Fixed Point Free Involutions on the 3-Sphere

In 1, P. A. Smith has proved that the set of fixed points of a periodic homeomorphism T of the 3-sphere S3 onto itself is an i-sphere imbedded in S3, i = -1, 0, 1, or 2, where the -1-sphere denotes the empty set. In trying to prove that an involution (map of period 2) on S3 is conjugate, in the group of all homeomorphisms of S3 onto itself, to an orthogonal involution, it is natural to divide the problem into four cases, as the dimension, n, of the fixed point sphere is -1, 0, 1, or 2, respectively. R. H. Bing, 2, has settled the case n=2 in the negative, unless one assumes that the 2-sphere of fixed points is tamely imbedded. Regarding the case n= 1, Montgomery and Samelson, 3, have shown that, if T is semi-linear, the circle of fixed points cannot be a knotted torus knot of type (p, 2), and moreover, that if the circle of fixed points is not knotted, then the involution, T, is conjugate to an orthogonal involution. This latter result suggests an approach for the case n = -1. An invariant set will mean a set mapped onto itself by the involution. In the absence of fixed points, the invariant circles become important. In fact, we have: