title: $\mathcal{Z}$-compactifiable manifolds which are not pseudo-collarable
authors: Shijie Gu
text: 
  𝒵-compactifiable manifolds which are not pseudo-collarable]𝒵-compactifiable manifolds which are not pseudo-collarable
  
  Department of Mathematical Sciences
  
  Central Connecticut State University, New Britain, CT, 06053
  sgu@ccsu.edu
  
  
  
  
  
  It is shown that there exist 𝒵-compactifiable manifolds
  with noncompact boundary which fail to be pseudo-collarable.
  
  [
      Shijie Gu
      September 19th, 2021.
  =========================
  
  
  
  § INTRODUCTION
  
  An m-manifold M^m with (possibly empty) boundary is completable if there exists a compact manifold M^m and a compactum C⊆∂M^m such that M^m\ C is homeomorphic to M^m. In this case M^m is called a (manifold) completion of M^m.[In literature, when M^m has one end, completing a manifold is also known as finding a collar of the end.] A full characterization for completable manifolds, in the piecewise-linear (PL), topological (TOP) and smooth (DIFF) categories, was achieved through combined efforts of topologists over the past five decades; see <cit.>, <cit.>, <cit.> and <cit.>.
  
  
  
  
  
  [Manifold Completion Theorem] An
  m-manifold M^m (m≠ 4,5) is completable if and only if
  
  
  
    * M^m is inward tame,
  
  
    * M^m is peripherally π_1-stable at infinity,
  
  
    * σ_∞(M^m)∈lim{K_0(π_1(N))| N a clean neighborhood of
  infinity} is zero, and
  
  
    * τ_∞(  M^m)  ∈lim^1{*Wh(π_1(N))| N a clean neighborhood
  of infinity} is zero.
  
  
  When m =1, the characterization is trivial. For m = 2,3, Condition (<ref>) alone would be necessary and sufficient. See <cit.> and <cit.>. The Manifold completion theorem fails in dimension 4. Counterexamples were discovered in <cit.> and <cit.>. In TOP, Theorem 1.1 holds in dimension 5 provided Condition
  (<ref>) is strengthened to ensure the existence of arbitrarily small neighborhoods
  of infinity with stable peripheral pro-π_1 groups that are "good" in the language of <cit.>.
  
  
  Although Condition (<ref>) is necessary in order for such a completion to exist, such condition is too rigid to characterize many exotic examples related to current research trends in topology and geometric group theory. For instance, the exotic universal covering spaces produced by Davis <cit.> are not completable (because Condition (<ref>) fails) yet their ends exhibit nice geometric structure. More examples such as (open) manifolds that satisfy Conditions (<ref>), (<ref>) and (<ref>) but not Condition (<ref>) can be found in <cit.>. Is there a way to characterize manifolds sharing the structure with these manifolds at infinity? When M^m is an open manifold (or more generally, a manifold with compact boundary), by weakening Condition (<ref>), Guilbault <cit.> initiated a program to answer this question, which can be viewed as a natural generalization of Siebenmann's collaring theorem for manifolds with compact boundary. Define a manifold neighborhood of infinity N in a manifold M^m to be a homotopy collar provided FrN ↪ N is a homotopy equivalence. A pseudo-collar is a homotopy collar which contains arbitrarily small homotopy collar neighborhoods of infinity. A manifold is pseudo-collarable if it contains a pseudo-collar neighborhood of infinity. The idea of pseudo-collars and a detailed motivation for the definition are nicely exposited in <cit.>. Through a series of papers <cit.>, a characterzation for pseudo-collarable manifolds with compact boundary (hence, finitely many ends) was provided. In a recent work, the author <cit.> developed a complete characterization for high-dimensional manifolds with (possible noncompact) boundary.
  
   An m-manifold M^m (m≥ 6) is pseudo-collarable iff each of the
  following conditions holds:
  
  
  
  
    *  M^m is inward tame,
  
  
    *  M^m is peripherally perfectly π_1-semistable at infinity, and
  
  
    *  σ_∞(M^m)∈lim{K_0(π_1(N))| N a clean neighborhood of
  infinity} is zero. 
  
  
  The dimension restriction m≥ 6 is only required for the sufficiency of Theorem <ref>, see <cit.>. Compared to Theorem <ref>, when m≤ 3, pseudo-collarability is just the ordinary collarability.
  When all of the groups in Condition (<ref>) involved are "good" in
  the sense of Freedman and Quinn, the conclusion holds for m=5 in TOP. Just like Theorem 
  <ref>, Theorem <ref> fails in dimension 4. Counterexamples are again the ones from  <cit.> and <cit.>.
  
  
  Theorem <ref> captures the pseudo-collar structure of the ends, however, it is unclear if pseudo-collarability guarantees a compactification called 𝒵-compactification[It is an open question that whether pseudo-collarability and Condition (<ref>) of Theorem <ref> are
  sufficient for manifolds to be 𝒵-compactifiable, cf. <cit.> and <cit.>.], which is a generalization of completion intended for preserving the homotopy type of the original space.  A compactification
  X=X⊔ Z of a space X is a 𝒵-compactification if, for every open set U⊆X,
  U\ Z↪ U is a homotopy equivalence, where ⊔ denotes a disjoint union. The motivation first came from the modification of the manifold completion applied to Hilbert cube manifolds in <cit.>. This compactification has been proven to be useful in both
  geometric group theory and manifold topology, for example, in attacks on the Borel and Novikov Conjectures. An arguably major open problem about 𝒵-compactification is the existence of a characterization of 𝒵-compactifiable manifolds by completely removing Condition (<ref>) from Theorem <ref>, cf. <cit.>, <cit.> and <cit.>.
  
  Are Conditions (<ref>), (<ref>) and (<ref>) of Theorem <ref> sufficient for manifolds to be 𝒵-compactifiable?
  
  In dimensions ≤ 3, Condition (<ref>) ensures that 𝒵-compactifications are just completions. Thus, Question <ref> is aimed at 𝒵-compactifications of high-dimensional manifolds.  Although it's unknown whether these conditions are sufficient, in <cit.>, Guilbault and the author provided a best possible result saying that
  an m-manifold M^m (m≥ 5) satisfies Conditions (<ref>), (<ref>) and (<ref>) of Theorem <ref>, if and only if M^m × [0,1] admits a 𝒵-compactification.
  
  
  Clearly, completable manifolds are both pseudo-collarable and 𝒵-compactifiable. It is known that various manifolds such as Hilbert cube manifolds (satisfying Conditions (<ref>), (<ref>) and (<ref>)) and Davis's manifolds are both pseudo-collarable and 𝒵-compactifiable (see <cit.> or <cit.>) but not completable. Nevertheless, the relationship between pseudo-collarable manifolds and 𝒵-compactifiable manifolds are not well-understood. Here we answer the following open question in negative.
  
  <cit.>
  Are 𝒵-compactifiable manifolds pseudo-collarable?
  
