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Page i Acknowledgement Page ii Table of Contents Page iii Page iv Abstract Page v Page vi Chapter 1. Introduction: The fixed point problem for nonseparating plane continua Page 1 Page 2 Page 3 Page 4 Page 5 Page 6 Chapter 2. Embeddings and prime end structure of chainable continua Page 7 Page 8 Page 9 Page 10 Page 11 Page 12 Page 13 Page 14 Page 15 Page 16 Page 17 Page 18 Page 19 Page 20 Page 21 Page 22 Page 23 Page 24 Page 25 Page 26 Page 27 Page 28 Page 29 Page 30 Page 31 Page 32 Page 33 Page 34 Page 35 Page 36 Page 37 Page 38 Page 39 Page 40 Page 41 Page 42 Page 43 Page 44 Page 45 Page 46 Page 47 Page 48 Page 49 Page 50 Page 51 Page 52 Chapter 3. Principal embeddings of atriodic plane continua Page 53 Page 54 Page 55 Page 56 Page 57 Page 58 Page 59 Page 60 Page 61 Page 62 Page 63 Page 64 Page 65 Page 66 Page 67 Page 68 Page 69 Page 70 Page 71 Page 72 Page 73 Page 74 Page 75 Page 76 Page 77 Page 78 Page 79 Page 80 Page 81 Page 82 Page 83 Page 84 Page 85 Page 86 Chapter 4. Inequivalent embeddings and prime ends Page 87 Page 88 Page 89 Page 90 Page 91 Page 92 Page 93 Page 94 Page 95 Page 96 Page 97 Page 98 Page 99 Page 100 Page 101 Page 102 Page 103 Page 104 Page 105 Page 106 Page 107 Page 108 Page 109 Page 110 Page 111 Page 112 Page 113 Page 114 Page 115 Page 116 Page 117 Page 118 Page 119 Page 120 Page 121 Page 122 Page 123 Page 124 Page 125 Page 126 Page 127 Page 128 Page 129 Page 130 Page 131 Page 132 Page 133 Page 134 Page 135 Page 136 Page 137 Page 138 Page 139 Page 140 Page 141 Page 142 Page 143 Page 144 Chapter 5. The prime end structure of quotient spaces Page 145 Page 146 Page 147 Page 148 Page 149 Page 150 Page 151 Page 152 Page 153 Page 154 Page 155 Page 156 Page 157 Page 158 Page 159 Page 160 Page 161 Page 162 Page 163 Page 164 Page 165 Page 166 Page 167 Page 168 Page 169 Page 170 Page 171 Page 172 Page 173 Page 174 Page 175 Page 176 Page 177 Page 178 Page 179 Page 180 Page 181 Page 182 Page 183 Page 184 Page 185 Page 186 Page 187 Page 188 Page 189 Page 190 Page 191 Page 192 Page 193 Page 194 Page 195 Page 196 Page 197 Page 198 Page 199 Page 200 Page 201 Page 202 Page 203 Page 204 Page 205 Page 206 Page 207 Page 208 Page 209 Page 210 Page 211 Page 212 Page 213 Page 214 Page 215 Page 216 Page 217 Page 218 Page 219 Page 220 Page 221 Page 222 Chapter 6. Principal embeddings of plane continua and extendable homeomorphisms Page 223 Page 224 Page 225 Page 226 Page 227 Page 228 Page 229 Page 230 Page 231 Page 232 Page 233 Page 234 Page 235 Page 236 Page 237 Page 238 Page 239 Page 240 Page 241 Page 242 Page 243 References Page 244 Page 245 Page 246 Biographical sketch Page 247 Page 248 Page 249 Page 250 
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EMBEDDINGS OF PLANE CONTINUE AND THE FIXED POINT PROPERTY BY JOHN CLYDE MAYER DISSERTATION PRESENTED TO THE GRADUATE COUNCIL OF THE UNIVERSITY OF FLORIDA IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE OF DOCTOR OF PHILOSOPHY UNIVERSITY OF FLORIDA 1982 ACKNOWLEDGEMENT Grateful acknowledgment is made for the untiring assistance and direction of the author's advisor, Beverly L. Brechner, in bringing this study to completion. TABLE OF CONTENTS ACKNOWLEDGEMENT ......................................... ii ABSTRACT ............................................... v 1 INTRODUCTION: THE FIXED POINT PROBLEM FOR NONSEPARATING PLANE CONTINUE .................. 1 1.1 Motivation ................................ 1 1.2 Embeddings and Prime End Structure..... 3 1.3 LakeofWada Channels.................... 4 2 EMBEDDINGS AND PRIME END STRUCTURE OF CHAINABLE CONTINUA............................. 7 2.1 Introduction............................ 7 2.2 Embeddings of the Knaster UContinuum.. 9 2.3 Principal Embeddings of Chainable Continua.............................. 19 2.4 Nonprincipal Embeddings of Chainable Continua.............................. . 23 2.5 NPrincipal Embeddings of Chainable Continua.............................. . 31 3 PRINCIPAL EMBEDDINGS OF ATRIODIC PLANE CONTINUA ...................................... 53 3.1 Introduction........... ....... ........ 53 3.2 The Xodic Continuum.................... 54 3.3 LakeofWada Channels................... 84 4 INEQUIVALENT EMBEDDINGS AND PRIME ENDS........ 87 4.1 Introduction............................ 87 4.2 The Sin 1/x Continuum................... 88 4.3 The Knaster UContinuum................. 99 4.4 Uncountably Many Embeddings of Uncountably Many Continua............. 127 5 THE PRIME END STRUCTURE OF QUOTIENT SPACES.... 145 5.1 Introduction........................... 145 5.2 Prime Ends and Quotient Spaces......... 148 5.3 Accessibility ........................... 172 5.4 The Psuedo Arc......................... 178 5.5 Chainable Continua with End Subcontinua 185 5.6 Inaccessibility......................... 195 6 PRINCIPAL EMBEDDINGS OF PLANE CONTINUE AND EXTENDABLE HOMEOMORPHISMS...................... 223 6.1 Introduction............................ 223 6.2 Principal Embeddings of Triodic Continue .............................. 224 6.3 Extendable Homeomorphisms of Principally Embedded Chainable Continua.............................. 233 REFERENCES.............................................. 245 BIOGRAPHICAL SKETCH..................................... 247 Abstract of Dissertation Presented to the Graduate Council of the University of Florida in Partial Fulfillment of the Requirements for the Degree of Doctor of Philosophy EMBEDDINGS OF PLANE CONTINUE AND THE FIXED POINT PROPERTY By John Clyde Mayer May 1982 Chairman: Beverly L. Brechner Major Department: Mathematics A longoutstanding problem in plane topology is the following: Do nonseparating plane continue have the fixed point property for continuous maps (Problem 107 in the Scottish Problem Book)? A number of partial solutions are known. One important result is a theorem due independently to H. Bell and K. Sieklucki which has the following consequence (pointed out by B. Brechner): A nonseparating plane continuum admitting a fixed point free map must have a LakeofWada channel in its boundary in every embedding into the plane. This consequence, which seems to reveal much about the structure of any potential counterexample to the fixed point property, motivates the studies of embeddings of plane continue in this essay. The principal tool we utilize to study such embeddings is prime end theory. v Major divisions of this dissertation are devoted to (1) Embeddings of chainable continue with (one or more) LakeofWada channels, and theorems concerning the conditions under which a chainable continuum can be (re)embedded without a LakeofWada channel. (2) An example of an atriodic nonchainable plane continuum, embedded with a LakeofWada channel, of which type any minimal treelike counterexample to the fixed point property (should one exist) is likely to be. (3) Embeddings of triodic treelike continue with (multiple) LakeofWada channels. (4) Inequivalent embeddings of plane continue and the prime end structures of such embeddings, including the fact that the sin 1/x continuum and the Knaster Ucontinuum each have uncountably many inequivalent embeddings with the same prime end structure and the same set of accessible points. (5) The relationship between the prime end structures of a treelike indecomposable continuum and the quotient continuum induced by shrinking each of a collection of proper subcontinua to a point. (6) The utilization of the latter to extend theorems of W. Lewis concerning the pseudo arc, and of J. Krasinkiewicz and S. Mazurkiewicz concerning accessibility of points, subcontinua, and composants of indecomposable plane continue. CHAPTER 1 INTRODUCTION: THE FIXED POINT PROBLEM FOR NONSEPARATING PLANE CONTINUE 1.1. Motivation A longstanding problem in plane topology is the following: Do nonseparating plane continue have the fixed point property for continuous maps? For homeomorphisms? This question, Problem #107 in the Scottish Problem Book, credited to Sternbach, has received attention from a number of mathematicians. A survey of fixed point problems and theorems, including the above question, has been made by R.H. Bing [1969]. A number of partial solutions to the problem are known. Chainable continue, known to be planar [Bing, 1951], have the fixed point property for continuous maps [Hamilton, 1951]. A closed disk in the plane has the fixed point property by the wellknown Brouwer Fixed Point Theorem. Early known results also include that nonseparating Peano continue have the fixed point property [Borsuk, 1932], as do one dimensional nonseparating arcwise connected treelike continue (that is trees, dendrites, and the like) [Borsuk, 1954]. More recently Hagopian showed that nonseparating arcwise connected continue [1971] and uniquely arcwise connected plane continue [1979] have the fixed point property. 2 Of particular interest, however, is the Cartwright LittlewoodBell Theorem ([Cartwright and Littlewood, 1951] and [Bell, 1967] ) showing that nonseparating plane continue have the fixed point property for extendable homemorphisms. It has been pointed out by B. Brechner in conversation that this result extends to essentially extendable homeomorphisms: those homeomorphisms of a plane continuum which can be extended to the plane under some reembedding of the continuum, even if not extendable in the given embedding. (See Brechner [1981].) This latter result suggests that questions of embeddings of continue into the plane may be of some significance in making further progress on the problem. In the more general case of continuous maps, an important result is a consequence of a theorem due independently to H. Bell [1978] and K. Sieklucki [1968] : a nonseparating plane continuum admitting a fixed point free map must have a LakeofWada channel in its boundary for every embedding into the plane (Embedding corollary 2.5 [Brechner and Mayer, 1981 ].) (For the original ThreeLakes ofWada construction see Hocking and Young [1961].) The choice of topics in this essay is motivated by the BellSieklucki Theorem and the Embedding Corollary, which seem to reveal much about the structure of any nonseparating plane continuum that could admit a fixed point free map. For instance, if it were the case that every nonseparating nondegenerate plane continuum with a LakeofWada channel in its boundary could be reembedded in 3 the plane so that the channel disappeared, then the fixed point property would follow. In [1980a] B. Brechner and J. Mayer define the concept of a principal continuum: a nonseparating continuum that has a LakeofWada channel in its boundary in every embedding into the plane. Thus one question of significance to the fixed point problem (asked in [1980a ]) is: Do there exist principal continue? 1.2 Embeddings and Prime End Structure. Our primary interest is in embedding problems, and our primary tool in investigating and comparing embeddings will be their prime end structures. Prime ends, a concept whose original development is due to Caratheodory [1913] are a means of studying and organizing the approaches to the boundary of a simply connected plane domain. While the complement of a nonseparating continuum in the plane E is not simply connected, the complement of such a continuum in 2 the sphere S is simply connected. As we can always insure that the embedding of a continuum into S misses the point at infinity, there is no practical difference for our 2 2 purposes in whether embeddings are in E or S Ursell and Young [1951] have pointed out that prime end theory can be developed just as well in the latter context. Through prime end theory we can identify the set of prime ends of a nonseparating continuum with the set of points on the boundary of the unit circle B in E (or S ). For basic concepts and references for prime end theory, we refer the reader to Brechner [1978]. A sketch of essential concepts is included herein where needed in the various chapters. (See 2.2.8 and 4.2.1.) 