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RCD*(K,N) spaces are semi-locally simply~connected

  • Jikang Wang ORCID logo EMAIL logo
Veröffentlicht/Copyright: 4. Oktober 2023

Abstract

It was shown in [A. Mondino and G. Wei, On the universal cover and the fundamental group of an RCD * ( K , N ) -space, J. reine angew. Math. 753 2019, 211–237] that any RCD * ( K , N ) space ( X , d , 𝔪 ) has a universal cover. We prove that for any point x X and R > 0 , there exists r < R such that any loop in B r ( x ) is contractible in B R ( x ) ; in particular, X is semi-locally simply connected and the universal cover of X is simply connected. This generalizes the earlier work in [J. Wang, Ricci limit spaces are semi-locally simply connected, preprint 2021] that any Ricci limit space is semi-locally simply connected.

Funding statement: The author is supported by the Fields Institute for Research in Mathematical Sciences.

Acknowledgements

The author would like to thank Xingyu Zhu for helpful discussions about RCD * spaces and Jiayin Pan, Jaime Santos-Rodríguez, Sergio Zamora-Barrera for some comments to simplify the proof. The author would also thank the referee for some helpful suggestions.

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Received: 2023-04-17
Revised: 2023-06-28
Published Online: 2023-10-04
Published in Print: 2024-01-01

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