On canonical constructions. III

In the second paper in this series [2; 3], it was demonstrated that under suitable topological conditions on a space S (which are in particular always satisfied for a manifold), the problem of reconstructing S from its group G of homeomorphism onto itself may be reduced to the problem of reconstructing the points of S from G. If S can be reconstructed then every automorphism of G is inner, and it was shown in this way that every automorphism of G is in fact inner when S is the unit interval; this is the theorem of Fine and Schweigert [I]. The present paper is devoted principally to proving the FineSchweigert theorem for the disc. Given a space S topologically equivalent to the interior of the unit circle in the plane, the point set of S will be canonically reconstructed from the group G of homeomorphisms of S onto itself. A principal tool is the classical theorem that if g is an orientation-preserving homeomorphism of finite order of the disc S onto itself, then S may be mapped homeomorphically on the interior of the unit circle in such a way that g becomes a rotation, [4]. Since a homeomorphism of odd order is necessarily orientation-preserving, it follows that an element of odd order (different from one) of the group G of homeomorphisms of S onto itself has precisely one fixed point, which will be called its center. If x is any point of S, let x' denote the set of all elements of G of finite odd order (different from one) having x for center. The collection S' of all sets x' is in canonical one-one correspondence with the point set of S. Our problem will be at an end if S' can be exhibited as a structure derived from the purely algebraic structure G. To this end it is sufficient to be able to decide within the structure of G when two elements of odd order (different from one) have distinct centers. This occupies the remainder of the paper. Henceforth, when an element of G is said to be of odd order, it will be understood that the order is different from one.