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arXiv · 2207.10029

Almost splitting maps, transformation theorems and smooth fibration theorems

Hongzhi Huang·Xian-Tao Huang
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Abstract

In this paper, we introduce a notion, called generalized Reifenberg condition, under which we prove a smooth fibration theorem for collapsed manifolds with Ricci curvature bounded below, which gives a unified proof of smooth fibration theorems in many previous works (including the ones proved by Fukaya and Yamaguchi respectively). A key tool in the proof of this fibration theorem is the transformation technique for almost splitting maps, which originates from Cheeger-Naber (\cite{CN}) and Cheeger-Jiang-Naber (\cite{CJN21}). More precisely, we show that a transformation theorem of Cheeger-Jiang-Naber (see Proposition 7.7 in \cite{CJN21}) holds for possibly collapsed manifolds. Some other applications of the transformation theorems are given in this paper.

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Hongzhi Huang, Xian-Tao Huang. 2022-07-20. Almost splitting maps, transformation theorems and smooth fibration theorems. https://arxiv.org/abs/2207.10029

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