Branched coverings and equivariant smoothings of 4-manifolds (Q1747823)

Some new families of finite group actions on 4-manifolds are constructed, from the viewpoint of cyclic branched coverings. Examples: (1) The equivariant tangent bundle reduces to a \(G\)-vector bundle, but the manifold is not equivariantly smoothable, (2) There are nondiffeomorphic equivariant smoothings which correspond to the same \(G\)-vector structure on the equivariant tangent bundle. For a closed topological \(n\)-manifold \(M\), where \(n\neq 4\), by the Kirby-Siebenmann theory, suitably defined equivalence classes of smooth structures on \(M\) are classified by vector bundle structures on the topological tangent bundle, whose fibers are homeomorphic to \(\mathbb{R}^n\). A similar result in its \(G\)-equivariant version, for a locally linear finite group \(G\) action on manifolds, was proved by \textit{R. Lashof} and \textit{M. Rothenberg} [Proc. Symp. Pure Math., Vol. 32, Part 1, 211--266 (1978; Zbl 0407.57018)]. It is shown that 4-dimensional components of the fixed point set obstruct \(G\)-vector bundle structures. Examples (1) include relatively recent results, e.g. \textit{K. Kiyono}'s example: nonsmoothable cyclic group actions on connected sums of two or more copies of \(S^2\times S^2\), see [Algebr. Geom. Topol. 11, No. 3, 1345--1359 (2011; Zbl 1231.57023)], see also [\textit{X. Liu} and \textit{N. Nakamura}, Topology Appl. 155, No. 9, 946--964 (2008; Zbl 1143.57021)] for nonsmoothable finite group actions on 4-manifolds. Some related topics are considered: actions on exotic 4-spaces (\(\mathbb{R}^4\)), actions on higher dimensional manifolds whose fixed point set contains 4-dimensional components.