R^{2}-IRREDUCIBLE UNIVERSAL COVERING SPACES OF P^{2}-IRREDUCIBLE OPEN 3-MANIFOLDS

ROBERT MYERS

Abstract. An irreducible open 3-manifold W is R^{2}-irreducible if it contains no non-trivial planes, i.e. given any proper embedded plane Π in W some component of W − Π must have closure an embedded halfspace R^{2} × [0, ∞). In this paper it is shown that if M is a connected, P^{2}-irreducible, open 3-manifold such that π_{1}(M) is finitely generated and the universal covering space M̃ of M is R^{2}-irreducible, then either M̃ is homeomorphic to R^{3} or π_{1}(M) is a free product of infinite cyclic groups and fundamental groups of closed, connected surfaces other than S^{2} or P^{2}. Given any finitely generated group G of this form, uncountably many P^{2}-irreducible, open 3-manifolds M are constructed with π_{1}(M) ≅ G such that the universal covering space M̃ is R^{2}-irreducible and not homeomorphic to R^{3}; the M̃ are pairwise nonhomeomorphic. Relations are established between these results and the conjecture that the universal covering space of any irreducible, orientable, closed 3-manifold with infinite fundamental group must be homeomorphic to R^{3}.

1. Introduction

Suppose M is a connected, P^{2}-irreducible, open 3-manifold with π_{1}(M) finitely generated and non-trivial. It is easy to construct examples of such M for which the universal covering space M̃ is not homeomorphic to R^{3}. Start with any 3-manifold N satisfying the given conditions. Let U be a Whitehead manifold, i.e. an irreducible, contractible, open 3-manifold which is not homeomorphic to R^{3}(see e.g. [16], [4]). Choose end-proper embeddings of [0, ∞) in each of N and U. (A map between manifolds is end-proper if pre-images of compact sets are compact; it is ∂-proper if the pre-image of the boundary is the boundary; it is proper if it has both these properties. These terms are applied to a submanifold if its inclusion map has the corresponding property.) Let X and Y be the exteriors of these rays. (The exterior of a submanifold is the closure of the complement of a regular neighborhood of it.) ∂X and ∂Y are each planes. We identify them to obtain a P^{2}-irreducible open 3manifold M with π_{1}(M) ≅ π_{1}(N). Let p : M̃ → M be the universal covering map. Then M̃, p^{−1}(X), and p^{−1}(Y) are P^{2}-irreducible [5]. Each component Ỹ of p^{−1}(Y)

1991 Mathematics Subject Classification. Primary: 57M10; Secondary: 57N10, 57M60. Key words and phrases. 3-manifold, open 3-manifold, P^{2}-irreducible, R^{2}-irreducible, universal

covering space, contractible open 3-manifold, Whitehead manifold.

Research at MSRI is supported in part by NSF grant DMS-9022140.

has interior Ũ homeomorphic to U and so contains a compact, connected subset J which does not lie in a 3-ball in Ũ. If M̃ were homeomorphic to R^{3} then J would lie in a 3-ball B in M̃. Standard general position and minimality arguments applied to ∂B and ∂Ỹ would then yield a 3-ball B′ in Ũ containing J, a contradiction. Alternatively, one could use the Tucker Compactification Theorem [14] to obtain a compact polyhedron K in Ũ such that some component V of Ũ − K has non-finitely generated fundamental group. But this is impossible since the union of V and M̃−Ũ is a component of M̃ − K whose fundamental group is isomorphic to π_{1}(V).

In this example ∂Ỹ is a non-trivial plane in M̃, i.e. a proper plane Π such that no component of M̃ − Π has closure homeomorphic to R^{2} × [0, ∞) with Π = R^{2} × {0}. This paper shows that it is harder to find examples if one rules out this behavior by requiring that M̃ be R^{2}-irreducible in the sense that, in addition to being irreducible, it contains no non-trivial planes.

Define a closed surface group to be the fundamental group of a closed, connected 2-manifold.

Theorem 1. Let M be a connected, P^{2}-irreducible, open 3-manifold with π_{1}(M) finitely generated. If the universal covering space M̃ of M is R^{2}-irreducible, then either

(1) M̃ is homeomorphic to R^{3} or

(2) π_{1}(M) is a free product of infinite cyclic groups and infinite closed surface groups.

The second possibility can be disjoint from the first.

Theorem 2. Suppose G is a free product of finitely many infinite cyclic groups and infinite closed surface groups. Then there is a P^{2}-irreducible open 3-manifold M such that π_{1}(M) ≅ G and M̃ is an R^{2}-irreducible Whitehead manifold. Moreover, for each given G there are uncountably many such M for which the M̃ are pairwise non-homeomorphic.

This generalizes an example of Scott and Tucker [12] for which G is infinite cyclic. These results have a bearing on the following well-known problem.

Conjecture 1 (Universal Covering Conjecture). Let X be a closed, connected, irreducible, orientable 3-manifold with π_{1}(X) infinite. Then the universal covering space X̃ of X is homeomorphic to R^{3}.

P^{2}-IRREDUCIBLE OPEN 3-MANIFOLDS 3

Since there are only countably many homeomorphism types of closed 3-manifolds Theorem 2 implies that there must exist uncountably many R^{2}-irreducible Whitehead manifolds M̃ which cover open 3-manifolds M with π_{1}(M) ≅ G but cannot cover a closed 3-manifold. This generalizes a result of Tinsley and Wright [13] which shows that there must exist uncountably many non-R^{2}-irreducible Whitehead manifolds M̃ which cover open 3-manifolds M with π_{1}(M) infinite cyclic but cannot cover a closed 3-manifold. Unfortunately this argument does not provide any specific such examples. Specific examples of non-R^{2}-irreducible Whitehead manifolds M̃ which cover open 3manifolds M with π_{1}(M) infinite cyclic or, more generally, a countable free group, but cannot cover a closed 3-manifold are given in [9] and [10], respectively. At the time of this writing the problem of providing specific examples of R^{2}-irreducible Whitehead manifolds which non-trivially cover other open 3-manifolds but cannot cover a closed 3-manifold is still open.

One can make several conjectures related to Conjecture 1. We consider the selection below. In all of them G is assumed to be a finitely generated group of covering translations acting on a Whitehead manifold W with quotient a 3-manifold M.

Conjecture 2. G is a free product of infinite cyclic groups and fundamental groups of ∂-irreducible Haken manifolds.

Conjecture 3. G is a free group or contains an infinite closed surface group.

Conjecture 4. If W is R^{2}-irreducible, then G is a free product of infinite cyclic groups and infinite closed surface groups.

A proper plane Π in W is equivariant if for each g ∈ G either g(Π) = Π or Π ∩ g(Π) = ∅.

Conjecture 5 (Special Equivariant Plane Conjecture). If G is not a free product of infinite cyclic groups and infinite closed surface groups, then W contains a non-trivial equivariant plane.

