Curvature and symmetries of closed four-manifolds with a lower curvature bound
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This thesis details homeomorphism classi cation theorems for closed, simply connected, Riemannian 4-manifolds admitting an isometric circle action. We proceed by imposing progressively weaker curvature conditions, which results in a slight weakening of the classi cation. In particular, we consider positive, non-negative, and almost non-negative sectional curvature. For positive curvature, the homeomorphism classi cation is obtained by Hsiang and Kleiner in  and the equivariant di eomorphism classi cation by Grove and Searle in  and Grove and Wilking in . For non-negative sectional curvature, we look to the independent work of Kleiner in his thesis , and Searle and Yang in  for the homeomorphism classi cation, which was then improved to a di eomorphism classi cation by work of Galaz-Garc a , and Grove and Wilking in . For almost non-negative curvature, the recent work of Harvey and Searle in  gives the di eomorphism classi cation, which coincides with the classi cation for non-negatively curved manifolds. In fact, they show that almost non-negative curvature in this setting implies the existence of an S1-invariant metric of non-negative curvature.
Thesis (M.S.)-- Wichita State University, College of Liberal Arts and Sciences, Dept. of Mathematics, Statistics, and Physics