4 A continuum with a LakeofWada channel in its boundary must have a particular kind of prime end associated with it (Theorem 2.9 in Brechner and Mayer [1980a].) The arrangement of points on the boundary of the unit circle and their correspondence to kinds of prime ends determine the prime end structure of a continuum in S Thus we can detect LakeofWada channels through prime and structure. Furthermore, it is shown in Brechner and Mayer [1980a] that if two embeddings of a continuum into S are equivalent (in the sense that there is some homeomorphism of S onto itself carrying one embedding onto the other) then the two embeddings have the same prime end structure (Theorem 2.11 of [1980a].) However, in passing from the embedding itself to the prime end structure, some information is lost. The converse of the above theorem fails, as we show in Chapter 4. 1.3 LakeofWada Channels. In succeeding chapters, a LakeofWada channel running throughout the entire boundary of a continuum is called a simple dense canal, a concept due to Sieklucki [1968] and reiterated in Brechner and Mayer [1980a], Definition 2.1. The precise definition is given herein in Section 2.2.9. Roughly, a simple dense canal (sdc) is a ray in the complement that converges to the entire boundary of a nonseparating continuum in S in such a way that the ray goes down an evernarrowing LakeofWada channel in the boundary. We call an embedding of a continuum with a sdc in the complement a principal embedding. 5 In Chapter 2 we show that chainable continue, if 2 indecomposable, can always be embedded in S with a sdc. (No continuum with decomposable boundary can be embedded with a sdc in the boundary.) Also, if an indecomposable chainable continuum has at least one endpoint, it can be embedded without a sdc, that is, nonprincipally embedded. Chapter 2 also includes a wealth of examples of chainable continue embedded with one or more sdc's, including countably many and uncountably many. Therein and throughout liberal use is made of figures and diagrams to graphically represent the continue constructed. For us, the study of embeddings of plane continue has a significant visual component, at least on the intuitive level. In Chapter 3 we present an example of an atriodic nonchainable continuum in S embedded with a sdc. The example presented is of the type of treelike plane continuum most likely to admit a fixed point free map, if any do. The likelihood is suggested by the properties the example shares with the treelike continue admitting fixed point free maps recently constructed by D. Bellamy [1980]. Bellamy's example is trodic and nonplanar. The FugateMohler modification of his example is atriodic, but it is not known if it is planar or if it is nonplanar. In Chapter 4 we show that both the sin 1/x continuum and the Knaster Ucontinuum (or buckethandle) have uncountably many inequivalent embeddings into the plane with the same prime end structure. This shows that an equivalence of prime end structures, even between 6 homeomorphic continue, cannot be "lifted" to an equivalence of the embeddings. In Chapter 5 we investigate the relationship between the prime end structure of a nonseparating continuum in S2 and the prime end structure of certain quotient continue in 2 S formed by shrinking subcontinua of the given continuum to points. This enables us to extend certain theorems in Chapter 2; in particular, we show that no chainable continuum with an end subcontinuum (not equivalent to endpoint) is principal. We are also able to extend theorems of Mazurkiewicz and Krasinkiewicz regarding accessibility of composants of indecomposable plane continue. In Chapter 6 we give several examples of triodic 2 continue embedded in S with sdc's. Such continue as we construct exhibit periodic homemorphisms interchanging the channels. By extending examples of W. Lewis [1981] we show that certain chainable continue also have this property. Of the articlesreferenced above, Brechner [1978] and Brechner and Mayer [1980a] are essential background to the topics and treatment of this essay. With that proviso, the following chapters are relatively independent, though all interconnected by the common themes of the fixed point problem and of embeddings of plane continue. Essential definitions, theorems, and concepts are reviewed in each chapter to the extent that they will be utilized in the development of that chapter. This results in some replication of key definitions and important theorems. CHAPTER 2 EMBEDDINGS AND PRIME END STRUCTURE OF CHAINABLE CONTINUE 2.1 Introduction. In this chapter we apply several concepts arising out of a promising approach to the fixed point problem for nonseparating plane continue to chainable continue and their embeddings into the plane. The concepts of principal embedding and principal continuum were introduced by Brechner and Mayer [1980a, 1980b]. In the present chapter they are applied to show two main results: 2.1.1 Theorem. Every indecomposable chainable continuum can be principally embedded into E. 2.1.2 Theorem. No chainable continuum with at least one endpoint is principal. It follows from a theorem proved independently by Bell [1978] and Sieklucki [1968] that any nonseparating continuum X which admits a fixed point free map contains a minimally invariant indecomposable subcontinuum X0 in its boundary, and furthermore, the continuum X0 formed by the union of X0 and its bounded complementary domains, if any, must have the property that every embedding of it into the plane has a LakeofWada channel. We call a LakeofWada channel a simple dense canal (sdc). (See 2.2.9 for precise 8 definition.) Such an embedding is principal. (See Embedding Corollary 2.5 of Brechner and Mayer [1980a].) The concept of a simple dense canal is due to Sieklucki [1968]. In Section 2, embeddings, both principal and nonprincipal, of the Knaster Ucontinuum are discussed. Lemmas 2.2.5 and 2.2.6 are instrumental in the proofs of Theorems 2.1.1 and 2.1.2. In Section 2.3, Theorem 2.1.1 is proved. The principal embedding of the Ucontinuum discussed in 2.2.11 is the model for the principal embedding of any indecomposable chainable continuum. In Section 2.4, Theorem 2.1.2 is proved. Techniques similar to those of Section 2.3 are used, but with the nonprincipal embedding of the Ucontinuum discussed in 2.2.7 and 2.2.8 as the model. We also raise several questions regarding strenghthenings of Theorems 2.1.1 and 2.1.2. In Section 2.5, examples are given for each n, 1 L n 0o and n = 2 of a chainable continuum M embedded with n exactly n simple dense canals. These continue have exactly one endpoint and can be reembedded without sdc's in accordance with Theorem 2.1.2. Similar examples of chainable continue with multiple endpoints and with no endpoints are given. We make extensive use of "inverselimitwithembedding" diagrams, introduced in Brechner and Mayer [1980a] and here discussed in 2.2.7, in presenting these examples. 9 In what follows, all spaces are metric and all continue are compact connected spaces, and for convenience, nondegenerate. 2.2 Embeddings Of The Knaster UContinuum. Our technique in proving Theorems 2.1.1 and 2.1.2 will be to show that any indecomposable chainable continuum can be embedded in the plane as the intersection of a defining sequence (Dii_ of chains of open disks such that the pattern that D. follows in Di1_ can be consolidated to the pattern that one chain of a defining sequence for the Knaster Ucontinuum follows in its predecessor. (Since we shall often refer to sequences indexed by the positive integers, we shall write them as (Dii., omitting the limits on the index.) The consolidation can be done in such a way that the diameter of the links of the consolidations goes to zero. That some such consolidations of defining sequences for any indecomposable continuum exist follows from Bellamy [1981] and Rogers [1970], but for the diameter of the links to go to zero requires a stronger condition than indecomposability; chainability suffices. The definitions of chains, links, and consolidations of chains are standard. (See Bing, [1948].) 2.2.1 Definition. A defining sequence (G.]. for a continuum X is a sequence of open covers of X such that for each i, (1) G. is 1 coherent, (2) Gi+. closure refines G., (3) the mesh of G. is i less than 1/2 (4) G. minimally covers X, and (5) X =iQGi. * (G. denotes the union of the links of G..) This definition 10 is due to Ingram and Cook [1967], though we have strengthened condition (3). 2.2.2 Theorem. A continuum X is indecomposable iff there is a defining sequence [G.i.i for X such that for each i there is a j>i such that if G. = L1U L2,L1 and L2 being coherent * subcollections of G., then either L1 or L2 meets every set in G. [Ingram and Cook, 1967]. 2.2.3 Definition. In order to describe an embedding of a chainable continuum as the intersection of a defining sequence of chains of open disks in the plane, we make use of the notion of the pattern that one chain follows in another which it refines due to Bing [1948] Roughly, a pattern is a map which tells us in which links of the containing chain the links of the refining chain sit. Thus for chains C = C(l), C(2), ... C(m)] and D = [D(1), D(2), ... D(n)], with D refining C, the pattern D follows in C is a set of ordered thh pairs ((xl,,Y), ... (Xnyn ) which indicates that the xi link of D is a subset of the yith link of C. A pattern is not necessarily unique, since a link of D may lie in the intersection of two links of C. Consecutive yi 's may not differ by more than one. We say a pattern reverses when for i > 2, from x. to x i+, both yi+l < Y. and yi Yi_i or both Yi+ >yi and yi and yi S yi.,. A reversal in a previously constant or increasing (decreasing) immediately preceding part of the pattern represents a bend in the chain. Because patterns are not unique, the same chaining may correspond to 11 both a pattern with and a pattern without a reversal. Such "pseudobends" occur in the intersection of two links of the containing chain. 