Conjecture 6 (Equivariant Plane Conjecture). If W contains a non-trivial plane, then it contains a non-trivial equivariant plane.

These conjectures are related as follows.

Theorem 3. (4) ⇐ (1) ⇔ (2) ⇔ (3) ⇔ (5) ⇐ (4 + 6)

Theorems 1 and 3 are proven in section 2. Theorem 2 is proven in sections 3–7. Section 3 presents a modified version of the criterion used by Scott and Tucker [12] for showing that a 3-manifold is R^{2}-irreducible. Sections 4 and 5 treat, respectively, the special cases in which G is an infinite cyclic group and an infinite closed surface group. The constructions and notation of these special cases are used in section 6, which treats the general case. Section 7 shows how to get uncountably many M with non-homeomorphic M̃ for each group G.

2. The proofs of Theorems 1 and 3

Lemma 2.1. Let M be a connected, P^{2}-irreducible, open 3-manifold. Let Q be a compact, connected, 3-dimensional submanifold of M such that ∂Q is incompressible in M and π_{1}(Q) is not an infinite closed surface group. Let p : M̃ → M be the universal covering map and G the group of covering translations. Let Q̃ be a component of p^{−1}(Q). Then

(1) Each component of p^{−1}(∂Q) is a plane.

(2) There is no component Π of ∂Q̃ which is invariant under the subgroup G_{0} of G consisting of those covering translations which leave Q̃ invariant.

(3) If each component of ∂Q̃ is a trivial plane, then M̃ is homeomorphic to R^{3}.

Proof. (1) follows from the incompressibility of ∂Q in M.

Suppose S is a component of ∂Q and Π is a component of p^{−1}(S) which is invariant under G_{0}. Since the restriction of p to Q̃ is the universal covering space of Q and the restriction of G_{0} to Q̃ is the group of covering translations we have that π_{1}(S) → π_{1}(Q) is an isomorphism, contradicting our assumption on π_{1}(Q). This establishes (2).

We now prove (3). Suppose that each component Π of ∂Q̃ bounds an end-proper halfspace H_{Π} in M̃. Let K_{Π} be the closure of the component of M̃ − Π which does not contain int Q̃.

Assume that for all such Π we have H_{Π} = K_{Π}. Then M̃ is the union of Q̃ and an open collar attached to ∂Q̃, hence M̃ is homeomorphic to int Q̃. Since Q is Haken, the Waldhausen Compactification Theorem [15] implies that Q̃ is homeomorphic to a closed 3-ball minus a closed subset of its boundary, hence int Q̃ is homeomorphic to R^{3}, and we are done.

Thus we may assume that for some Π we have H_{Π} ≠ K_{Π}. Then H_{Π} ∩ K_{Π} = Π and H_{Π} ∪ K_{Π} = M̃. Now G_{0} has an element g such that g(Π) ≠ Π. Since Q̃ ⊆ H_{Π} and g(Q̃) = Q̃ we must have g(K_{Π}) ⊆ H_{Π}. Since R^{2} × [0, ∞) is R^{2}-irreducible (see e.g. [8]) it follows that K_{Π} is homeomorphic to R^{2} × [0, ∞). Thus M̃ is homeomorphic to R^{3}.

Proof of Theorem 1. By passing to a covering space of M, if necessary, we may assume that π_{1}(M) is indecomposable with respect to free products and is neither an infinite cyclic group nor an infinite closed surface group. Let C be the Scott compact core [11] of M, i.e. C is a compact, connected, 3-dimensional submanifold of M such that π_{1}(C) → π_{1}(M) is an isomorphism. The conditions on π_{1}(M) imply that ∂C is incompressible in M. We thus can apply Lemma 2.1 with Q = C to finish the proof.

Proof of Theorem 3. We first show that (1) ⇒ (2) ⇒ (3) ⇒ (1). If (1) is true, then M must be non-compact; this follows from the fact that if M were closed and non-orientable, then it would be Haken and so have universal covering space homeomorphic to R^{3}. Let C be the Scott compact core for M. Since M is irreducible we may assume that no component of ∂C is a 2-sphere; it follows that C is irreducible. If C is ∂-irreducible, then we are done. If C is not ∂-irreducible, then there is a finite set of compressing disks for ∂C in C which express C as a ∂-connected sum of 3-balls and ∂-irreducible Haken manifolds, thus yielding (2). Clearly (2) ⇒ (3). Suppose (3) is true and M is closed. If G is free, then M is by [2, Theorem 5.2] a connected sum of 2-sphere bundles over S^{1}, hence is not aspherical, hence W is not contractible. If G contains an infinite closed surface group, then by a result of Hass, Rubinstein, and Scott [1] W is homeomorphic to R^{3}.

Clearly Theorem 1 and the fact that M cannot be closed and non-orientable show that (1) ⇒ (4).

We now show that (1) ⇒ (5). Let C be the Scott compact core of M. Then the assumptions on G imply that there is a set of compressing disks for ∂C in C such that some component Q of C split along this collection of disks satisfies the hypotheses of Lemma 2.1. Thus any component of the pre-image of ∂Q is an equivariant non-trivial plane.

We next show that (5) ⇒ (1). Assume M is closed. If π_{1}(M) is a free product of infinite cyclic groups and infinite closed surface groups, then we apply (3) to obtain (1). If π_{1}(M) is not such a group, then the existence of an equivariant plane, together with the compactness of M, implies that M is Haken, and so (1) follows by Waldhausen [15].

Finally we show that (4 + 6) ⇒ (1). If W is R^{2}-irreducible, then (4) implies the hypothesis of (2), hence implies (1). If W is not R^{2}-irreducible, then (6) implies as before that M is Haken, thus (1) holds.

3. Nice quasi-exhaustions and R^{2}-irreducibility

We shall reformulate a criterion due to Scott and Tucker [12] for a P^{2}-irreducible open 3-manifold to be R^{2}-irreducible. A proper plane Π in an open 3-manifold W is homotopically trivial if for any compact subset C of W the inclusion map of Π is end-properly homotopic to a map whose image is disjoint from C.

Lemma 3.1. Let W be an irreducible, open 3-manifold, and let Π be a proper plane in W. If Π is homotopically trivial, then Π is trivial.

Proof. This is Lemma 4.1 of [12].

Lemma 3.2. Let W be a connected, irreducible, open 3-manifold, and let {C_{n}}_{n≥1}, be a sequence of compact 3-dimensional submanifolds of W such that C_{n} ⊆ int C_{n+1} and

(1) each C_{n} is irreducible,

(2) each ∂C_{n} is incompressible in W − int C_{n},

(3) if D is a proper disk in C_{n+1} which is in general position with respect to ∂C_{n} such that ∂D is not null-homotopic in ∂C_{n+1}, then D ∩ ∂C_{n} has at least two components which are not null-homotopic in ∂C_{n} and bound disjoint disks in D.

