Symmetry groups of four-manifolds
Abstract
Description
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.
This is a substantially revised and shortened version. The argument has been streamlined by a more natural organization of the "minimal bad" cases and a much simplified treatment of rank 2 nonabelian groups (such as the eight-element dihedral group) using results from "Four-manifolds which admit Z_p x Z_p actions" (math.GT/0002187). The main result has also been sharpened slightly by appeal to the G-signature theorem, and now covers manifolds with the cohomology of CP^2 # CP^2. 15 pages, LaTeX
This is a substantially revised and shortened version. The argument has been streamlined by a more natural organization of the "minimal bad" cases and a much simplified treatment of rank 2 nonabelian groups (such as the eight-element dihedral group) using results from "Four-manifolds which admit Z_p x Z_p actions" (math.GT/0002187). The main result has also been sharpened slightly by appeal to the G-signature theorem, and now covers manifolds with the cohomology of CP^2 # CP^2. 15 pages, LaTeX