2.2.4 Definitions. A pattern alone does not determine an embedding. In order to distinguish various embeddings of the same pattern, we define a pair of notions due to Brechner [1978] and expand some upon them. Let C = [C(l), ... C(m)) and D = (D(l), ... D(n)3 be chains of open disks in the plane with D refining C. By open disk we mean a convex open set with simple closed curve boundary. We assume chain C has been "straightened out" by some orientationpreserving homeomorphism of the plane so that C(l) is on the left and C(m) is on the right with the nerve of C parallel to the horizontal axis. Thus the terms "above" and "below" used below are unambiguous. Suppose D(l) is a subset of C(k). Then D is descending in C provided that from C(k), D follows a pattern in C in such a way that each time the pattern reverses (a bend), the subsequent subchain of D is below the preceding subchain (the bend is downward); D is ascending in C provided that from C(k), D follows a pattern in C in such a way that each time the pattern reverses, the subsequent subchain of D is above the preceding subchain (the bend is upward). If D follows in C a pattern which contains no reversals, then we say D is straight through C. If D consists of two straight through subchains, one from (and contained in) the first (last) link of C to (and contained 12 in) the last (first) link of C, and the other reversing in the last (first) link of C and returning to the first (last) link of C, then we say D is twotoone in C. If D can be consolidated to a chain following a twotoone pattern in C, then we say D is at least twotoone in C. By extension of the above, if D consists of n straight through subchains running from endlink to endlink of C, then we say D is ntoone in C. A chain D following a pattern in C which can be consolidated to an ntoone pattern is said to be at least ntoone in C. By the term Ulike we denote any at least 2 toone pattern or any consolidation of such a pattern to a 2ntoone pattern. We also call a chainable continuum Ulike if it has a defining sequence of at least 2 toone chain covers. The thrust of Lemmas 2.2.5 and 2.2.6 is that all indecomposable chainable continue are Ulike. In what follows we uniformly use "n" subscripted by the index of the chain cover to denote the last link of the chain. Thus C.i(n.) denotes the last link of chain C.. That open disk chains can be used to produce an embedding of chainable continuum X into the plane, that is, guarantee the existence of a homeomorphism of X into the plane, is a consequence of Theorem 4 of Bing [1951] . 2.2.5 Lemma. Let X be an indecomposable chainable continuum. Then there is a defining sequence [Di.i of chain covers for X such that for each i>l, there exist positive integers kl 13 (2) D (k2) C D. (ni1), and (3) subchain D.(klk3) is minimal with respect to properties (1) and (2). Hence, D. is at least twotoone in D._I. Proof. Let (Cj.3 be a defining sequence of chain covers for X. Our proof proceeds by progressively selecting subsequences of the C.'s until we have a subsequence which 3 almost meets conditions (1) and (2) above. A certain sequence of consolidations of the members of this final subsequence is then shown to meet conditions (1) through (3). Select from CiC.3. a subsequence C aa such that for all 3 3 a a a, the mesh of C is less than 1/2a+l and the closure of any a two consecutive links of Ca+l is contained in one link of C It then follows from Theorem 2.2.2 that for each a, a there is a b>a such that Cb consists of two subchains Cb(l"yb) and Cb(ybnb) such that both subchains meet every link of C Furthermore, as the C 's are minimal covers, we a a may assume that C (yb) is a subset of an endlink of Ca and that at least one of the subchains contains a link which is a subset of the other endlink of C. a If C (yb) C Ca(n a) then let c = b. Otherwise, apply Theorem 2.2.2 again to find a c>b such that C consists of c two subchains c (l,yc) and C (yc ,nc) such that both subchains meet every link of Cb (and so every link of Ca) D a and C (y ) is a subset of an endlink of C In either case, we may now find positive integers rc C (r ,s ,t ) such that both C (r ) and C c(t ) meet C (1) and c c c c c c c c a at least one of them is a subset of C (1), and C (s ) is a subset of Cc(n). subset of C (n ). a a 14 By repeatedly following the above procedure, we may select a subsequence (Ci.i of (C aa such that for all i>l, there are positive integers u C. (u.,v.,w.) such that both Ci(ui) and C (wi) meet Cl(1), at least one of them being a subset of C _l(1), and C.(v.) is a subset of C. l(ni 1). We will produce the sequence (Di.i by consolidating each C. to a chain D. while preserving closure refinement. 1 1 Let D1 = C. Assume (Dil]1 have been defined. It follows 1* i i=l from our selection of the mesh of each C that C.(1)U C (2) lies in a single link of CjI, and so in a single link of the consolidation of Cj_1, namely DI. Let Dj(1) = C. (1)U Cj (2), and D (m) = Cj (m+l)' for 2 < m < njl. The mesh of C. guarantees that the mesh of Dj is less than 1/23. Closure refinement is preserved, since D (1) is a subset of I a single link of DjI. Thus [D.i is a defining sequence for X. For each i > 1, the triple u., v., w. corresponds to a triple r. Di_1 that the original triple had to Ci_1 except that Di(ri) and Di(ti) are now both subsets of D._ (1). Let k1 be the i si" last link of subchain D.(r.,s.) in D_.(1). Let k = s.. 1 1 1 2 1 i Let k3 be the first link of D (si,ti) in Dil(l). Then [Di3i i satisfies conditions (1) through (3). It can then be shown that for each i, Di+1 can be consolidated to a twotoone chain in D. Hence, D.i+ is at least twotoone in D.. QED 15 Note that the selection of the k.'s in Lemma 2.5 (and J in Lemma 2.2.6 to follow) is not unique. 2.2.6 Lemma. Let X be an indecomposable chainable continuum. Then there is a defining sequence Di3i of chain covers for X such that for each i>l, and for each positive even integer i i k+i_ uhta m, there exist positive integers k (1) Di(k')UDi(k') U ... U Di(k'+) C Di(l), (2) Di(ki)U 1 1 3 i mll i2 Di(k )U ... U D m(kl) C Di(ni), and (3) subchain Di ( i D (kk +) is minimal with respect to properties (1) and (2). Hence, D. is at least mtoone in Dil, and furthermore, X is Ulike. Proof. Let (Di]. be a defining sequence of chain covers for X satisfying the conditions of Lemma 2.2.5. Then for all j > 1, D. is at least twotoone in D . Let p be chosen so that 2p < m < 2P. Let CDi}i be the subsequence th of CDI. chosen by selecting the first, the p+1 and at the nth stage, the (nl)p+lth member of CD.]j.. Then for each i > 1, D. will be at least 2Ptoone in D. i. But as 2P is no less than m, each D. is at least mtoone in D. i. That CD.]i satisfies conditions (1) through (3) can then be shown. QED A version of Lemma 2.2.6 can be proved for odd m, though with ki for odd j in D i(1) and k' for even j in 3 3 i D (n._), 1 < j < m+l. That is, D (km+) lies in the last link of D not the first.l link of D il'1 not the first. 16 2.2.7 Standard Embedding of the Knaster Ucontinuum. We can describe what we will call the standard embedding of the Knaster Ucontinuum by a defining sequence [Ci}i of open disks in the plane such that C i+1 follows in C. a descending twotoone pattern of embedding. This 1 twotoone pattern and descending embedding is schematically represented by the "inverselimitwithembedding" diagram in Figure 2.1. The infinite repetition of the indicated patternwithembedding produces the standard embedding of the Ucontinuum. We suppose C. to be straightened out to 1 guide us in embedding C i+1 in Ci. A point p of a chainable continuum X is an endpoint if, and only if, for some defining sequence for X, p is the intersection of first links. Note that p = .Ql1Ci(1) is the only endpoint of the Ucontinuum. 2.2.8 Prime End Structure Of The Standard Embedding Of The Knaster UContinuum. Basic definitions concerning prime ends and further references may be found in Brechner [1978] Figure 2.2. represents the only accessible composant of the standard embedding of the Ucontinuum. A composant is accessible if it contains at least one accessible point. In this case the composant is accessible at every point, by an argument similar to the proof of Lemma 2.4.1.1. Note (Qi.i is a chain of crosscuts of E U such that Q. p. This chain of crosscuts defines a prime end E. By I(E) we denote the impression of E, the intersection of the closures of the bounded domains of Q.iU U, over all i. (These domains are 17 towered.) It can be shown that I(E)=U, and for any prime end F distinct from E, such as that defined by [Ri.i in Figure 2.2, I(F) is a single point. The existence of at least one prime end E with I(E)=U is necessitated by U's indecomposability [Brechner, 1978] . By P(E) we denote the set of principal points of prime end E, the set of point to each of which some chain of crosscuts defining E converges. It can be shown that P(E)=[p], the endpoint of U, for E defined by [Qi.i above. Such a prime end is of the second kind: I(E) nondegenerate and P(E) degenerate. Any other prime and F of U is of the first kind: I(F)=P(F), both degenerate. For any prime end E, each of I(E) and P(E) is a continuum (or degenerate) with P(E) a subcontinuum of I(E). 2.2.9 Definition A simple dense canal (sdc) of the continuum X is a ray D C E2X (a 11 continuous image of [0, )) such that (1) DD=Bd X, (2) at each point of D there is a transverse crosscut to X, and (3) diameter of such crosscuts goes to zero as we go to infinity on D. A crosscut Q is transverse to D at a point d if, and only if, QfnD = (d3, and for a sufficiently small open disk neighborhood U of D, each component of UQ contains exactly one component of (Un D) d. 2.2.10 Theorem. A nonseparating plane continuum X has a simple dense 2 canal iffthere exists a prime end E of E X such that I(E) = Bd X = P(E). (Theorem 2.9, [Brechner and Mayer, 1980a].) 2.2.11 Definitions. A prime end E such that I(E) = P(E), both nondegenerate, is of the third kind. So a sdc corresponds to a prime end of the third kind for which I(E) = Bd X. For treelike continue, the necessary and sufficient condition reduces to I(E) = X = P(E). Such embeddings of nonseparating continue are termed principal embeddings. If all the embeddings of X into the plane are principal, then X is called a principal continuum ( [Brechner and Mayer, 1980a], Definition 2.6). In virtue of the standard embedding, the Knaster Ucontinuum is not a principal continuum. 2.2.12 Principal Embeddings of the UContinuum. In Remark 3.5 of Brechner and Mayer [1980a], we indicated that the Ucontinuum has a principal embedding. Consider the defining sequence for the Ucontinuum formed by taking each oddnumbered chain of the standard defining sequence. Let that be [Ci. Note that the pattern one oddnumbered chain of the standard sequence follows in another is fourtoone. Embed the Ucontinuum according to the inverselimitwithembedding diagram of Figure 2.3(a) repeated infinitely. The hatched line in Figure 2.3(a) represents an initial arc of a ray D extendable in such a way as to be dense in U. Transverse crosscuts to U exist at each point of D, and as they can be confined to a single link, their diameter goes to zero. Hence D is a simple dense canal in this embedding of U. Note also that any prime end not corresponding to D is such that its impression 19 is degenerate. Furthermore, in this embedding endpoint p and its composant are inaccessible. Figure 2.3(b) illustrates another principal embedding of the Ucontinuum, but with the endpoint accessible. This embedding has two prime ends whose impression is all of U, but only one of them corresponds to a sdc. An endcut to p corresponds to the other prime end. It should be noted that the embeddings represented by Figures 2.1, 2.3(a), and 2.3(b) of the Ucontinuum are inequivalent. This is evident in comparing the differing accessibility of endpoint p in Figures 2.1 and 2.3(a). That 2.3(b) is also not equivalently embedded with respect to either of the other is a consequence of the differing prime end structures. (See Theorem 2.11 of Brechner and Mayer [1980a].) 