On the dimensions of certain spaces of homeomorphisms

Abstract

INTRODUCTION. This paper studies the dimensions of the groups of all homeomorphisms of certain metric continua. It is divided into the following (overlapping) parts: ?2: Continua with nonzero dimensional groups of homeomorphisms, ?3: Applications of Theorem 2.1, ?4: 1-dimensional continua, ?5: Continua with finite, positive dimensional groups of homeomorphisms. In ?2, we establish sufficient conditions for the group of all homeomorphisms of a continuum to be nonzero dimensional, and thus obtain a large class of continua with this property-the (to be defined) locally setwise homogeneous continua. It is well known and easy to show (see Theorem 1.1) that the group of homeomorphisms of an arc is oo-dimensional. The component of the identity in this case is large. However it is not hard to show that the groups of all homeomorphisms of the universal plane curve and universal curve are totally disconnected. It therefore seems likely that they are zero-dimensional. In fact, such a result was announced (erroneously) by R. D. Anderson [1], who called it to my attention and suggested the general problem of this paper to me. His argument was for total disconnectivity instead of 0-dimensionality and is given in Theorem 1.2. However, corollaries of our main theorem of ?2 show that each of these groups is at least 1-dimensional. In ?3 the above corollaries are obtained, as well as an interesting corollary of the proof of Theorem 2.1. This corollary asserts that the group of those homeomorphisms of Sn, n > 1, which carry a fixed, countable, dense subset of St onto itself, is at least 1-dimensional. In ?4, we investigate the 1-dimensional continua. Our main result asserts that