RCD空间中修正的第一贝蒂数的上界和环面稳定性结果
An upper bound on the revised first Betti number and a torus stability result for RCD spaces
摘要：
我们证明了一个上界的abelianised修订的基本群（称为“修订的第一贝蒂数“）的紧凑$RCD^{*}(K,N)$空间的秩，在相同的精神，著名的Gromov-Gallot上界的第一个贝蒂数的光滑紧凑黎曼流形与里奇曲率以下有界。当合成的Ricci下界足够接近（负）零，并且修正的第一贝蒂数的上述上界饱和（即等于$N$的整数部分，记为$\lfloor N \rfloor$），则我们建立了一个环面稳定性结果，说明该空间作为度量测度空间是$\lfloor N \rfloor$ -可求正的，证明了有限覆盖与$\lfloor N \rfloor$ -维平坦环面mGH-接近，并证明了当$N$是整数时，空间本身与平坦环面bi-Hölder同胚.这第二个结果扩展到类的非光滑$RCD^{*}(-δ, N)$空间的一个著名的环面稳定性定理的Colding（后来完善的Cheeger-Colding）。
英文摘要：
We prove an upper bound on the rank of the abelianised revised fundamental group (called "revised first Betti number") of a compact $RCD^{*}(K,N)$ space, in the same spirit of the celebrated Gromov-Gallot upper bound on the first Betti number for a smooth compact Riemannian manifold with Ricci curvature bounded below. When the synthetic lower Ricci bound is close enough to (negative) zero and the aforementioned upper bound on the revised first Betti number is saturated (i.e. equal to the integer part of $N$, denoted by $\lfloor N \rfloor$), then we establish a torus stability result stating that the space is $\lfloor N \rfloor$-rectifiable as a metric measure space, and a finite cover must be mGH-close to an $\lfloor N \rfloor$-dimensional flat torus; moreover, in case $N$ is an integer, we prove that the space itself is bi-Hölder homeomorphic to a flat torus. This second result extends to the class of non-smooth $RCD^{*}(-δ, N)$ spaces a celebrated torus stability theorem by Colding (later refined by Cheeger-Colding).
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
