映射类群对正数量曲率度量的作用
The action of the mapping class group on metrics of positive scalar curvature
摘要：
本文给出了高维流形$M$的正数量曲率度量空间$\mathcal{R}^+(M)$上的映射类群$π_0(\mathrm{Diff}(M))$作用的刚性定理.这个结果适用于大量的情况，例如单连通$6$ -流形和高维球面。我们的证明相当直接，使用了参数化莫尔斯理论、$2$ -指数定理和球体上某些度量的计算的结果。我们还给出了单连通$7$ -维$\mathrm{Spin}$ -流形的非平凡性准则和作用分类。
英文摘要：
We present a rigidity theorem for the action of the mapping class group $π_0(\mathrm{Diff}(M))$ on the space $\mathcal{R}^+(M)$ of metrics of positive scalar curvature for high dimensional manifolds $M$. This result is applicable to a great number of cases, for example to simply connected $6$-manifolds and high dimensional spheres. Our proof is fairly direct, using results from parametrised Morse theory, the $2$-index theorem and computations on certain metrics on the sphere. We also give a non-triviality criterion and a classification of the action for simply connected $7$-dimensional $\mathrm{Spin}$-manifolds.
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