Fixing ϵ > 0, a complete Riemannian n-manifold M is called ϵ-collapsed, if every unit ball in M has a volume < ϵ. In Riemannian geometry, interplays between a collapsing geometry and topology has been an important component, and complexities in topology of a collapsed M is linked to a bound on curva ture. Around 1980-2000, collapsed manifolds of bounded sectional curvature was intensively studied by Cheeger-Fukaya-Gromov and many others, which has found several applications.

In this talk, we will survey a development in the last decade in investigating collapsed manifolds of Ricci curvature bounded below and universal covering of every unit ball in M is not collapsed.

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