  
  
  It is worth noting that Question <ref> is related to an older open question posed in <cit.> asking if
  a 𝒵-compactifiable open manifold can fail to be pseudo-collarable. However, Question <ref> is formulated in a more general setting with the manifolds with non-empty boundary taken into account, which is essential to our examples.
  
  The following theorem is the main result of this paper.
  
  There exist 𝒵-compactifiable manifold M^m (m≥ 4) with noncompact boundary which fail to be pseudo-collarable.
  
  
  
  About the organization of this paper: In Sections <ref>-<ref>, we recapitulate definitions
  and results from earlier papers <cit.> and <cit.> which
  provide preliminaries about neighborhoods of infinity, ends and peripheral perfect semistability condition. The main content is <ref>, where we prove Theorem <ref>. The starting point of our 4-dimensional examples is based on the construction of a contractible open 3-manifold W^3 which cannot be embedded in any compact 3-manifolds, see  <cit.> and <cit.>. After introducing W^3, we use the interaction of knot theory and combinatorial group theory to show that W^3 × [0,1) is a desired example. More specifically, in <ref>, we prove that the space W^3 × [0,1) is 𝒵-compactifiable. In <ref>, we show that W^3 × [0,1) does not satisfy Condition (<ref>) of Theorem <ref>. Finally, in <ref>, we produce examples of dimension greater than four, using the structure of W^3 × [0,1).
  
  
  
  
  § CONVENTIONS AND NOTATIONS 
  
  Manifolds are assumed to be PL. Equivalent results in the DIFF and TOP categories may be obtained in the usual ways. Unless stated
  otherwise, an m-manifold M^m is permitted to have a boundary, denoted
  ∂ M^m. We denote the manifold interior by
  *intM^m. For A⊆ M^m, the point-set
  interior will be denoted *Int_M^mA and the
  frontier by Fr_M^mA. A closed manifold is a compact
  boundaryless manifold, while an open manifold is a non-compact
  boundaryless manifold.
  
  A q-dimensional submanifold Q^q⊆ M^m is
  properly embedded if it is a closed subset of M^m and Q^q
  ∩∂ M^m=∂ Q^q; it is locally flat if each
  p∈*intQ^q has a neighborhood pair homeomorphic to
  (
  
  
  ℝ
  ^m,
  
  
  ℝ
  ^q) and each p∈∂ Q^q has a neighborhood pair
  homeomorphic to (
  
  
  ℝ
  _+^m,
  
  
  ℝ
  _+^q). By this definition, the only properly embedded codimension
  0 submanifolds of M^m are unions of its connected components; a more
  useful variety of codimension 0 submanifolds are the following: a codimension
  0 submanifold Q^m⊆ M^m is clean if it is a closed subset
  of M^m and Fr_MQ^m is a properly embedded locally flat
  (hence, bicollared) (  m-1)-submanifold of M^m. In that
  case, M^m\ Q^m is also clean, and Fr_M^mQ^m is a clean 
  codimension 0 submanifold of both ∂ Q^m and ∂(M^m\ Q^m).
  
  When the dimension of a manifold or submanifold is clear, we will often omit
  the superscript; for example, denoting a clean codimension 0 submanifold
  simply by Q. Similarly, when the ambient space is clear, we denote
  (point-set) interiors and frontiers by *IntA and Fr A.
  
  For any codimension 0 clean submanifold Q⊆ M^m, let ∂
  _MQ denote Q∩∂ M^m and *int_M
  Q=Q∩*intM^m; alternatively, ∂_MQ=∂
  Q\int(Fr Q) and *int_MQ=Q\∂ M^m.
  Note that *int_MQ is a m-manifold and ∂(
  *int_MQ)  =*int( Fr Q).
  
  A metric space Y is said to be an absolute neighborhood retract (abbreviated as ANR) if, for
  each closed subset A of a metric space X, every map f: A → Y has a continuous extension F: U → Y
  defined on some neighborhood U of A in X. Every topological manifold is an ANR <cit.>.
  
  
  
  § ENDS, *PRO-Π_1, AND THE PERIPHERALLY PERFECTLY SEMISTABLE
  CONDITION
  
  
  We recall the definition of the peripherally perfectly semistability developed in <cit.>, which are essential for this paper. Since all the examples we shall construct in <ref> satisfy the inward tameness (Condition (<ref>)) and the Wall finiteness obstruction (Condition (<ref>)), we omit the illustrations of those conditions. Readers are referred to <cit.> or <cit.> for details.
  
  
   §.§ Neighborhoods of infinity, partial neighborhoods of infinity, and ends.
   Let M^m be a connected manifold. A clean neighborhood of infinity
  in M^m is a clean codimension 0 submanifold N⊆ M^m for which
  M^m\ N is compact. Equivalently, a clean neighborhood
  of infinity is a set of the form M^m\ C where C is a
  compact clean codimension 0 submanifold of M^m. A clean compact exhaustion of
  M^m is a sequence {  C_i}  _i=1^∞ of clean
  compact connected codimension 0 submanifolds with C_i⊆*Int_M^mC_i+1 and ∪ C_i=M^m. By letting
  N_i=M^m\ C_i we obtain the corresponding
  cofinal sequence of clean neighborhoods of infinity. Each such N_i
  has finitely many components {  N_i^j}  _j=1^k_i. By
  enlarging C_i to include all of the compact components of N_i we can
  arrange that each N_i^j is noncompact; then, by drilling out regular
  neighborhoods of arcs connecting the various components of each Fr
  _M^mN_i^j (thereby further enlarginging C_i), we can
  arrange that each Fr_M^mN_i^j is connected. An N_i with
  these latter two properties is called a 0-neighborhood of infinity. For
  convenience, most constructions in this paper will begin with a clean compact
  exhaustion of M^m with a corresponding cofinal sequence of clean
  0-neighborhoods of infinity.
  
  Assuming the above arrangement, we let N_1^j,1=N_1^j and N_i+1^j,i+1= N_i+1^l⊆ N_i^j,i, where l ∈{1,…, k_i+1}. An end of M^m is determined by a
  nested sequence of components ε=(  N_i^j,i)
  _i=1^∞ of the N_i; each component is called a neighborhood
  of ε. In 3.3, we discuss components {N^j} of a neighborhood of infinity
  N without reference to a specific end of M^m. In that situation, we will refer to the
  N^j as partial neighborhoods of infinity for M^m (partial 0-neighborhoods if N is a
  0-neighborhood of infinity). Clearly every noncompact clean connected codimension
  0 submanifold of M^m with compact frontier is a partial neighborhood of infinity with respect to an appropriately chosen compact C; if its frontier is connected it is a partial 0-neighborhood of infinity.
  
