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Journal of the London Mathematical Society Advance Access published online on April 15, 2009
Journal of the London Mathematical Society, doi:10.1112/jlms/jdp002
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© 2009 London Mathematical Society

The diffeomorphism group of a K3 surface and Nielsen realization

Jeffrey Giansiracusa

Mathematical Institute, Oxford University
24–29 Saint Giles’
Oxford
OX1 3LB
United Kingdom

The Nielsen realization problem asks when the group homomorphism Diff (M) -> {pi}0 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 B{pi}0 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.

2000 Mathematics Subject Classification 57R70 (14J28; 58D27; 57S05).

Received April 28, 2008; revised December 2, 2008;
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