Involutions on S^4

TL;DR

This paper classifies locally linear involutions on S^4, showing they are linear under certain conditions, and explores the uniqueness of equivariant tubular neighborhoods, with applications to knot theory.

Contribution

It proves that involutions with 1-dimensional fixed points are linear if they admit an equivariant tubular neighborhood, and establishes an equivariant Schoenflies theorem.

Findings

Involutions with 1D fixed points are linear if they have an equivariant tubular neighborhood.

Equivariant tubular neighborhoods of 1D fixed points are not unique.

Strongly negative amphichiral knots with trivial Alexander polynomial are equivariantly topologically slice.

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.