Involutions on S^4

Abstract

This paper studies locally linear involutions on S^4. Our main theorem shows that any such involution with a 1-dimensional fixed-point set is necessarily linear, provided the fixed-point set admits an equivariant tubular neighborhood. The proof combines modified surgery theory with an equivariant version of the Schoenflies theorem, which we establish here. We also show that equivariant tubular neighborhoods of 1-dimensional fixed-point sets, when they exist, are not unique, in contrast to the nonequivariant case. Our results combine with earlier work to provide a classification of all locally linear involutions on S^4. As a further application, we obtain that strongly negative amphichiral knots with trivial Alexander polynomial are equivariantly topologically slice with respect to the linear action, strengthening a previous result of the first two authors. Finally, we also prove that when the fixed-point set is 2-dimensional, the involution is linear if and only if the fixed-point set is an unknotted 2-knot.