2.3 Principal Embeddings Of Chainable Continua. We now have sufficient tools to show that any indecomposable chainable continuum can be principally embedded. We will use the pattern to Figure 2.3(a) to produce an infolded Ulike embedding of any indecomposable chainable continuum. 2.3.1 Proof Of Theorem 2.1.1. Let X be an indecomposable chainable continuum. By Theorem 4 of Bing [1951], X can be realized as the intersection of a defining sequence of chains of open disks in the plane. However, we want a particular kind of embedding, one with a sdc in the complement. 20 There is a defining sequence C.3.ii for X satisfying the following conditions: (1) C. is taut: that is, any pair of nonadjacent links 1 of C. is a positive distance apart. 1 (2) If a link of C. (other than an endlink) lies entirely in Ci1 (k), for any k, 1 < k < ni_,11 then that link is contained in a subchain of C. of nine links 1 lying entirely in C_ (k). (3) C. is at least fourtoone in Cil; that is, for each i > 1, there exist positive integers r. < s. < ti < u. < v. such that: 1 1 (a) Ci(ri)U Ci (ti)UCi (vi) C Ci1(l), (b) Ci(si)U Ci(ui) C Cil(nil), (c) subchain C. (ri,v.) is minimal. Conditions (1) and (2) follow from the fact that X is compact. Condition (3) follows from Lemma 2.2.6. We may think of the C.covers abstractly. Then X can be embedded in the plane by specifying a particular defining sequence D.i.i of chains of open disks in the plane whose intersection can be shown to be homeomorphic to X. Such a homeomorphism is defined in Theorem 11 of Bing [1948], which Bing uses to prove Theorem 4 of 1951 1. What is required to apply Theorem 11 is that D. follow in D i_ the pattern that C. follows in C. I. In producing the embedding below, we think of the D.chains semiabstractly as chains of topological open 1 disks with as yet no particular embedding in the plane. We then specify how these open disks are to be situated in the 21 plane to produce the desired embedding of X. Figure 2.4(a) is an example of the pattern that some D. might follow in D i_. For convenience, we have written the pattern descending, but at the moment think of it abstractly, with no particular embedding. In carrying out our embedding, we make use of an auxiliary sequence of open disk chains [Ui)i, such that Ui refines D il exactly fourtoone and contains D. in a particular way. Chain U.(l,m.) consists of four straight through subchains Ui(l,hi), Ui(hi,ji), Ui(Jilki) and Ui(ki,mi) with 1 < h. i < ji < ki < mi, and links Ui(l), Ui (ji), and Ui(mi) lie in Di_1(l), while links Ui(hi) and Ui(ki) lie in Di l(nil). Furthermore, Ui(l,hi) contains D. (l,ri,si), Ui(hi,ji) contains Di(si,ti), Ui(ji,ki) contains D. (t.,u.), and Ui(k.,m.) contains Di(ui,v.,ni). The links of D. numbered r., s., ti, u., v. lie in the links of U. numbered 1, hi, ji, ki, mi' respectively. In placing a link of D. in a link of U., we preserve the pattern that D. follows in D I. That is, if Di (a) C Ui (b) and Ui (b)C Dil(c), then D (a) C D l(c) in the original pattern. Note that D.(1) may be in any link of U.(l,h.), and Di(n.) may be in any link of Ui(ki,mi), depending upon what the original pattern was. Suppose that Di_ has been realized as a chain of open disks in the plane. Then U. is embedded in D i_ with the infolded patternwithembedding of Figure 2.4(b). We have placed no restriction on the mesh of U. other than that U. refine Di_. There could be as few as four links of U. (one 22 from each subchain) in an interior link of D.i, essentially subdividing that interior link into four links, and as few as two links of U. in D i(ni), just Ui(hi) and Ui(ki). However, in order to keep the links of U. open disks, we need at least five links in D i_(1): Ui(l), Ui(mi), and a threelink subchain with U. (j.) in the middle. Such a U.chain meeting these minimal requirements is illustrated 1 in Figure 2.5. We now embed subchains D.(r.,si), D.(si,ti), Di(tiui), and D.(u.,v.) descending in Ui(l,hi), Ui(hi,ji), Ui(Ji,ki), and U.(k.,m.), respectively. This patternwith embedding is shown for our example in Figure 2.4(c). Compare also Figure 2.5. Note that links of D. numbered r.i, s.i, t.i, u.i, v. lie in links of U. numbered 1, hi, ji, ki, mi, respectively, and that the pattern D. follows in D._1 is preserved, as can be seen by comparing Figures 2.4(a) and 2.4(d). Since there are at least nine links of D. in the subchain lying in 1 D il(1) that contains D.(t.), we can require that Di(ti) have four links of D. both preceding and following it lying 1 in D. i(1). Hence there are enough links for D. to refine U.i(j.1, j.+l) C Dil(1), and still have the diameter of links of D. less than half the diameter of links of Di.1 The initial and final subchains of D. are embedded with 1 D.(l,r.) descending U.(l,h.) and above D (r.,s.) in 1 1 1 1 1 1l 1 Ui(l,hi), and Di(vini) ascending in U (ki,mi) and above D.(u.,v.) in U.(k.,m.). The resulting patternwith embedding diagram for our examples, showing how D. now sits in D. i, is illustrated in Figure 2.4(d). How the chains 23 D i_ (straightened out), U., and D. might appear in the plane is shown in Figure 2.5. A sdc D may then be constructed stagebystage as we illustrated with the principal embedding of the Ucontinuum. An arc of the canal in our example is shown as the hatched line in both Figures 2.4(d) and 2.5. That D.i(n.) may not lie in D i_(1), in contrast to the situation with the Ucontinuum, can only make this arc of the canal proportionately longer. QED 2.4 Nonprincipal Embeddings Of Chainable Continua. For a decomposable chainable continuum, every embedding into the plane is nonprincipal, as a sdc in X implies Bd X is indecomposable ( [Brechner and Mayer, 1980a] Theorem 2.9.) For an indecomposable chainable continuum, we can get an embedding using the twotoone standard, rather than the infolded fourtoone, embedding pattern for the Ucontinuum as our model. However, while the standard embedding of the Ucontinuum has no sdc, we have as yet been able to produce such an embedding (without a sdc) of an indecomposable chainable continuum, in general, when the continuum has at least one endpoint. Various examples (see our 2.5.7) of chainable continue without endpoints have proved reembeddable without a sdc, but the general proof eludes us, though we conjecture that it can be done. 2.4.1 Proof Of Theorem 2.1.2 Let X be an indecomposable chainable continuum with endpoint p. We can find a defining sequence (Ci]i of chain covers of X satisfying the following conditions: 24 (1) p is in Ci (1)Ci (2); hence Ci (1) C Cil(1). (2) Ci is taut. (As defined in 3.1(1).) (3) If a link of C. (other than an endlink) lies 1 entirely in Ci_1(k), for any k, 1 < k n i, then that link is contained in a subchain of C. of nine links lying entirely in C. i(k). (4) C. is at least twotoone in Cil; that is, for each i > 1, there exist positive integers r. < S. < t. such that: (a) Ci (ri)UCi (ti) C C i(1), (b) Ci (si) C Cil (ni1), (c) subchain C. (r.,s.,t.) is minimal. (5) Subchain C.(l,r.) is minimal; that is, no link of Ci(l,ri) lies in Ci1 (ni.1). Condition (1) holds because p is an endpoint of X. See Section 5 of Bing [1951]. Conditions (2) and (3) follow from the fact that X is compact. Condition (4) is a consequence of Lemma 2.2.5. Since first links are towered, we may choose the first subchain satisfying condition (4), thus satisfying condition (5). As in the proof of Theorem 2.2.1, our procedure for embedding X is to specify a defining sequence [D.i) of chains of open disks in the plane such that D. follows in 1 Di_1 the pattern that C. follows in C i. We think of (D.i.i semiabstractly, as chains of topological open disks with no particular embedding. Figure 2.6(a) indicates how some D. 1 might sit in Di , written descending for convenience. In some cases it could be that r.=l, though not in the example 25 we illustrate in Figure 2.6(a). Also note that s. is not 1 uniquely determined: any of the three "loop ends" of D. 1 that lie in Di l(n _I) in Figure 2.6(a) could contain Di (si). In carrying out our embedding we make use of an auxiliary sequence U.i.i of chains of open disks, such that U. refines Di_1 exactly twotoone and contains D. in a particular way. Chain U.i(l,m.) consists of two straight through subchains U.(l,k.) and U.(k.,m.) with 1 < k. < mi, and links Ui(l) and Ui(mi) lie in Di_l(1), while link Ui(k.) lies in D l(n. il). Furthermore, U. (l,k.) contains D. (l,r.,s.) and U. (k.,m.) contains D. (s.,t.,n.). The links of D. numbered 1, r., s., t. lie in the links of U. numbered 1, 1, k., mi., respectively. In placing a link of D. in a link of Ui, we preserve the pattern that Di follows in D. . That is, if Di (a) C U. (b) and U. (b) C D. l(c), then D. (a) C Dil(c) in the original pattern. Note that D.i(1) may be in any link of Ui(l,hi), and Di(ni) may be in any link of U.(k.,m.), depending upon what the original pattern was. Suppose that Di_1 has been realized as a chain of open disks in the plane. We embed U. in Dil following the descending patternwithembedding of Figure 2.6(b). We then embed subchains D.(l,r.,s.) and D.(s.,t.) descending in Ui(l,k.) and Ui (ki.,mi), respectively. This patternwith embedding is shown for our example in Figure 2.6(c). Note that links of D. numbered 1, ri, s., t. lie in links of U. numbered 1, 1, ki, mi., respectively, and that that pattern Di follows in Di_ is preserved. We embed the final 26 subchain D.(t.,n.) in U.(k.,m.) descending and below D.(s.,t.) in U. (k.,m.). Note that D. is descending throughout U.. Figure 6(d) illustrates the resulting 1 embedding of D. in Di_. Note that D (l,s.) is descending in Di_1 and that the initial subchain of D i+ will be descending in D.(l,si). Figure 2.7 represents the first three stages of a typical continuum X embedded according to the above procedure. The intervening U. chains have been omitted. 1 The heavier line is the minimal subchain D3(r3,s3,t3). It is the embedding of X constructed above that we claim has no simple dense canal. We prove this claim by showing that only one prime end E, defined by a chain of crosscuts converging to endpoint p = i1D.i (1) = in1Ui(1), is such that I(E) = X. But, as p is accessible with respect to E, P(E) = (p), so E is not of the third kind, so does not correspond to a sdc. We will show that every prime end F distinct from E is such that I(F) is a proper subcontinuum of X, so even if F is of the third kind, it does not correspond to a sdc. (It is possible for a point, even an endpoint, to be accessible with respect to one prime end and also be a principal point of a distinct prime end, including a prime end corresponding to a sdc, as is illustrated in Figure 2.3(b) for a principal embedding of the Ucontinuum. Example 2.5.1 also has this property. Hence we cannot simply assume that a chain of crosscuts defining prime end F does not converge to p.) 27 Our proof is similar to those of Theorems 4.3.1 through 4.3.3 of Brechner [1978]. It will be convenient and aid comparison to divide our proof into three lemmas roughly corresponding to Brechner's theorems. 