Robert Myers

Excellent 1-manifolds in compact 3-manifolds

Abstract Necessary and sufficient conditions are given for a compact, properly embedded 1-manifold J in a compact, connected 3-manifold M to be homotopic rel∂ J to a 1-manifold K which is excellent in the sense that its exterior is P 2 -irreducible and boundary irreducible and boundary irreducible and has the property that every properly embedded incompressible surface of zero Euler characteristic is boundary parallel. Moreover when this is the case there is a ribbon concordance from K to J and there are infinitely many such K with nonhomeomorphic exteriors. this theorem is then applied is then applied to extend results of the author on relative homology cobordisms and characterizations of certain 3-manifolds by their sets of knot groups to the nonorientable case; in the course of proving the latter result an extension to nonorientable 3-manifolds of Johannsons theorem on the determination of boundary irreducible, anannular Haken manifolds by their fundamental groups is developed. In addition the Hass-Thompson characterization of closed, orientable 3-manifolds of Heegaard genus at most one is extended to bounded and/or nonorientable 3-manifolds.

Contractible open 3-manifolds which are not covering spaces

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

CONTRACTIBLE OPEN 3-MANIFOLDS WHICH NON-TRIVIALLY COVER ONLY NON-COMPACT 3-MANIFOLDS

Suppose

End reductions, fundamental groups, and covering spaces of irreducible open 3-manifolds

M

Contractible open 3-manifolds with free covering translation groups☆

is a closed, connected, orientable, \irr\ \3m\ such that

ON COVERING TRANSLATIONS AND HOMEOTOPY GROUPS OF CONTRACTIBLE OPEN N-MANIFOLDS

G=\pi_1(M)

Compactifying sufficiently regular covering spaces of compact 3-manifolds

is infinite. One consequence of Thurstons geometrization conjecture is that the universal covering space

UNCOUNTABLY MANY ARCS IN S3 WHOSE COMPLEMENTS HAVE NON-ISOMORPHIC, INDECOMPOSABLE FUNDAMENTAL GROUPS

\widetilde{M}

Splitting homomorphisms and the Geometrization Conjecture

of

Free involutions on lens spaces

M