Then any proper plane in W can be end-properly homotoped off C_{n} for any n. Proof. This is Lemma 4.2 of [12].

The precise criterion we shall use is as follows. Lemma 3.3. Let W be a connected, irreducible, open 3-manifold. Suppose that for each compact subset K of W there is a sequence {C_{n}}_{n≥1} of compact 3-dimensional submanifolds such that C_{n} ⊆ int C_{n+1} and

(1) each C_{n} is irreducible,

(2) each ∂C_{n} is incompressible in W − int C_{n} and has positive genus,

(3) each C_{n+1} − int C_{n} is irreducible, ∂-irreducible, and anannular,

(4) K ⊆ C_{1}. Then W is R^{2}-irreducible. Proof. Let D be a disk as in part (iii) of Lemma 3.2. If every component of D ∩ ∂C_{n} is null-homotopic in ∂C_{n}, then one can isotop D so that D ∩ C_{n} = ∅ and hence ∂C_{n+1} is compressible in C_{n+1} − int C_{n}. If only one component α of D ∩ ∂C_{n} is not null-homotopic in ∂C_{n}, then ∂D ∪ α bounds an annulus A which can be isotoped so that A∩∂C_{n} = α, hence C_{n+1} −int C_{n} is not anannular. If no two of the components of D ∩ ∂C_{n} which are not null-homotopic in ∂C_{n} bound disjoint disks in D, then these components must be nested on D. We can isotop D to remove null-homotopic components and then intermediate annuli to again get an incompressible annulus joining ∂C_{n+1} to ∂C_{n}. Now apply Lemma 3.2 and then Lemma 3.1.

Let {C_{n}} be a sequence of compact, connected 3-dimensional submanifolds of an irreducible, open 3-manifold W such that C_{n} ⊆ int C_{n+1} such that W − int C_{n} has no compact components. This will be called a quasi-exhaustion for W. A quasiexhaustion for W whose union is W is an exhaustion for W. A quasi-exhaustion is nice if it satisfies conditions (1)–(3) of Lemma 3.3. Thus that lemma can be rephrased by saying that if every compact subset of W is contained in the first term of a nice quasi-exhaustion, then W is R^{2}-irreducible.

We shall need some tools for constructing Whitehead manifolds with nice quasiexhaustions. Define a compact, connected 3-manifold Y to be nice it is is P^{2}- irreducible, ∂-irreducible, and anannular, it contains a two-sided proper incompressible surface, and it is not a 3-ball; define it to be excellent if, in addition, every connected, proper, incompressible surface of zero Euler characteristic in Y is ∂-parallel. So in particular an excellent 3-manifold is anannular and atoroidal while a nice 3manifold is anannular but may contain a non-∂-parallel incompressible torus.

A proper 1-manifold in a compact 3-manifold is excellent if its exterior is excellent; it is poly-excellent if the union of each non-empty subset of the set of its components is excellent.

Lemma 3.4. Every proper 1-manifold in a compact, connected 3-manifold whose boundary contains no 2-spheres or projective planes is homotopic rel ∂ to an excellent proper 1-manifold.

Proof. This is a special case of Theorem 1.1 of [6].

Define a k-tangle to be a disjoint union of k proper arcs in a 3-ball.

Lemma 3.5. For all k ≥ 1 poly-excellent k-tangles exist.

Proof. This is Theorem 6.3 of [7].

We shall also need the following criterion for gluing together excellent 3-manifolds to get an excellent 3-manifold.

Lemma 3.6. Let Y be a compact, connected 3-manifold. Let S be a compact, proper, two-sided surface in Y. Let Y′ be the 3-manifold obtained by splitting Y along S. Let S′ and S′′ be the two copies of S which are identified to obtain Y. If each component of Y′ is excellent, S′, S′′, and (∂Y′)−int (S′ ∪S′′) are incompressible in Y′, and each component of S has negative Euler characteristic, then Y is excellent.

Proof. This is Lemma 2.1 of [6].

4. The infinite cyclic case

In [12] Scott and Tucker described an R^{2}-irreducible Whitehead manifold which is an infinite cyclic covering space. This section gives a general procedure for constructing such examples. The construction introduced here will be incorporated into that for the general case in section 6.

Let P_{n} = D_{n} × [0, 1], where D_{n} is the disk of radius n. We call P_{n} a pillbox. Identify D_{n} × {0} with D_{n} × {1} to obtain a solid torus Q_{n}. Let R_{n} be a solid torus and H_{n} a 1-handle D × [0, 1] joining ∂D_{n} × (0, 1) to ∂R_{n}. Let V_{n} = P_{n} ∪ H_{n} ∪ R_{n} and M_{n} = Q_{n} ∪ H_{n} ∪ R_{n}. We call V_{n} an eyebolt. We embed M_{n} in the interior of M_{n+1} as follows.

We choose a collection of arcs θ_{0}, θ_{1}, α_{0}, α_{1}, α_{2}, β_{1}, β_{2}, γ_{1}, γ_{2}, δ_{1}, δ_{2}, ε in P_{n+1}which satisfy certain conditions described below. θ_{0}, θ_{1}, and α_{0} meet in a common endpoint in int P_{n+1} but are otherwise disjoint. The other endpoints of θ_{0} and α_{0} lie in (int D_{n+1}) × {0}; that of θ_{1} lies in (int D_{n+1}) × {1}. We let θ = θ_{0} ∪ θ_{1}. All the other arcs are proper arcs in P_{n+1}which are disjoint from each other and from θ ∪ α_{0}. γ_{1}, β_{2}, and δ_{2} run from (int D_{n+1}) × {0} to itself. γ_{2}, β_{1}, and δ_{1}run from (int D_{n+1}) × {1} to itself. α_{1} runs from (int D_{n+1}) × {0} to (int D_{n+1}) × {1}. α_{2} runs from (int D_{n+1}) × {1} to int (P_{n+1} ∩ H_{n+1}). ε runs from int (P_{n+1} ∩ H_{n+1}) to itself. We denote the image in Q_{n+1} of an arc by the same symbol, relying on the context to distinguish an arc in P_{n+1} from its image in Q_{n+1}. We require that θ be a simple closed curve in Q_{n+1} and that α_{0} ∪ β_{1} ∪ γ_{1} ∪ δ_{1} ∪ α_{1} ∪ β_{2} ∪ γ_{2} ∪ δ_{2} is an arc consisting of subarcs which occur in the given order. We require that any union of these arcs which contains α_{0} and at least one other arc has excellent exterior in P_{n+1}, and that the same is true for any union of these arcs which contains neither θ_{0}, θ_{1}, nor α_{0}. This can be achieved as follows. Note that the exterior of α_{0} in P_{n+1} is a 3-ball B. Choose a poly-excellent 11-tangle in B and then slide its endpoints so that exactly two of the arcs meet a regular neighborhood of α_{0}. Extend them to meet α_{0} in the desired configuration.

