The diffeomorphism group of a K3 surface and Nielsen realization
The diffeomorphism group of a K3 surface and Nielsen realization
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K3 面的微分同胚群和 Nielsen 实现
DOI：
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发表时间：
2007
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通讯作者：
Jeffrey Giansiracusa
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作者：
Jeffrey Giansiracusa
Nielsen实现问题是指群同态Diff (M)→π0 Diff (M)何时允许一个截面。对于闭曲面M， Kerckhoff证明了在任意有限子群上都存在一个截面，而Morita证明了如果属足够大，则在整个映射类群上不存在截面。我们在4维证明了该类型的第一个不存在性定理：如果M是一个光滑的、闭向的4流形，它包含一个K3曲面作为连通的和，那么在整个映射类群上不存在截面。这是通过证明在Bπ0 Diff (M)的有理上同调中的某些障碍是非零来实现的。我们通过证明当拉回到K3曲面上的爱因斯坦度量的模空间时，它们是非零的来检测这些类。
The Nielsen realization problem asks when the group homomorphism Diff (M) → π0 Diff (M) admits a section. For M, a closed surface, Kerckhoff proved that a section exists over any finite subgroup, but Morita proved that, if the genus is large enough, then no section exists over the entire mapping class group. We prove the first nonexistence theorem of this type in dimension 4: if M is a smooth, closed‐oriented 4‐manifold that contains a K3 surface as a connected summand, then no section exists over the whole of the mapping class group. This is done by showing that certain obstructions lying in the rational cohomology of Bπ0 Diff (M) are nonzero. We detect these classes by showing that they are nonzero when pulled back to the moduli space of Einstein metrics on a K3 surface.