  
  
  
   §.§ The fundamental group of an end
   For each end ε, we will define the fundamental group at
  ε; this is done using inverse sequences. Two inverse sequences of
  groups and homomorphisms A_0α_1⟵
  A_1α_2⟵A_3α_3
  ⟵⋯ and B_0β_1⟵
  B_1_β_2⟵B_3β_3
  ⟵⋯ are pro-isomorphic if they contain
  subsequences that fit into a commutative diagram of the form
  
      A_i_0      ^λ_i_0+1,i_1       A_i_1      ^λ_i_1+1,i_2       A_i_2      ^λ_i_2+1,i_3       A_i_3   ⋯
                                          
             B_j_0      ^μ_j_0+1,j_1       B_j_1      ^μ_j_1+1,j_2       B_j_2      ^μ_j_2+1,j_3      ⋯
  
  An inverse sequence is stable if it is pro-isomorphic to a constant
  sequence C*id⟵
  C*id⟵C*id
  ⟵⋯. Clearly, an inverse sequence is pro-isomorphic to
  each of its subsequences; it is stable if and only if it contains a
  subsequence for which the images stabilize in the following manner
  
      A_0      ^λ_1       A_1      ^λ_2       A_2      ^λ_3       A_3   ⋯
                                          
            Im( λ_1)       ^≅      Im( λ _2)       ^≅      Im( λ_3)       ^≅      ⋯
  
  where all unlabeled homomorphisms are restrictions or inclusions.
  
  Given an end ε=(  N_i^k_i)  _i=1^∞,
  choose a ray r:[1,∞)→ M^m such that r(
  [i,∞))  ⊆ N_i^k_i for each integer i>0 and form
  the inverse sequence
  
      π_1(  N_1^k_1,r(  1)  )  λ
      _2⟵π_1(  N_2^k_2,r(  2)  )
      λ_3⟵π_1(  N_3^k_3,r(
      3)  )  λ_4⟵⋯
  
  where each λ_i is an inclusion induced homomorphism composed with
  the change-of-basepoint isomorphism induced by the path .  r|
  _[  i-1,i]. We refer to r as the base ray and the
  sequence (<ref>) as a representative of the
  “fundamental group at ε based at r
  ” —denoted *pro-π_1(
  ε,r). We say the fundamental
  group at ε is stable if (<ref>) is a
  stable sequence. 
  
  If {H_i,μ_i} can be chosen so that each μ_i is an epimorphism,
  we say that our inverse sequence is semistable (or Mittag-Leffler, or pro-epimorphic).
  In this case, it can be arranged that the restriction maps in the bottom row of (<ref>) are
  epimorphisms. Similarly, if {H_i,μ_i} can be chosen so that each μ_i is a monomorphism,
  we say that our inverse sequence is pro-monomorphic; it can then be arranged that the
  restriction maps in the bottom row of (<ref>) are monomorphisms. It is easy to see that an
  inverse sequence that is semistable and pro-monomorphic is stable.  A nontrivial (but standard) observation is that both semistability and stability of ε do
  not depend on the base ray (or the system of neighborhoods if infinity used to
  define it). See <cit.> or <cit.>.
  
  Recall that a commutator element of a group H is an element of the form x^-1y^-1xy
  where x,y ∈ H; and the commutator subgroup of H; denoted [H,H] or H^(1), is the subgroup generated by all of its commutators. The group H is perfect if H = [H,H]. An inverse sequence of groups is perfectly semistable if it is pro-isomorphic to an inverse
  sequence.
  
  
      G_0 λ_1 G_1 λ_2G_2 λ_3⋯
  
  of finitely generated groups and surjections where each (λ_i) perfect. 
  
  
  
   §.§ Relative connectedness, relatively perfect semistability, and the peripheral perfect semistability condition.
  
  
  Let Q be a manifold and A⊆∂ Q. We say that Q is
  A-connected at infinity if Q
  contains arbitrarily small neighborhoods of infinity V for which A∪ V
  is connected.
  
  If A⊆∂ Q and Q is A-connected at infinity: let {  V_i} be a cofinal sequence
  of clean neighborhoods of infinity for which each A∪ V_i is connected;
  choose a ray r:[1,∞)→*IntQ such that r(
  [i,∞))  ⊆ V_i for each i>0; and form the inverse
  sequence
  
      π_1(  A∪ V_1,r(  1)  )  μ
      _2⟵π_1(  A∪ V_2,r(  2)  )
      μ_3⟵π_1(  A∪ V_3,r(  3)
      )  μ_4⟵⋯
  
  where bonding homomorphisms are obtained as in (<ref>). We
  say Q is A-perfectly π_1-semistable at infinity  (resp. A-π_1-stable at infinity) if (<ref>) is perfectly semistable (resp. stable).
  Independence of this property from the choices of {  V_i}
  and r follows from the traditional theory of ends by applying <cit.>. 
  
   Let M^m be an manifold and ε be an end of M^m
  
  
    *  M^m is peripherally locally connected at infinity if it contains arbitrarily small
  0-neighborhoods of infinity N with the property that each component N^j is ∂_M N^j-connected at infinity.
  
    * M^m  is peripherally locally connected at ε if ε has arbitrarily small 0-neighbor-hoods P that are ∂_M P-connected at infinity.
   
  
  
  An N with the property described in Condition (<ref>) above will be called a strong 0-neighborhood of infinity for M^m, and a P with the property described in Condition (<ref>)
  will be called a strong 0-neighborhood of ε. More generally, any connected partial
  0-neighborhood of infinity Q that is ∂_M Q-connected at infinity will be called a strong
  partial 0-neighborhood of infinity.
  
  Since every inward tame manifold M^m is peripherally locally connected at infinity <cit.>, that condition plays less prominent role than the next definition.
  
  Let M^m be a manifold and ε an end of M^m.
  
  
    * M^m is peripherally perfectly π_1-semistable at infinity if it contains arbitrarily small strong
  0-neighborhoods of infinity N with the property that each component N^j is ∂_M N^j-perfectly π_1-semistable at infinity.
  
  
    * M^m is peripherally perfectly π_1-semistable at ε if ε has arbitrarily small strong 0-neighborhoods P that are ∂_M P-perfectly π_1-semistable at infinity.
  
   
  
  
  If M^m contains arbitrarily small 0-neighborhoods of infinity N with
  the property that each component N^j is ∂_M N^j-perfectly semistable at infinity, then those components provide arbitrarily small neighborhoods of the ends satisfying the necessary perfectly semistable condition. Thus, it's easy to see peripheral perfect π_1-semistability at infinity implies peripheral perfect π_1-semistability at each end.
  
  
  A locally finite polyhedron P is inward tame if it contains
  arbitrarily small polyhedral neighborhoods of infinity that are finitely
  dominated. 
  
  
  
  
  
  
  
  
  
  
  
  
  Let M^m be inward tame and peripherally locally connected at infinity. Then every strong
  partial 0-neighborhood of infinity Q⊆ M^m is ∂_M Q-π_1-semistable.
  