Next let κ_{1}, κ_{2}, and κ_{3} be product arcs in H_{n+1} joining (int D)×{0} to (int D)×{1}. Let R_{n} ⊆ int R_{n+1} be any null-homotopic embedding. Let λ_{1} and λ_{2} be disjoint proper arcs in R_{n+1} −int R_{n} with λ_{1} joining int (H_{n+1} ∩R_{n+1}) to itself and λ_{2} joining int (H_{n+1} ∩ R_{n+1}) to ∂R_{n}. We require λ_{1} ∪ λ_{2} to be excellent in R_{n+1} − int R_{n}. We also require that these arcs, together with ε, fit into an arc whose subarcs form the sequence κ_{1}, λ_{1}, κ_{2}, ε, κ_{3}, λ_{2} and that κ_{1} meets α_{2} in a common endpoint.

Now we embed P_{n} in P_{n+1} as a regular neighborhood of the arc θ so that the two disks of P_{n} ∩ (D_{n+1} × {0, 1}) are identified to give an embedding of Q_{n} in Q_{n+1}. Note that these embeddings are not consistent with the product structures. From the discussion above we have an arc ω in M_{n+1} − int (Q_{n} ∪ R_{n}) running from ∂Q_{n} to ∂R_{n}. We embed H_{n} as a regular neighborhood of ω. We change notation slightly by now letting α_{0} be the old α_{0} minus its intersection with the interior of Q_{n}.

We let M be the direct limit of the M_{n} and let p : M̃ → M be the universal covering map. p^{−1}(Q_{n}) = p^{−1}(P_{n}) is the union of pillboxes P_{n,j} = D_{n} × [j, j + 1] meeting along the D_{n} × {j} to form D_{n} × R. Note that this embedding is not the product embedding. p^{−1}(R_{n}) is a disjoint union of solid tori R_{n,j}. p^{−1}(H_{n}) is a disjoint union of 1-handles H_{n,j} joining ∂D_{n} × (j, j + 1) to ∂R_{n,j}; these are regular neighborhoods of lifts ω_{j} of ω. p^{−1}(M_{n}) = p^{−1}(V_{n}) is the union of p^{−1}(P_{n}), p^{−1}(H_{n}), and p^{−1}(R_{n}). It is the union of eyebolts V_{n,j} = P_{n,j} ∪ H_{n,j} ∪ R_{n,j} meeting along the D_{n} × {j}. M̃ is the nested union of the p^{−1}(M_{n}).

Let Σ_{n}^{m} = ∪_{j=−m}^{m}V_{n,j} and Λ_{n}^{m} = P_{n,−(m+1)} ∪P_{n,m+1}. Let Φ_{1}^{m} = ∅, and, for n ≥ 2, let Φ_{n}^{m} = ∪_{j=m+2}^{m+n}(P_{n,−j} ∪P_{n,j}). Note that Λ_{n}^{m} and Φ_{n}^{m} (for n ≥ 2) are each disjoint unions of two 3-balls, Λ_{n}^{m} ∩ Σ_{n}^{m} is a pair of disjoint disks, and (for n ≥ 2) so is Λ_{n}^{m} ∩ Φ_{n}^{m}. Define C_{n}^{m} = Σ_{n}^{m} ∪ Λ_{n}^{m} ∪ Φ_{n}^{m}.

Lemma 4.1. {C_{m}^{m}} is an exhaustion for M̃. Each C^{m} is a nice quasi-exhaustion.

Proof. C_{n}^{m} ⊆ int C_{n+1}^{m}, and C_{n}^{m} ⊆ C_{n}^{m+1}. A given compact subset K of M̃ lies in some p^{−1}(M_{n}) and thus in a finite union of V_{n,j} and hence in some Σ_{n}^{m} ⊆ C_{n}^{m} ⊆ C_{q}^{q}, where q = max{m, n}. Thus {C_{m}^{m}} is an exhaustion for M̃.

C_{n}^{m} is a cube with 2m + 1 handles. Let Y = C_{n+1}^{m} − int C_{n}^{m}. We will show that Y is excellent by successive applications of Lemma 3.6.

Consider a P_{n+1,j} contained in C_{n+1}^{m}. If |j| < m, then it meets C_{n}^{m} in a regular neighborhood of the union of the j^{th} copies of all the arcs in P_{n+1}. Thus Y ∩ P_{n+1,j} is excellent, and Lemma 3.6 implies that the union of these Y ∩ P_{n+1,j} is excellent. For |j| ≥ m some care must be taken so that one is always gluing excellent 3-manifolds along surfaces of the appropriate type. Note that Y ∩ (P_{n+1,m} ∪ P_{n+1,m+1} ∪ ⋯ ∪ P_{n+1,m+n−1} ∪ P_{n+1,m+n}) is equal to the exterior of the m^{th} copy of all the arcs but β_{1} and δ_{2} in P_{n+1,m} together with the exterior of the (m + 1)^{st} copy of β_{2}, δ_{2}, and θ in P_{n+1,m+1}, the exterior of the j^{th} copy of θ in P_{n+1,j} for m + 1 < j < m + n, and the 3-ball P_{n+1,m+n}. This space is homeomorphic to the exterior of the m^{th} copy of all the arcs but β_{1}, δ_{1}, and θ_{1} in P_{n+1,m+1} together with the exterior of the (m + 1)^{st} copy of β_{2} and δ_{2} in P_{n+1,m+1}, and the 3-ball consisting of the union of the P_{n+1,j} for which m + 1 < j ≤ m + n. This can be seen by taking the arc consisting of the m^{th} copy of θ_{1} and the j^{th} copy of θ for m < j < m + n and retracting it onto the endpoint in which it meets the rest of the graph. This space is then excellent by Lemma 3.6. Similar remarks apply for j ≤ −m, so these spaces can be added on to get that Y ∩ ∪_{j=−(m+n)}^{m+n}P_{n+1,j} is excellent.

We fill in the remainder of Y by adding the exteriors of the j^{th} copies of κ_{1}, κ_{2}, and κ_{3} in H_{n+1,j} and λ_{1} ∪ λ_{2} in R_{n+1,j} − int R_{n,j} for |j| ≤ m. Since the first of these spaces is a product the union of the two spaces is homeomorphic to the second space, and Lemma 3.6 applies to complete the proof that Y is excellent.