2.4.1.1 Lemma. All accessible points of X are in the composant C of endpoint p. p Proof. Assume D1 is straightened out and let R and S be rays in the plane drawn respectively upward and downward from endpoint p of X so that R (respectively, S) meets Bd D.i(1) in exactly one point and otherwise meets no other link of DI. Then RUS separates E into right and left halfplanes with X contained in the closure of the right halfplane, and only p in X in the closure of the left halfplane. (See Figure 2.7.) Let q be any accessible point of X distinct from p. There are two homotopy classes of crosscuts in the right halfplane minus X from p to q: those like T1 in Figure 2.7 which go around X from p to q, and those like T2 in Figure 2.7 which do not go around X. Let A be a crosscut from p to q which, for some k, is homotopic with fixed endpoints in the right halfplane minus X to a crosscut which, except for short terminal segments in Dk(l) and the link of Dk containing q, lies entirely above Dk in the right halfplane when Dk is straightened out. We can do this because A can be chosen so that its image under a straighteningout homeomorphism lies above some straightened out Dk chain. We may assume that for all i, A meets only those links of D. 1that contain p and q. that contain p and q. 28 Let K be the subcontinuum of X irreducible between p and q. Let C. be the subchain of D. which minimally covers 1 1 K. The first links of (Ci.i are towered on p. Since A enters a link of Dk from above, it meets a link of Dk+l (lSk+l). But Dk+l (1,Sk+l) is descending in Dk and no link of Dk+l lies above D k+l(,1sk+l ). All subsequent initial subchains of CDi]i=k+2 that lie in Dk+l(l,sk+l ) are descending in their immediately containing chain, and at each stage, no subsequent subchain lies above them. Hence, the last links of CC.i.i are towered on q. Though C.(1) is in every case the first link of Di, the last link of C. is not the last link of D. for all i > k. Indeed, iQC.i will be at least some fixed distance d from any point of X in D k(n ). So K = nC.#X. Hence, p and q lie in a proper k+l k+l i=l i subcontinuum of X, and so are in the same composant. QED (2.4.1.1) Endpoint p is accessible, and therefore, a prime end E corresponding to an endcut to p is defined by any chain of crosscuts (Q.]). which cuts R and S in exactly one point each (or is homotopic to such a chain). Any such chain converges to p, so p is the only principal point of E. In Figure 2.7, Q1 and Q2 illustrate such a chain. Since E has only a single principal point, E does not correspond to a sdc. 2.4.1.2 Lemma. If F is a prime end of X distinct from E, then I(F) is a proper subcontinuum of X (indeed, of C .) Proof. Let F be a prime end of X distinct from E and [Pi]i a chain of crosscuts defining F. If some subchain (P.]. of [P.i. is such that P. intersects R and S exactly 11 J 29 once each (or is homotopic with fixed endpoints in E X, to such a crosscut), then (P.]. defines prime end E. So there is a J, such that for all i > J, P. is homotopic with fixed endpoints in E X, to a crosscut that does not intersect RU S. We may assume that J=l, and that (P.i.i is a chain of crosscuts, none of which intersect RUS. In Figure 2.7, P1 and P2 illustrate such a chain of crosscuts. P1U X separates the right halfplane of E and the bounded domain cut off by P U X contains P. for all i > 1. Because of the Ulike embedding, we may straighten out chains without disturbing RUS, so that P1 may be seen to be homotopic to a crosscut which lies above Uk, and therefore above Dk, in the right halfplane, for some k. We may assume P1 meets only those links of U. and D. that contain the endpoints of P1. Then, as in the proof that X contains only one accessible composant, the continuum irreducible between the endpoints of P1 is a subcontinuum of the accessible composant C of X, and so is properly contained in X. Since the domains cut off by a chain of crosscuts are towered, I(F) is properly contained in X. QED(2.4.1.2) In Figure 2.7, PI and P2 are the first two crosscuts in a chain defining some prime end F distinct from E. Note that if the bending of subsequent chains Di., for i > 1, is back toward endpoint p and descending, as in D2 and D3, then the continuum irreducible between the endpoints of P1 may contain endpoint p. This is the case if X is the pseudo arc [Brechner, 1978]. 30 2.4.1.3 Lemma. There is only one prime end E of X such that I(E) = X, but X contains no sdc. Proof. From Lemma 2.4.1.2 it follows that no prime end F distinct from E corresponds to a sdc. We have already observed that E does not correspond to a sdc. As X is indecomposable, some prime end must be such that its impression is X. Only E qualifies, so I(E) = X, but X contains no sdc. QED(2.4.1.3) With Lemma 2.4.1.3 we conclude the proof of Theorem 2.1.2, having shown that every chainable continuum with at 2 least one endpoint can be nonprincipally embedded in E It would be welcome if the condition that the continuum have an endpoint could be eliminated. 2.4.2 Question. Can every chainable continuum be nonprincipally embedded? If so, then the fixed point property for chainable continue would follow as a corollary to Embedding Corollary 2.5 of Brechner and Mayer [1980a]. 2.4.3 Question. Can every indecomposable treelike plane continuum be principally embedded? 2.4.4 Question. Can every treelike plane continuum be nonprincipally embedded? If so, then all treelike plane continue have the fixed point property. We conjecture, though we have no proof, that the atriodic nonchainable continuum in Mayer, [1980b] 31 and in Chapter 3 is a principal continuum. If our conjecture is true, then the answer to 2.4.4 is no. 2.5 NPrincipal Embeddings Of Chainable Continua. An embedding of continuum X into the plane with exactly n sdc's is termed an nprincipal embedding. If every embedding of X into the plane contains at least n sdc's, then X is an nprincipal continuum ([Brechner and Mayer, 1980a], Definition 2.6.) In this section we present examples of chainable continue with nprincipal embeddings. Details of the proofs that our examples have the properties we mention are left to the reader. Figures are provided for the most part, and results stated are generally evident from the figures and the techniques used in the proofs of Theorems 2.1.1 and 2.1.2. 2.5.2 Example A chainable continuum M can be constructed for each n n > 0 embedded with exactly n sdc's. The Knaster Ucontinuum in an embedding in which twotoone chains are alternately ascending and descending is M0. For each n > 0, M is defined by a sequence of chains (of open disks in the n plane), each member of which consists of 2(n+l) straight through subchains (though not all straight through the entire containing chain). A pair of consecutive subchains comprises a loop, which consists of a straight through chain from the first link of the containing chain to a designated link, then returning straight through to the first link. There is one "longest" loop which is straight through from end to end of the containing chain, succeeded by n "shorter" 32 loops, of graduated "length." The embedding alternates ascending and descending versions of the above pattern. Figure 2.8(a) illustrates the inverselimitwithembedding diagram for M2. Note the alternation of an ascending with a descending pattern. The infinite repetition of this pair of embedding patterns produces M2. Note that the bends of C3 occur in bend or end links of C2 that lie in C1(1). It can be shown that M2 (and Mn for any n) has exactly one endpoint which is the limit of all bend and end links. To show that there is just one endpoint, show that every other point is an interior point of an arc; that is, is in an interior link at every stage of a defining subsequence of straight through subchains. Figure 2.8(c) illustrates several stages in the chain construction of M2 in the plane. There, and in Figure 2.8(a), the lines marked with differing symbols represent the two sdc's. Note that a segment of a sdc passing through links of C2 and between upper and lower subchains of C3 meets every link of C1. In this fashion one can show the canal is dense in M2. Figure 2.8(b) shows how to reembed M2 with no sdc's in accordance with Theorem 2.1.2. In the 2principal embedding of M2 (and the nprincipal embedding of any M ) the endpoint is the only accessible point of its composant. This is because the chains are alternately ascending and descending, and so "block" endcuts to other points of the endpoint composant. The endpoint is also a principal point of every prime end corresponding to in Figure 2.10 of a chainable continuum M embedded with c c 33 one of the sdc's. Note that an endcut to the endpoint also corresponds to a prime end whose impression is all of M2 2.5.2 Example. We can modify the construction in Example 2.5.1 to produce a chainable continuum Mo, embedded with a countable infinity of sdc's. At each stage in the sequence of chains defining Mo,.there is one more loop than in the immediately preceding stage. Figure 2.9 illustrates the inverselimitwithembedding diagram for the U.principal embedding. Note that chains are alternately ascending and descending. In accordance with Theorem 2.1.2, M,.can be reembedded with no sdc's. The required reembedding is similar to that of M2 in Figures 2.8(b). 2.5.3 Example. A chainable continuum with c = 2 sdc's can be constructed in a fashion similar to that of Example 2.5.2 by adding at each stage an additional loop for every loop of the preceding stage. We illustrate a different construction sdc's. In general, chain Ci contains 2il +1 loops, as illustrated. Chains are uniformly descending, but alternate starting in link 1, link n. i+, link n. i+2, link 1, every four stages. Only three of the sdc's are illustrated by the lines marked with differing symbols in Figure 2.10. That the number of sdc's is 20can be seen by noting that they can be represented as the branches of an infinite binary tree. By reembedding Mc with the longest loop on the outside at each stage, all sdc's are eliminated. This 34 particular reembedding does not follow the proof of Theorem 2.1.2. 2.5.4 Example. The special embedding P of the pseudo arc is produced by alternating descending with ascending crooked chains between opposite endpoints p and q of P In [1978] s Brechner describes this embedding and conjectures that each accessible point of P lies in a different composant. In [1980] Lewis shows this conjecture to be true. The endpoints of any crosscut of P thus lie in different s composants. Hence the continuum irreducible between the endpoints is Ps It follows that for any prime end E of P s I(E)=P . 2.5.4.1 Theorem. The special embedding P of the s pseudo arc contains uncountably many sdc's. Proof. Let (Ci.i be a defining sequence of crooked, alternately ascending and descending, open disk chain covers for the special embedding P of the pseudo arc. We show that a binary tree can be constructed dense in Ps Each branch corresponds to a sdc. We first show that a ray can be constructed dense in P and forming a sdc, and then show that countably many branch points can be introduced to the ray, turning it into an infinite tree. Suppose ray R enters link C.i(a.) of chain C.. Without loss of generality we assume R enters C. from above (viewing Ci as straightened out) and that C. is descending in Ci.. Since C i+l is crooked in C., there is a (crooked) loop extending from Ci(l) to Ci(ai) and back to C.