AROUND THE BING-BORSUK CONJECTURE

JONATHAN HEINDL

SUBMITTED IN PARTIAL FULFILLMENT

OF THE REQUIREMENTS FOR THE

DEGREE OF MASTER OF SCIENCE IN MATHEMATICS

NIPISSING UNIVERSITY

SCHOOL OF GRADUATE STUDIES

NORTH BAY, ONTARIO

© Jonathan Heindl October 2012 Page | iv

Abstract

In this paper we examine a problem known as the Bing-Borsuk conjecture. This conjecture states that every n-dimensional, n ∈ N, homogeneous compact ANR is a topological n-manifold. This was proposed in 1965 by R. H. Bing and K. Borsuk, who also proved the n < 3 cases. However, it is still unknown if this holds for all n ∈ N. We examine the central results related to this conjecture and the progress that has been made toward solving it. Further, we examine why the Bing-Borsuk conjecture is stronger than the Poincaré conjecture. In other words, the n = 3 case of the Bing-Borsuk conjecture implies the Poincaré conjecture. The Poincaré conjecture (recently proven by Grigori Perelman in [44], [45], [46]) states that every homotopy 3-sphere is homeomorphic to a 3-sphere, and is one of the most famous and significant results in topology. This demonstrates why the Bing-Borsuk conjecture is of fundamental importance in topology and mathematics in general. This also displays the difficulty of solving the Bing-Borsuk conjecture, given that the Poincaré conjecture took nearly a century to prove. Page | v

Table of Contents

Abstract……………………………………………………………………………………………………………………iv

Table of Contents………………………………………………………………………………………………………v

1 Introduction…………………………………………………………………………………………………………..1

1.1 Basic Definitions and Theorems…………………………………………………………….....1

1.2 Algebraic Topology………………………………………………………………………………….6

1.3 Simplicial Complexes……………………………………………………………………………..13

2 The Bing-Borsuk Conjecture………………………………………………………………………………...17

2.1 Manifold Properties……………………………………………………………………………….17

2.2 Construction of Bing’s Dogbone Space…………………………………………………....21

2.3 The Bing-Borsuk Conjecture and Related Results…………………………………....24

2.4 A Special Case…………………………………………………………………………………….….26

2.5 An Equivalent Conjecture……………………………………………………………………....28

2.6 Modified Bing-Borsuk Conjecture……………………………………………………….….29

3 The Poincaré Conjecture……………………………………………………………………………………...30

3.1 The Poincaré Conjecture is Equivalent to no Fake 3-Cells……………………….30

3.2 The Bing-Borsuk Conjecture implies the Poincaré Conjecture……………...... 34

3.2.1 The Construction of the Space K………………………………………….…..35

3.2.2 K is an Absolute Neighbourhood Retract………………………………....36

3.2.3 Some Definitions and Lemmas……………………………………………...... 37

3.2.4 K is Homogeneous……………………………………………………………….....43 Page | vi

3.2.5 K is not a 3-Manifold……………………………………………………………...... 45

3.2.6 K Has Dimension 3……………………………………………………….……..…...47

Bibliography……………………………………………………………………………………………………….…..48

Page | 1

Chapter 1. Introduction

Section 1.1. Basic Definitions and Theorems

In this paper we assume that all spaces are separable metric spaces, unless stated otherwise.

Definition 1.1.1. An n-manifold is a separable metric space such that each point has a � neighbourhood homeomorphic to Euclidean n-space � .

Here we say that a manifold is closed if it is connected, compact, and without boundary.

Definition 1.1.2. A space X is said to be homogeneous if for any two points �1, �2 ∈ X, there is a homeomorphism of X onto itself taking �1 to �2.

Definition 1.1.3. An embedding of X in Y is a map f : X → Y that maps X homeomorphically to the subspace f (X ) in Y.

Definition 1.1.4. A continuum is a nonempty compact connected metric space.

Definition 1.1.5. A space X is said to be locally compact at x if there is some compact subspace C of X that contains a neighbourhood of x. If X is locally compact at each of its points, X is said simply to be locally compact.

Definition 1.1.6. A space X is said to be locally contractible if, for every point y ∈ X and for every neighbourhood U of y, there exists a neighbourhood V of y and a continuous map f : V × [0, 1] → U such that f (x, 0 ) = x and f (x, 1 ) = y for every point x ∈ V.

� Definition 1.1.7. A space X is said to be locally k-connected, L� (k ≥ 0 ), if for every point x ∈ X and every neighbourhood U ⊂ X of x, there exists a neighbourhood V ⊂ U of x such that the inclusion-induced homeomorphisms ��≤�(V ) → ��≤� (U ) are trivial. Clearly, locally contractible spaces (such as manifolds) are locally k-connected for all k.

Definition 1.1.8. A collection A of subsets of a space X is said to have order n +1 if some point of X lies in n +1 elements of A, and no point of X lies in more than n +1 elements of A. Page | 2

Definition 1.1.9. Given a collection A of subsets of X, a collection B is said to refine A, or to be a ′ ′ ′ ′ refinement of A, if for each element � of B there is an element � of A such that � ⊂ � .

Definition 1.1.10. A space X is said to be finite dimensional if there is some integer n such that for every open cover A of X, there is an open cover B of X that refines A and has order at most n +1.

Definition 1.1.11. The Lebesgue covering dimension (or topological dimension) of X is defined to be the smallest value of n for which the above statement holds; we denote this by dim X.

Theorem 1.1.12 (K. Menger, G. Nöbeling, L. Pontrjagin, G. Tolstowa, S. Lefschetz, W. Hurewicz). Every 2�+1 separable metric space of dimension n can be embedded in � .

Definition 1.1.13. A retraction of X onto A is a map r : X → X such that r (X ) = A and r |A = ���. Here we say that A is a retract of X.

Definition 1.1.14. A subspace A ⊂ X is said to be a neighbourhood retract of X if there exists an open set U such that A ⊂ U ⊂ X and A is a retract of U.

Definition 1.1.15. A space X is said to be an absolute (neighbourhood) retract (shortly A(N)R) if for every normal space Y that contains X as a closed subspace, X is a (neighbourhood) retract of Y.

Definition 1.1.16. A space X is called an absolute (neighbourhood) extensor (shortly A(N)E) if, whenever Y is a separable metric space and A is a closed subspace of Y, any continuous function f : A → X can be extended to a continuous function from (some neighbourhood of A in)Y into X.

Theorem 1.1.17 [9]. A subspace A of a space X is a retract of X if and only if, for every space Y, every continuous map f : A → Y has a continuous extension f : X → Y.

Theorem 1.1.18 [9]. A space X is an A(N)R for metric spaces if and only if X is an A(N)E for metric spaces.

Definition 1.1.19. A Euclidean neighbourhood retract (ENR) is a locally compact, locally contractible � subset X of Euclidean n-space � . Page | 3

Lemma 1.1.20. Let f : X → Y be a bijective continuous function. If X is compact and Y is Hausdorff, then f is a homeomorphism.

Theorem 1.1.21. Every compact manifold, with or without boundary, is an ENR.

� Proof: � is locally contractible, so it naturally follows that every manifold is also locally � contractible. Hence we need only show that a manifold can be embedded in � , for some k. Here it suffices to consider the case where the manifold M is closed. For if M is not closed, we can embed it in the closed manifold obtained from two copies of M by identifying their boundaries.

Thus, assume M is a closed manifold. By compactness, there exist finitely many closed balls � � �� whose interiors cover M, where n is the dimension of M. Let ��: M → � be the quotient map � collapsing the complement of the interior of �� to a point. These ��’s are the components of a map � � f : M → (� ) . This map is injective since if x and y are distinct points in M with x in the interior � of �� , then ��(x) ≠ ��(y).

� �+1 Next we take the product of the standard embeddings � ↪ � , giving us an embedding � � � � (� ) ↪ � . Composing f with this embedding, we obtain a continuous injection M ↪ � . Recalling � that M is compact, it follows (from Lemma 1.1.20) that we have an embedding of M into � . □

� Theorem 1.1.22 [29]. A compact subspace C of � is a retract of some neighbourhood if and only if C is locally contractible.

Lemma 1.1.23 [25]. A space X is an ANE for spaces with dimension less than or equal to n if and �−1 only if X is L� .

Theorem 1.1.24. Every Euclidean neighbourhood retract (ENR) is an absolute neighbourhood retract (ANR).

Proof: First we consider the non-compact case. Suppose X is a non-compact n-dimensional ENR. By definition, X is locally compact hence we can take the one-point compactification of X and embed �+1 this as a closed subspace in � . Then by Theorem 1.1.22, X is an ANR. Page | 4

Now suppose that X is a compact ENR such that dim X = n. By Theorem 1.1.12, X can be 2�+1 embedded (as a closed subspace) in � . By definition of ENR, X is locally contractible hence X is 2�+1 L� . By Lemma 1.1.23, X is an ANE for all spaces with dimension less than or equal to 2n +1. By Theorem 1.1.18, X is also an ANR for these spaces. It remains to show that X is an ANE for all spaces. Then (again by Theorem 1.1.18) X will be an ANR and the proof will be complete.

2�+1 Without loss of generality, we can embed X in � . As discussed above, X is an ANR in 2�+1 2�+1 � so we can find a neighbourhood U of X in � and a retraction r : U → X. Now suppose Y is a metric space, A ⊂ Y is a closed subspace, and f : A → X is a continuous map. Next, we use the fact 2�+1 2�+1 that � is an absolute extensor. Hence we can find a map �:̅ Y → � that extends f. Then −1 �̅ (U ) = V is an open neighbourhood of A in Y and r ⃘ �|̅ �: V → X extends f. Therefore X is an ANE and the proof is complete. □

Theorem 1.1.25. A closed manifold is an ANR.

Proof: This result follows directly from Theorem 1.1.21 and Theorem 1.1.24. □ The following lemma regarding Euclidean space is well known:

Lemma 1.1.26. For any two points inside a closed disk in Euclidean space, there is a homeomorphism of the disk that takes one point to the other and is identity on the boundary.

This result immediately implies our next lemma:

Lemma 1.1.27. Suppose we have two points that lie inside an open subset of a manifold, and that � this open subset is homeomorphic to � . Then there is a homeomorphism of the manifold that takes one point to the other.

Lemma 1.1.28. Assume that every point x ∈ X has a neighbourhood U such that for each u ∈ U, there is a homeomorphism h of X such that h(x) = u. Then each orbit of the homeomorphism group of X is open. In particular, if X is connected then X is homogeneous.

Page | 5

Theorem 1.1.29. A closed manifold is homogeneous.

Proof: Suppose we have a closed manifold M and a homeomorphism f : ܴ௡ ՜ M onto an open subset containing x = f (0 ). Now, f (ܤ௡) is a compact neighbourhood of X, and since X is Hausdorff, it follows that f (ܤ௡) is closed. Lastly, we can apply Lemma 1.1.27 and then apply Lemma 1.1.28. The result immediately follows and the proof is complete. ᇝ





















Page | 6

Section 1.2. Algebraic Topology

Here we introduce some necessary definitions and results related to algebraic topology and homotopies.

Definition 1.2.1. If f and ݂ᇱ are continuous maps of the space X into the space Y, we say that f is homotopic to ݂ᇱ if there is a continuous map F : X × I ՜ Y such that

F (x, 0 ) = f (x) and F (x, 1 ) = ݂ᇱ(x)

for each x. The map F is called a homotopy between f and ݂ᇱ. If f is homotopic to ݂ᇱ, we write .f ؄ ݂ᇱ. If f ؄ ݂ᇱ and ݂ᇱ is the constant map, we say that f is nullhomotopic

If we think of the parameter t as representing time, then F represents a continuous deformation of the map f to the map ݂ᇱ as t goes from 0 to 1.

Definition 1.2.2. A path in X from ݔ଴ to ݔଵ is a continuous map f : [0, 1] ՜ X such that f (0) = ݔ଴ and

.f (1) = ݔଵ. Here ݔ଴ is called the initial point and ݔଵ is called the final point

Definition 1.2.3. A space X is said to be path-connected if every pair of points of X can be joined by a path in X.

Definition 1.2.4. Two paths f and ݂ᇱ, mapping the interval I = [0, 1] into X, are said to be path

homotopic if they have the same initial point ݔ଴ and the same final point ݔଵ, and if there is a continuous map F : I × I ՜ X such that

F (s, 0 ) = f (s) and F (s, 1 ) = ݂ᇱ(s),

,F (0, t ) = ݔ଴ and F (1, t ) = ݔଵ

ᇱ ᇱ I. We call F a path homotopy between f and ݂ , and denote this by f ӉӋ ௣ ݂ . The first א for each s, t condition says that F represents a continuous deformation of the path f to the path ݂ᇱ, and the second condition says that the end points of the path remain fixed during the deformation.