It remains to show that each ∂C_{n}^{m} is incompressible in M̃ − int C_{n}^{m}. Since each C_{n+s+1}^{m}−int C_{m+s}^{m} is ∂-irreducible we have that ∂C_{n}^{m} is incompressible in C_{n+q}^{m}−int C_{n}^{m} for each q ≥ 1. p^{−1}(M_{n+q}) is the union of C_{n+q}^{m} and the sets C^{−} = ∪_{j<−m}V_{n+q,j} and C^{+} = ∪_{j>m}V_{n+q,j}. We have that C^{−} ∩ C_{n+q}^{m} and C^{+} ∩ C_{n+q}^{m} are disjoint disks, while C^{−} ∩ C^{+} = ∅. It follows that ∂C_{n}^{m} is incompressible in p^{−1}(M_{n+q}) − int C_{n}^{m}. Since M̃ is the nested union of the p^{−1}(M_{n+q}) over all q ≥ 1 we have the desired result.

5. The surface group case

Let F be a closed, connected surface other than S^{2} or P^{2}. Let n ≥ 1. Regard F as being obtained from a 2k-gon E, k ≥ 2, by identifying sides s_{i} and s′_{i}, 1 ≤ i ≤ k. This induces an identification of the lateral sides S_{i} = s_{i} × [−n, n] and S′_{i} = s_{i} × [−n, n] of the prism P_{n} = E × [−n, n] which yields Q_{n} = F × [−n, n]. Let R_{n} be a solid torus and H_{n} a 1-handle D × [0, 1]. Let V_{n} = P_{n} ∪ H_{n} ∪ R_{n}, where P_{n} ∩ R_{n} = D × {1} is a disk in ∂R_{n}, and P_{n} ∩ R_{n} = ∅. We again call V_{n} an eyebolt. It is a solid torus whose image under the identification is M_{n} = Q_{n} ∪ H_{n} ∪ R_{n}, a space homeomorphic to the ∂-connected sum of F × [−n, n] and a solid torus.

We define an open 3-manifold M by specifying an embedding of M_{n} in the interior of M_{n+1} and letting M be the direct limit. The inclusion [−n, n] ⊆ [−(n + 1), n + 1] induces P_{n} ⊆ P_{n+1} and hence Q_{n} ⊆ Q_{n+1}. We let R_{n} ⊆ int R_{n+1} be any nullhomotopic embedding. Again the interesting part of the embedding will be that of H_{n} in M_{n+1}. It will be the regular neighborhood of a certain arc ω in M_{n+1}−int (Q_{n}∪R_{n}) joining ∂Q_{n} to ∂R_{n}.

The arc ω is the union of 4k + 7 arcs any two of which are either disjoint or have one common endpoint. The 4k + 2 arcs α_{0}, α_{i}, β_{i}, γ_{i}, δ_{i}, 1 ≤ i ≤ k, and ε lie in E × [n, n + 1] and are identified with their images in Q_{n+1}; the three arcs κ_{1}, κ_{2}, and κ_{3} lie in H_{n+1}, and the two arcs λ_{1} and λ_{2} lie in R_{n+1}. These arcs will have special properties to be described later. We first describe their combinatorics. The arcs in P_{n+1} are all proper arcs in E × [n, n + 1]. α_{0} runs from (int E) × {n} to int S_{1}. For 1 ≤ i < k, α_{i} runs from int S_{i} to int S_{i+1}. α_{k} runs from int S′_{k} to int (P_{n+1} ∩ H_{n+1}). For 1 ≤ i ≤ k, β_{i} and δ_{i} each run from int S′_{i} to itself, while γ_{i} runs from int S_{i} to itself. These arcs are chosen so that under the identification their endpoints match up in such a way as to give an arc which follows the sequence α_{0}, β_{1}, γ_{1}, δ_{1}, α_{1}, …, β_{k}, γ_{k}, δ_{k}, α_{k}. We require ε to run from int (P_{n+1} ∩ H_{n+1}) to itself. κ_{1}, κ_{2}, and κ_{3} are product arcs in H_{n+1} lying in (int D) × [0, 1]. λ_{1} and λ_{2} are proper arcs in R_{n+1} − int R_{n}, with λ_{1} running from int (H_{n+1} ∩ R_{n+1}) to itself and λ_{2} running from int (H_{n+1} ∩ R_{n+1}) to ∂R_{n}. These arcs are chosen so as to fit together into the sequence κ_{1}, λ_{1}, κ_{2}, ε, κ_{3}, λ_{2} with the endpoint of κ_{1} other than κ_{1} ∩ λ_{1} being the same as the endpoint of α_{k} other than α_{k} ∩ δ_{k}. This gives ω.

We now describe the special properties required of these arcs. We require that α_{0} ∪ β_{1} ∪ γ_{1} ∪ δ_{1} ∪ α_{1} ∪ ⋯ ∪ β_{k} ∪ γ_{k} ∪ δ_{k} ∪ ε be a poly-excellent (4k + 2)-tangle in E × [n, n + 1] and λ_{1} ∪ λ_{2} to be an excellent 1-manifold in R_{n+1} − int R_{n}.

We now consider the universal covering map p : M̃ → M. Our goal is to construct a sequence {C^{m}} of nice quasi-exhaustions whose diagonal {C_{m}^{m}} is an exhaustion for M̃.

The universal covering space F̃ of F is tesselated by copies E_{j} of E. We fix one such copy E_{1}. We inductively define an exhaustion {F_{m}} for F̃ as follows. F_{1} = E_{1}. F_{m+1} is the union of F_{m} and all those E_{j} which meet it. Each F_{m} is a disk (which we call a star). The inner corona I_{m} of F_{m} is the annulus F_{m+1} − int F_{m}. Each vertex on ∂F_{m} lies in either one or two of those E_{j} contained in F_{m}. Each E_{j} in I_{m} meets meets F_{m} in either an edge or a vertex; in both cases it meets exactly two adjacent E_{ℓ} of I_{m}, and each of these intersections is an edge. For n ≥ 2 we define the outer n-corona O_{n}^{m} to be the annulus F_{m+n} − int F_{m+1}; we define O_{1}^{m} = ∅. Let σ_{2} be a proper arc in F_{2} consisting of three edges of the polygons in F_{2}. . Inductively define a proper arc σ_{m+1} in F_{m+1} by adjoining to σ_{m} two arcs spanning I_{m} which are edges of polygons in I_{m}. Thus each σ_{m} is an edge path in F_{m} splitting it into two unions of polygons F′_{m} and F′′_{m}.

We now consider the structure of M̃. For n ≥ 1, p^{−1}(Q_{n}) = p^{−1}(P_{n}) is the union of prisms P_{n,j} = E_{j} × [−n, n] meeting along their lateral sides to form F̃ × [−n, n]. p^{−1}(R_{n}) is a disjoint union of solid tori R_{n,j}. p^{−1}(H_{n}) is a disjoint union of 1-handles H_{n,j} running from E_{j} × {n} to ∂R_{n,j}; these are regular neighborhoods of lifts ω_{j} of ω. Now p^{−1}(M_{n}) = p^{−1}(V_{n}) is the union of p^{−1}(P_{n}), p^{−1}(H_{n}), and p^{−1}(R_{n}). It can be expressed as the union of the eyebolts V_{n,j} = P_{n,j} ∪ H_{n,j} ∪ R_{n,j} meeting along the lateral sides of the P_{n,j}. Finally M̃ is the nested union of the p^{−1}(M_{n}).

