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This is a slightly altered version of the authors thesis from 2014. In thefirst main part we show that the quotient space of a compact, simply connectedand nonnegatively curved Riemannian 4-manifold by an effective, isometriccircle-action admits an approximation in Gromov-Hausdorff topology by smooth,positively curved Riemannian 3-manifolds. In the second main part we show thata compact, nonnegatively curved and fixed point homogeneous manifold isdiffeomorphic to the unit normal bundles of two smooth closed submanifoldsglued together along their boundaries. As a corollary we show that a compactand simply connected torus manifold admitting an invariant metric ofnonnegative curvature is rationally elliptic.

$\mathsf S^1$-actions on $4$-manifolds and fixed point homogeneous manifolds of nonnegative curvature