On \(\mathbb{Z}_3\)-actions on spin 4-manifolds (Q1688141)

Let \(X\) be a closed, simply-connected, smooth, spin \(4\)-manifold whose intersection form is isomorphic to \(2k (-E_8) \oplus l H\), where \(H\) is the hyperbolic form. The main theorem of the article is that if a pseudofree topological \(\mathbb{Z}_3\)-action on \(X\) is locally linear, then \(\mathrm{Sign}(g, X) \equiv -k \pmod{3}\). Its proof is based on the properties of the Kirby-Siebenmann invariant and the Rochlin invariant. The authors also construct an example of nonsmoothable locally linear \(\mathbb{Z}_3\)-actions on certain elliptic surfaces \(E(n)\) with \(n \equiv 2 \pmod{6}\). To construct these locally linear \(\mathbb{Z}_3\)-actions, they use the realization theorem of \textit{A. L. Edmonds} and \textit{J. H. Ewing} [Am. J. Math. 114, No. 5, 1103--1126 (1992; Zbl 0766.57020)], and the mod \(p\) vanishing theorem of the Seiberg-Witten invariants [\textit{F. Fang}, Int. J. Math. 9, No. 8, 957--973 (1998; Zbl 0922.57013)] is used to give a constraint on smooth \(\mathbb{Z}_3\)-actions.