Let Σ_{n}^{m} be the union of those V_{n,j} such that E_{j} is in the star F_{m}. Let Λ_{n}^{m} be the union of those P_{n,j} such that E_{j} is in the inner corona I_{m}. Let Φ_{n}^{m} be the union of those P_{n,j} such that E_{j} is in the outer n-corona O_{n}^{m}. Note that Λ_{n}^{m} and Φ_{n}^{m} (for n ≥ 2) are solid tori, Λ_{n}^{m} ∩ Σ_{n}^{m} is an annulus which goes around Λ_{n}^{m} once longitudinally and consists of those lateral sides of the prisms in Σ_{n}^{m} which lie on ∂Σ_{n}^{m}, and (for n ≥ 2) Λ_{n}^{m} ∩ Φ_{n}^{m} is an annulus which goes around each of these solid tori once longitudinally. We now define C_{n}^{m} = Σ_{n}^{m} ∪ Λ_{n}^{m} ∪ Φ_{n}^{m}.

Lemma 5.1. {C_{m}^{m}} is an exhaustion for M̃. Each C^{m} is a nice quasi-exhaustion.

Proof. Note that C_{n}^{m} ⊆ int C_{n+1}^{m}, and C_{n}^{m} ⊆ C_{n}^{m+1}. Suppose K is some compact subset of M̃. Then K lies in some p^{−1}(M_{n}) and thus in a finite union of V_{n,j} and hence in some Σ_{n}^{m} ⊆ C_{n}^{m} ⊆ C_{q}^{q}, where q = max{m, n}. Thus {C_{m}^{m}} is an exhaustion for M̃.

Each C_{n}^{m} is a cube with handles, so is irreducible. The number of handles is at least one, so ∂C_{n}^{m} has positive genus. Let Y = C_{n+1}^{m} − int C_{n}^{m}. We will prove that Y is excellent by successive applications of Lemma 3.6. Let P_{n+1,j}^{+} and P_{n+1,j}^{−} denote, respectively, E_{j} × [n, n + 1] and E_{j} × [−(n + 1), −n].

Consider a P_{n+1,j} contained in Σ_{n+1}^{m}. It meets C_{n}^{m} in P_{n+1,j} together with regular neighborhoods of certain arcs in P_{n+1,j}^{+}. These arcs consist at least of the j^{th} copies of the α_{i}, the γ_{i}, and ε which are part of the lift ω_{j} of ω. If another prism P_{n+1,ℓ} in Σ_{n+1}^{m} meets P_{n+1,j} in a common lateral side, then either ω_{j} or ω_{ℓ} will meet this side; in the latter case this contributes a β_{i} and δ_{i} to the subsystem of arcs in P_{n+1,j}^{+}. Since the full system of arcs was chosen to be poly-excellent this subsystem of arcs is excellent and so has excellent exterior Y ∩ P_{n+1,j}^{+}. Let U′ be the union of those Y ∩ P_{n+1,j}^{+} such that E_{j} ⊆ F′_{m}. This space can be built up inductively by gluing on one Y ∩ P_{n+1,j}^{+} at a time, with the gluing being done along either a disk with two holes (when P_{n+1,j} is glued along one lateral side) or a disk with four holes (when P_{n+1,j} is glued along two adjacent lateral sides). No component of the complement of this surface in the boundary of either manifold is a disk, hence this surface is incompressible in each manifold. It follows that U′ is excellent. Similar remarks apply to the space U′′ associated with F′′_{m}.

Next consider a P_{n+1,j}^{+} contained in Λ_{n+1}^{m}. If E_{j} ⊆ F_{m+1} and meets F′_{m} in an edge of E_{ℓ} ⊆ F′_{m}, then either ω_{ℓ} misses P_{n+1,j}^{+} or meets it in copies of β_{i} and δ_{i}. Thus enlarging U′ by adding Y ∩ P_{n+1,j}^{+} either adds a 3-ball along a disk in its boundary, giving a space homeomorphic to U′ or gives a new excellent 3-manifold. We adjoin all such Y ∩ P_{n+1,j}^{+} to U′. Then we consider those E_{j} which meet F′_{m} in a vertex. Then P_{n+1,j}^{+} = Y ∩ P_{n+1,j}^{+}, and one can successively adjoin these 3-balls along disks in their boundaries. We denote the enlargement of U′ from all these additions again by U′. Similar remarks apply to U′′.

Now (F′_{m+n+1} − int F′_{m+1}) × [n, n + 1] is a 3-ball which meets U′ in a disk, so we adjoin it to U′ to get a new U′ homeomorphic to the old one. We then adjoin the 3-ball (F′_{m+n+1} − int F′_{m+n}) × [−n, n] which meets this space along a disk to obtain our final U′. The same construction gives U′′.

Now U′ and U′′ are each excellent. U′ ∩ U′′ is an annulus with a positive number of disks removed from its interior corresponding to its intersection with arcs passing from F′_{m}×[n, n+1] to F′′_{m}×[n, n+1]. No component of the complement of this surface in ∂U′ or in ∂U′′ is a disk; this corresponds to the fact that F′_{m} × {n}, F′′_{m} × {n}, F′_{m}×{n+1}, and F′′_{m}×{n+1} each meet some ω_{j}. Thus this surface is incompressible in both U′ and U′′, so U′ ∪ U′′ is excellent.

Finally we add on the Y ∩ R_{n,j} for E_{j} ⊆ F_{m} to U′ ∪ U′′ to conclude that Y is excellent.

It remains to show that each ∂C_{n}^{m} is incompressible in M̃−int C_{n}^{m}. First note that since each C_{n+s+1}^{m}−int C_{n+s}^{m} is ∂-irreducible we must have that ∂C_{n}^{m} is incompressible in C_{n+q}^{m} − int C_{n}^{m} for each q ≥ 1. Now consider the set

M̃_{n+q} = p^{−1}(M_{n+q}) ∪ (F̃ × [−(n + q + 1), −(n + q)]).

It can be obtained from C_{n+q}^{m} as follows. First add the solid tori R_{n+1,j} ∪ H_{n+q,j} in p^{−1}(M_{n+q}) for which E_{j} ⊆ F_{m+q+n}; these meet C_{n+q}^{m} in disks. Then add

(F_{m+q+n} × [−(n + q + 1), −(n + q)]) ∪ (F̃ − int F_{m+q+n}) × [−(n + q + 1), n + q]).

This is a space homeomorphic to R^{2} × [0, 1] which meets C_{n+q}^{m} in the disk

(F_{m+q+n} × {−(n + q)}) ∪ ((∂F_{m+q+n}) × [−(n + q), n + q]).

