Symmetry groups of four-manifolds

Abstract

If a (possibly finite) compact Lie group acts effectively, locally linearly, and homologically trivially on a closed, simply-connected four-manifold with second Betti number at least three, then it must be isomorphic to a subgroup of S^1 x S^1, and the action must have nonempty fixed-point set. Our results strengthen and complement recent work by Edmonds, Hambleton and Lee, and Wilczynski, among others. Our tools include representation theory, finite group theory, and Borel equivariant cohomology.