(2), and the 35 loop is followed by a subchain returning to Ci(ai). Extend R through links C.i(2,a.) (R is moving in reverse order) passing between the lower part of the loop and the returning subchain, and entering a link C i+1 (a i+) sitting in C.i(2). This situation, with only a few of the bends of C i+ shown, is diagramed in Figure 2.11(a). Now C i+2 is ascending in C i+. In link C i+l(a i+) we can find a loopend, where the loop of Ci+2 extends from C i+l(n. i+) to C i+l(a i+) and back to C i+l(n i+l), and the loop is followed by a subchain returning to C i+l(ai+ ) (all in reverse order). We can extend R through links C i+l(ai+ ,ni+ 1) between links of the lower part of the loop and the returning subchain, and entering a link C i+2(a i+2) that lies in C i+l(n 1). Note that R reaches within twice the mesh of C of p and within 1 twice the mesh of C i+1 of q. Proceeding in this fashion, we can extend R alternately toward p and q, and so that the closure of R includes P We have constructed R so that s crosscuts of decreasing diameter exist at every point, as we go to infinity on R; hence R constitutes a sdc in P . s To see that there are uncountably many sdc's in P we modify our construction above. Assume that above we pick the first loop we come to that fits the described conditions. (For descending chains this will be the first loop satisfying the conditions in the order on the chain; for ascending chains it will be the last loop in that order.) Note that in link C i+l (ai+l 1) and below the loop ending in Ci+l1(ai+.) there is a loop, possibly consisting of our previously designated returning subchain and a subchain 36 from C i+l(a.i+11) to C i+l(n. i+l), with still another returning subchain back to C i+l(a i+l) below it. We may introduce a branch point to R in link C i+l(a i+l) and extend a branch R' from R at that point to C i+l (a i+l1) and then further extend R' through links Ci1 (ai+ 1 ni+1l) between the loop and the returning subchain and entering a link C i+2(bi+2 ) that sits in C i+l(n i+l). This link of Ci+2 will be distinct from that which R entered, and R and R' follow separate "channels" between links of C i+2.* We may repeat this branching maneuver in both C i+2(a i+2) and Ci+2(b i+2). Figures 2.11(b) and (c) indicate one branching and the continuation of the rays in separate channels thereafter. Figure 2.11(d) illustrates an entire crooked refinement (with minimal bending). Each link (3 to n.l) of 1 the containing chain is the entrance to a distinct channel, from top (the hatched lines) and bottom (where we show an example of branching). By repeatedly branching, we can extend R so as to form a binary tree dense in Ps so Ps has uncountably many sdc's. QED 2.5.5 Example. Theorem 3.1 of Brechner and Mayer [1980a] shows that the threepoint continuum has embeddings both with and without sdc's. Continue with multiple sdc's can be produced with exactly two endpoints, with uncountably many endpoints (the pseudo arc), or with a number of endpoints proportional to the number of sdc's. We describe one series of examples from the many possible below. 37 The Knaster Scontinuum (like the Ucontinuum in Example 2.5.1) can be used as a basis for constructing continue with multiple sdc's. Figure 2.12(a) is the standard embedding of the Scontinuum. It has exactly two endpoints, whose composants are accessible at every point, and no sdc. Figure 2.12(b) is an embedding of the Scontinuum with one sdc. (Two embedding patterns alternate.) We can produce a continuum Z embedded with n exactly 2n sdc's and having exactly two endpoints. The embedding of the Scontinuum with alternate chains ascending and descending is Z For Z we introduce n extra loops of graduated length in both the first and last of the three straight through subchains defining Z0 in a manner similar to that of Example 2.5.1. The endpoints of Z are opposite endpoints, since there is a definnig sequence for Z such n that each chain is a chain from one endpoint to the other. For a continuum with a pair of opposite endpoints it is enough to embed it with all chains between the two endpoints descending to eliminate all sdc's. The proof is similar to that of Theorem 2.1.2; show that all but the prime ends corresponding to endcuts to the opposite endpoints have properly contained impressions. To produce continue with n sdc's and two endpoints for odd n, we need only eliminate one of the loops from each stage of the construction of Zk, k = (n+l)/2. Other variations on the Scontinuum are also possible. 2.5.6 Example. Bellamy [1980] has constructed an example of an indecomposable chainable continuum with no endpoints. Figure 2.13(a) is an example of such a continuum N discovered independently by us. Continuum N has no sdc in the given embedding, though it does have a simple canal, dense in a proper subcontinuum of N. Figure 2.13(c) represents several stages in the chain construction of N in the plane. The simple canal, not dense in N, is illustrated by the hatched line in both Figures 2.13(a) and (c). That N has no endpoints follows from the fact that every point of N is an interior point of an arc, that is, is in the intersection of a tower of straight through subchains, and always in an interior link. The proper subcontinuum N of N in which the indicated canal is dense is the intersection of the subchains C.i(l,p.) which are towered. Inspection reveals N to be homeomorphic to the Ucontinuum with u p =.Ql1Ci(pi) as its endpoint. (Note that Ci(p.) is the link in which Ci+l (1) and Ci+l (pi+) sit.) Though the canal in N is dense in Nu, it is not a sdc of Nu because there are not transverse crosscuts at each point to N . One composant of N, inaccessible in the given embedding, consists of the union of N with a ray (a 11 u continuous image of [0,)) at their common endpoint p. If N is shrunk out of N, the quotient space is the u Ucontinuum. The endpoint of the Ucontinuum corresponds to the nondegenerate element of the decomposition. (These properties of N were pointed out by Beverly Brechner in discussion.) 39 Figure 2.13(b) illustrates a modified construction producing continuum N' in which the hatched line represents a canal that is dense in N'. Note that link C. (p.) now lies in the second straight through subchain. Note that the C. (l,p.) subchains are not towered in the sequence defining N', and indeed, can be used to define an increasing tower of proper subcontinua whose limit is N'. 2.5.7 Example. Continuum N, like the U and Scontinua, can be used as a basis for constructing continue embedded with multiple sdc's. For each n > 0, we can construct a continuum N  n embedded with exactly n sdc's and having no endpoints. Embedding N with alternate chains ascending and descending produces N0. Figure 2.14(a) illustrates N2. Compare Figures 2.14(a) and 2.8(a) for the similarity in the constructions of M2 and N2* Figure 2.14(b) shows how N2 may be reembedded without a sdc. An endcut to p =.Ql1C. i(pi) corresponds to the only prime end whose impression is N2. Any of the continue N can be similarly reembedded so as to have no sdc. As is the case with N = No, each continuum N contains a composant which is the union of a Ucontinuum subcontinuum N and a ray at their common endpoint p. If N is shrunk out of Nn the quotient space is continuum Mn of Example 2.5.1. Note that in the nonprincipal embedding of N2 given by Figure 2.14(b), the composant containing N is accessible, but N itself is accessible only at its endpoint p. In the 2principal embedding of N2 given by Figure 40 2.14(a), the composant containing N is again accessible, u and now N is accessible, though not at every point (Nu is indecomposable), and in particular, not at point p. In the manner of Examples 2.5.2 and 2.5.3 we can also construct chainable continue N oand N with wo and c sdc's, ^o c respectively. Any of our examples can be constructed with crooked rather than straight through chains, thus producing embeddings of the pseudo arc with n sdc's, for each n > 0. 41 C2 Figure 2.1 Figure 2.2 42 o1 1 n1 cD U2 (a) C1 1 n1 no2 (b) Figure 2.3 DiSi D^ I . tiK 1Ui  u i 13~ ri 1 hi i ki m ni (a) 1 nii mi I 1 nj1 Figure 2.4 T% Ui1 ui J. Figure 2.5 I ni1 Di1 ri D I ti ni (a) Di_i ni1 Ui i 1____________ Ui ri I s i D i I n=ti 1 1 Di I t si Figure 2.6 Figure 2.7 1 V bl I 1 a bl n1 n2 1 a2 a 12a bc2 b2a 1 b2 n2 C 1 2 b2 n2 a 3 n3 3 n a ________a______ (a) (b) Figure 2.8 n 2 b0 n c1 1 b11 nI C b 21 [ . 2 C 2 b21 1 a2_ b 21 n2 C2 C2 1 1 a2 _'__02 b3C3 nn c3 n44 3 31 32 1 W Figure 2.9 1 bnl 21 21 .._________________ __ b22 :702 2 21 nb212 b22 Cn 1 22  2 b 31 32 c b33 3 b b34 n3 1 b31 b3 b 43 b34 n3 c3 .. . b43 04 b 46 F e47 S24 Figure 2.10 R l C. 1 R' R b+2 Rfni+2 b,=a,.2 ''l^ 5!/' 1,2 7=ni Figure 2.11 C1+21 ci i+i 1 '' * *  ^ ni+l C.1.4 1 ai+l1 ni+l1 ci+2 E .. .. .. J 1 Ci+l In i + 1 ^JiiiZI ^^^ [__^ ^ 2 an1 2 ai2 5i 6f 51 \  1 "1 1I n __________3_________________ 02 2 (a) 02 1 n 2 C3  3 f 1 3 Figure 2.12 1 Pi n 1 ____l C 1 P1 nl P2_________ 02____________________ n2 2 P2 n2, P2 nn2 n2 SP32 ___2___' _P2 S3 __ 1________ 11 n3 n3 P3 (c) Figure 2.13 1 p1 1q r, n1 1 P2q2 2 r2 n2 II S (a) (a) Figure 2.14 1 Pl q1 r1 nl CHAPTER 3 PRINCIPAL EMBEDDINGS OF ATRIODIC PLANE CONTINUE 3.1 Introduction Several known examples of tree like continue which admit fixed point free maps are atriodic and nonchainable, and each proper subcontinuum is an arc ([Bellamy, 1981] , [Oversteegen and Rogers, 1980 ] ). In [1980a] Brechner and Mayer show that if there is an indecomposable nonseparating plane continuum which admits a fixed point free map, it must have a LakeofWada channel in every embedding. This is a consequence of independent results of Bell [1978] and Sieklucki [1968 ] In this chapter an example is given of a treelike, atriodic, nonchainable, indecomposable nonseparating plane continuum each of whose proper subcontinua is an arc which has an embedding with a LakeofWada channel. The construction of the example is based upon an example of Ingram's [1972] Ingram's example is a treelike, atriodic, nonchainable, nonseparating plane continuum each of whose proper subcontinua is an arc. However, since it has an embedding with no LakeofWada channel, it has the fixed point property ([Brechner and Mayer, 1980a], Theorem 4.1). Both Ingram's example and ours are proved nonchainable by showing they have a positive span. 54 We construct our example as an inverse limit of X's, so will refer to it as the Xodic continuum. The resulting continuum, X, is homemorphic to a continuum defined as the intersection of a defining sequence (in the sense of Ingram and Cook [1967 ]) of treecovers, and we make use of both constructions in our proofs. We were informed by C. Hagopian that he has a similar example as an inverse limit of X's, but with exactly two LakeofWada channels. In Section 3.2 we construct the example, X, and prove that it is atriodic and nonchainable. The main theorems of this section are 3.2.4 showing that X is atriodic, and 3.2.7 showing that X is nonchainable. The most complex theorem is 3.2.6 showing that X has properties sufficient to guarantee positive span. In Section 3.3 we show that X has an embedding in the plane with a LakeofWada channel. We also raise some questions about X and the fixed point problem for nonseparating plane continue. All spaces are metric and distance functions are as usual for spaces and their products. All functions (maps) are continuous. 3.2 The Xodic Continuum. We shall define the Xodic continuum X in two ways. It is evident that the continue so defined are homeomorphic. 2 The second definition will be such that X C E and so provides an embedding of X in the plane. 