Lastly add all the remaining solid tori R_{n+q,j} ∪ H_{n+q,j}, where E_{j} ⊆ F̃ − int F_{m+q+n}; these do not meet C_{n+q}^{m}. This description shows that C_{n+q}^{m} ∩ (M̃_{n+q} − int C_{n+q}^{m}) consists of (finitely many) disjoint disks, and therefore ∂C_{n}^{m} is incompressible in M̃_{n+q} − int C_{n}^{m}. Finally since M̃ is the nested union of the M̃_{n+q} over all q ≥ 1 we have that ∂C_{n}^{m} is incompressible in M̃ − int C_{n}^{m}.

6. The general case

Suppose G_{1}, …, G_{k} are infinite cyclic groups and infinite closed surface groups. For i = 1, …, k let P_{n}^{i} be a pillbox or prism, as appropriate, with quotient Q_{n}^{i} a solid torus or product I-bundle over a closed surface, respectively. We let H_{n}^{i} be a 1-handle attached to P_{n}^{i} as before. We let R_{n} be a common solid torus to which we attach the other ends of all the H_{n}^{i}. The union of the Q_{n}^{i} and H_{n}^{i} with R_{n} is called M_{n}. As before we choose arcs in the P_{n}^{i}, H_{n}^{i}, and R_{n} and use them to define an embedding of M_{n} into the interior of M_{n+1}.

The choice of arcs in R_{n+1} − int R_{n}, as well as the embedding R_{n} ⊆ int R_{n+1}, requires some discussion, since we will want this family λ of arcs to be poly-excellent. Choose a poly-excellent (2k+2)-tangle λ^{+} in a 3-ball B, with components λ_{t}^{i}, 1 ≤ i ≤ k + 1, t = 1, 2. Construct a graph in B by sliding one endpoint of each λ_{2}^{i}, 1 ≤ i ≤ k, so that it lies on int λ_{2}^{k+1}. Thus these λ_{2}^{i} now join ∂B to distinct points on int λ_{2}^{k+1}; all the other λ_{t}^{i} still join ∂B to itself. Now choose disjoint disks E_{1} and E_{2} in ∂B such that E_{t} meets the graph in ∂λ_{t}^{k+1} ∩ int E_{t}. Glue E_{1} to E_{2} so that B becomes a solid torus R_{n+1} and λ_{1}^{k+1} ∪λ_{2}^{k+1} becomes a simple closed curve. The regular neighborhood of this simple closed curve is our embedding of R_{n} in the interior of R_{n+1}. Clearly R_{n} is null-homotopic in R_{n+1}. By Lemma 3.6 its exterior is excellent as is the exterior of the union of R_{n} with any of the λ_{t}^{i}, 1 ≤ i ≤ k, t = 1, 2.

Let p : M̃ → M be the universal covering map. Then p^{−1}(R_{n}) consists of disjoint solid tori whose union separates p^{−1}(M_{n}) into components with closures L_{n}^{i,μ}, where L_{n}^{i,μ} is a component of p^{−1}(Q_{n}^{i} ∪ H_{n}^{i}). Let Z_{n}^{i,μ} be the union of L_{n}^{i,μ} and all those components of p^{−1}(R_{n}) which meet it. Then Z^{i,μ} = ∪_{n≥1}Z_{n}^{i,μ} is an open subset of M̃ which has a family {C^{i,μ,m}} of quasi-exhaustions as previously described. We will develop from these families an appropriate family {C^{m}} of quasi-exhaustions of M̃.

We start by choosing a component R̂_{1} of p^{−1}(R_{1}). For each n there is then a unique component R̂_{n} of p^{−1}(R_{n}) which contains R̂_{1}. We define C_{n}^{1} to be the union of R̂_{n} and the (finitely many) C_{n}^{i,μ,1} which contain it; this space is a solid torus which meets each of these C_{n}^{i,μ,1} in a 3-ball which is either a pillbox and a 1-handle or a prism and a 1-handle. Suppose C_{n}^{m} has been defined and that it is the union of the C_{n}^{i,μ,m} for which C_{n}^{m} ∩ L_{n}^{i,μ} ≠ ∅. We define C_{n}^{m+1} in two steps. We first take the union C′ of all the C_{n}^{i,μ,m} such that C_{n}^{i,μ,m} ⊆ C_{n}^{m}. This is just the union of the n^{th} elements of the (m + 1)^{st} quasi-exhaustions for those Z_{n}^{i,μ} such that {i, μ} is in the current index set. The second step is to enlarge the index set by adding those {i, ν} for which C′ ∩ L_{n}^{i,ν} ≠ ∅ and then adjoin the C_{n}^{i,ν,m+1} to C′ in order to obtain C_{n}^{m+1}. One can observe that the L_{n}^{i,μ} and p^{−1}(R_{n}) give p^{−1}(M_{n}) a tree-like structure and that the passage from C_{n}^{m} to C_{n}^{m+1} goes out further along this tree.

Lemma 6.1. {C_{m}^{m}} is an exhaustion for M̃. C^{m} is a nice quasi-exhaustion.

Proof. Again we have C_{n}^{m} ⊆ int C_{n+1}^{m} and C_{n}^{m} ⊆ C_{n}^{m+1} with the result that {C_{m}^{m}} is an exhaustion for M̃.

As regards the excellence of C_{n+1}^{m} − int C_{n}^{m} we note that the only thing new takes place in those components of p^{−1}(R_{n+1}) contained in C_{n+1}^{m}. Instead of two arcs λ_{1} and λ_{2} as before we have λ_{1}^{i} and λ_{2}^{i} as i ranges over some non-empty subset of {1, …, k} We then apply the poly-excellence of the full set of λ_{t}^{i}.

The incompressibility of ∂C_{n}^{m} in M̃ − int C_{n}^{m} follows as before. We first note that ∂C_{n}^{m} is incompressible in C_{n+q}^{m} − int C_{n}^{m} for each q ≥ 1. Now define M̃_{n+q} to be the union of p^{−1}(M_{n+q}) and, for each of the surface group factors G_{i} of G, the copy F̃^{i,μ} × [−(n + q + 1), −(n + q)] of F̃^{i} × [−(n + q + 1), −(n + q)] contained in Z^{i,μ}, where F̃^{i} is the universal covering space of the surface F^{i} with π_{1}(F^{i}) ≅ G_{i}. Then the exterior of C_{n+q}^{m} in M̃_{n+q} meets it in a collection of disjoint disks, from which it follows that ∂C_{n}^{m} is incompressible in M̃_{n+q} − int C_{n}^{m}, thus is incompressible in M̃ − int C_{n}^{m}.

