Dehn twists and the Nielsen realization problem for spin 4-manifolds (Q6593012)

Recently, \textit{D. Baraglia} and \textit{H. Konno} [Proc. Am. Math. Soc. 151, No. 9, 4079--4087 (2023; Zbl 1520.57035)] and \textit{B. Farb} and \textit{E. Looijenga} [J. Differ. Geom. 127, No. 2, 505--549 (2024; Zbl 1545.57022)]) proved that the smooth Nielsen realization problem fails for the smooth 4-manifold underlying every \(K3\)-surface. In the present paper, ``we prove that for a closed oriented smooth spin 4-manifold \(X\) with nonzero signature, the Dehn twist about a (+2)- of \((-2)\)-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 problem in the topological and the smooth category for connected sums of copies of \(K3\) and \(S^2 \times S^2\). The main ingredient of the proofs are \textit{Y. Kato}'s 10/8-type inequality for involutions [Topology Appl. 307, Article ID 107868, 13 p. (2022; Zbl 1495.57021)] and a refinement of it.''\N\N(The Nielsen realization problem is interesting also for mapping class groups generated by Dehn twists about 2-spheres in connected sums of 3-manifolds, see a recent preprint by \textit{L. Chen} and \textit{B. Tshishiku} [``Nielsen Realization for sphere twists on 3-manifolds'', Preprint, \url{arXiv:2204.04820}] and a paper by the reviewer [Rend. Ist. Mat. Univ. Trieste 53, Paper No. 15, 7 p. (2021; Zbl 1498.57026)] and its Erratum [\url{arXiv:2105.04901}]).