 Page | 7

Figure 1.2.5. A path homotopy F between two paths.

Theorem 1.2.6. The relations ̴̲ and ̴̲ � are equivalence relations.

Lemma 1.2.7 (The pasting lemma). Let X = A ∪ B, where A and B are closed in X. Let f : A → Y and g : B → Y be continuous maps. If f (x) = g(x) for every x ∈ A ∩ B, then f and g combine to give a continuous function h : X → Y, defined by setting h(x) = f (x) if x ∈ A, and h(x) = g(x) if x ∈ B.

Definition 1.2.8. If f is a path in X from �0 to �1, and g is a path in X from �1 to �2, we define the product f ∗ g of f and g to be the path h given by the equation

�(2�) ��� � ∈ [0, 1� ], h(s) = � 2 1 ��2� – 1� ��� � ∈ [ �2 , 1].

The function h is well-defined and continuous by the pasting lemma; it is a path in X from �0 to �2. This function can be thought of as a path whose first half is the path f and whose second half is the path g.

Theorem 1.2.9. The product operation on paths induces a well-defined operation on path homotopy classes, defined by the equation

[f ] ∗ [g ] = [f ∗ g ]. Page | 8

Definition 1.2.10. Let X be a space and ݔ଴ a point of X. A path in X that begins and ends at ݔ଴ is

.called a loop based at ݔ଴

Definition 1.2.11. The set of path homotopy classes of loops based at ݔ଴, with the operation *, is

.(called the fundamental group of X relative to the base point ݔ଴. This is denoted by ߨଵ(X, ݔ଴ Sometimes this group is called the first homotopy group of X.

The reason we consider loops is to guarantee that this set is closed under the operation *. Without loops, the product of two paths may not be defined (when the final point of the first path is not the initial point of the second path). Therefore the set of path homotopy classes of loops satisfies the axioms for a group.

= (Definition 1.2.12. Let Ƚ be a path in X from ݔ଴ to ݔଵ, and let ߙത be the path defined by Ƚഥ(s Ƚ(1 – s)(this is called the reverse of Ƚ). We define the map

(Ƚෝ: ߨଵ(X, ݔ଴ሻ՜ ߨଵ(X, ݔଵ

by the equation

.[ሾȽכሾf ሿכȽෝ([f ]) = [Ƚഥሿ

,This map is well-defined because the operation * is well-defined. If f is a loop based at ݔ଴

.(Ƚ) is a loop based at ݔଵ. Thus Ƚෝ maps ߨଵ(X, ݔ଴) into ߨଵ(X, ݔଵ כ f)כ then clearly Ƚഥ

Theorem 1.2.13. The map Ƚෝ is a group isomorphism.

Corollary 1.2.14. If X is path-connected and ݔ଴, ݔଵ are two points of X, then ߨଵ(X, ݔ଴) is isomorphic

.(to ߨଵ(X, ݔଵ

Thus if X is path-connected, the group ߨଵ(X, ݔ଴) is (up to isomorphism) independent of the

.choice of basepoint ݔ଴. In this case, we simply write ߨଵ(X ) to represent the fundamental group of X  Definition 1.2.15. A space X is said to be simply connected if it is path-connected and the fundamental group of X is trivial.

Definition 1.2.16. Let f : X ՜ Y and g : Y ՜ X be continuous maps. Then f and g are called g : Y ל f : X ՜ X is homotopic to the identity map of X, and f ל homotopy equivalences if the map g ՜ Y is homotopic to the identity map of Y. Each is said to be a homotopy inverse of the other. Two spaces that are homotopy equivalent are said to have the same homotopy type. Page | 9

Definition 1.2.17. A space having the homotopy type of a point is called contractible. In other words, the identity map of the space is nullhomotopic (homotopic to a constant map).

Definition 1.2.18. A retraction of X onto A is a map r : X → X such that r (X ) = A and r |A = ���.

Definition 1.2.19. Suppose A is a subspace of X. We say that A is a deformation retract of X if the identity map of X is homotopic to a map that carries all of X into A, such that each point of A remains fixed during the homotopy. This means that there is a continuous map H : X × I → X such that H(x, 0 ) = x, H(x, 1 ) ∈ A for all x ∈ X, and H(a, t ) = a for all a ∈ A. The homotopy H is called a deformation retraction of X onto A.

Definition 1.2.20. For a map f : X → Y, the mapping cylinder �� is the quotient space of the disjoint union (X × I ) ∪ Y obtained by identifying each (x, 1 ) ∈ X × I with f (x) ∈ Y.

It is a general fact that the mapping cylinder �� deformation retracts to the subspace Y by sliding each point (x, t ) along the segment {x} × I ⊂ �� to the endpoint f (x) ∈ Y.

Figure 1.2.21. The mapping cylinder �� associated with a map f : X → Y.

Definition 1.2.22. For a pair (X, A) we say that A has a mapping cylinder neighbourhood in X if there exists a closed neighbourhood N containing a subspace B (thought of as the boundary of N ) with N – B being an open neighbourhood of A, such that there exists a map f : B → A and a homeomorphism h : �� → N with h |A ∪B = ��� ∪ �.

Page | 10

Figure 1.2.23. A mapping cylinder neighbourhood of A in X.

Suppose we are given a map f : X → Y, a subspace A ⊂ X, and a homotopy of f |A. Now we ask: when is it possible to extend this to a homotopy of the given f ? We say that the pair (X, A) has the homotopy extension property if this extension problem can always be solved.

Definition 1.2.24. (X, A) has the homotopy extension property if for any space Y, every pair of maps X × {0} → Y and A × I → Y that agree on A × {0} can be extended to a map X × I → Y.

Theorem 1.2.25. A pair (X, A) has the homotopy extension property if and only if X × {0} ∪ A × I is a retract of X × I.

Proof: For the first direction, suppose (X, A) has the homotopy extension property and (for notation sake) let Y = X × {0} ∪ A × I. Then the identity map X × {0} ∪ A × I → X × {0} ∪ A × I extends to a map X × I → X × {0} ∪ A × I. Hence by definition, X × {0} ∪ A × I is a retract of X × I.

For the other direction, suppose X × {0} ∪ A × I is a retract of X × I. Here we will only consider the case where A is closed, as the other case is very technical (see [29]). Note that both X × {0} and A × I are closed. Hence by the pasting lemma, any two maps X × {0} → Y and A × I → Y that agree on A × {0} combine to give a continuous map X × {0} ∪ A × I → Y. Then we can compose the map X × {0} ∪ A × I → Y with a retraction X × I → X × {0} ∪ A × I, giving us an extension X × I → Y. Therefore (X, A) has the homotopy extension property. □ Page | 11

Theorem 1.2.26. A pair (X, A) has the homotopy extension property if A has a mapping cylinder neighbourhood in X.

Proof: Suppose A has a mapping cylinder neighbourhood in X. First note that I × I retracts onto I × {0} × ∂I × I, hence B × I × I retracts onto B × I × {0} ∪ B × ∂I × I. It follows that we have a retraction of �� × I onto �� × {0} ∪ (A ∪B ) × I. Thus (��, A ∪B ) has the homotopy extension property. Recall that �� is homeomorphic to N, hence (N, A ∪B ) also has the homotopy extension property.

Now suppose we have a map X → Y and a homotopy of its restriction to A. We can take the constant homotopy on X – (N – B ) and then extend this over N by applying the homotopy extension property for (N, A ∪B ) to the given homotopy on A. Then we take the constant homotopy on B, and this completes the proof. □

Theorem 1.2.27 [29]. Suppose (X, A) and (Y, A) satisfy the homotopy extension property, and f : X → Y is a homotopy equivalence with f |A = ���. Then f is a homotopy equivalence that is constant on A.

Corollary 1.2.28. If (X, A) satisfies the homotopy extension property and the inclusion A ↪ X is a homotopy equivalence, then A is a deformation retract of X.

Proof: Apply the above theorem to the inclusion A ↪ X. □

Theorem 1.2.29. A map f : X → Y is a homotopy equivalence if and only if X is a deformation retract of the mapping cylinder ��. Hence two spaces X and Y are homotopy equivalent if and only if there exists a third space Z containing both X and Y as deformation retracts.

Proof: Consider the diagram below. Here the maps i and j are the inclusion maps and r is the canonical retraction, so f = ri and i ≃ jf.

Page | 12

Now, j and r are homotopy equivalences, hence we have that f is a homotopy equivalence if and only if i is a homotopy equivalence. This follows from the fact that the composition of two homotopy equivalences is a homotopy equivalence and a map homotopic to a homotopy equivalence is a homotopy equivalence.

Further, the pair (��, X ) satisfies the homotopy extension property by Theorem 1.2.26 (using the neighbourhood X × [0, ½] of X in ��). Therefore we can apply Corollary 1.2.28 to ��, and this completes the proof.

□

Page | 13

Section 1.3. Simplicial Complexes

௡. Then the setܴ א Definition 1.3.1. Suppose S ؿ ܴ௡ is a linear subspace and b

{ S א b + S = {b + x : x

is called an affine subspace of ܴ௡ parallel to S. Here we define the dimension of b + S to be the dimension of S.

Definition 1.3.2. We say that the set {ݒ଴, … , ݒ௞} is affinely independent if it is not contained in any affine subspace of dimension strictly less than k.

௡ :the following are equivalent , ܴ א It is well known that for any k +1 distinct points ݒ଴, … , ݒ௞

,The set {ݒ଴, … , ݒ௞} is affinely independent(1)

,The set {ݒଵ – ݒ଴, … , ݒ௞ – ݒ଴} is linearly independent(2) ௞ ௞ .௞ = 0ܿ =ڮ = If ܿ଴, … , ܿ௞ are real numbers such that σ௜ୀ଴ ܿ௜ݒ௜ = 0 and σ௜ୀ଴ ܿ௜ = 0, then ܿ଴(3)

௡ Definition 1.3.3. Let {ݒ଴, … , ݒ௞} be an affinely independent set of k +1 points in ܴ . The simplex

spanned by them, denoted by [ݒ଴, … , ݒ௞], is the set

௞ ௞ ,{ݒ଴, … , ݒ௞] = {σ௜ୀ଴ ݐ௜ ݒ௜: ݐ௜ ൒ 0 and σ௜ୀ଴ ݐ௜ = 1]

with the subspace topology.

௞ ݒ଴, … , ݒ௞], the numbers ݐ௜ are called the barycentric] א Definition 1.3.4. For any point x = σ௜ୀ଴ ݐ௜ ݒ௜

coordinates of x with respect to [ݒ଴, … , ݒ௞]. It follows that these coordinates are uniquely determined by x.

Definition 1.3.5. Each of the points ݒ௜ is called a vertex of the simplex. The dimension of the simplex is k (the dimension is one less than the number of vertices). A k-dimensional simplex is called a k-simplex.

In more geometric terms, a simplex is the generalization of the notion of a triangle or tetrahedron to arbitrary dimension. A 2-simplex is a triangle (including its interior), a 3-simplex is a tetrahedron, and a 4-simplex is a pentachoron. A single point may be considered a 0-simplex, and a line segment may be considered a 1-simplex. Page | 14

Definition 1.3.6. An n-dimensional open (closed) cell is a topological space that is homeomorphic to an n-dimensional open (closed) ball.

It is well known that every k-simplex is a closed k-cell.

Definition 1.3.7. Let ɐdenote a k-simplex. Each simplex spanned by a nonempty subset of the vertices of ɐ is called a face of ɐ. The faces that are not equal to ɐ itself are called its proper faces. The 0-dimensional faces of ɐ are just its vertices, while the 1-dimensional faces are called its edges.

The (k –1)-dimensional faces of a k-simplex are called its boundary faces. We define the boundary of ɐ to be the union of its boundary faces, and its interior is ɐ minus its boundary. An open k-simplex is the interior of a k-simplex.

Definition 1.3.8. A (Euclidean) simplicial complex is a collection K of simplices in Euclidean space ܴ௡, satisfying the following conditions:

,K, then every face of ɐ is in K א If ɐ(1) (2)The intersection of any two simplices in K is either empty or a face of each, (3)K is a locally finite collection.