7. Uncountably many examples

We now describe how to get uncountably many examples for a given group G. We will use a trick introduced in [7]. Let {X_{n,s}} be a family of exteriors of non-trivial knots in S^{3} indexed by n ≥ 2 and s ∈ {0, 1}; they are chosen to be anannular, atoroidal, and pairwise non-homeomorphic. (One such family is that of non-trivial, non-trefoil twist knots.) One chooses a function φ(n) with values in {0, 1}, i.e. a sequence of 0’s and 1’s indexed by n, and constructs a 3-manifold M[φ] by embedding X_{n,φ(n)} in M_{n} − int M_{n−1} so that ∂X_{n,φ(n)} in incompressible in M_{n} − int M_{n−1} (but is compressible in M_{n}). The idea is to do this in such a way that for “large” compact sets C in M̃[φ] one has components of p^{−1}(X_{n,φ(n)}) which lie in M̃ − C and have incompressible boundary in M̃ − C for “large” values of n; moreover, every knot exterior having these properties should be homeomorphic to some X_{n,φ(n)}. Thus if M̃[φ] and M̃[ψ] are homeomorphic one must have φ(n) = ψ(n) for “large” n. One then notes that there are uncountably many functions which are pairwise inequivalent under this relation.

We proceed to the details. First assume φ is fixed, so we can write s = φ(n). The most innocuous place to embed X_{n,s} is in R_{n} − int R_{n−1} since this space is common to all our constructions. Recall that this space contains arcs λ_{1}, λ_{2} or, if G is a non-trivial free product, arcs λ_{1}^{i}, λ_{2}^{i}, 1 ≤ i ≤ k; call this collection of arcs λ. We wish X_{n,s} to lie in the complement of λ in such a way that it is poly-excellent in R_{n} − int (R_{n−1} ∪ X_{n,s}). We revise the construction of λ from section 6 as follows. Let B_{0} and B_{1} be 3-balls. Choose disjoint disks D_{r} and D′_{r} in ∂B_{r}. Let ζ_{r} be a simple closed curve in ∂B_{r} − (D_{r} ∪ D′_{r}) which separates D_{r} from D′_{r}. Let A_{r} and A′_{r} be the annuli into which ζ_{r} splits the annulus ∂B_{r} − int (D_{r} ∪ D′_{r}), with the notation chosen so that A_{r}∩D_{r} = ∅. Let τ_{r} be a poly-excellent (4k+4)-tangle in B_{r} which is the union of (2k + 2)-tangles ρ_{r} and ρ′_{r} satisfying the following conditions. Each component of ρ_{0} runs from int D_{0} to int A′_{0}. Each component of ρ_{1} runs from int A′_{1} to int D′_{1}. Each component of ρ′_{0} runs from int D′_{0} to itself. Each component of ρ′_{1} runs from int D′_{1} to int D_{1}. We then glue A′_{0} to A′_{1} and D′_{0} to D′_{1} so as to obtain a space homeomorphic to a 3-ball minus the interior of an unknotted solid torus contained in the interior of the 3-ball. The 2-sphere boundary component is D_{0} ∪ D_{1}; the torus boundary component is A_{0} ∪ A_{1}. The gluing is done so that the endpoints of the arcs match up to give a system λ^{+} of 2k + 2 arcs. Each arc in this system consists of an arc of ρ_{0} followed by an arc of ρ_{1} followed by an arc of ρ′_{0} followed by an arc of ρ′_{1}. We then glue X_{n,s} to this space along their torus boundaries so as to obtain a 3-ball B. We then apply the construction of section 6 to λ^{+} to get a poly-excellent system λ of arcs in R_{n} − int R_{n−1}. It is easily seen that this 3-manifold is nice and that ∂X_{n,s} is, up to isotopy, the unique incompressible non-∂-parallel torus in it; ∂X_{n,s} is also, up to isotopy, the unique incompressible torus in the exterior K_{σ} of any non-empty union σ of components of λ.

Lemma 7.1. If M̃[φ] and M̃[ψ] are homeomorphic then there is an index N such that φ(n) = ψ(n) for all n ≥ N.

Proof. Consider M̃. Y = C_{n}^{m} − int C_{n−1}^{m} contains copies of K_{σ} for various choices of σ. The closure of the complement in Y of these copies consists of excellent 3manifolds which meet the copies along incompressible planar surfaces. It follows that the various copies of ∂X_{n,s} in Y are, up to isotopy and for n ≥ 3, the unique incompressible tori in Y. The incompressibility of ∂C_{n}^{m} in M̃ − int C_{n}^{m} implies that these tori are also incompressible in M̃ − int C_{n−1}^{m}.

Suppose T is an incompressible torus in M̃ − int C_{n−1}^{m}. Then T lies in M̃_{n+q} for some q ≥ 0. The exterior of C_{n+q}^{m} in M̃_{n+q} consists of copies of D×R and R^{2}×[0, 1] to which disjoint 1-handles have been attached. It meets C_{n+q}^{m} in a set of disjoint disks. It follows that T can be isotoped into C_{n+q}^{m} − int C_{n−1}^{m}. Since ∂C_{n+u}^{m} for 1 ≤ u < q is not a torus it is easily seen that T can be isotoped into some C_{v}^{m} − int C_{v−1}^{m} and thus is isotopic to some copy of ∂X_{v,φ(v)}. Thus any knot exterior X incompressibly embedded in M̃ − int C_{n−1}^{m} is homeomorphic to some X_{v,φ(v)}.

Now consider two different functions φ and ψ. We will show that if M̃[φ] and M̃[ψ] are homeomorphic then there is an N such that φ(n) = ψ(n) for all n ≥ N. Suppose h : M̃[φ] → M̃[ψ] is a homeomorphism. Distinguish the various submanifolds arising in the construction of these two manifolds by appending [φ] and [ψ], respectively. For n ≥ 2 there are incompressibly embedded copies X̃_{n,φ(n)} of X_{n,φ(n)} in M̃[φ] − int C_{1}^{1}[φ]. There is an index ℓ such that h(C_{1}^{1}[φ]) ⊆ int C_{ℓ}^{ℓ}[ψ]. By construction ∪_{n≥2}X̃_{n,φ(n)} is end-proper in M̃[φ], so there is an index N such that for all n ≥ N we have h(X̃_{n,φ(n)}) ⊆ M̃[ψ] − int C_{ℓ}^{ℓ}[ψ]. Since h(∂X̃_{n,φ(n)}) is incompressible in M̃[ψ] − int h(C_{1}^{1}[ψ]) it is incompressible in the smaller set M̃[ψ] − int C_{ℓ}^{ℓ}[ψ]. Thus it is homeomorphic to X_{v,ψ(v)} for some v > ℓ. Since the knot exteriors are pairwise non-homeomorphic we must have n = v and φ(n) = ψ(v) = ψ(n).

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Department of Mathematics, Oklahoma State University, Stillwater, OK 74078

E-mail address: myersr@math.okstate.edu