searle-Yang]). Utilizing Freedman's topological classification of simply connected 4-manifolds, one concludes that such a manifold is homeomorphic to S^4 or CP^2 (adding S^2xS^2 and CP^2 + -CP^2 when the euler characteristic is 4). Combining work of Fintushel (going back about 35 years) with Perelman's resolution of the Poincare conjecture actually yields the classification up to diffeomorphism. Further geometry including a knot characterization provides a more direct classification up to equivariant diffeomorphism (joint work with Wilking).