Figure 1.3.9. On the left is an example of a valid simplicial complex, while the right shows an invalid arrangement of simplices.

Definition 1.3.10. If K is a simplicial complex in ܴ௡, the dimension of K is defined to be the maximum dimension of the simplices in K. Page | 15

′ ′ Definition 1.3.11. A subset � ⊂ K is said to be a subcomplex of K if whenever σ ∈ � , every face of σ ′ is in � . A subcomplex is (naturally) a simplicial complex itself. For any k ≤ n, the set of all simplices of K of dimension at most k is a subcomplex called the k-skeleton of K.

Definition 1.3.12. The barycenter or “center of gravity” of the simplex [�0, … , ��] is the point b = � ∑�=0 �� �� whose barycentric coordinates �� are all equal, ie. �� = 1/(k +1) for each i.

Definition 1.3.13. The barycentric subdivision of [�0, … , ��] is the decomposition of [�0, … , ��] into k-simplices [b, �0, … , ��−1] where, inductively, [�0, … , ��−1] is a (k – 1)-simplex in the barycentric subdivision of a face [�0, … , ���, … , ��]. The induction starts with the case k = 0 when the barycentric subdivision of [�0] is defined to be just [�0] itself.

The next two cases k = 1, 2 and part of the case k = 3 are shown in the figure below. It follows from the inductive definition that the vertices of simplices in the barycentric subdivision of [�0, … , ��] are exactly the barycenters of all the n-dimensional faces [��0, … , ���] of [�0, … , ��] for 0 ≤ n ≤ k. When n = 0 this gives the original vertices �� since the barycenter of a 0-simplex is itself. The barycenter of [��0, … , ���] has barycentric coordinates �� = 1/(n +1) for i = �0, … , �� and �� = 0 otherwise.

Figure 1.3.14. The barycentric subdivision of a 1-simplex, a 2-simplex, and (partially) a 3-simplex.

� Definition 1.3.15. Given a simplicial complex K in � , the union of all the simplices in K with the � subspace topology inherited from � is a topological space denoted by |K | and called the polyhedron of K.

Definition 1.3.16. Given a space X, a homeomorphism between X and the polyhedron of some simplicial complex is called a triangulation of X. Any space that admits a triangulation is said to be triangulable. Page | 16

In the 1950s, Edwin Moise proved that every 3-manifold is triangulable [26].

� � Definition 1.3.17. An affine map F : � → � is any map of the form F (x) = c +A(x), where A is a � linear map and c is some fixed vector in � .

Definition 1.3.18. Suppose K and L are simplicial complexes. A simplicial map from K to L is a continuous map f : |K | → |L | whose restriction to each simplex σ ∈ K agrees with an affine map taking σ onto some simplex in L.

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Chapter 2. The Bing-Borsuk Conjecture

Section 2.1. Manifold Properties

Definition 2.1.1. An n-dimensional (n ∈ N ) locally compact Hausdorff space X is called a Z- � � homology n-manifold if for every point x ∈ X and all k ∈ N, ��(X, X – {x}; Z ) ≅ ��(� , � – {0}; Z ). Trivially, every topological manifold is a homology manifold.

Definition 2.1.2. An n-dimensional topological space X is called a generalized n-manifold (n ∈ N ) if X is an ENR Z-homology n-manifold. It follows that every topological n-manifold is a generalized n-manifold.

Every generalized (n ≤ 2)-manifold is known to be a topological n-manifold [60]. On the other hand, for every n ≥ 3 there exist totally singular generalized n-manifolds, i.e. generalized manifolds that are not locally Euclidean at any point (see [18], [19], [50]).

Definition 2.1.3. A topological space X is said to have the invariance of domain property if for every pair of homeomorphic subsets U, V ⊂ X, U is open if and only if V is open.

Brouwer proved a century ago that every topological n-manifold has the invariance of domain property [12]. However, this property alone is insufficient to characterize manifolds. It was proven that any homology n-manifold has the invariance of domain property [59]. Further, as mentioned above, for n ≥ 3 there exist homology n-manifolds that are not topological n-manifolds. Therefore there exist non-manifolds that have this property.

Definition 2.1.4. An n-dimensional compact space X is called a Cantor n-manifold if whenever X can be expressed as the union X = �1 ∪ �2 of its proper closed subsets, then dim(�1 ∩ �2) ≥ n – 1.

Urysohn introduced this notion in 1925 and proved that every topological n-manifold is a Cantor n-manifold [57], [58]. More fundamental results were established by Aleksandrov in 1928 [1]. In 1993, Krupski proved a more general result:

Theorem 2.1.5 [34]. Every generalized n-manifold is a Cantor n-manifold. Page | 18

Definition 2.1.6. If f, g : X → Y are maps and Y has a bounded metric, then we define the distance between these two maps to be:

dist(f, g) = sup{dist(f (x), g(x)) | x ∈ X }.

Definition 2.1.7. A space X is said to have the disjoint (k, m)-cells property (k, m ∈ N ) if for each � � ′ � ′ � pair of maps f : � → X and g : � → X and every � > 0, there exist maps � : � → X and � : � → ′ ′ ′ � ′ � X such that dist(f, � ) < �, dist(g, � ) < �, and � (� ) ∩ � (� ) = ∅.

It is well known that topological manifolds of dimension n have the disjoint (k, m)-cells property for k +m +1 ≤ n (see [53]).

Definition 2.1.8. A map f : X → Y between two spaces is called proper if f is closed and the preimage of every compact set in Y is compact in X.

Definition 2.1.9. A proper onto map f : M → X between two spaces is said to be cell-like if for every −1 −1 point x ∈ X, the point-inverse � (x) contracts in any neighbourhood of itself (� (x) has the shape of a point).

Definition 2.1.10. A space X is said to be resolvable if there exists a proper, cell-like, surjective map f : M → X defined on some n-manifold M. In this case, the map f is called a (cell-like) resolution of X. Trivially, every topological manifold is resolvable.

A natural way in which a generalized manifold may arise is as the image of a cell-like map defined on a manifold. The result below demonstrates the crucial importance of cell-like maps in geometric topology (proven for n ≤ 2 by Wilder [60], for n = 3 by Armentrout [3], for n = 4 by Quinn [47], and for n ≥ 5 by Siebenmann [55]).

Theorem 2.1.11 (Cell-like Approximation theorem). For every ε > 0, every n ∈ N, and every cell-like � � � � map f : � → � between topological n-manifolds � and � , there exists a homeomorphism � � � h : � → � such that dist(f (x), h(x)) < ε for every x ∈ � .

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The fact that not all resolvable generalized manifolds are manifolds has been known since the mid 1950’s, when Bing constructed his famous Dogbone space as the cell-like image of a map 3 defined on � (see [5], [7]). Although we do not prove this fact, we do construct Bing’s Dogbone space in the next section.

Generalized manifolds have been a subject of intense studies since the 1960’s [50]. In the mid 1970’s, Cannon recognized that the disjoint (2, 2)-cells property plays a key role in characterizing manifolds of dimension n ≥ 5. The disjoint (2, 2)-cells property is often referred to as the disjoint disks property:

Definition 2.1.12. A space X is said to have the disjoint disks property (DDP) if, for every ε > 0 and 2 ′ ′ 2 ′ 2 every pair of maps f, g : � → X, there exist maps � , � : � → X with disjoint images � (� ) ∩ ′ 2 ′ ′ � (� ) = ∅ such that dist(f, � ) < � and dist(g, � ) < �.

Theorem 2.1.13 (Edwards). For n ≥ 5, topological n-manifolds are precisely the n-dimensional resolvable spaces with the disjoint disks property.

An analogous result for 3-manifolds was proven in the early 1980’s by Daverman and Repovš [21], [22] (whereas only partial results are known in dimension 4, see [4], [21]). Before we examine this result, we require the following definition.

Suppose K is a polyhedron and z ∈ K. Impose a triangulation on K, and regard T as a simplicial complex whose underlying point set equals K. Subdivide T if necessary, so that z corresponds to a vertex of T.

Definition 2.1.14. For such a K topologically embedded as a closed subset of a generalized 3-manifold X, X \K is said to have free local fundamental group at z ∈ K, abbreviated as 1-FLG at z, if for each sufficiently small neighbourhood U of z there exists another neighbourhood V of z with z ∈ V ⊂ U. Further, if W is any connected open set with z ∈ W ⊂ V , then for each nonempty ′ ′ ′ component � of W – K the (inclusion-induced) image �1(� ) → �1(� ) is a free group on m – 1 ′ ′ generators. Here � denotes the component of U \K containing � and m is the number of ′ “components” of St(z) – z whose images meet �����, where St(z) denotes the simplicial star of z in the complex T. Page | 20

Then X \K is said to be 1-FLG in X if it is 1-FLG in X at each point of K.

Definition 2.1.15. A space X is said to have the spherical simplicial approximation property (SSAP) if 2 2 for each μ : � → X and each ε > 0, there exists a map ψ : � → X and a finite topological 2-complex 2 �� ⊂ X such that dist(ψ, μ) < ε, ψ(� ) ⊂ ��, and X \�� is 1-FLG in X.

Theorem 2.1.16 (Daverman, Repovš). Topological 3-manifolds are precisely the 3-dimensional resolvable spaces with the simplicial spherical approximation property.

Conjecture 2.1.17 (Generalized Manifolds Resolution conjecture). Every generalized (n ≥ 3 )- manifold has a resolution.

In dimension 3, the Generalized Manifolds Resolution conjecture implies the Poincaré conjecture and only special cases are known [50]. In higher dimensions, this conjecture turns out to be false: there exist non-resolvable generalized n-manifolds for every n ≥ 6 [16]. In 2007, Bryant et al. further strengthened this result with the following DDP theorem [17].

Theorem 2.1.18 (Bryant, Ferry, Mio, Weinberger). There exist non-resolvable generalized n-manifolds with the disjoint disks property, for every n ≥ 7.

Hence, generalized manifolds may possess nice general position properties. However, the majority of properties listed above are known to be insufficient by themselves to characterize manifolds. Homogeneity is the remaining single possibility. Is this property strong enough to characterize manifolds? Are there other combinations of these properties that characterize manifolds which have not yet been discovered? These questions lead us to the Bing-Borsuk conjecture.

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Section 2.2. Construction of Bing’s Dogbone Space

Before we construct Bing’s Dogbone space (see [5] for more details) we must first cover some definitions relating to decomposition spaces.

Definition 2.2.1. A decomposition G of a topological space X is a partition of X. Explicitly, G is a subset of the power set of X, and its elements are pairwise disjoint nonempty sets that cover X.

Associated with any decomposition G of a space X is the decomposition space having underlying point set G but denoted as X |G in order to emphasize the distinction between the generating partition and the resultant space.

Definition 2.2.2. A decomposition G of a space X is said to be upper semicontinuous if each g ∈ G is closed in X and if, for each g ∈ G and each open subset U of X containing g, there exists another ′ open subset V of X containing g such that every � ∈ G intersecting V is contained in U.

Definition 2.2.3. An arc is a space that is homeomorphic to the closed unit interval.

3 3 Definition 2.2.4. A closed set X in � is said to be tame if there is a homeomorphism h of � onto 3 itself such that h(X ) is a polyhedron (with simplices in � ).

3 Definition 2.2.5. A closed set in � that is not tame is called wild.

3 Bing’s Dogbone space is an upper semicontinuous decomposition G of � , such that the 3 elements of G are points and tame arcs. However, this decomposition of � is topologically 3 different from � . The nondegenerate elements of G are the components of a closed set �0. We define �0 as the intersection of a decreasing sequence of closed sets A, ∑ ��, ∑ ���, ⋯ . First we describe A and then construct this sequence below.

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(1)Suppose A is a solid sphere with two handles (see Figure 2.2.6 below). This is a polyhedron. It is helpful to think of A as an elongated figure 8 together with all points whose distances from it are within some positive number ߝ.

(2)Now suppose ܣଵ, ܣଶ, ܣଷ, ܣସ are four polyhedra imbedded in A as shown below. It is

geometrically clear that all the ܣ௜’s are topologically equivalent to A. We require that the

upper loops of the ܣ௜’s are linked together, as are the lower loops. We call ܣଵ, ܣଶ, ܣଷ, ܣସ the A’ s defined at the first stage.

(3)Then ܣ௜ଵ, ܣ௜ଶ, ܣ௜ଷ, ܣ௜ସ are imbedded in ܣ௜ just as ܣଵ, ܣଶ, ܣଷ, ܣସ were imbedded in A. At this ଶ stage there are 4 ܣ௜௝’s defined.

