Kleiner showed in his thesis that a simply-connected smooth 4-manifold of positive sectional curvature with a non-zero Killing field (or equivalently, an isometric effective S^1 action) is homeomorphic to S^4 or CP^2. I'll present a proof in Lee Kennard's lecture notes on group actions. By applying Donaldson-Freedman theory, it suffices to show that the Euler characteristic of M is less than 4. By a classical result of Conner and Kobayashi, the Euler characteristic of M is the same as that of the fixed point set under the S^1 action. We then investigate the topology of the S^1 fixed point set, concluding the result using some facts about group actions, Wilking's connectedness lemma and a version of Toponogov's comparison theorem.

### Graduate Student Geometry-Topology Seminar

Monday, January 30, 2017 - 3:15pm

#### Yuhang Liu

University of Pennsylvania