Symmetry groups of four-manifolds

If a (possibly finite) compact Lie group acts effectively, locally linearly, and homologically trivially on a closed, simply connected four-manifold M with b2(M)⩾3, then it must be isomorphic to a subgroup of S1×S1, and the action must have nonempty fixed-point set. sults strengthen and complement recent work by Edmonds, Hambleton and Lee, and Wilczyński, among others. Our tools include representation theory, finite group theory, and Borel equivariant cohomology.