Dehn twists and the Nielsen realization problem for spin 4–manifolds

We prove that, for a closed oriented smooth spin 4-manifold X with non-zero signature, the Dehn twist about a (+2)-or (-2)-sphere in X is not homotopic to any finite order diffeomorphism.In particular, we negatively answer the Nielsen realization problem for each group generated by the mapping class of a Dehn twist.We also show that there is a discrepancy between the Nielsen realization problems in the topological category and smooth category for connected sums of copies of K3 and S 2 × S 2 .The main ingredients of the proofs are Y.Kato's 10/8-type inequality for involutions and a refinement of it.