K3曲面的微分同胚群与Nielsen实现
几何拓扑
2017-11-15 v3 代数拓扑
微分几何
摘要
Nielsen实现问题探讨的是从 Diff(M) 到 pi_0 Diff(M) 的群同态何时存在截面。对于M为闭曲面的情况，Kerckhoff证明了在任何有限子群上都存在截面，但Morita证明了如果亏格足够大，则在整个映射类群上不存在截面。我们证明了四维情形下此类首个不存在性定理：如果M是一个包含K3曲面作为连通和项的光滑闭定向四维流形，则在整个映射类群上不存在截面。这是通过证明位于 B(pi_0 Diff(M)) 的有理上同调中的某些阻碍类非零来实现的。我们通过证明这些类在拉回到K3曲面上的爱因斯坦度量模空间时非零，来检测这些类。
引用
@article{arxiv.0705.4545,
title = {The diffeomorphism group of a K3 surface and Nielsen realization},
author = {Jeffrey Giansiracusa},
journal= {arXiv preprint arXiv:0705.4545},
year = {2017}
}
备注
20 pages, published version. ERRATUM: This paper is withdrawn. As pointed out by Bena Tshishiku, Borel's theorem on the cohomology stable range for arithmetic groups is incorrectly quoted and applied in this paper; in the case relevant to K3 surfaces the range is unfortunately zero. Hence the proof of the part of Theorem 1.1 referring to K3 surfaces is fundamentally broken. See arXiv:1711.03139