(4)We continue in this way: four ܣ௜௝ଵ, ܣ௜௝ଶ, ܣ௜௝ଷ, ܣ௜௝ସ are imbedded in ܣ௜௝ in the same manner. Generally, for each integer n, we construct four figure 8’s in each of the 4௡ A’ s defined at the ݊௧௛ stage.

ڮ ת ௜௝௞ܣ σ ת ௜௝ܣ σ ת ௜ܣ σ ת ଴ = Aܣ We could have placed the A’ s so that each component of(5) is an arc that does not intersect any horizontal plane in two points. We shall suppose that

this is the case. Each element of G is either a point in the complement of ܣ଴ or a component

of ܣ଴. Each element of G in ܣ଴ is a tame arc and the collection of such tame arcs corresponds to a Cantor set in the decomposition space. Page | 23



Figure 2.2.6. Here we have A (a solid sphere with two handles) and ܣଵ, ܣଶ, ܣଷ, ܣସ are four polyhedra imbedded in A.

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Section 2.3. The Bing-Borsuk Conjecture and Related Results

Bing and Borsuk proved in 1965 that for n < 3, every n-dimensional homogeneous compact ANR is a topological n-manifold [8]. They also conjectured that this holds in all dimensions:

N, homogeneous compact ANR א Conjecture 2.3.1 (Bing-Borsuk conjecture). Every n-dimensional, n is a topological n-manifold.

Jakobsche proved in 1978 that in dimension 3, the Bing-Borsuk conjecture implies the Poincaré conjecture [32], [33]. In the next chapter, we give an exposition of this proof. Given the difficulty of proving the Poincaré conjecture [42], it is understandable why the Bing-Borsuk conjecture remains unsolved.

In 1970 Bredon proved the following theorem [10], [11].

N ) and for some (and א Theorem 2.3.2 (Bredon). If X is an n-dimensional homogeneous ENR (n

௞(X, X – {x}; Z ) are finitely generated, then X is a Z-homologyܪ X, the groups א hence all) points x n-manifold.

In 1976 Lysko proved the following result [38].

Theorem 2.3.3 (Lysko). Let X be a connected finite-dimensional homogeneous ANR. Then X is a Cantor manifold and it possesses the invariance of domain property.

In 1985 Seidel proved a similar result in the case of locally compact, locally homogeneous separable ANR’s [54]. Next, we quote the following result by Krupski [34].

Theorem 2.3.4 (Krupski). Let X be a locally compact homogeneous space. Then:

(1)If X is an ANR of dimension n > 2, then X has the disjoint (0, 2)-cells property, (2)If dim X = n > 0, X has the disjoint (0, n – 1 )-cells property, and X is an Lܥ௡ିଵ space, then

local homologies satisfy ܪ௞(X, X – {x}) = 0 for k < n and ܪ௡(X, X – {x}) ് 0.

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� Definition 2.3.5. A space X is said to be acyclic in dimension n ∈ N if � (X; Z ) = 0.

Definition 2.3.6. If X is a connected space and A ⊂ X, we say that A separates X if X \A is not connected.

In 2003 Yokoi established the following algebraic property of n-dimensional homogeneous ANR’s which is also possessed by topological n-manifolds [61].

Theorem 2.3.7 (Yokoi). Let X be an n-dimensional homogeneous ANR continuum which is cyclic in dimension n. Then no compact subset of X, acyclic in dimension n –1, separates X.

These partial results, demonstrating that homogeneity implies several other manifold properties, indicate why the Bing-Borsuk conjecture could be true.

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Section 2.4. A Special Case

Definition 2.4.1. Let X be a topological space. A coordinate chart (or just chart) on X is a pair (U, φ), where U is an open set of X and φ : U → �� is a homeomorphism from U to an open subset �� = � φ(U ) ⊂ � . By definition, each point of a manifold is contained in the domain of some chart.

Definition 2.4.2. An atlas for a topological space X is a collection {(��, ��)} of charts on X such that � ∑ ��� = X. If the range of each chart is � , then X is an n-manifold.

Definition 2.4.3. Suppose that (��, ��) and (��, ��) are two charts for a manifold M such that −1 ��∩ �� is nonempty. The transition map ��,� is the map defined on ��(��∩ ��) by ��,� = �� ∘ �� .

Definition 2.4.4. Given two manifolds M and N, a bijective map f from M to N is called a −1 diffeomorphism if both f : M → N and its inverse � : N → M are differentiable. If these functions � are r -times continuously differentiable, then f is called a � - diffeomorphism.

� � Definition 2.4.5. A subset K ⊂ � is said to be � - homogeneous if for every pair of points x, y ∈ K � 1 there exist neighbourhoods ��, �� ⊂ � of x and y, respectively, and a � - diffeomorphism h : (��, �� ∩ K, x) → (��, �� ∩ K, y). This notation means that h maps �� onto ��, �� ∩ K onto �� ∩ K, −1 and h(x) = y. Here h and ℎ have continuous first derivatives.

� Definition 2.4.6. A � - manifold is a topological manifold with an atlas whose transition maps are all k -times continuously differentiable.

In 1996 Repovš et al. proved the following result which in some sense can be considered a smooth version of the Bing-Borsuk conjecture [52].

Theorem 2.4.7 (Repovš, Skopenkov, Scepin). Let K be a locally compact (possibly non-closed) � 1 1 � subset of � . Then K is � - homogeneous if and only if K is a � - submanifold of � .

This theorem does not work for arbitrary homeomorphisms. A counterexample is the 3 1 Antoine Necklace: a wild Cantor set in R which is clearly homogeneously (but not C - 3 homogeneously) embedded in R [2]. Page | 27

Definition 2.4.8. Given spaces X and Y, a map f : X → Y is called Lipschitz continuous if there exists a real constant K ≥ 0 such that, for all �1, �2 ∈ X, ��(f (�1), f (�2)) ≤ K ��(�1, �2).

Further, Theorem 2.4.7 does not even work for Lipschitz homeomorphisms (homeomorphisms that are Lipschitz continuous). Malesic and Repovš proved in 1999 that there 3 exists a Lipschitz homogeneous wild Cantor set in R [39]. Their result was later strengthened by Garity et al. [27].

Definition 2.4.9. A space X is said to be topologically rigid if any other space Y which is homotopy equivalent to X is also homeomorphic to X.

Theorem 2.4.10 (Garity, Repovš, Zeljko). There exist uncountably many rigid Lipschitz 3 homogeneous wild Cantor sets in R .

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Section 2.5. An Equivalent Conjecture

Daverman and Husch were able to determine an equivalent conjecture to the Bing-Borsuk conjecture [20].

Definition 2.5.1. A proper map p : E ՜ B between locally compact separable metric ANR’s satisfies the approximate homotopy lifting property for a space X if whenever h : X × I ՜ B and H : X × {0} ՜ E are maps such that p ƕH = h | X × {0} and ߚ is a cover of B, h extends to a map H : X × I ՜ E such that h and p ƕH are ߚ-close.

Definition 2.5.2. A surjective map p : E ՜ B between locally compact, separable metric ANR’s E and B is said to be an approximate fibration if p has the approximate homotopy lifting property for every space X.

Definition 2.5.3. An embedding f of a space X is said to be a nice embedding if f can be extended to an embedding of X × (-ɂ, ɂ) for some ߝ > 0.

The below conjecture is equivalent to the Bing-Borsuk conjecture:

Conjecture 2.5.4. Suppose that a space X is nicely embedded in ܴ௡ା௠, for some m ൒ 3. Further, suppose that X has a mapping cylinder neighbourhood N = ܥఝ of a map ɔ : μ ՜ X, with mapping cylinder projection Ɏ : N ՜ X. Then Ɏ : N ՜ X is an approximate fibration.

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Section 2.6. Modified Bing-Borsuk Conjecture

Recall that Bryant et al. proved in 1996 that there exist nonresolvable generalized n-manifolds for every n ൒ 6 [16]. Based on earlier work by Quinn, these nonresolvable generalized manifolds must be totally singular (have no points with Euclidean neighbourhoods) [48]. Moreover, in 2007 Bryant et al. strengthened their result to show that there exist nonresolvable generalized n-manifolds with the disjoint disks property, for every n ൒ 7 [17]. Based on these results, the following conjecture was proposed:

Conjecture 2.6.1 (Bryant, Ferry, Mio, Weinberger). Every generalized (n ൒ 7)-manifold satisfying the disjoint disks property, is homogeneous.

Note that if Conjecture 2.6.1 is true, then the Bing-Borsuk conjecture is false for n ൒ 7. In 2002 Bryant suggested the following modified Bing-Borsuk conjecture [14].

Conjecture 2.6.2 (Modified Bing-Borsuk conjecture). Every homogeneous (n ൒ 3)-dimensional ENR is a generalized n-manifold.

A further change was posed by Quinn and is based on a modification of the homogeneity property itself [49].

Definition 2.6.3. A space X is said to be homologically arc-homogeneous provided that for every path

X × I, X × I – Ȟ(Ƚ)) is)כܪ X × 0, X × 0 – (Ƚ(0), 0)) ՜)כܪ Ƚ : [0, 1] ՜ X, the inclusion induced map an isomorphism, where Ȟ(Ƚ) denotes the graph of Ƚ.

The following is a conjecture proposed by Quinn [49]. This was proven in 2006 by Bryant, and so far is the strongest result relating to the Bing-Borsuk conjecture [15].

Theorem 2.6.4 (Bryant). Every n-dimensional homologically arc-homogeneous ENR is a generalized n-manifold.

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Chapter 3. The Poincaré Conjecture

Here we will examine why the Bing-Borsuk conjecture is stronger than the Poincaré conjecture. In other words, the 3-dimensional case of the Bing-Borsuk conjecture implies the Poincaré conjecture. To achieve this result, first we show that the Poincaré conjecture is equivalent to the statement that there are no fake 3-cells. Then we show that the existence of a fake 3-cell implies the existence of a 3-dimensional homogeneous compact ANR which is not a manifold. These two results combine to give us our desired result.

Section 3.1. The Poincaré Conjecture is Equivalent to the Statement that there are no Fake 3-Cells

Definition 3.1.1. A homotopy 3-sphere is a closed 3-manifold which has the homotopy type of a 3-sphere.

Definition 3.1.2. A fake 3-sphere is a homotopy 3-sphere that is not homeomorphic to a 3-sphere.

Definition 3.1.3. A fake 3-cell is a compact contractible 3-manifold which is not homeomorphic to a closed 3-cell.

Here we also assume that the boundary of a fake 3-cell is homeomorphic to the 2-sphere.

Conjecture 3.1.4 (The Poincaré conjecture). Every homotopy 3-sphere is homeomorphic to a 3- sphere.

Before we prove the main result of this section, we must first cover some important definitions and theorems.

Definition 3.1.5. A local orientation of an n-manifold M at a point x is a choice of generator �� of the infinite cyclic group ��(M, M – {x}).

Definition 3.1.6. An orientation of an n-manifold M is a function x ↦ �� assigning to each x ∈ M a local orientation �� ∈ ��(M, M – {x}), satisfying the ‘local consistency’ condition below.

� This condition means that each x ∈ M has a neighbourhood � ⊂ M containing an open ball B of finite radius about x such that all the local orientations �� at the points y ∈ B are the � � images of one generator �� of ��(M, M – B ) ≈ ��(� , � – B ) under the natural maps ��(M, M – B ) → ��(M, M – {y}). If an orientation exists for M, then M is called orientable. Page | 31

Theorem 3.1.7 (Poincaré duality). If M is a closed orientable n-manifold, then there are �−� isomorphisms ��(M; Z ) ≈ � (M; Z ) for all k.

Theorem 3.1.8 (Hurewicz Theorem). If a space X is (n – 1)-connected for n ≥ 2, then ���(X ) = 0 for i < n and ��(X ) ≈ ��(X ).

Theorem 3.1.9 (Whitehead’s Theorem #1). Suppose that X and Y are connected CW-complexes and f : X → Y is a map such that �∗: ��(X ) → ��(Y ) is an isomorphism for all n. Then f is a homotopy equivalence.

Theorem 3.1.10 (Whitehead’s Theorem #2). Suppose X and Y are connected CW-complexes such that dim X, dim Y ≤ m and f : X → Y is a map such that �∗: ��(X ) → ��(Y ) is an isomorphism for all n ≤ m. Then f is a homotopy equivalence.