  
  By <cit.>, Q\∂ M^m is inward tame. Using <cit.>, proving Q⊆ M^m is ∂_M Q-π_1-semistable is 
  equivalent to showing that Q\∂ M^m has semistable pro-π_1. The latter follows from a slight modification
  of <cit.> (or a fact that "degree 1 maps between manifolds induce
  surjections on fundamental groups" as suggested in <cit.>).
  
  
  
  
  § 𝒵-COMPACTIFIABLE BUT NOT PSEUDO-COLLARABLE MANIFOLDS WITH BOUNDARY
  
  We shall prove Theorem <ref> by focusing on the fundamental group at infinity of certain contractible manifolds with noncompact boundary.
  
  
  
   §.§ 𝒵-compactification of W^3 × [0,1)
  
  Before constructing the contractible open 3-manifold W^3, we first give a general result to guarantee that the product of W^3 with [0,1) is 𝒵-compactifiable.
  
   
  Let M be a contractible open manifold. Then M × [0,1) is 𝒵-compactifiable.
  
  
  The proof of Proposition <ref> is based on the next result.
  
  <cit.>
  
  Suppose that X is a compactum and Z ⊂ X a closed subset such that
  
  
    *  Int Z = ∅,
  
    *      X = n < ∞,
  
    *   for every k = 0, 1, …, n, every point z ∈ Z and every neighborhood U of z, there exists a neighborhood V of z such that α: S^k → V \ Z extends to α̃: B^k+1→ U \ Z and
  
    *     X \ Z is an ANR.
  
  Then X is an ANR and Z ⊂ X is a 𝒵-set.
  
  
  
  The strategy is to show the one-point compactification M × [0,1) ⊔{∞} is a 𝒵-compactification. To that end, we verify Conditions (<ref>)-(<ref>) in Lemma <ref>.  Verifying Conditions (<ref>) and (<ref>) are trivial. The standard ANR theory shows that M × [0,1) is an ANR. To check Condition (<ref>), it suffices to show that for each neighborhood U of infinity we may choose some neighborhood V of infinity in U which contracts in itself. Let us pick a compactum C⊂ M and a number a∈ (0,1) such that K = C × [0,a] containing M× [0,1)\ U.  Then V =  M× [0,1) \ K = (M \ C) × [0,1) ∪ M × [a,1) is the desired neighborhood of infinity. To see that, we observe that there exists an obvious deformation retract of V onto M ×{b}, where b∈ [a,1). Since M ×{b} deformation retracts to a point in itself, the composition of the above two deformation retractions provides a contraction of V in itself.
  
  
  
  
   §.§ The fundamental group at infinity of W^3 × [0,1)
  
  
  The result below and the necessity of Theorem <ref> imply that the only way to see that W^3 × [0,1) is not pseudo-collarable is to show that the peripherally perfect semistability at infinity fails.
  
  <cit.>
  Let M be a connected open manifold. If M has finite homotopy type, then M × [0, 1) is one-ended and inward tame, with σ_∞(M × [0,1))=0.
  
  
  First, we prepare some algebraic devices.
  
  
  A group G is said to be hypoabelian if the following equivalent conditions are satisfied:
  
  
    * G contains no nontrivial perfect subgroup.
  
    * The transfinite derived series[The transfinite derived series of a group is an extension of its derived series such that the successor of a given subgroup is its commutator subgroup, and subgroups at limit ordinals are given by intersecting all previous subgroups.] terminates at the identity.
  
  
  
  
  
  
  
    * Every abelian group is hypoabelian.
  
  
    * Solvable groups, residually solvable groups and free groups are hypoabelian.
  
  
    * Every right-angled Artin group is hypoabelian.
  
  
    * The Baumslag-Solitar groups BS(1,n) are solvable, thus, hypoabelian.
  
  
  
  
  Hypoabelianity is closed under a number of operations:
  
  
  
  
    * Free products of hypoabelian groups are hypoabelian.
  
  
    * Every extension of a hypoabelian group by a hypoabelian group is hypoabelian.
  
  
    * Let H be a subgroup of a group G. Since conjugate subgroups are isomorphic, if H is hypoabelian, so are the conjugates of H.
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  A split HNN-extension G of a group B is an HNN-extension G = ⟨ B,t | t^-1η(a)t = ι(a) ⟩, where a∈ A such that one of the injections η,ι: A ↪ B (say η) splits, that is, B is a split extension N ⟶ B  ϕη⟵ A.
  
  
  
  
  A group G is a split amalgamated free product iff G may be expressed as an amalgamated free product B∗_A C, where one of the injections A ↪ B or A ↪ C splits.
  
  
  The results below show that hypoabelianism can be preserved under delicate group operations.
  
  <cit.>
  
  Split amalgamated free products of hypoabelian groups are hypoabelian.
  
  
  Given a nonempty set of groups {G_λ|λ∈Λ} together with a group H which
  is isomorphic with a subgroup H_λ of G_λ by means of monomorphism ϕ_λ: H → G_λ. There is a frequently used term known as the free product of the G_λ's with the amalgamated subgroup H. Roughly speaking, this is the largest
  group generated by the G_λ's in which the subgroups H_λ are identified with H by ϕ_λ. Such groups are known as 
  generalized free products. Readers are referred to <cit.> for a more explicit description.
  
  Let A be a hypoabelian group. Then there exists a transfinite derived series of A terminates at some ordinal α. We call all derived subgroups a filtration of A. Let B be a second hypoabelian group with a transfinite derived series terminates at some ordinal β, and let a∈ A, b∈ B be two non-identity elements. There exist ordinals m∈{0,1,…,α} and n∈{0,1,…,β} such that the generators a ∈ A^(m)\  A^(m+1) and
  b ∈ B^(n)\ B^(n+1), where A^(m) and B^(n) denote the m-th
  and n-th derived subgroups of A and B, respectively. Then we say the filtrations of A and B are compatible if cosets aA^(m+1) and
  bB^(n+1) have the same order in factor groups A/A^(m+1) and B/B^(n+1), respectively. 
  
  
  
  The generalized free product of two hypoabelian groups A and B amalgamating a cyclic subgroup is hypoabelian, provided that the filtrations of A and B are compatible.
  
  
  The proof uses a similar argument in the proof of <cit.>. Let a ∈ A, b ∈ B be
  two non-identity elements. Assuming the notions above, we establish the amalgamated product G = {A∗ B; a = b} of A and B
   with elements a and b identified. Let
  C be the central product of A/A^(m+1) and B/B^(n+1) amalgamating aA^(m+1) with
  bB^(n+1):
  
      C = {A/A^(m+1)× B/B^(n+1);aA^(m+1)=bB^(n+1)}.
  
  Define a homomorphism ϕ: G → C sending a_i to (a_i A^(m+1), B^(n+1))
  and b_j to (A^(m+1), b_j B^(n+1)), where a_i,b_j are generators of A,B, respectively. Let K = ϕ and D = ⟨ a ⟩ = ⟨ b ⟩. Then K ∩ D = 1.
  By <cit.>, K is actually a free product of conjugates of subgroups of A and B, and a free group. Thus, K is
  hypoabelian. Since G is an extension of a hypoabelian group by a hypoabelian group, G is also hypoabelian.
  
