Abstract

The almost splitting map is a powerful tool in studying manifolds with Ricci curvature uniformly bounded from below and their Gromov-Hausdorff limits. In this talk, we will introduce a new transformation theorem for almost splitting maps, which drops the non-collapsed assumption in the one obtained by Cheeger, Jiang and Naber.  We will also introduce some applications of the new transformation theorem, one of which is a new smooth fibration theorem. More precisely, we introduce a notion, called the generalized Reifenberg condition, under which we prove a smooth fibration theorem for collapsed manifolds with Ricci curvature lower bound. This gives a unified proof of smooth fibration theorems in many previous works, including the classical ones by Fukaya and Yamaguchi respectively. The new transformation theorem is a key tool in its proof. Some other applications of the transformation theorem will be introduced in this talk. This talk is based on my joint work with Hongzhi Huang.