The two above theorems directly imply the following:

Theorem 3.1.11 (Whitehead). If X is a connected 3-manifold such that �1(X ), �2(X ), and �3(X ) are all trivial, then X is contractible.

Next, we require some well-known results:

Theorem 3.1.12 (Universal coefficient theorem for cohomology). If a chain complex C of free abelian � groups has homology groups ��(C ), then the cohomology groups � (C; G ) of the cochain complex Hom(��, G ) are determined by split exact sequences

ℎ � 0 → Ext(��−1(C ), G ) → � (C; G ) → Hom(��(C ), G ) → 0.

Theorem 3.1.13 (Mayer-Vietoris sequence). For a pair of subspaces A, B ⊂ X such that X is the union of the interiors of A and B, the following sequence is exact:

� � � ⋯ → ��(A ∩ B ) → ��(A) ⊕ ��(B ) → ��(X ) → ��−1(A ∩ B ) → ⋯

⋯ → �0(X ) → 0.

Theorem 3.1.14 (Seifert-van Kampen Theorem). Let X = U ∪ V, where U and V are open in X. Assume that U, V, and U ∩ V are path connected and let �0 ∈ U ∩ V. Let H be a group, and let

�1: �1(U, �0) → H and �2: �1(V, �0) → H Page | 32

be homomorphisms. Let �1, �2, �1, �2 be the homeomorphisms induced by inclusion.

′ ′ If �1 ∘ �1 = �2 ⃘ �2, then there is a unique homomorphism � : �1(X, �0) → H such that � ⃘ �1 = �1 ′ and � ⃘ �2 = �2.

Lemma 3.1.15 [30]. A 3-manifold M is a homotopy 3-sphere if and only if M is closed, connected, and simply connected.

Proof: For the first direction, suppose we have a 3-manifold M that is a homotopy 3-sphere. It is well known that a 3-sphere is closed, connected, and simply connected and these properties are invariant under homotopies. It follows that M is closed, connected, and simply connected.

The next direction is less trivial. Suppose now that M is a 3-manifold that is closed, connected, and simply connected. Firstly, M is simply connected implies that �1(M ) = 1. Note that �1(M ) is an abelianization of �1(M ), hence �1(M ) = 0. Since M is connected, the Hurewicz Theorem implies that �2(M ) = 0 and hence �2(M ) = 0.

Now define B as a 3-cell in M and let C = ����\����.

Next we can apply the Seifert-van Kampen Theorem to C. Here we take U = C and note that 2 2 B ∩ C = � . So we take V to be a thickening of this � by some fixed ε > 0. There is a deformation 2 retraction of V onto � , hence V is simply connected. It follows that U ∩ V is simply connected (hence it is obviously path-connected). Applying Theorem 3.1.14, we get that �1(C ) = 1.

Then by Theorem 3.1.13 (Mayer-Vietoris sequence) it follows that �2(C ) = 0, and thus ��(C ) = 0 for all q ≥ 1. Then by repeatedly applying the Hurewicz Theorem we get that ��(C ) = 0 for all q. Then by Theorem 3.1.11 (Whitehead), C is contractible.

We already know that �1(M ) = �2(M ) = 0. Further, M is simply connected hence �1(M ) = 2 3 0. Then by Poincaré duality it follows that � (M ) = � (�) = �1(�) are all trivial. By Theorem 3.1.12, we get �2(�) = 0 and �3(�) = Z. Applying the Hurewicz Theorem again, it follows that 3 3 �3(�) = Z. Then define f : � → M to be the map that takes � to the generator of π3(�). Then 3 �∗: ��(� ) → ��(M ) is an isomorphism for all n ≤ 3. Thus f is a homotopy equivalence by Theorem 3.1.10 (Whitehead’s Theorem #2). Therefore M is a homotopy 3-sphere, and the proof is complete.

□

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Now we can prove the main result of this section:

Theorem 3.1.16. The Poincaré conjecture is equivalent to the statement that there are no fake 3- cells.

Proof: First we prove that the Poincaré conjecture implies that there is no fake 3-cell. This of course is the same as proving that a fake 3-cell implies a fake 3-sphere. Thus, suppose we have a fake 3-cell F. Now take a 3-cell (call this B ) and glue F and B along their boundaries. The result M is clearly not a 3-sphere: for if it were a 3-sphere, this would immediately contradict the fact that F is a fake 3-cell. Hence to show that M is a fake 3-sphere, we must show that it is a homotopy 3-sphere. By Lemma 3.1.15, this is reduced to showing that M is simply connected.

Hence we can apply Theorem 3.1.14 (Seifert-van Kampen Theorem) again. Take U to be a thickening of B and take V to be a thickening of F. Then U and V are simply connected. Further, 2 2 recall that the boundary of both F and B is � . Hence U ∩ V is a thickening of � . Thus there is a 2 deformation retraction from U ∩ V to � , which implies that U ∩ V is simply connected. Applying Theorem 3.1.14, it follows that M is simply connected.

For the next direction, we must prove that a fake 3-sphere implies the existence of a fake 3-cell.

Suppose we have a fake 3-sphere M. Cut a 3-cell out of M and call it �1. Then let �2 = 2 ����\�����1�. Clearly the intersection of �1 and �2 is � . Similar to the proof of Lemma 3.1.15, we can use the Seifert-van Kampen Theorem to prove that �2 is simply connected. Hence it follows that �1(�2) = 0.

2 3 Then by Poincaré duality it follows that � (�2) = � (�2) = �1(�2) = 0. By Theorem 3.1.12, we get that �2(�2) = �3(�2) = 0. Applying the Hurewicz Theorem, it follows that �2(�2) = �3(�2) = 0. Also, recall that �2 is simply connected hence �1(�2) is trivial. Thus �1(�2), �2(�2), and �3(�2) are all trivial, so we can apply Theorem 3.1.11 (Whitehead). Thus �2 is contractible.

At this point we know that �2 is a compact contractible 3-manifold. So to prove that �2 is a fake 3-cell, we must show that it is not a 3-cell. But this follows immediately by definition. For if �2 were a 3-cell, then M would be a 3-sphere, which is a contradiction. Therefore �2 is a fake 3-cell and the proof is complete.

□

Page | 34

Section 3.2. The Bing-Borsuk Conjecture implies the Poincaré Conjecture

Recall our central results:

The Poincaré conjecture. Every homotopy 3-sphere is homeomorphic to a 3-sphere.

The Bing-Borsuk conjecture. Every n-dimensional homogeneous compact ANR is an n-dimensional manifold.

The main goal of this section is the following theorem and an exposition of its proof:

Theorem 3.2.1 [32]. If there exists a fake 3-cell F, then there exists a 3-dimensional homogeneous compact ANR-space K which is not a manifold.

This theorem, combined with the main result from Section 3.1, proves that the 3-dimensional case of the Bing-Borsuk conjecture implies the Poincaré conjecture.

Proof: During this proof, we assume the existence of a fixed fake 3-cell F. Recall from Section 1.3 that in the 1950’s, Edwin Moise proved that every 3-manifold is triangulable. Hence we can assume that F has a given triangulation with a fixed orientation.

The existence of this fake 3-cell F then allows us to construct a 3-dimensional homogeneous compact ANR-space K which is not a manifold. We divide this process into the following steps:

3.2.1 The construction of the space K. 3.2.2 K is an absolute neighbourhood retract. 3.2.3 Some definitions and lemmas. 3.2.4 K is homogeneous. 3.2.5 K is not a 3-manifold. 3.2.6 K has dimension 3.

Page | 35

Step 3.2.1. The Construction of K

Here we inductively construct a sequence of 3-manifolds �� and maps ���: �� → ��. Then we obtain K = lim{��, ���} as the limit of this sequence. Define �1 to be the boundary of a 4-simplex and fix on it an orientation. Now suppose we have a manifold ��−1 with a given triangulation and an orientation. We use the existence of a fake 3-cell F to perform a construction on ��−1. This construction will produce the next manifold ��.

Now, let σ denote a closed 3-simplex of ��−1. Take the second barycentric subdivision of σ, ′ ′ ′ choose one of its closed 3-simplices � such that � ⊂ Intσ, and remove the interior of � .

′ Next take a copy �� of a fake 3-cell F and construct the space (σ\Int� ) ∪ ℎ��� where ℎ�: ′ ′ ∂�� → ∂� is a simplicial homeomorphism which reverses the orientation. Note that both ∂� ⊂ ��−1 and ∂�� have the same triangulation as the boundary of a 3-simplex, and have orientations induced by the orientations of ��−1 and ��, respectively.

We perform this construction for all 3-simplices of ��−1: for each closed 3-simplex σ, there is a corresponding fake 3-cell �� which we attach to σ in the specified way. This produces the next ′ manifold ��. Note that the orientations induced on σ\Int� by the orientations of ��−1 and �� are equal for each σ. Hence the orientation of this new manifold �� is completely determined by the orientation of the previous manifold ��−1. The triangulation of �� is determined by the second ′ barycentric subdivision on each set of the form σ\Int� , and it is a fixed triangulation of F on each set ��.

Next we define ��,�−1: �� → ��−1 as follows: for each 3-simplex σ of ��−1 we take ′ ′ ′ ��,�−1|(σ\Int� ) = ��σ\Int� and on each �� ⊂ �� we let ��,�−1|��: �� → � be any simplicial map ′ which, restricted to ∂�� = ∂� , is equal to the identity map. Also note that there exists a 2 neighbourhood of ∂�� that is homeomorphic to � × I, and this guarantees the existence of the map ��,�−1|��. Finally take ��� = ��+1,� · ��+2,�+1 ··· ��,�−1 for j < i. By ��: K → ��, we shall denote the natural projection of K = lim{��, ���} into ��.

Page | 36

Step 3.2.2. K is an Absolute Neighbourhood Retract

For this section, we need two well-known results:

,Theorem 3.2.2 [29]. If (X, A) is a CW pair, then X × {0} ׫A × I is a deformation retract of X × I hence (X, A) has the homotopy extension property.

Theorem 3.2.3 [29]. If the pair (X, A) satisfies the homotopy extension property and A is contractible, then the quotient map q : X ՜ X |A is a homotopy equivalence.

Proof: Since dim = 3, we can embed in 7-dimensional Euclidean space ଻. Let { , , … , } ܭ௜ ܭ௜ ܴ ߪଵ ߪଶ ߪ௡೔ denote the family of 3-simplices of , and let { , , … , } denote a family of 7-simplices in ଻ ܭ௜ ߪതଵ ߪതଶ ߪത௡೔ ܴ such that for 1 ൑ j ൑ k ൑ ݊௜:

(1) ߪഥఫ contains ߪ௝ as a 3-dimensional face,

,ߪ௞ for j ് k ת ߪത௞ = ߪ௝ ת ߪഥఫ (2)

(3) diamߪ௝ = diamߪഥఫ.

Now let = … . Following from the definition of and , it is clear that . ௜ܤ௜ ؿܭ ௜ܤ ௜ܭ ௜ ߪതଵ ׫ ߪതଶ ׫ ׫ ߪത௡೔ܤ

Further, note that ܭ௜ାଵ can be considered to be a subpolyhedron of ܤ௜.

Let denote the fake 3-cell attached in the construction of instead of the removed ܨఙೕ ܭ௜ାଵ simplex ᇱ. In fact, can be realized as a subpolyhedron of = . It is also ఙೕܨ ת ఙೕ ߪ௝ܨŠ–Šƒ–μ —• ఙೕ ߪഥఫܨ ߪ௝ ଵ clear that each simplex of ܨ has diameter less than diamߪ (since ܨ is obtained by taking the ఙೕ ଶ ௝ ఙೕ

second barycentric subdivision of ߪ௝).

ǥـ ଷܤ ـ ଶܤ ـ ଵܤ ௜ and hence we can assume thatܤ ௜ାଵ ؿܤ ௜ାଵ ؿܭ Moreover, we can see that ஶ ௜ is homeomorphic to K. By Theorem 3.2.2 and Theorem 3.2.3, itܤ ௜ୀଵځ Then it is easy to check that follows that ܭ௜ and ܭ௜ାଵ have the same homotopy type. Consequently, so do ܤ௜ and ܤ௜ାଵ (since we can find a deformation retraction from ܤ௜ to ܭ௜). Hence there exists a retraction ݎ௜: ܤ௜ ՜ ܤ௜ାଵ (by [56]).