  
  
  
  
  
  
  Without the compatibility, we may not conclude that the cyclic amalgamation is hypoabelian even though A and B are assumed to be solvable or nilpotent. The proofs of <cit.> and <cit.> are flawed since both the arguments rely on the compatibility in the central product structure. It follows from <cit.> that there are torsion-free metabelian groups whose commutator quotients are nontrivial with prime exponent for any odd prime p. Choosing groups A and B with different primes p, say 3 and 5, gives infinite order elements a∈ A and b∈ B (not in the commutator subgroup) that have different orders modulo the commutator subgroup. Hence, the central product structure fails.
  
  
  
  
  
  
  
  
  
  
  The following lemma quoted from <cit.> indicates the strategy we will use to prove the main theorem.
   
  Let 
  
      G_0 ← G_1 ← G_2 ←⋯
  
  be an inverse sequence of groups with surjective but non-injective bonding homomorphism. Suppose each G_i is a hypoabelian group. Then the inverse sequence is not perfectly semistable.
  
  
  Now, we reproduce a famous example originially proposed by Bing, i.e., a 3-dimensonal contractible open manifold W^3,[Haken <cit.> first proved that W^3 embed in no compact 3-manifold. Recently, the author <cit.> showed that W^3 cannot even embed in a much more general compact space — a compact, locally connected and locally 1-connected 3-dimensional metric space.] see <cit.>, <cit.> or <cit.>.
  
  Let {T_l|l = 0,1,2,…} be a collection of disjoint solid tori standardly embedded in S^3. Let the solid torus T_l' be embedded in IntT_l as in Figure <ref>. The cube C_l contains a tamely embedded thickened trefoil knot with the interior of a standard ball removed. In this paper, the choice of such a knot is not unique. Alternating (or more generally adequate) knots of nonzero writhe number and whose knot groups are hypoabelian can be applied to create more options for K (look ahead to Corollary <ref>). To produce more examples, one simply removes a standard ball from a thickened knot K and use the resulting space to replace the trefoil-knotted part in C_l. Let the oriented simple closed curves α_l, β_l, γ_l and δ_l be as shown in Figure <ref>. The curves α_l and β_l are transverse in ∂ T_l, and meet at the point q_l ∈∂ T_l. In a similar fashion, the curves γ_l and δ_l are transverse in ∂ T_l', and meet at the point p_l ∈∂ T_l'. For l ≥ 1, let L_l = T_l \IntT_l'. Define an embedding h_l+1^l: T_l → T_l+1 so that T_l is carried onto T_l+1' with h_l+1^l(α_l) = δ_l+1 and h_l+1^l(β_l) = γ_l+1. W^3 is the direct limit of the T_l's and denoted as W^3 = lim_l→∞(T_l,h_l+1^l). That is equivalent to viewing W^3 as the quotient space: ⊔_l T_l  W^3, where ⊔_l T_l is the disjoint union of the T_l's and q is the quotient map induced by the relation ∼ on ⊔_l T_l. If x∈ T_i and y∈ T_j, then x ∼ y iff there exists a k larger than i and j such that h_k^i(x) = h_k^j(y), where h_t^s = h_t^t-1∘ h_t-1^t-2∘⋯∘ h_s+2^s+1∘ h_s+1^s for t > s. Let ι_l: T_l ↪⊔_l T_l be the obvious inclusion map. The composition q ∘ι_l embeds T_l in W^3 as a closed subset. The injectivity follows from the injectivity of h_k+1^k. It is closed since for j > l the set h_j^l(T_l) is closed in T_j. Let T_l^* denote q ∘ι_l(T_l). T_l^* is embedded in T_l+1^* just as the way h_l+1^l(T_l) (= T_l+1') is embedded in T_l+1. 
  Hence, Figure <ref> can be viewed as a picture of the embedding of T_l^* in T_l+1^*. In general, for k > l, T_l^* is embedded in T_k^* just as h_k^l (T_l) is embedded in T_k. 
  
  
  
  
  
  W^3 as constructed above is an contractible open connected 3-manifold.
  
  
  See the proof of <cit.>.
  
  
  To show that the fundamental group at infinity of W^3 × [0,1) is not perfectly semistable, we start from a specific cofinal sequence of neighborhoods {N_j}_j=0^∞ of infinity of W^3, where N_j = ∪_l=j^∞ L_l^* and L_l^* = T_l+1^*\Int T_l^*.  Taking the product of N_j with [0,1) we define V_j = N_j × [0,1) ∪ W^3 × [1-1/j+2,1). Then {  V_j}_j= 1^∞ is a cofinal sequence of clean neighborhoods of infinity for which each N_0 ∪ V_j is connected. Form an inverse sequence in which baserays are suppressed
  
      π_1(  N_0 ∪ V_1)  μ
      _2⟵π_1(  N_0 ∪ V_2)
      μ_3⟵π_1(  N_0∪ V_3)  μ_4⟵⋯
  
  Since V_j deformation retracts onto W^3 ×{1-1/j+2} and W^3 is contractible, π_1(  N_0 ∪ V_j)≅π_1(  L_0^* ∪⋯∪ L_j-1^*/∂ T_j^*). Thus, Sequence (<ref>) is pro-isomorphic to the following sequence
  with basepoints suppressed
  
      π_1(  L_0^*/∂ T_1^* )  μ
      _2⟵π_1(  L_0^* ∪ L_1^* /∂ T_2^* )
      μ_3⟵π_1( L_0^* ∪ L_1^* ∪ L_2^*/∂ T_3^* )  μ_4⟵⋯
  
  To better understand each term in Sequence (<ref>), we consider the knot complement K_j defined as K_j = S^3 \Inth_j^0 (T_0) for j ≥ 1. That is, K_j is obtained by sewing the solid torus S^3 \IntT_j to T_j \Inth_j^0 (T_0) along ∂ T_j. By the construction of W^3, the pair (T_j^*, T_0^*) is homeomorphic to (T_j, h_j^0 (T_0)).
  Thus, 
  
      π_1 ( (T_j \Int h_j^0 (T_0)) /∂ T_j  ) ≅π_1( L_0^* ∪⋯∪ L_j-1^*/∂ T_j^* ).
   
  By <cit.> knot group π_1(K_j) is isomorphic to π_1 ( (T_j \Int h_j^0 (T_0)) /∂ T_j  ).  Hence, Sequence (<ref>) is pro-isomorphic to a sequence of knot groups
  
  
      π_1(  K_1 )  μ
      _2⟵π_1(  K_2 )
      μ_3⟵π_1( K_3 )  μ_4⟵⋯
  
  The details about the construction of knot space K_j can be found in <cit.>. It's clear that π_1(K_1) is isomorphic to π_1(S^3 \ K). As one option, we can choose a fibered knot for K. Recall that a knot k ⊂ S^3 is fibered if its exterior admits a locally trivial fibration over S^1.
  
