Vol. 3, No. 1, 2021

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Characteristic classes of bundles of K3 manifolds and the Nielsen realization problem

Jeffrey Giansiracusa, Alexander Kupers and Bena Tshishiku

Vol. 3 (2021), No. 1, 75–92

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, and prove that the homomorphism Diff(K) π0 Diff(K) does not split. One of the two methods of proof uses a result of Franke on the stable cohomology of arithmetic groups that strengthens work of Borel, and may be of independent interest.

characteristic classes, K3 surfaces, arithmetic groups, cohomology
Mathematical Subject Classification 2010
Primary: 19J35, 57R20
Secondary: 11F75, 14J28
Received: 3 August 2019
Revised: 25 November 2019
Accepted: 10 December 2019
Published: 20 May 2020
Jeffrey Giansiracusa
Department of Mathematics
Swansea University
United Kingdom
Alexander Kupers
Department of Mathematics
Harvard University
Cambridge, MA
United States
Bena Tshishiku
Department of Mathematics
Harvard University
Cambridge, MA
United States
Department of Mathematics
Brown University
Providence, RI
United States