Abstract: There are many interactions between geometry and topology. In this talk I will discuss one such example, which concerns the notion of positive isotropic curvature introduced by Micallef and Moore in 1988. I’ll mainly focus on the topological implications of this curvature condition. First I’ll briefly survey some of the previous works by various authors on Riemannian manifolds with positive isotropic curvature, including those by Brendle, Chen-Tang-Zhu, Fraser, Hamilton, et al. Then I’ll introduce my recent work on the topological classification of compact manifolds of dimension n ≥ 12 with positive isotropic curvature. The main tool is Ricci flow with surgery, which was used by Perelman to attack the Poincare conjecture and Thurston’s geometrization conjecture. Techniques from topology are also extensively used.