55 3.2.1 Inverse Limit Definition of X. Let X1 be the union of intervals [1,1] on the coordinate axes in the xyplane. For convenience we designate (0,1) as A, (1,0) as B, (1,0) as C, (0,i) as D, and (0,0) as 0. X1 is then the identification of four intervals, [OA ] [OB ] [OC ] [OD ] at a single point 0. By P mean the point (0,P) and similarly for B, C, and D. q q Thus [0C)] denotes the interval on the xaxis from (0,0) to q (. ,0). q Let f : X X1 be a map carrying A A onto OB, OR (orderreversing and proportionally) AA onto OA, OP (orderpreserving and proportionally) AA 3 2 onto OA, OR A 2A 2 2A onto OC, OP 2A 5A 2 A onto OC, OR 3 6 5A SA onto OD, OP B 0O onto OB, OR 3 BB A 2 onto 07, OP B 2B A 3 onto OA, OR B onto OC, OP 0 C onto OB, OR C C onto OD, OP D onto OB, OR D D onto OC, OP A schematic diagram of f is given in Figure 3.1. For each i, let X. = X and fi = f. We define X by: X = lim [ X.,f.3 Let f : X X be defined by n f f1 = f ff 2 fn 3.2.2 Defining Sequence for X. For each n, let Tn be a collection of open disks in the plane, E such that (1) T n+ strongly refines Tn; that is, for each L in T n+ there is some M C T such that L C M. (2) The mesh of T is less than 1/2n. n (3) T is a coherent collection of four subchains with exactly one junction link designated as follows: 57 (a) The junction link is 0 . (b) The four endlinks are An, Bn, Cnt and D . (c) The four subchains are 0 A 0 B n, OC and 0D. nn (4) T n+ follows in T the pattern suggested by function f, including orientation with respect to the plane. (That is, if T were "straightened out" by some n2 orientationpreserving homemorphism of E2, then Tn+l would sit in T exactly as Figure 3.1 suggests.) n Then X = R T. n=l n Figure 3.2 illustrates the first three stages in the construction of X. The third stage is represented by its nerve. Since Tn+l follows the pattern of f in Tn, we may conveniently refer to links of T on analogy with points of X That is, A n+/3 is a link of chain 0 n+A n+ that n n+l n+l n+l A corresponds to point (0,h) of Xn. Note An+ /3 is a link of T n+ sitting in link A of T Similarly, link B n+/2 of n+1 n n n+1 T n+l is a subset of link A /3 of T We could have defined n+1 n n the pattern that Tn+i follows in T directly in terms of such intermediate links and the chains between them. We require our chains have as few bends as possible. (Too many bends and we might get pseudoarcs as subcontinua.) We will call each T (which is a tree cover of X) an Xod cover of n  X. An open cover of X by subsets of X is derived from the above by letting T' be the collection of open sets of X such n that L' is in T' iff L' = L n X for some L is in T n n 58 2 It is clear from our definition that X C E in the construction of 3.2.2. We will refer to this embedding of X in what follows. 3.2.3 Definition of the Span of a Function. The span, of, of function f:XY is the least upper bound of all numbers for which there is a connected subset Z. of XxX such that I 1(Z ) = f2(Z ) and d(f(x),f(y)) > efor all (x,y) in Z The span, oX, of space X is the span of the identity function on X (Ingram [1972] due to Lelek [1968 ] ). 3.2.4 Theorem Let X be the continuum defined by Xod tree covers in 3.2.2. Then X is atriodic. Proof. We will show every proper subcontinuum of X is chainable. That X is then atriodic follows from Theorem 3 of Ingram [1968]. Let H be a proper subcontinuum of X. Let [T n be n n=l the defining sequence of treecovers for X, defined in 3.2.2. For each n, let F be that subcollection of T that covers H minimally. Then H = QiF For some N, for all n~ln k > N, Fk # Tk, for otherwise H = M. Since H is a continuum, each F must be coherent. We consider two cases: n If for some subsequence [Fj] of (F n) n F does not include 0., then H = jiF. is chainable as each F. is a chain. 59 So assume there is no such subsequence. Then for some J, for all k > J, Ok is in Fk* Let k be chosen as the maximum of N,J. Then Ok+3 is in Fk+3. Now 0k+3 C Bk+2 in Fk+2. But 0k+2 is in F k+2. Since Fk+2 is coherent, chain Ok+2Bk+2 C Fk+2. Observe that k+2B k+2 C k++l k+U k+1 Ak+l/3 U k+l Ck+l Ok+lBk+l C 0kBk U Ok+Ak+l/3 U OkCk, Ok+ A k+/3 C OkBk U OkAk, Ok+lCk+l C OkBk U OkDk "k+lk+l kk "kk Hence, 0 k+2Bk+2 meets every element of Tk. Since k+2B k+2 C Fk+2 and Fk contains Fk+2' Fk = Tk. But k > N, so Fk # Tk,a contradiction. Therefore, H is chainable, and by Ingram's theorem noted above, atriodic. QED Figure 3.3 shows how 03B3 sits in T. To show that X is nonchainable we will use Theorem 4 of Ingram [1972]. 3.2.5 Theorem. If X = lim [ Xi,fi] with each X. compact, and for E> 0, 1 of, >E, for each n, then oX > 0. Lelek [1968] observes that oX > 0 implies X is nonchainable. To apply 3.2.5 we require the following as a lemma. The proof is an application of the method of proof of Theorem 2 in Ingram U972]. 3.2.6 Theorem. There exists a sequence Z1, Z2 . of subcontinua of X1xX1 such that for each n, 1(Zn) = 52(Zn) = XI' fxf(Zn+l) l 1 = Zn Z = Zn, and if (p,q) is in Z1, then d(p,q) > ,' for all n. 60 Proof. Our proof is by induction on n. Z is the union of the following twenty subcontinua of X1xX I: m1=([OB] m3=([OB] m5=([OC] m7=([OC] mg=([ OD] mll=( [OD] m13= ( [OC] ml5=([OD] m17=([OD] m 19=([o] x WA]) U (B] x [OA] ) m2=([OA] x x [0])U (CB] x [o]) im4=([O] x [B x [A) U (tC3 x [OA] ) m6=([OA] x x [3)) U( 3C] x [0o]) m8=([o] x [C x WA]) U ((D] x [OA] ) m10=([OA] x [A3)U((D] x [0) m)12= ( O] x x (B) U (CC) x [OB] ) m14=([OB] x (B)) U ((D] x [OB] ) m16=([OB] x (C3) U ((D] x [OC] ) m18=([OC] x [A) U (0 x [A] ) m=([5A x WA) U (L[03 x [5AI m20 [A]) Each of in. (1 < i < k (1 j 10) below as kj (1 <_ j _< 10) below as [B]) U ([A] x [OB)) I) U ()] x [OB] ) (C]) U ((A) x [OC]) x (D])U(LA] x [OD] ) LD])U(t] x [OD] ) x (C3)U(tB] x [OC] x (D3)U(LB] x [OD] x (D])U((C] x [OD] x (O]XJU(CA] x [o3]) 20) is a continuum. We designate unions of certain m. containing a 1 common point; hence each k. is a contain 3 illustrate ZI. mI1 U m5 U m9 U mn19 mi2 U m6 U m10 Um20 mi2 U m4 U m13U m115 m U m3 U m 14U m 16 m6 U m8 U ml4U m17 m5 U m7 U m 83U m18 ml 0U m2U m16OU m18 m 9 U m11U m15U mi17 (O,A) (A, 0) (O,B) (B,O) (O,C) (CO) (0,D) (D,0) uum. Figures 5 and 6 mI n mi5 f m9n m19 m 2 n m 6 n mlo n m20 m 2 n m4 n m13 n m15 mI fl m3 fl m14n m16 m6 n m8 n m14n m17 inm5 n m7 n m13 n m18 min0 m12 n Am6 mi8 m9 in ml n m15nml17 61 k = m3 U m7 U min (A,) is in m nm n ml 9 3 13 3 1 k1o0= m4 U m U inm12 (,o) is in m4 f n8 m12 Observe that m1 C k 1n k4, m16C k4 n k7, m10 C k7nki, m6 C k2 n k5, m17 C k5 n kg, m 5 C k8 n k , 8 m13 C k3nk6. Hence, U k. is a continuum and further (k9U 13 3 j = jl D 9 8 20 k) CU k = Um.. Hence Z = Ui m. is a continuum. We 10 j= j i= iimi further observe that For each odd i, (1 < i < 19), mi = mi+1 so 1 = zI r 1(inm1Unlmlnml3U mi17) = T2(mlU inm10Um13OU m17) = XI, (B,C) is in m 14, (C,B)is in m13, (B,A) is in mi, (A,B) is in m2, (B,D) is in m16, (D,B) is in m 15 1 If (p,q) is in ZI, then d(p,q) > 1 We adopt the convention that < t,u > denotes a continuum M such that 7I1(M) = t and 7T2(M) = u and < t,u >1 = . Note that Z1 is the union of 20 such continue in the same order as in the induction hypothesis below. If < t,u > is a subcontinuum of Z and v,w are subarcs of X1 such that fiv maps onto t and fl maps onto u, both homemorphically, then L = (f1 Iv x f11w )(< t, u >) is a continuum such that TI (L) = v and 72(L) = w. We denote this continuum as L( as L, and call it the lifting of < t,u > with respect to v and w as defined in Ingram [1971]). Induction hypothesis. Z is a continuum of X XX1 such that (a) r1(Z ) = r2(Zn) = X. (b) Z is the union of twenty continue denoted by n < OB,OA > < OA,OB > < OB,A > < 0O,OB > < OC,OA > < OA,OC > < OC, A > < O,OC > < OD,OA > < OA,OD > < OD,OA > < AOD > < OC,OB > < OB,OC > < OD,OB > < OB,OD > A 5A 5A A < OD,OC > < OC,OD > < OA,A > < AO > where < t,u > is a continuum M such that 7I (M) = t and r2(M) = u. c) < t,u >I =< u,t > d) There are five points in Xl, Z as follows: 5A5A A x is in A, with (xl,0) in and (O,xI) in 2B A x2 is in 3 B, with (x2,O)in and (O,x2) in CA x is in CC, with (x3,O0) in and (O,x3) in X4 is in D, with (x4,0) in and (O,X4) in 3, x is in with (x,O) in and (O,x5) in e) There are three points in X1, Zn as follows: z is in OC with (B,z ) in z2 is in OA with (B,z2) in z3 is in OD with (B,z3) in Base Case. Observe that Z1 meets all of the above conditions. Induction. Our induction will be to construct Zn+l by lifting Zn so Zn+l satisfies (a)(e) and fxf(z n+) = Z . n nin+l n By a. (1 < i < 20) we will denote the twenty continue whose union is Zn+l* Furthermore, < t,u >' will denote a continuum which "corresponds" to < t,u > of Z This proof closely follows Ingram's method in [1972 ]1, as does the inductive hypothesis above. Continue a1, a2, a3, and a4. = 1 A 5A B 2B 11AA 1 A 5A. BB 11AA, L3 ( L5( 64 S1 BOB, 00, B A2A) U LO (2OBA 00 LA 5A) U 1 B 5A L9 ( a2 = That a is a continuum follows from (x3,0) is in so (fl x3)2B A is in L 1 L so ',B(3 1 fl2 (O,xI) is in so (2B, f1 (xl)) is in L2nL1 There is a y in 5A, with (A, y) in , so (,f 111A (y)) is in L31fLl A 5A (O,xI) is in <0,5AA> n B h,IA A L so (B, j11 A (x1)) is in L L There is a y in OB, with (y,A) in Sl n1 1 so (fI ( (y),) is in L5flL 65 (x2,O) is in so (f1 IO3(x.2,) is in L6nLl There is a y in OB with (y,C) in 1B 2A 1 N L1 so (f I1o (y),2) is in L71 1l (x2,O) is in so (f oI (x2), 5 ) is in L nLl So a2 is also a continuum. Also Tr1(a1) = OB & 72(al) = OC & fxf (aI) C Zn 1(a2) = OC & 7i2(a2) = OB & fxf (a2) C Zn 5 a = a3 = P OO'=1 iLwhr i i  a = The proof that a1(a2) is a continuum contains the proof that a3(a4) is a continuum. Further 7l(a3) = OB & n2(a3) = & fxf (a3) C Zn (a4) = & 2(a4) = OB & fxf (a4) C Zn 66 Continue a5, a6, a7, and a8. 5 C A 5( a5= 5 C AA 5 C AA L3( 5 C A 2A 5 C 2A5A L5( 5 C 5A L7( 1 a6 = That a5 is a continuum follows from (x4,0) is in 1 C A5 5 so (f l C x() ,) is in L5nflL 2o( (4) 6 L1 L2 (O,x1) is in so (j, flli3(x)) is in L 2 L3 2 63 12 3 There is a y in OB with (y,A) in 1C A isi 5 so (fioC(y),A) is in L3 nL4 (x2,O0) is in so (f 1O(x2) ) is in L 4 L5 7 '4 5 67 There is a y in OB with (y,C) in so (f1loC(y),2) is in L5 2 is iL5 L6 (x2,O) is in so (fiCO(x2),") is in L 5AL So a6 is also a continuum. Also TT 1(a5) = OC & 7T 2(a5) = OA & fxf (a5) C Zn T 1 (a6) = OA & iT 2(a6) = OC & fxf (a6) C Zn a = 7 < = >il a = That a7(a8) is a continuum follows from a5(a6). Further I1(a8)= OC & IT2(a) =O & fxf Tr1(a.) = 2& T2(a.) = OC & fxf 1 < i < 3. the proof for (a7) C Zn (a8) C Zn 68 Continue a9, a10, all and a12. a = 9 1 3 2, 9 2 AA ~( 9 ^D A2A. QnA L5( L9 ( a = That a9 is a continuum follows from (x3,0) is in so (fliD(x3),) in LInL9 (O,x ) is in s Df 1AA 9n so (, f (x)J ) in L2L3 There is a y in OB with (y,A) in so (f1 1o(y),2 ) in L9 L9 (x2,0) is in 11 D A 9 9 so (f 2(x2),2) in L4 5 69 There is a y in OB with (y,C) in ,l iD, 2A. 9 9 so (f 1lO(y) ,7) is in L.nLg (x2,O) is in 1 Dflo 2 5A 9 9 so (f 2Ox) A,5) is in L6L So a10 is also a continuum. Also 7T (a9) = OD & T 2(a9) = OA & fxf (a9) C Zn l (al) = OA &Tr 2(a10) = OD & fxf (a10) C Zn 3 a = a12 = <0 ,OD>' = a That a11 (a12) is a continuum follows from proof for a9(a10). Also l(all) = OD & 72(al) = O & fxf (a1) C Zn Ta A (a) OD & fxf (a ) C Z 1Tl(a12) = O & 2(a12) = OD & fxf (a12) C Zn Continue a13 and a14 13 C B 13 A C BBu a 3 13 A C BB 13 A COB, B>, 2B)U L 3 ( 13 C2B L 5 OBO>O=,) 1 a14 = That a13 is a continuum follows from (x4,O) is in ,,lC,, .B, LI 13n 3 so (f lC(x),) is in L 3nL 3 so4 3 1 2 (O,x5) is in C f 1BB 13 13 so ( 32f ljx5)) is in L2 L 3 There is a y in OB with (y,) in so (flo(y) ',) is in L3 n L43 (x2,O) is in 03> so (fl C (x2) 2B 13 13 = ,x ) 2 L4 5 L So a14 is also a continuum. Also 1l(al) = OC & T2(a13) = OB & fxf (a13) C Zn T 1(a1) = OB & 72(a14) = OC & fxf (a14) C Zn Continue a15 and a 16. a a15 1 L15 OBO>O D BB) U L15 OBA D B 2BU 15 D 2B L 5( 1 a16 = That a15 is a continuum follows from (x3,O) is in so (f1 DD(x3), B) is in L15 n L15 (O,x5) is in D fi B 15 15 so (D,f f i B(x5)) is in L2 1n L3 There is a y in OB with (y,A) in 1D B 15 15 so (f lo(y),2) is in L3 fl L4 (x2,O) is in so (f1IOE(x ),2) is in L5 L15 So a16 is also a continuum. Also T(a15) = OD & I2(a15) = OB & fxf (a15) C Zn T (a 6) = OB & 7T2(a16) = OD & fxf (a16) C Zn 72 Continue a17 and a18. a17 = 1 , 17 ( 3 23 a18 = That a17 is a continuum follows from (x3,0) is in so (fiID(x3) ,C) is in L 17 n L 1 (O,x4) is in ,D 1 _, ., . 