| We consider orientation-preserving, locally linear, involutions on oriented, closed, simply connected, topological 4-manifolds. We look at three different types of involutions: (1) fixed-point free involutions, (2) involutions with isolated fixed points, and (3) involutions whose fixed-point sets consist of a single, simply embedded 2-sphere.;We determine all the possible homeomorphism types of 4-manifolds M which support involutions of type (1), (2), and (3). We describe the fixed-point set of a locally linear involution in terms of a congruence condition and bounds of a function of the mod 2 Euler characteristic and the Euler number of the fixed-point set. In the special case of involutions of type (2), we determine the possible number of fixed points. We also demonstrate the existence of an infinite family of oriented, closed, simply connected, topological 4-manifolds with vanishing Kirby-Siebenmann obstructions which admit no locally linear involutions. Finally, we give the necessary and sufficient conditions for a free involution to be smoothable. |
