Abstract: We consider an even-dimensional manifold of positive curvature that admits an effective isometric torus action. Some topological properties of the manifold can be recovered depending on the dimension of the acting torus. First, I will present a survey of the classical results in this area. We will then discuss recent developments, focusing particularly on our new result concerning the Euler characteristic of 16-dimensional manifolds that admit a 3-dimensional torus action. This work is joint with Burkhard Wilking.
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