The construction of certain 0-dimensional transformation groups

Abstract

The purpose of this paper is to demonstrate the power of the techniques of construction introduced in [4] and [6] and hopefully, to encourage new research on the Hilbert-Smith Conjecture. The constructions make use of the functor described in [6] which is useful when a great deal of structure, or much computation is involved, e.g., infinite transformation groups or complicated local cohomology groups. We give an example in Part I of a 1-dimensional space X (1 for simplicity; similar constructions work for any n) and a free action of the p-adic group A, on X such that A1. dim X/A, = m =dim X+ 1: B. Hm (U) =Zp-, for any connected open subset U of X/Ap. Property B is hard to achieve; and this indicates the power of these techniques (e.g., see Lemma 1.3.2). To see why one is interested in such properties, recall the famous and as yet unsolved HILBERT-SMITH CONJECTURE. If a compact group G acts freely(l) on a manifold, then G is a Lie group. To prove this conjecture, it would suffice, [5], [7] or [1], to show that no p-adic group can act freely on a manifold. Thus in the past, researchers have looked for whatever surprising consequences they could find from the assumption that a p-adic group A, acts freely on an n-manifold X. In 1940, P. A. Smith [5] found, AO. dim X/A, $dim X. Later, C. T. Yang [7] found A2. dim XnI/A,=m=dim X+2 (or oo); and B. Hm(U) =Z,o, U any connected open subset of X/Ap. These two properties were also proved in [1] as easy (at least in the free case) consequences of the computation (*) Hq(BAp) = Z if q = O, =Zp if q=2, = 0 otherwise, where BAP is the classifying space of Ap. The proof is as follows: using (*), a wellknown spectral sequence has Zpas a corner term, and out pop A2 and B. Thus,