  
  Fibered knot groups are hypoabelian.
  
  
  By <cit.>, the commutator subgroup of the fibered knot group is finitely generated, i.e., free. 
  
  
  The Alexander polynomial of fibered knot is monic <cit.> <cit.> <cit.>, i.e., the coefficient of the highest degree term of the normalized Alexander polynomial is a unit ± 1. By the duality of the Alexander polynomial, its lowest degree term is also ± 1. This criterion is sufficient for alternating knots <cit.> and prime knots up to 10 crossings <cit.>. In general, the converse is not true. 
  The Alexander polynomial of any fibered knot is also the Alexander polynomial of infinitely many nonfibered knots.
  
  
  
  
  
  
  
  
  
  
  
  Next, we briefly discuss the concepts of crossing number, writhe and linking number and the connections among them. Crossing number is the minimal number
  of simple self-intersections which appear in a knot diagram or link diagram, i.e., a planar picture (regular projection) of a knot or link, of a given type.
  The signs of crossings in a knot diagram (equipped with an orientation) are defined by the (right-hand) rules in Figure <ref>.
  
  The writhe of a knot k, denoted by w(k), is the total number of positive crossings minus the total number of negative crossings. The writhe is the same for
  either orientation. Suppose we have an oriented link diagram with two components C_1 and C_2. Let n_1 (resp. n_2) be the sum of the signs of crossings between a pair of components C_1 and C_2 of a link in which the over arc belongs to C_1 (resp. C_2). Then the linking number, denoted lk(C_1,C_2), is equal to (n_1+n_2)/2. This is the same labeling used to compute the writhe of a knot, though in this case only crossings that involve both C_1 and C_2 are counted. The other seven equivalent definitions of the link number can be found in <cit.>. Consider a closed ribbon with disjoint boundary components C_1,C_2 and the axis of the ribbon C. Then the relationship between the linking number and the writhe of C can be expressed by the White formula which was discovered independently by Cǎlugǎreanu<cit.>, Fuller<cit.>, Pohl<cit.>, and White<cit.>:
  
      lk(C_1,C_2) = t(C_1,C_2) + w(C),
  
  where t(C_1,C_2) is the number of (full) twists of C_1 and C_2.
  
  
  We invoke two important techniques of constructing knots to analyze π_1(K_j) (j≥ 2).  The first is the twisted Whitehead doubling.
  
  
  Let K_P be a knot in S^3 and V_P an unknotted solid torus in S^3 with K_P⊂ V_P ⊂ S^3. Assume K_P is not contained in a 3-ball of V_P. Let K_C ⊂ S^3 be another knot and let V_C be a tubular neighborhood of K_C in S^3. A homeomorphism h: V_P → V_C which maps a meridian of S^3 \IntV_P onto a longitude of V_C and K_P onto a knot K_W=h(K_P). We say K_C is a companion of any knot K_W constructed (up to knot type) in this manner, and K_W is called a satellite of K_C. If h is faithful, meaning that h takes the preferred longitude and meridian of V_P, respectively, to the preferred longitude and meridian of V_C, consider K_P ⊂ V_P as in Figure <ref>,
  we say K_W is an untwisted Whitehead double of K_C. By unclasping and reconnecting K_W shown in Figure <ref>, we obtain a link, which is the disjoint union of two boundary components of a closed ribbon. The "preferred longitude" means that the link number of those two components is zero. If K_W contains no twists, by the White formula, this is equivalent to say that K_C has writhe zero. Note that a Whitehead doubling only consists of three actions: clasp (-.2
      < g r a p h i c s >
  ), twist (-.2
      < g r a p h i c s >
   or -.2
      < g r a p h i c s >
  )
  and doubling of the crossings (-.2
      < g r a p h i c s >
   or -.2
      < g r a p h i c s >
  ) of K_C. In this situation, the first two do not contribute to the link number. Instead of using the White formula, one can obtain the same conclusion through such an observation. In the notion of Rolfsen <cit.>, the link number is also known as the twisting number. If the linking number is nonzero, then K_W is a twisted Whitehead double. For instance, the satellite knot in Figure <ref> is a 3-twisted Whitehead double of a trefoil knot, where the twisting arises from the writhe of a trefoil knot. In this case, h is not faithful. The pair (V_P, K_P) is called
  pattern of K_W. 
  
  
  
  
  The second tool is based on a type of connected sum of a pair of manifolds (M_1^m,N_1^n)#(M_2^m,N_2^n), where N_i^n is a locally flat submanifold of M_i^m. Treat the above pair as (S^3, k_1) # (S^3,k_2) where k_i are tame knots. Removing a standard ball pair (B_i^3,B_i^1) from (S^3,k_i) and gluing the resulting pairs by a homeomorphism h: (∂ B_2^3,∂ B_2^1) → (∂ B_1^3,∂ B_1^1) to form the pair connected sum. For convenience, we use k_1 # k_2 other than pairs of manfolds. See <cit.> for details. Let 𝒦_1 (=K) be a knot corresponding to the knot space K_1, and denote a twisted Whitehead double of 𝒦_1 by 𝒦_1^Wh. Let 𝒦_2 = 𝒦_1^Wh#𝒦_1. Then an argument similar to <cit.> implies that 
  π_1(S^3 \𝒦_2) ≅π_1(K_2). Likewise, one can further find a knot 𝒦_3 on 3rd stage which is a connected sum of a twisted Whitehead double of 𝒦_2 and 𝒦_1. By iteration, a knot 𝒦_j can be viewed as 𝒦_j-1^Wh#𝒦_1 so that π_1(S^3\𝒦_j)≅π_1(K_j). Using Lemma <ref>, Sequence (<ref>) above not being perfectly
  semistable is equivalent to π_1(K_j) being hypoabelian, thus, our next step is to show that each knot group in Sequence (<ref>) is hypoabelian.
  
  
  
  
  The group of the link complement of the Whitehead link (which is isomorphic to π_1(V_P\ K_P) as shown in Figure <ref>) is hypoabelian.
  
  
  
  Consider a (unique) epimorphism ϕ from π_1(V_P\ K_P) to the infinite cyclic group ⟨ t ⟩
  sending each meridian of V_P to t. Since the Whitehead link is fibered <cit.>, the kernel ϕ of ϕ is isomorphic to the fundamental group of the fiber surface, which is a free group of finite rank, thus, hypoabelian. Then
  π_1(V_P\ K_P) is a ension of two hypoabelian groups, hence, hypoabelian.
  
  
  
  
  Let K_C be a non-trivial knot, K_W be a satellite knot and (V_P,K_P) be the pattern. Suppose the knot group of K_C and π_1(V_P \ K_P) are hypoabelian, and that the Alexander polynomial of K_W is nontrivial. Then the knot group of K_W is hypoabelian.
  
