Some topological results of Ricci limit spaces

Pan, Jiayin; Wang, Jikang

doi:10.1090/tran/8549

Some topological results of Ricci limit spaces
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by Jiayin Pan and Jikang Wang PDF

Trans. Amer. Math. Soc. 375 (2022), 8445-8464 Request permission

Abstract:

We study the topology of a Ricci limit space $(X,p)$, which is the Gromov-Hausdorff limit of a sequence of complete $n$-manifolds $(M_i, p_i)$ with $\mathrm {Ric}\ge -(n-1)$. Our first result shows that, if $M_i$ has Ricci bounded covering geometry, i.e. the local Riemannian universal cover is non-collapsed, then $X$ is semi-locally simply connected. In the process, we establish a slice theorem for isometric pseudo-group actions on a closed ball in the Ricci limit space. In the second result, we give a description of the universal cover of $X$ if $M_i$ has a uniform diameter bound; this improves a result by Ennis and Wei [Differential Geom. Appl. 24 (2006), pp. 554-562].

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Some topological results of Ricci limit spaces
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