Nielsen realization problem

Abstract: The study of the geometry of the classical Teichmiiller spaces was begun in 1959 by Kravetz [9]. The starting point was the classical theorem of TeichmUller on extremal quasiconformal maps between compact Riemann surfaces. The TeichmUller theorem was used to argue that with respect to the Teichmiiller metric, TeichmUller space is straight and that it has negative curvature. In turn, negative curvature was used to show that every finite subgroup of the Teichmiiller modular group has a fixed point. This latter statement is known to be equivalent to the Nielsen Realization Problem which conjectures that every finite subgroup of the mapping class group of a surface can be realized by a finite subgroup of the group of homeomorphisms. Recently Linch [10] found a mistake in Kravetz's curvature arguments so that problem and consequently also the fixed point problem were reopened. The main result in this paper is that Teichmiiller space does not have negative curvature. This result was announced in [14]. TeichmUller's theorem exhibits a close relationship between extremal quasiconformal maps and quadratic differentials. This in turn leads to the characterization of a geodesic through a point in TeichmUller space as all extremal maps determined by a fixed quadratic differential on the underlying surface. An attempt here is made again to study the geometry of Teichmiiller space this time using the class of geodesic rays determined by quadratic differentials with closed horizontal trajectories. Strebel has studied these particular differentials extensively, and two of his results are crucial for this paper. The first describes how the critical trajectories of a differential with closed trajectories partition the Riemann surface into ringdomains each of which is swept out by freely homotopic closed trajectories, the trajectories in different ringdomains not being freely homotopic. The corresponding

Abstract: 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.

Abstract: The Nielsen realization problem asks when the group homomorphism Diff(M) �¨ �I0Diff(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�I0Diff(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.