  
  
  Denote the various groups π_1(S^3\ K_C), π_1(S^3 \ K_W) and π_1(V_P\ K_P) by G_C, G_W and H, respectively. By Seifert-van Kampen theorem, G_W = H ∗_π_1(∂ V_P) G_C. The amalgamated free product of H and G_C with subgroups π_1(∂ V_P) ⊂ H and π_1(∂ V_C)⊂ G_C identified via a homeomorphsim h as described in Definition <ref>.
  
  Regard H as the fundamental group of the complement of a 2-component link K_P ∪ m_P, where m_P is a meridian of V_P. Let λ,t be the homotopy class of the longitude l_P and meridian m_P of V_P, respectively, in H. Using a Wirtinger presentation after replacing all meridional generators of m_P except t by element of the first commutator subgroup H^(1) we obtain 
  
      H = ⟨ t, ũ_i, λ | R̃_j(ũ_i^t^ν,λ) ⟩,
  
  where ũ_i∈ H^(1), ũ_i^t^ν=  t^νũ_i t^-ν, ν∈ℤ, R̃_j(ũ_i^t^ν,λ) represents a  relator list of this Wirtinger presentation; i,j are finite indices. The augmentation ϕ: G_W →ℤ induces a homomorphism ϕ: H →ℤ by sending t→ 1 and λ,ũ_i → 0. Let ℛ be ϕ. Apply Reidemeister-Schreier method to see that the presentation of the commutator subgroup G_W^(1) of G_W can be written as a free product with amalgamation:
  
      G_W^(1) = ℛ∗_F(∏_ϱ^∗ t^ϱ G_C^(1) t^-ϱ),
  
  where G_C^(1) is the commutator subgroup of G_C and ϱ ranges over ℤ. When the winding number n of K_P in V_P is zero, F is a free group of infinite rank generated by λ^t^ϱ = t^ϱλ t^-ϱ. Otherwise, F is a free group of rank n and ϱ = 0, 1, …, n-1.
  A detailed proof can be found in <cit.>. By presentations (7) and (8) in <cit.>, we obtain a presentation of ℛ
  
      ℛ = ⟨ũ_i^t^ϱ, λ^t^ϱ | R̃_j^t^ϱ(ũ_i^t^ν,λ)⟩,
  
  where R̃_j^t^ϱ(ũ_i^t^ν,λ)=  t^ϱR̃_j(ũ_i^t^ν,λ) t^-ϱ is a relator. The second group in the amalgamation G_W^(1) has a presentation (derived from the presentation of G_C^(1) <cit.>)
  
      ∏_ϱ^∗ t^ϱ G_C^(1) t^-ϱ = ⟨û_k^t^ϱ, λ^t^ϱ | R̂_ι^t^ϱ(û_k^t^ϱ), (λ^t^ϱ)^-1ŵ^t^ϱ(û_k^t^ϱ) ⟩,
  
  where û_k, ŵ(û_k) ∈ G_C^(1),  û_k^t^ϱ = t^ϱû_k t^-ϱ, R̂_ι^t^ϱ(û_k^t^ϱ)=  t^ϱR̂_ι(û_k^t^ϱ) t^-ϱ represents a relator,
  and ŵ^t^ϱ(û_k^t^ϱ) = t^ϱŵ(û_k^t^ϱ)t^-ϱ; k is a finite index.
  
  To see that G_W is hypoabelian, it suffices to show that the third derived series of G_W is hypoabelian. Note that the derived series of H (resp. G_C) does not stabilize at the third commutator subgroups H^(3) (resp. G_C^(3)). In fact, <cit.> implies that λ vanishes at G_W^(3). Dropping λ and using generators ṽ_κ∈ H^(3) and v̂_τ∈ G_C^(3) (while keeping λ in G_W^(2)), we obtain 
  
      G_W^(3) =  ⟨ṽ_κ^λ^ϱ, v̂_τ^λ^ϱ | r̃_ζ^λ^ϱ(ṽ_κ^λ^μ), r̂_ς^λ^ϱ(v̂_τ^λ^μ)⟩,
  
  where μ∈ℤ,  ṽ_κ^λ^ϱ =λ^ϱṽ_κλ^-ϱ,  r̃_ζ^λ^ϱ(ṽ_κ^λ^μ) = λ^ϱr̃_ζ(ṽ_κ^λ^μ)λ^-ϱ represents a relator, v̂_τ^λ^μ = λ^ϱv̂_τλ^-ϱ and r̂_ς^λ^ϱ(v̂_τ^λ^μ) = λ^ϱr̂_ς(v̂_τ^λ^μ)λ^-ϱ represents a relator; κ, τ, ζ and ς are finite indices. Since G_C^(3),
  λ G_C^(3)λ^-1, …, have disjoint sets of generators and so do H^(3), λ H^(3)λ^-1, …, G_W^(3) can be rewritten as a free product of free products
  
      G_W^(3) = (∏_ϱ^∗λ^ϱ H^(3)λ^-ϱ)∗(∏_ϱ^∗λ^ϱ G_C^(3)λ^-ϱ).
  
  It follows from the hypotheses that H^(3) and G_C^(3) are hypoabelian. The free product is hypoabelian, for the conjugates of H^(3) and G_C^(3) are hypoabelian.
  
  
  
  
  
    * The requirement that the Alexander polynomial of K_W is nontrivial in Lemma <ref> cannot be omitted. Removing it might cause counterexamples such as the untwisted Whitehead double of any nontrivial knot. It's well-known that the knot group of an untwisted Whitehead double of any nontrivial knot is perfect.
  
  
    *  That the Alexander polynomial of K_W is nontrivial alone cannot guarantee that the knot group of K_W is hypoabelian. Here is an example suggested by Professor Ian Agol <cit.>. Consider the (p,q) cable of K, denoted K_p,q, is a statellite knot with pattern the (p,q) torus knot, T_p,q. More precisely, K_p,q is the image of a torus knot living on the boundary of a tubular neighborhood of K. Thus, p is the number of times K_p,q traverses the longitudinal direction of K, and q is the meridional number. By <cit.>, the Alexander polynomial of a cable knot is determined by
  
      Δ_K_p,q(t) = Δ_T_p,q(t)·Δ_K(t^p).
  
  Let K be an untwisted Whitehead double of a nontrivial knot. Since K has trivial Alexander polynomial,  G^(1) = G^(2). Therefore, when one takes a cable of this knot, its derived series will agree with that of torus knot T_p,q. Because the knot group of any T_p,q is hypoabelian, the transfinite derived series G_p,q^(i) (i ≥ 0) of the knot group of K_p,q terminates at the commutator subgroup of the pattern knot at G_p,q^(ω), where ω is the first ordinal. 
  
  
    * It's impossible to weaken the hypothese that the knot group of K_C and π_1(V_P \ K_P) are hypoabelian in Lemma <ref> to the Alexander polynomials of K_C and K_P are nontrivial. One emulates the example in (<ref>) above to produce a cable of a cable of an untwisted Whitehead double of an nontrivial knot such that the patterns are torus knots.
  