௜(ߪഥሻؿఫ ߪഥఫ. This condition implies thatݎ ௜, we getܤ We can also claim that for each ߪഥఫ ؿ ஶ ௜, which provesܤ ௜ୀଵځ ଵ ՜ܤ : ଵ} is a sequence of maps that converge to the retraction rݎ ·…·௜ିଵݎ ·௜ݎ} that K is an ANR.

ᇝ Page | 37

Step 3.2.3. Some Definitions and Lemmas

Suppose M is a 3-manifold. Let Z denote the family of 3-cells contained in the interior of M and let S(Z ) denote the sum of all the interiors of 3-cells in Z. Further, the pair (M, Z ) is orientable means .Z א for every Zwe have a fixed orientation on M and induced orientations on μ, Z, and μ

Definition 3.2.4. A family Z of 3-cells in the interior of a given 3-manifold M is called good if:

,ଶ = ׎ܼ ת For every ܼଵ and ܼଶ in Z such that ܼଵ ് ܼଶ, we have ܼଵ(1) ,Z is a tame 3-cell in M, i.e. (M, Z ) is homeomorphic to a polyhedral pair א Each Z(2) ,Z : diamZ > ߝ} is finite א Z is a null-family in M, i.e. for every ߝ > 0 the set {Z(3) (4)S(Z ) is dense in M.

Theorem 3.2.5 [40]. Suppose G is an upper semi-continuous decomposition of ܴଷ into points and tame 3-cells such that G is a null-family. Further, let ܩଵ be the set of non-degenerate elements of G כ and let ܩଵ be the union of the elements of ܩଵ. If ߝ is a positive number and U is an open set ଷ כ ,G, (diam݂ఌ[g]) < ߝ א ଵ, there exists a homeomorphism ݂ఌ of ܴ onto itself such that if gܩ containing ଷ .U ), then ݂ఌ(x) = x – ܴ) א and x

א Lemma 3.2.6. Let T be a good family of 3-cells in M, let ࢀଵ be a finite subfamily of T, and let u :H(M ) such that א H(M ). Then for any ߝ > ߜ > 0 there exists a v

,ࢀଵ א v |T = u |T for any T(1)

,T \ࢀଵ with diamT < ߝ and diamu(T ) < ߝ א diamv(T ) < ߜ for any T(2) (3)dist(v, u) < ߝ, .dist(ݒିଵ, ݑିଵ) < ߝ(4)

Proof:

.{ T \ܶଵ: diamT < ɂ and Ɂ ൑ diamu(T ) < ɂ א Let ࢀ଴ = {T

Recall that T is a good family of 3-cells in M, hence T is a null-family. It follows that for any

ࢀ଴, it is possible to find an open T - saturated neighbourhood ்ܷ homeomorphic to an open א T ଷ subset of ܴ . This means that any cells in T that intersect ்ܷ are contained in ்ܷ. Further, we have that diam < and diamu( ) < . Moreover, we can require that = for any , మ ׎ ܶଵ்ܷ ת ߝ ்ܷ ߝ ்ܷభ ்ܷ ᇱ ᇱ .ࢀଵ א ܶ ,ࢀ଴ א ׎ for any T = ܶ ת ்ܷ ࢀ଴, ܶଵ ്ܶଶ, and thatא ଶܶ Page | 38

3 ′ Then by Theorem 3.2.5 (considering �� as a subset of � ) there is a homeomorphism � ′ 0 0 such that � (x) = x for x ∈ M \⋃ ���∈� . Also note that we can select these ��’s so that ⋃ ���∈� is

0 contained in the interior of M. Hence the boundary of M is contained in M \⋃ ���∈� , which implies ′ ′ that � (x) = x for all x ∈ ∂M. Thus � ∈ H(M ).

′ 0 Again by Theorem 3.2.5, we have that diam(u� (T )) < � for any T ∈ T with T ⊂ ⋃ ���∈� . ′ ′ ′ 0 Finally, we set v = u� . Recall that �� ∩ � = ∅ for any T ∈ �0, � ∈ �1, hence �1 ⊂ M \⋃ ���∈� . ′ ′ Thus � (x) = x for all x contained in �1, so it follows that v |T = uv |T = u |T for any T ∈ �1. Using similar arguments, we see that v satisfies the remaining conditions of the lemma.

□

�−1 � Theorem 3.2.7 (The Annulus Theorem). Let f, g : � → � be disjoint, locally flat imbeddings with �−1 � �−1 f (� ) inside the bounded component of � – g(� ). Then the closed region A bounded by �−1 �−1 �−1 f (� ) and g(� ) is homeomorphic to � × [0, 1].

Lemma 3.2.8. Let M and N be orientable 3-manifolds, h : M → N an orientation preserving homeomorphism, Y and F two good families of 3-cells contained in the interior of M and N � respectively, and for every (Y, Z ) ∈ Y × Z, let �� : ∂Y → ∂Z be an orientation-preserving ′ homeomorphism. Then there exists a bijective function p : Y → Z and a homeomorphism ℎ : ′ ′ �(�) M \S(Y ) → N \S(Z ) such that ℎ |∂M = h |∂M, and ℎ |∂Y = �� for every Y ∈ Y.

Proof: Without loss of generality, we can assume that M = N, h = ���, and diamM ≤ 1. It is well � � known that each �� can be extended to a homeomorphism �� : Y → Z for Y ∈ Y and Z ∈ Z. Let � = � {�� :Y ∈ Y, Z ∈ Z } and let H(M ) be the set of all homeomorphisms of M which are the identity on ∂M. Further, define

−� −� �� = {Z ∈ Z : diamZ ≥ 2 } and �� = {Y ∈ Y : diamY ≥ 2 }.

For any f ∈ H(M) and any family T of subsets of M, define f (T ) = {f (T ): T ∈ T }.

Now we shall inductively construct homeomorphisms ��, �� ∈ H(M ), for n = 1, 2, … such that the following conditions are satisfied:

Page | 39

−1 (��) If Y ∈ ��, then there is a Z ∈ Z such that ��(Y ) = ��(Z ) and �� ��|Y ∈ �,

′ −1 (��) If Z ∈ ��, then there is a Y ∈ Y such that ��(Y ) = ��(Z ) and �� ��|Y ∈ �,

−� −1 (��) diam��(Y ) < 2 for every Y ∈ Y \(�� ∪ �� ��(��)),

′ −� −1 (��) diam��(Z ) < 2 for every Z ∈ Z \(�� ∪ �� ��(��)),

−1 (��) ��|Y = ��−1|Y for every Y ∈ ��−1 ∪ ��−1��−1(��−1),

′ −1 (��) ��|Z = ��−1|Z for every Z ∈ ��−1 ∪ ��−1��−1(��−1),

−�+2 −1 −1 −�+2 (��) dist(��, ��−1) ≤ 2 , dist(�� , ��−1) ≤ 2 ,

′ −�+3 −1 −1 −�+3 (��) dist(��, ��−1) ≤ 2 , dist(�� , ��−1) ≤ 2 .

It follows that f = lim�� and g = lim�� are in H(M ).

′′ −1 ′′ Now set ℎ = � f. Due to the conditions listed above, we will see that ℎ satisfies the ′ ′′ specifications of the lemma. Following from conditions (��) and (��) , ℎ is a homeomorphism ′′ ′′ ′ ′′ such that ℎ (M \S(Y )) = M \S(Z ) and ℎ |Y ∈ � for every Y ∈ Y. So we can take ℎ = ℎ |M \S(Y ) ′′ and p(Y ) = ℎ (�) for every Y ∈ Y. Thus we have a bijective function p : Y → Z and a ′ ′ homeomorphism ℎ : M \S(Y ) → M \S(Z ). Recall that f, g ∈ H(M ) and h = ��� so clearly ℎ |∂M = ′′ ′ p(Y) h |∂M. Further, since p(Y ) = ℎ (�) it follows that h |∂Y = φY for every Y ∈ Y. Therefore all we need to do is perform this construction, and the lemma will be proven.

First we set �0 = �0 = ���. Then suppose that for some n ≥ 1, ��−1 and ��−1 are already constructed.

−�+1 Now we can use Lemma 3.2.6 with � = 2 , � = min{diam��−1(Y ): Y ∈ ��\��−1}, T = Z, −1 �1 = ��−1 ∪ ��−1��−1(��−1) and u = ��−1. Note that (��) guarantees the condition that � > � > 0. Hence we get a new homeomorphism v ∈ H(M ) such that: v |Z = ��−1|Z for Z ∈ �1, diam(v(Z )) < −�+1 −1 −1 −�+1 � for Z ∈ T \�1, dist(v, ��−1) < 2 , and dist(� , ��−1) < 2 .

−1 Using similar reasoning as in the proof of Lemma 3.2.6, given Y ∈ ��\(��−1��−1(��−1) ∪ ��−1), we can use (��−1) to get a Y - saturated neighbourhood �� of Y which is homeomorphic to 3 −�+1 −�+1 an open subset of � and satisfies diam�� < 2 and diam��−1(��) < 2 . Further, we require −1 that the ��’s are pairwise disjoint and miss the elements of ��−1 ∪ ��−1��−1(��−1). Page | 40

Note that ��−1(Y ) is not contained in any element of v(Z ) and hence there is a �� ∈ Z \ −1 ��−1\ ��−1��−1(��−1) with v(��) ⊂ ��−1(Y ). Since every Y ∈ Y and Z ∈ Z is a tame cell in M, it follows from the Annulus Theorem that we can find a w ∈ H(M ) such that: w(x) = x if x is in none −1 of the ��’s, w ��−1(Y ) = v (��) and � v ��−1|Y ∈ � where Y runs over ��\��−1\ −1 −�+2 ��−1��−1(��−1). Applying Lemma 3.2.6 again with � = 2 , � = min{diamv(Z ): Z ∈ ��\��−1}, −1 T = Y, �1 = �� ∪ ��−1��−1(��−1) and u = w ��−1, we get a homeomorphism �� ∈ H(M ) such that: −�+1 ��|Y = w ��−1|Y for Y ∈ �1, diam(��(Y )) < � for every Y ∈ Y \�1, dist(��, w ��−1) < 2 , and −1 −1 −�+1 dist(�� , (���−1) ) < 2 . It is easy to see that with v in place of ��, conditions (��), (��), (��), and (��) are satisfied.

−1 Similarly, given Z ∈ ��\��−1\�� v (��−1), the set v (Z ) is not contained in any member of −1 ��(Y ) and hence there is a �� ∈ Y \��\�� v (��−1) with ��(��) ⊂ v (Z ). Again by the Annulus Theorem and Lemma 3.2.6 (as in the construction of w and �� above), we get �� ∈ H(M ) such that −1 ′ ′ ′ ′ ��|Z = v |Z for Z ∈ ��−1 ∪ � ��(��). Therefore conditions (��) , (��) , (��) , and (��) are satisfied and the proof is complete.

□

3 Theorem 3.2.9 [40]. Let G be an upper semi-continuous decomposition of � into compact continua such that G is a null-family and each non-degenerate element of G is a tame 3-cell. Then G is 3 homeomorphic to � .

Lemma 3.2.10. If Z is a good family of 3-cells in the interior of a 3-manifold M, and G is a decomposition of M whose non-degenerate elements are exactly the elements of Z, then M |G is homeomorphic to M. ′ Proof: For a given triangulation of M, clearly we are able to construct a good family of 3-cells � ′ such that the 2-skeleton of this triangulation misses the elements of � . Hence by Lemma 3.2.8, we can assume that there is a triangulation � of M such that the 2-skeleton of � misses all the elements of Z.

Now let �: M → M |G denote the projection map. Then by Theorem 3.2.9, it follows that for every 3-simplex � of �, there is a homeomorphism ℎ�: � → �(�). Further, we have that ℎ�(x) = �(x) for each x ∈ ��. This allows us to “glue” the ℎ�’s together, which gives us our desired homeomorphism h : M → M |G.

□ Page | 41

௜ܭ ఙ ؿܨ ௜ andܭ ௜ିଵ. Letܭ Definition 3.2.11. Take a fixed natural number i and a fixed 3-simplex ߪ of

be the sets described in Step 3.2.1, and recall that ݂௜: lim(ܭ௜, ݂௜௝) ՜ ܭ௜ is the natural projection. Then ିଵ we define a fake 3-sponge as ݂௜ (ܨఙ), and denote this by ܨ෨. Further, we define the boundary of ܨ෨ as ିଵ ߲ܨ෨ = ݂௜ (߲ܨఙ).

