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Characteristic classes of bundles of \(K3\) manifolds and the Nielsen realization problem. (English) Zbl 1442.19013

Summary: Let \(K\) be the \(K3\) manifold. In this note, we discuss two methods to prove that certain generalized Miller-Morita-Mumford classes for smooth bundles with fiber \(K\) are nonzero. As a consequence, we fill a gap in a paper of the first author J. Giansiracusa [J. Lond. Math. Soc., II. Ser. 79, No. 3, 701–718 (2009; Zbl 1171.57033)], and prove that the homomorphism \(\operatorname{Diff}(K) \to \pi_0 \operatorname{Diff} (K)\) does not split. One of the two methods of proof uses a result of J. Franke [Prog. Math. 258, 27–85 (2008; Zbl 1243.11066)] on the stable cohomology of arithmetic groups that strengthens work of A. Borel [Ann. Sci. Éc. Norm. Supér. (4) 7, 235–272 (1974; Zbl 0316.57026)], and may be of independent interest.

MSC:

19J35 Obstructions to group actions (\(K\)-theoretic aspects)
57R20 Characteristic classes and numbers in differential topology
11F75 Cohomology of arithmetic groups
14J28 \(K3\) surfaces and Enriques surfaces
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