The group of homeomorphisms of a solenoid

Let X be a topological space. An n-mean on X is a continuous function ,u: XX which is symmetric and idempotent. In the first part of this paper it is shown that if X is a compact connected abelian topological group, then X admits an n-mean if and only if H1(X, Z) is n-divisible where Hm(X, Z) is m-dimensional Cech cohomology with integers Z as coefficient group. This result is used to show that if Ia is a solenoid and Aut(la) is the group of topological group automorphisms of Ya, then Aut( (a) is algebraically Z2 x G where G is 1to, Zn, or D 1 Z. For a given Y., the structure of Aut(Ia) is determined by the n-means which a admits. Topologically, Aut(la) is a discrete space which has two points or is countably infinite. The main result of the paper gives the precise topological structure of the group of homeomorphisms G(a) of a solenoid Ya with the compact open topology. In the last section of the paper it is shown that G(a) is homeomorphic to Yax 12 x Aut(Xa) where 12 is separable infinite-dimensional Hilbert space. The proof of this result uses recent results in infinite-dimensional topology and some techniques using flows developed by the author in a previous paper. Introduction. Let X be a topological space. An n-mean for n > 2 on X is a continuous function 1i: Xn -X having the property that yu(x, * A., n) t(x T(l) *9 **, xr(n)) for any permutation 7n of II, * **, nI and p(x, x, * * *, x) = x for all x in X. We say simply that ti is symmetric and idempotent, respectively. Aumann showed that the circle T does not admit an n-mean for any n in [1]. In Eckmann [31 and Eckmann, Ganea, and Hilton [4] this result was extended to show that many other spaces do not support n-means. In particular, in [4] it is shown that if X is a compact connected polyhedron and X admits an n-mean for some n, then X is contractible. For the most part the above authors have devoted their efforts to widening the class of spaces known not to admit an n-mean. In the first section of this paper we show that a large class of compact connected abelian groups admit n-means for various n's. We give necessary and sufficient conditions that a compact connected abelian topological group H admit an n-mean. Among the equivalent conditions we show that H admits an n-mean if and only if Presented to the Society, January 19, 1972; received by the editors September 28, 1971. AMS 1970 subject classifications. Primary 57E05, 22B05.