It immediately follows that if we take any other natural number ݅ᇱ and simplex ߪᇱ, we get a pair ( ෨’, ෨’) homeomorphic to ( ෩, ෨). Further, | = for j i. Hence ିଵ| : ෨ ܨ ߲ܨ ܨ ߲ܨ ݂௜௝ ߲ܨఙ ݅݀பி഑ ൒ ݂௜ ߲ܨఙ ߲ܨఙ ՜ ߲ܨ is a homeomorphism of 2-spheres which induces an orientation on ߲ܨ෨.

be a family of metric spaces where כܯ Definition 3.2.12. For every orientable 3-manifold M we let :if and only if there exists a good family Z of 3-cells in M such that כܯ א X

,෨௓ܨ ܈א௓ڂ X = M \S(Z ) ׫(1)

,෨௓ܨ ෨ ՜ܨ :Z there is a homeomorphism ݃௓ א For every Z(2) (3) ෨ ෨ = for and (M \S(Z )) ෨ = Z = ( ෨) where Z Z, and moreover א ܨ௓ ߲ ݃௓ ߲ܨ ת ௓మ ׎ ܼଵ ് ܼଶܨ ת ௓భܨ

݃௓|߲ܨ෨: ߲ܨ෨ ՜ ߲Z is an orientation-reversing homeomorphism,

.෨௓ > ߝ} is finiteܨZ: diam א X is equipped with a metric in which for every ߝ > 0 the set {Z(4)

Lemma 3.2.13. If M and N are orientable 3-manifolds, h : M ՜ N is an orientation-preserving כ כ then there is a homeomorphism ݄ത: ܺଵ ՜ ܺଶ such that ݄ത |߲M , ܰ א and ܺଶ ܯ א homeomorphism, ܺଵ = h |߲M.

Proof: According to definition 3.2.12, we have two good families of 3-cells: Y in M and Z in N such

෨௓. Also by definition 3.2.12, we have twoܨ ࢆא௓ڂ ෨௒ and ܺଶ = N \S(Z ) ׫ܨ ࢅא௒ڂ that ܺଵ = M \S(Y ) ׫

Y × Z, we א ( Then for any (Y, Z .܈א෨௓}௓ܨ෨ ՜ܨ :ࢅ and {݃௓א෨௒}௒ܨ෨ ՜ܨ :families of homeomorphisms {݃௒ ௓ ௓ can use these two families to give us a homeomorphism ߮௒ : μ ՜ μ. We simply define ߮௒ = ିଵ (݃௓݃௒ )|߲Y.

Now we can apply Lemma 3.2.8. This gives us a bijective function p : Y ՜ Z and a homeomorphism ݄ᇱ: M \S(Y ) ՜ N \S(Z ). To get our desired homeomorphism ݄ത, we define ݄ത(x) = ᇱ = (Y. Set ݄ത(x א ෨௒ for each Yܨ M \S(Y ). Now all that is left is to define ݄ത on א x) for all x) ݄ ିଵ ෨௒ (this is well defined because p : Y ՜ Z is bijective). Lastly (by Lemmaܨ א ௣(௒)݃௒ )(x) for all x݃) 3.2.8) we know that ݄ᇱ|μ = h |μ, hence it follows that ݄ത|߲M = h |߲M. Therefore we have our

desired homeomorphism ݄ത: ܺଵ ՜ ܺଶ. ᇝ Page | 42

Lemma 3.2.14. Let P be a subpolyhedron of �� (in the fixed triangulation of ��) which is a −1 ∗ ∗ 3-manifold. Then �� (P ) ∈ � . In particular, we can take P = �� which gives us K ∈ �� for every natural n.

Proof: Following from the construction in Step 3.2.1, we know that �� ∩ ��+1 = ��\S(���) where ∞ � � �� is some finite family of 3-cells in ��. Set Z = ⋃�=1 �� and �� = {Z ∈ Z : Z ⊂ P }, then �� is a good family of 3-cells in P. Further, we have that ���|P \S(��) = ���\S(��) for all m ≥ n. Hence the −1 natural projection �� gives us a homeomorphism between �� (P \S(��)) and P \S(��).

−1 −1 � Then we can write �� (P ) = P \S(��) ∪ ⋃ ����∈� where ��� = �� (��). Here �� is a copy of �ℎ the fake 3-cell F attached in the � step of construction instead of the removed interior of the 3-cell Z ∈ ��. As mentioned above, ��� is homeomorphic to �� and ∂��� = ∂Z (this identification reverses the orientation for each Z ∈ ��).

It is easy to check that in any metric coincident with the topology of K, the set {Z ∈ ��: diam��� > �} is finite for every � > 0 (this follows from the fact that �� is a good family of 3-cells in −1 ∗ P ). All the conditions of definition 3.2.12 are satisfied, therefore �� (P ) ∈ � . □

Definition 3.2.15. Given two disjoint connected n-manifolds �1 and �2, their connected sum (denoted by �1 # �2) is the manifold constructed by deleting the interiors of closed n-balls �1 ⊂ �1 and �2 ⊂ �2 and identifying the resulting boundary spheres ∂�1 and ∂�2 via some homeomorphism between them.

Lemma 3.2.16. Suppose �0 is a copy of the homotopy 3-sphere. Then for every orientable 3- ∗ manifold M and every X ∈ � , we have X ∈ (M # �0)*.

Proof: By definition, there exists some good family Z of 3-cells in M such that X = M \S(Z ) ∪ 1 1 1 1 ⋃ ����∈� . Now apply Lemma 3.2.14 (with P = �� ), giving us ��� ∈ �� * for any �1 ∈ Z. Hence ��� = �0\S(Y ) ∪ ⋃ ����∈� where �0 is a copy of the fake 3-cell F, and Y is a good family of 3-cells in �0. Let T = Y ∪ Z \{�1}, hence we have X = ((M \Int�1) ∪ �0)\S(T ) ∪ ⋃ ����∈� . However, (M \Int�1) ∪ �0 = M # �0. Thus X = (M # �0)\S(T ) ∪ ⋃ ����∈� where T is a good family of 3-cells in M # �0. Therefore, by definition X ∈ (M # �0)* and the proof is complete.

□ Page | 43

Step 3.2.4. K is Homogeneous

Proof: Take two points x, y ∈ K. We can find two sequences {��}�∈� and {��}�∈� of open sets in K such that x = ⋂ ���∈� , y = ⋂ ���∈� , �� ⊃ ���+1, and �� ⊃ ���+1 for every i. Further, each of �� and �� −1 has the form �� (A) where A is some open 3-cell in �� for some natural j = j (i ). We also require −1 that j (1) = 1 and j (k ) < j (l ) for any natural k < l. Hence �1 and �1 have the form �1 (A) for ′ some open 3-cell A ⊂ �1. By Lemma 3.2.14, K \U1 ∈ (K1\A1)* and K \�1 ∈ (�1\�1)* for some open ′ 3-cells �1 and �1 in �1.

Next, we are going to use induction to prove the following statement:

For every n ∈ N, we can find a homeomorphism ℎ�+1: K \��+1 → K \��+1 such that ℎ�+1(∂���+1) = ∂���+1 and the homeomorphism ℎ�+1|∂���+1: ∂���+1 → ∂���+1 preserves orientation.

By Lemma 3.2.13, there is a homeomorphism ℎ1: K \�1 → K \�1 such that ℎ1(∂��1) = ∂��1 and ℎ1|∂��1: ∂��1 → ∂��1 is a homeomorphism preserving the orientation. Now we assume the statement is true for all n. That is, suppose we have constructed a homeomorphism ℎ�: K \�� → K \�� such that ℎ�(∂���) = ∂��� and ℎ�|∂���: ∂��� → ∂��� preserves orientation.

′ ′ For p = j (n) and q = j (n +1) we have open 3-cells ��, �� ⊂ ��, ��, and �� ⊂ �� such that

−1 −1 −1 −1 ���\��+1 = �� (�̅�)\ �� (��) = �� (��� (�̅�)\��) and

−1 −1 ′ ���\��+1 = �� (��� (�̅′�)\��).

Again using Lemma 3.2.14,

−1 −1 ′ ���\��+1 ∈ (��� (�̅�)\��)* and ���\��+1 ∈ (��� (�̅′�)\��)*.

From the way we constructed K, it follows that there are orientation-preserving −1 −1 ′ homeomorphisms from ��� (�̅�)\�� onto �1 # �2 # �1 # ... # ��1 and from ��� (�̅′�)\�� onto �1 # �2 # �1 # ... # ��2. Here �1 and �2 are 3-cells, each �� is a copy of the homotopy 3-sphere H, and �1, �2 are two natural numbers.

Page | 44

Now apply Lemma 3.2.13 and Lemma 3.2.16, so we get a homeomorphism ℎ��+1: ���\��+1 → ���\��+1 such that ℎ��+1|∂��� = ℎ�|∂��� and that ℎ��+1|∂��+1: ∂���+1 → ∂���+1 is an orientation- preserving homeomorphism. Set ℎ�+1(x) = ℎ�(x) for x ∈ K \�� and ℎ�+1(x) = ℎ��+1(x) for x ∈ ���\��+1. This gives us a homeomorphism ℎ�+1: K \��+1 → K \��+1 such that ℎ�+1(∂���+1) = ∂���+1 and the homeomorphism ℎ�+1|∂���+1: ∂���+1 → ∂���+1 preserves orientation. This completes our induction on n.

Thus we can define a homeomorphism h : (K, x) → (K, y) by setting h |(K \��) = ℎ�|(K \��) for each n ∈ N. Recalling that x = ⋂ ���∈� and y = ⋂ ���∈� , we have that h(x) = y. Therefore K is homogeneous and the proof is complete.

□

Page | 45

Step 3.2.5. K is not a 3-Manifold

Theorem 3.2.17 [3]. Suppose M is a 3-manifold and G is a cellular decomposition of M such that M |G is a 3-manifold N. Then M and N are homeomorphic.

Definition 3.2.18. A set A in an n-dimensional compact space X is said to be cellular if there exist ∞ n-cells �1, �2, … in X such that ��+1 ⊂ Int�� and ⋂�=1 �� = A.

� Lemma 3.2.19 [13]. Let Q be an n-cell. Suppose we have a map f :Q → � and that A ⊂ IntQ is the only inverse set under f. Then A is cellular in Q.

�−1 � Lemma 3.2.20 [13]. Let h be a homeomorphic embedding of � × I into � . Then the closure of �−1 either complementary domain of h(� ×½) is an n-cell.

Proof: Here we shall suppose that K is a 3-manifold and derive a contradiction. Recall that Lemma ∗ � 3.2.14 states K ∈ �� for every natural n. Hence by definition K = ��\S(��) ∪ ⋃ ����∈� where �� is a good family of 3-cells in ��. Further, there exists a family of homeomorphisms {��: �� → ���}�∈�� and for every � > 0 the set {Z ∈ ��: diam(���) > �} is finite. This implies that there exists a �0 ∈ Z � 0 and a 3-cell Q contained in K = ��\S(��) ∪ ⋃ ����∈� , such that ��� ⊂ IntQ. It immediately follows that the quotient space ��|�0 is homeomorphic to ��, hence �0: �� → ��|�0 is a homeomorphism.

Next we consider a family Y of 3-cells in ��|�0 defined as follows: Y ∈ Y if and only if there exists a Z ∈ Z \{�0} such that �0(Z ) = Y. It is easy to check that Y is a good family of 3-cells in ��|�0. Now we consider the space

0 � 0 K |��� = (��\S(��) ∪ ⋃ ����∈� )|��� ,

and the corresponding quotient map �1. Then we have

0 � 0 K |��� = �1(��\S(��)) ∪ ⋃�∈� \{� } �1(���).

We identify �1(��\S(��)) and �0(��\S(��)) = (��|�0)\S(Y ). Then the family {�1��: �� → �1(���)}�∈�\{�0} is a family of homeomorphisms satisfying definition 3.2.12. Thus K |���0 ∈ (��|�0)*. Page | 46

Recall that ��|�0 and �� are homeomorphic. It follows by Lemma 3.2.13 that K |���0 and K are homeomorphic. Since ���0 is contained in a 3-cell Q ⊂ K, Lemma 3.2.19 implies that ���0 is a cellular set in K.

Next we will prove that ��� is cellular for every Z ∈ ��. Take any �1 ∈ �� and consider the space

1 � 1 K \Int��� = (��\Int�1)\S(��\{�1}) ∪ ⋃�∈� \{� } ���.

