Contractible open 3-manifolds which are not covering spaces

Abstract

SUPPOSE M is a closed, aspherical 3-manifold. Then the universal covering space @ of M is a contractible open 3-manifold. For all “known” such M, i.e. M a Haken manifold [ 151 or a manifold with a geometric structure in the sense ofThurston [14], A is homeomorphic to R3. One suspects that this is always the case. This contrasts with the situation in dimension n > 3, in which Davis [2] has shown that there are closed, aspherical n-manifolds whose universal covering spaces are not homeomorphic to R”. This paper addresses the simpler problem of finding examples of contractible open 3manifolds which do not cover closed, aspherical 3-manifolds. As pointed out by McMillan and Thickstun [l l] such examples must exist, since by an earlier result of McMillan [lo] there are uncountably many contractible open 3-manifolds but there are only countably many closed 3-manifolds and therefore only countably many contractible open 3-manifolds which cover closed 3-manifolds. Unfortunately this argument does not provide any specific such examples. The first example of a contractible open 3-manifold not homeomorphic to R3 was given by Whitehead in 1935 [16]. It is a certain monotone union of solid tori, as are the later examples of McMillan [lo] mentioned above. These examples are part of a general class of contractible open 3-manifolds called genus one Whitehead manifolds. In this paper it is proven that none of these manifolds can cover a closed 3-manifold. In fact a stronger result is obtained: genus one Whitehead manifolds admit no non-trivial, fixed point free, properly discontinuous group actions. Thus they cannot non-trivially cover even another noncompact 3-manifold. There is some disagreement as to the proper definition of proper discontinuity. If X is a manifold, G is a group of homeomorphisms of X, and XEX, let G, be the isotropy subgroup of G at x, i.e. G, = (g E Gig(x) = x). G acts properly discontinuously on X if(i) for each point x E X there is an open neighborhood U of x such that U rig(U)) = 0 for every g E G\G, and (ii) a condition on G, which is in dispute. Some authors require that each G, be trivial (see [9], [13]). Under this definition the phrase “fixed point free” is redundant and G acts properly discontinuously if and only if the projection X-+X/G is a regular covering map. Other authors may merely require that each G, be finite (see [3]). This allows G to have elements of finite order with fixed points, such as those occurring in Kleinian groups. The second definition is of course implied by the first; it in turn implies that for every compact subset C of X the set (g E GIG n g(C) # a} is finite. This last condition is the working