  
  
  The following result answers a question posed in <cit.> and  <cit.> in affirmative.
  
  Let K_C be a non-trivial knot. If π_1(S^3\ K_C) is hypoabelian, then the knot group of a twisted Whitehead double K_W of K_C is hypoabelian.
  
  
  An easy exercise in <cit.> shows that the Alexander polynomial of K_W is nontrivial. Apply Lemma <ref> and Lemma <ref>.
  
  
  
  Let K_1 and K_2 be knots and G_1 and G_2 be the corresponding knot groups. If G_i is hypoabelian, then the 
  knot group of K_1 # K_2 is hypoabelian.
  
  
  Note that π_1(S^3 \ K_1 # K_2) can be written as a free product of G_1 and G_2 amalgmating an infinite cyclic group generated by 
  the homotopy class of meridian of K_1 and K_2. The result follows readily from Proposition <ref>. 
  
  
  
  
  
  
  
  
  W^3 × [0,1) is not peripherally perfectly π_1-semistable at infinity, hence, not pseudo-collarable.
  
  
  Iterating Lemmas <ref>, <ref>, Corollary <ref> and the hereditary of hypoabelianism under group extension for π_1(K_i)'s in Sequence (<ref>) shows that Sequences (<ref>) and 
  (<ref>) are pro-isomorphic to a sequence of hypoabelian groups. Furthermore, Lemma <ref> or a direct observation ensures that Sequence (<ref>) is semistable.  However, Lemma <ref> shows that the perfectness fails. Applying the necessity Theorem <ref> completes the proof.
  
  
  Let D be a link diagram, and x a crossing of D. Associated to D and x are two
  link diagrams, each with one fewer crossing than D, called the A-resolution and
  B-resolution of the crossing. See Figure <ref>. A knot or link diagram is A-adequate if, when all the crossings are resolved using A-resolution, each resulting simple closed curve runs through the remnants of each crossing at most once. B-adequacy is
  defined similarly using the B-resolution at each crossing. A diagram is adequate if it is both
  A- and B-adequate. It follows from a direct observation or <cit.> that adequacy is preserved under the "taking parallels" action such as the doubling action in a Whitehead double. An adequate knot is a generalization of an alternating knot. Many nice topological and geometric properties of adequate knots can be found in <cit.>.
  
  
  
  
  There are infinitely many W^3 × [0,1) (up to homeomorphism) which are not pseudo-collarable.
  
  
  It suffices to show that there are infinitely many distinct choices (up to ambient isotopy in S^3) for the knot K in Figure <ref>. Take a fibered adequate knot K with writhe ≥ 3, e.g., a trefoil knot.  Note that the connected sum of fibered knots is fibered, thus, the corresponding knot group is hypoabelian. By the definition of adequate knot, the connected sum of fibered adequate knots is fibered adequate, one can produce an infinite sequence {𝒦_i} of distinct fibered adequate knots by taking the i-fold connected sum of copies of K. We further employ the Tait conjectures <cit.> to assure that the writhe of each 𝒦_i is greater than or equal to 3i. Next, we shall show that W^3 × [0,1) constructed based on distinct pair (𝒦_i,𝒦_j) (i ≠ j) are not homeomorphic. Note that each compact subset of W^3 embeds in S^3. For a cube with handles R_l ⊂ W^3 at stage l in the construction, we take a homeomorphism of S^3 onto itself taking R_l onto a canonical position (e.g., see Figure <ref>). Since the Tait conjectures ensure the additivity of crossing numbers when taking connected sum, and the crossing number is a topological invariant, there is no ambient isotopy which moves 𝒦_i to 𝒦_j in S^3. Hence, the W^3 × [0,1) constructed based on 𝒦_i is not homeomorphic to the one constructed using 𝒦_j.
  
  
  
  
  
  
  
  
  
  
  
  
  
   §.§ High-dimensional examples
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  In this section, we shall produce examples of higher dimensions modeled on W^3 × [0,1) constructed in <ref>.
  
  The case m = 4 is just Propositions <ref> and <ref>. 
  
  For cases m >4, let's let W^3 be as in <ref>. For convenience,
  we denote W^3 × [0,1) by X. We shall show that X × S^(m-4) is 𝒵-compactifiable but not pseudo-collarable.
   Let X̂= X ⊔{∞} be the one-point compactification of X. By Proposition <ref>, X̂ is a 𝒵-compactification of X. We invoke an alternative definition of 𝒵-compactification that there exists a homotopy H: [0,1] ×X̂→X̂ such that H_0 = Id_X̂ 
  and H_t(X̂)⊆X̂ - {∞} for all t > 0. Subsequently, we define a homotopy H' = H ×Id:   [0,1] ×X̂× S^(m-4)→X̂× S^(m-4)
  such that H'_0 = Id_X̂× S^(m-4) and H'_t(X̂× S^(m-4))⊆ (X̂ - {∞})× S^(m-4) for all t > 0.
  Thus, X̂× S^(m-4) is a 𝒵-compactification of X × S^(m-4).
  
  To see the breakdown of pseudo-collarability, we will consider the fundamental group at infinity of X × S^(m-4). More specifically, it suffices to show that the peripherally perfect π_1-semistability fails. One just need to consider the same cofinal sequence (<ref>) of neighborhoods of infinity as depicted in Section <ref>, and take the product of Sequence (<ref>)  with S^(m-4). That gives the sequence (with baserays surpressed) as follows.
  
  
      π_1(  (N_0 ∪ V_1)× S^(m-4))  ⟵π_1(  (N_0 ∪ V_2) × S^(m-4))
      ⟵π_1(  (N_0∪ V_3) × S^(m-4)) ⟵⋯
  
  
  
  
  For m ≥ 6, Sequence (<ref>) is pro-isomorphic to Sequence (<ref>), which is an inverse sequence of hypoabelian groups. Hence, X × S^(m-4) is not peripherally perfectly π_1-semistable,
  using Lemma <ref>.
  
  For m = 5, Sequence (<ref>) is pro-isomorphic to
  
  
      π_1(N_0 ∪ V_1) ×ℤ⟵π_1(N_0 ∪ V_2) ×ℤ⟵π_1(N_0∪ V_3)×ℤ⟵⋯
  
  By the work in <ref>, π_1(N_0 ∪ V_j) and ℤ are hypoabelian, therefore, Sequence (<ref>) is an inverse sequence of hypoabelian groups. Apply Lemma <ref> and the necessity of Theorem <ref>.
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  § ACKNOWLEDGEMENTS
  
  I would like to thank Professor Craig Guilbault for introducing hypoabelian groups to me, and his marvelous instinct that W^3 × [0,1) is a counterexample. I also thank the anonymous referee for very helpful and constructive comments, especially, for pointing out the error in the proof of Proposition <ref> of the initial version of the paper.
  
  
  
  
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