Involutions on odd four-manifolds (Q1117510)

This paper shows that every closed, simply connected topological 4- manifold with odd intersection form of rank larger than one admits an involution. The involutions that are constructed are homologically trivial and have fixed point sets consisting of a 2-sphere and a collection of isolated points. These involutions are locally linear except perhaps at one of the isolated fixed points. It is known that if the Kirby-Siebenmann triangulation obstruction is nonzero, then any involution must have a fixed point where it is not locally linear [\textit{S. Kwasik} and \textit{P. Vogel}, Duke Math. J. 53, 759-764 (1986)]. In many cases it is shown that the constructed involutions are locally linear when the Kirby-Siebenmann obstruction vanishes. It remains an open question, reduced to a question about characteristic elements of quadratic forms, whether this is true in general. These results extend and depend upon earlier results of the author [Trans. Am. Math. Soc. 299, 155-170 (1987; Zbl 0623.57027)].