Now, ��\{�1} is a good family of 3-cells in ��\Int�1 hence by Lemma 3.2.14, K \Int���1 ∈ (��\Int�1)*. Similarly, K \Int���0 ∈ (��\Int�0)*. Note that ∂�1 ⊂ ���1 ⊂ K and ∂�0 ⊂ ���0 ⊂ K, hence −1 (��1|∂��)(��0 | ∂��) : ∂�0 → ∂�1 is a homeomorphism preserving orientation. Hence there exists a −1 homeomorphism h : ��\Int�0 → ��\Int�1 such that h |∂�0 = (��1|∂��)(��0| ∂��) . Thus by Lemma 3.2.13, there exists a homeomorphism ℎ�: K \Int���0 → K \Int���1 such that ℎ�|∂�0 = −1 (��1|∂��)(��0 | ∂��) .

′ ′ −1 This allows us to extend ℎ� to a homeomorphism ℎ : K → K. Simply set ℎ (x) = ��1��0 (x) for ′ x ∈ ���0. Clearly it follows that ℎ (���0) = ���1. Therefore ���1 is cellular in K for any �1 ∈ ��.

� Suppose �� is the decomposition of the space K = ��\S(��) ∪ ⋃ ����∈� whose non- ′ degenerate elements are the sets ��� for all Z ∈ ��; and let �� be the decomposition of �� whose non-degenerate elements form the good family ��. Then the quotient space K |�� is homeomorphic ′ ′ to ��|��. By Lemma 3.2.10, ��|�� is homeomorphic to �� and hence K |�� also is.

Recall that (to derive a contradiction) we have supposed K is a 3-manifold and �� was shown to consist of sets which are cellular in K. Hence we can apply Theorem 3.2.17. Then K is homeomorphic to K |��, which is homeomorphic to ��. Thus K is homeomorphic to �� for every natural n. This implies that �1 is homeomorphic to �2. But �1 is a 3-sphere and �2 contains the fake 3-cell F. This is impossible by Lemma 3.2.20, hence we have a contradiction. Therefore K is not a 3-manifold and the proof is complete. □

Page | 47

Step 3.2.6. K has dimension 3

� Theorem 3.2.21 [31]. If the inverse sequence S = {��, �� } consists of separable metric spaces �� such that dim�� ≤ n for i = 1, 2, … , then the limit X = lim�→∞ � satisfies the inequality dimX ≤ n.

Proof: First we show that dimK ≤ 3. Recalling that dim�� = 3 for every n and K = lim{��, ���}, the above theorem implies that dimK ≤ 3.

Next, to show that dimK ≥ 3, we simply note that the Bing-Borsuk conjecture is true for n < 3 (see [8]). Hence if dimK < 3, it would follow that K is a 3-manifold, which it isn’t. Hence we have that dimK ≤ 3 and dimK ≥ 3. Therefore dimK = 3, and the proof is complete. □

Page | 48

Bibliography

[1] P. S. Aleksandrov, Untersuchungen Uber Gestalt und Lage Abgeschlossener Mengen Beliebiger Dimension, Ann. of Math. 30 (1928): 101-187. Print. [2] M. L. Antoine, Sur la Possibilité d’ Etendre L’Homéomorphie de Deux Figures à Leur Voisinages, C. R. Acad. Sci. Paris 171 (1920): 661-663. Print. [3] S. Armentrout, Cellular Decompositions of 3-Manifolds that Yield 3-Manifolds, Mem. Amer. Math. Soc. 107, Amer. Math. Soc., Providence, R. I., (1971). Print. [4] M. Bestvina, R. J. Daverman, G. A. Venma, J. J. Walsh, A 4-Dimensional 1-LCC Shrinking Theorem, Geometric Topology and Geometric Group Theory, Milwaukee, WI, 1997, Topology Appl. 110 (2001): 3-20. Print. 3 [5] R. H. Bing, A Decomposition of � into Points and Tame Arcs such that the Decomposition Space 3 is Topologically Different from � , Ann. of Math. 65 (1957): 484-500. Print. [6] R. H. Bing, An Alternate Proof that 3-Manifolds can be Triangulated, Ann. of Math. 69 (1959): 37-65. Print. 4 [7] R. H. Bing, The Cartesian Product of a Certain Nonmanifold and a Line is � , Ann. of Math. 70 (1959): 399-412. Print. [8] R. H. Bing, K. Borsuk, Some Remarks Concerning Topologically Homogeneous Spaces, Ann. of Math. 81 (1965): 100-111. Print. [9] K. Borsuk, Theory of Retracts, Polish Scientific Publishers: Warszawa, 1966. Print.

[10] G. Bredon, Generalized Manifolds Revisited, Topology of Manifolds, (J. D. Cantrell and C. H. Edwards, Eds.), Proceeding of the University of Georgia. Print. [11] G. Bredon, Sheaf Theory, McGraw-Hill: New York, 1967. Print. [12] L. E. J. Brouwer, Beweis der Invarianz der Dimensionenzahl, Ann. of Math. 70 (1911): 61-165. Print. [13] M. Brown, A Proof of the Generalized Schoenflies Theorem, Bull. Amer. Math. Soc. 12 (1961): 812-814. Print. [14] J. L. Bryant, Reflections on the Bing-Borsuk Conjecture, Abstracts of the talks presented at the 19th Annual Workshop in Geometric Topology, June 13-15, (2002): 2-3. Print. [15] J. L. Bryant, Homologically Arc-homogeneous ENRs, Geometry & Topology Monographs 9 (2006): 1-6. Print. [16] J. L. Bryant, S. C. Ferry, W. Mio, S. Weinberger, Topology of Homology Manifolds, Ann. of Math. 143 (1996): 435-467. Print. Page | 49

[17] J. L. Bryant, S. C. Ferry, W. Mio, S. Weinberger, Desingularizing Homology Manifolds, Geom. Topol. 11 (2007): 1289-1314. Print. [18] J. W. Cannon, The Recognition Problem: What is a Topological Manifold?, Bull. Amer. Math. Soc. 84 (1978): 832-866. Print. [19] R. J. Daverman, Decompositions of Manifolds, Pure and Applied Mathematics 124, Academic Press, Inc., Orlando, FL, 1986. Print. [20] R. J. Daverman, L. S. Husch, Decompositions and Approximate Fibrations, Michigan Math. J. 31 (1984): 197-214. Print. [21] R. J. Daverman, D. Repovš, A New 3-Dimensional Shrinking Criterion, Trans. Amer. Math. Soc. 315 (1989): 219-230. Print. [22] R. J. Daverman, D. Repovš, General Position Properties that Characterize 3-Manifolds, Canad. J. Math. 44 (1992): 234-251. Print. [23] R. J. Daverman, T. L. Thickstun, The 3-Manifold Recognition Problem, Trans. Amer. Math. Soc. 358 (2006): 5257-5270. Print. [24] E. Duda, Brouwer Property Space, Duke Math. J. 30 (1963): 647-660. Print. [25] J. Dugundji, Absolute Neighbourhood Retracts and Local Connectedness in Arbitrary Metric Spaces, Comp. Math. 13 (1958): 229-246. Print. [26] M. E. Edwin, Affine Structures in 3-Manifolds: V. The Triangulation Theorem and Hauptvermutung, Ann. of Math. Second Series 56 (1952): 96-114. Print. [27] D. J. Garity, D. Repovš, M. Zeljko, Uncountably Many Inequivalent Lipschitz Homogeneous 3 Cantor Sets in � , Pacific J. Math. 222 (2005): 287-299. Print. [28] D. M. Halverson, D. Repovš, The Bing-Borsuk and the Busemann Conjectures, Mathematical Communications 13 (2008): 163-184. Print.

[29] A. Hatcher, Algebraic Topology. New York: Cambridge University Press, 2002. Print. [30] J. Hempel, 3-Manifolds, Ann. of Math. Studies: Princeton, 1976. Print. [31] W. Hurewicz, H. Wallman, Dimension Theory, Princeton: New Jersey, 1941. Print. [32] W. Jakobsche, The Bing-Borsuk Conjecture is Stronger than the Poincaré Conjecture, Fund. Math. 106 (1980): 127-134. Print. 3 [33] W. Jakobsche, D. Repovš, An Exotic Factor of � × R, Math. Proc. Camb. Phil. Soc. 107 (1990): 329-344. Print. [34] P. Krupski, On the Disjoint (0, n)-Cells Property for Homogeneous ANRs, Colloq. Math. 66 (1993): 77-84. Print. [35] K. Kuperberg, 2-Wild Trajectories, Auburn University (2005): 518-523. Print. Page | 50

[36] R. C. Lacher, Cell-Like Mappings and their Generalizations, Bull. Amer. Math. Soc. 83 (1977): 495-552. Print. [37] J. Lee, Introduction to Topological Manifolds. New York: Springer, 2011. Print. [38] J. M. Lysko, Some Theorems Concerning Finite Dimensional Homogeneous ANR-Spaces, Bull. Acad. Polon. Sci. Math., Astron. et Phys. 24 (1976): 491-496. Print. [39] J. Malesic, D. Repovš, On Characterization of Lipschitz Manifolds, New Developments in Differential Geometry, Budapest 1996, Kluwer Acad. Publ., Dordrecht, (1999): 265-277. Print. 3 3 [40] D. V. Meyer, A Decomposition of � into Points and a Null Family of Tame 3-Cells in � , Ann. of Math. 78 (1963): 600-604. Print. [41] E. A. Michael, Some Extension Theorems for Continuous Functions, Pacific J. Math. 3 (1953): 789-806. Print. [42] J. Morgan, G. Tian, Ricci Flow and the Poincaré Conjecture, Clay Mathematics Monographs 3, American Mathematical Society, Providence, R. I.; Clay Mathematics Institute, Cambridge, MA, (2007). Print. [43] J. Munkres, Topology. Second Edition. New Jersey: Pearson Education, Inc., 2000. Print. [44] G. Perelman, The Entropy Formula for the Ricci Flow and its Geometric Applications, arXiv.org: math.DG/0211159, 2002. [45] G. Perelman, Finite Extinction Time for the Solutions to the Ricci Flow on Certain Three- Manifolds, arXiv.org: math.DG/0307245, 2003. [46] G. Perelman, Ricci Flow and Geometrization of Three-Manifolds, arXiv.org: math.DG/0303109, 2003. [47] F. S. Quinn, Ends of Maps III: Dimensions 4 and 5, J. Diff. Geo. 17 (1982): 503-521. Print. [48] F. S. Quinn, An Obstruction to the Resolution of Homology Manifolds, Mich. Math. J. 34 (1987): 285-291. Print. [49] F. S. Quinn, Problems on Homology Manifolds, Exotic Homology Manifolds - Oberwolfach 2003, Geom. Topol. Monogr. 9, Geom. Topol. Publ., Coventry, (2006): 87-103. Print. [50] D. Repovš, The Recognition Problem for Topological Manifolds: A Survey, Kodai Math. J. 17 (1994): 77-108, 538-548. Print. 1 [51] D. Repovš, A. B. Skopenkov, E. V. Scepin, A Characterization of � - Homogeneous Subsets of the Plane, Boll. Unione Mat. Ital. A 7:3 (1993): 437-444. Print. 1 � 1 [52] D. Repovš, A. B. Skopenkov, E. V. Scepin, � -Homogeneous Compacta in � are � - � Submanifolds of � , Proc. Amer. Math. Soc. 124 (1996): 1219-1226. Print. Page | 51

[53] C. P. Rourke, B. J. Sanderson, Introduction to Piecewise-Linear Topology, Springer-Verlag, New York - Heidelberg, 1972. Print. [54] H. Seidel, Locally Homogeneous ANR-Spaces, Arch. Math. 44 (1985): 79-81. Print.

[55] L. C. Siebenmann, Approximating Cellular Maps by Homeomorphisms, Topology 11 (1972): 271-294. Print. [56] E. H. Spanier, Algebraic Topology, New York: McGraw-Hill, 1966. Print.

[57] P. S. Urysohn, Memoire sur Multiplicite Cantoriennes, Fund. Math. 7 (1925): 30-137. Print. [58] P. S. Urysohn, Works on Topology and Other Areas in Mathematics, Vol. 1, (in Russian), Moscow-Leningrad, 1951. Print. [59] J. Väisälä, The Invariance of Domain under Acyclic Mappings, Duke Math. J. 33 (1966): 679- 681. Print. [60] R. L. Wilder, Topology of Manifolds, Amer. Math. Soc. Coll. 32 (1949). Print. [61] K. Yokoi, Bubbly Continua and Homogeneity, Houston J. Math. 29 (2003): 337-342. Print.