17r~ 17 so D(1,f C(x4)) is in L 17 n L17 So a18 is also a continuum. Also Tl(a17) = OD & 12(a17) = OC & fxf (a17) iT (a 1) = OC & TT2(a1g) = OD & fxf (al1) Continue a19 and a20. A5A = L19 ( 5A A 1 a20= < A,0O3> = a19 That a19 is a continuum follows from (O,x4) is in (A i 5A A i i 19 19 so (6',f  A(x4)) is in L1 n L2 Cz n c z n )C >,O'A 73 So a20 is also a continuum. Also A 5A I (a) = 0 & (a 9) = :A & fxf (ae) C Z 1 19 3 19 6 9 5A A 7T a20) = 6 A & 72 (a = 0i & fxf (a) C Zn 1 20 6 2 (20) 3 (20) n 2 0 We now show Zn = iai is a continuum satisfying (a) n+l I= (e) of the induction hypothesis and such that fxf (Zn+) = Z Designate the following five points: x{= 5fl A = fl f1 2B( x = f1 C(z3) & x= f D(Zl) f 1 AA x = f (z2) These points will be shown below to satisfy (d) and will be used to show b. (1 < i < 10), defined below, are each continue. We explicitly define b. for odd i, and note 1 that for even i, b. = b. . 1 i11 b = a2 Ua6Ua 10Ua where L implies (x ,0) is in a2, L7 S^10 implies (x ,0) is in a L implies (x{,0) is in ae0, implies 7XO is in6,L 20 & L 2 implies (x{,0) is in a20. 74 b, = a U a a Ua where L1 implies (x',0) is in al, L 14 b3 =a1U a14U a16Ua3, where 51 implies~ ~1 implies (x',O) is in ael 16216 implies (x,O) is in a14, L imle5x,)i na6 & L3 implies (x,O) is in a3. b5 = a5Ua 3Ual8Ua7, where L5 implies (x,0) is in a5 , 13 18 L 13 implies (xpO) is in a13, L 1 implies (x,O) is in a 7 & implies (x,) is in a7 . L18, Lo) b7= a Ua1Ua1 U a where L9 implies (x.,O) is in a9, 1 5 177ill 1 L 15 implies (x,O) is in ae5, L 17 implies (x.,O) is in a1, & L 1 implies (x.,O) is in a1l. =4ipis(,Oisia4 8 b9 = a4Ua8Ua12, where L5 implies (x,,0) is in a , 2 implies (x,O) is in a8, & L3 implies (x,O) is in a12. So x{, x x3, xj, x satisfy (d). Observe that a2C bnb4, a13C b4 n b5, a8 C b5 n b8, a12 C b. n b9, a8C b9 n b6, a14C b6nb3, a1C b3 b2, ba9 C b2 n b, 2 0 and a1 C b n b Hence Z = a is a continuum, and is the union of twenty continue satisfying (b). is the union of twenty continue satisfying (b). 75 Note that T1(a1 U a13 U a17 U a10) = 2 (a1 U al3 U a17U a10) = XI, so (a) is also satisfied by Z n=. Furthermore, (e) is satisfied since There is a z' in OC with (B,z') in al4= in ae3 = There is a z in OA with (B,z2) in a = in a2 = There is a z' in OD in (B,z') in a 6= a15 = I As a. = a11 for odd i, 1 < i < 19, (c) is satisfied by i l Zn+1 Finally, we must show fxf (Zn+l) = Zn. Since fxf (a.) CZ for 1 < i < 20, we need only show Z C fxf (Z n+l). We observe that fxf (a3 U a13 U al5 U a18 U a20) includes A A Hence fxf (a4 a U 14 a6 U ae7 U ael 9) includes < > <5A A <9,OD>, 76 Thus Zn C fxf (Z n+l). Therefore, fxf (Z n+) = Z . n n+l n+l n Our inductive step is thereby completed. Since (p,q) 1 n 1 in Z1 implies d(p,q) > it follows that of > 3, for all n and the theorem is proved. QED Figure 3.5 represents Z1 C X XX1 with hatched and xed lines. Figure 3.7 represents Z2 C XlXX1 with hatched lines going to hatched lines and xed lines going to xed lines in Figure 5 under fxf. Two parts of Z3 are illustrated at the bottom of Figure 3.7. 3.2.7 Theorem. X is nonchainable. Proof. By Theorem 3.2.6,oX satisfies the conditions of Theorem 3.2.5. Hence we may conclude that OX > 0. Therefore, X is nonchainable. QED That X is nonseparating is evident. That X is indecomposable is a corollary to the proof of Theorem 3.2.3 by applying Ingram and Cook's criterion of indecomposability [1967] Another proof that X is indecomposable follows from Theorem 3.3.2 and a theorem of Sieklucki [1968] quoted in Lemma 2.2 of Brechner and Mayer [1980a]. It is evident from the construction of X that each proper subcontinuum, being chainable, is also an arc, since all subchains are relatively straight in the covers they refine. We observe that X has two and only two endpoints, e and f, such that e is the intersection of the tower 77 C . CDC C4 C D3 C C2C D1 and f is the intersection of the tower C . CC5C D4 C C3 C D2C C1 3.3. LakeofWada Channels. By LakeofWada channel we mean a simple canal, defined in Brechner and Mayer [1980a] which definition is due to Sieklucki [1968] (For the original LakeofWada construction see Hocking and Young [1961] pp. 143144.) 3.3.1 Definition. Let X be a nonseparating continuum and let D be a set homemorphic to [0,1) in E X, where a:D * [0,1) is a given homeomorphism. Then D will be called a simple canal in X if the following three conditions are satisfied: (1) D D C Bd X (2) For each p in D, there is a "bridge" to X; that is a crosscut to X, which (crosscut) is transverse to D, and intersects D at exactly one point. (3) If p. (i.e., a(pi) + 1), then there is a sequence of bridges (Q }i such that Q Pi D = (pi) and diam QPi 0. Pi If, in addition, condition (4) holds, where (4) is (4) D D = Bd X, we call D a simple dense canal (sdc) [Sieklucki, 1968]. 78 We can show that the embedding of X given in 3.2.2 has a sdc or LakeofWada channel in either of two ways: by 2 directly constructing the requisite ray in E X, or by defining a chain of crosscuts Q of E x that defines a prime end E of the third kind with I(E)=P(E)=X. (See Brechner [1978] for basic definition of prime ends.) For the equivalence of these methods see Brechner and Mayer [1980a], Theorem 2.9. An embedding for which there exists a sdc D in E X such that D D = Bd X, is termed principal in Definition 2.6 [Brechner and Mayer, 1980a] The embedding of X here given is consequently principal. The first four stages of the construction of a sdc D are illustrated in Figure 3.4. Figure 3.2 illustrates two stages of D (the "railroad tracks"). Alternately, let Qi be a crosscut of E X such that for all i > 2, the endpoints of Qi lie in C. and A., while Qi C D. i. Then either the odd or the even subsequence of (Q 3' defines a prime end E i i=2 such that I(E)=P(E)=X. For the even subsequence, Qi e, and for the odd subsequence, Q. f, where e and f are the endpoints of X. We thus have the theorem below. 3.3.2 Theorem. X can be embedded with a simple dense canal D in E2X such that D D = X. That is, the embedding of 3.2.2 is principal. 3.3.3 Corollary. X is indecomposable. Proof. Follows from 3.3.1 and Lemma 2.2 of Brechner and Mayer [1980a]. 79 If every embedding of X into the plane were such that E X contained a sdc, then X would be a principal continuum in the sense of Definition 2.6 of Brechner and Mayer [1980a]. As a principal continuum X would be a candidate for a nonseparating plane continuum admitting a fixed point free map. Thus two questions suggest themselves 3.3.4 Question. Is X a principal continuum? 3.3.5 Question. Does X have the fixed point property for continuous maps? In [1975] Lelek asks if there is an example of a nonchainable continuum with span zero. This author, in efforts to modify Ingram's example to produce a principally embedded, atriodic, nonchainable continuum considered several examples for which said author was not able to prove the span nonzero. On the other hand, the examples are principally embedded, and appear to be nonchainable, though atriodic. We show a schematic diagram for the bonding function for one such example as an inverse limit of T's in Figure 3.8. (For surjective span see Lelek [1977].) 3.3.6 Question. Does the continuum of Figure 3.8 have a span greater than zero? Surjective span greater than zero? 3.3.7 Question. Is the continuum of Figure 3.8 chainable? D Figure 3.1 Figure 3.2 Figure 3.3 Figure 3.4 .4 , V., B  C. C $ A B C. PB PB I B B a A C l(~ '~* C a 'p.' C CI C 0 0 4, B C D ~ at Figure 3.5 fll Part of X1 x X1 with part of Z1 marked Figure 3.6 ... i : . . . ___ . .. k , A A A " a~~a O a a Ce " 0 ^ ^  ' Sal A o C .o , D I l r ....... D ,L '' D  ,\ I I I i . . . .. .. ...     :,,,, '*' ' 0 0 0 B 0.' 0 C. 0 ~ ~ .. .. ,..t o '. *. o o .' A ,. f ^ ^ ^ o ot6 Figure 3.7 Figure 3.8 CHAPTER 4 INEQUIVALENT EMBEDDINGS AND PRIME ENDS 4.1 INTRODUCTION. In [1980a] Brechner and Mayer show that equivalent embeddings of a nonseparating plane continuum have the same prime end structure (Theorem 2.11). Though not explicitly stated, this fact has been used previously in the literature. For instance, M. Smith [1980] and W. Lewis [1980] have independently shown that there are uncountably many inequivalent embeddings of the pseudo arc into the plane. This result was achieved by exploiting different prime end structures (directly in Lewis' case, indirectly, in terms of differing accessibility of composants by Smith) to distinguish different embeddings. In this Chapter we show that the converse of Brechner and Mayer's theorem is false: there are inequivalent embeddings of a nonseparating continuum into the plane that have the same prime end structure, and indeed, that have the same set of accessible points. The following theorems stand in partial contrast to the methods of Smith and Lewis: 4.1.1 Theorem. There exist uncountably many inequivalent embeddings of the sin 1/x continuum into the plane with the same prime end structure and the same set of accessible points. 87 4.1.2 Theorem. There exist uncountably many inequivalent embeddings of the Knaster Ucontinuum (bucket handle) into the plane with the same prime end structure. Moreover, the set of accessible points in each of these embeddings is exactly the composant of the Ucontinuum that contains the endpoint of the Ucontinuum. In Section 4.2, we show how to construct uncountably many embeddings of the continuum formed by the sin 1/x curve plus its limit segment. Thereafter, we show these embeddings have the same prime end structure, but that any two are inequivalent, thus proving Theorem 4.1.1. In Section 4.3, we proceed similarly to prove Theorem 4.1.2, showing the Knaster Ucontinuum also has uncountably many inequivalent embeddings with the same prime end structure and the same set of accessible points. We first prove the theorem for two specific, easily visualizable, embeddings of the Ucontinuum. In Section 4.4, we indicate how Theorem 4.1.2 can be extended to each of the uncountable class of Utype Knaster continue identified by W.T. Watkins [1980] . 4.2 The Sin 1/x Continuum. The standard embedding of the sin 1/x continuum (Figure 4.1) consists of a ray R, the graph of (0,1] under the function y=sin 1/x in the xyplane, plus the limit segment [p,q] the interval [1,1] on the yaxis. The ray R consists of a number of loops, where a loop is a segment of R with exactly one peak and one trough in the standard 89 embedding. For simplicity, we will fix the set of loops as the segments of R between alternate successive points of zero amplitude in order from 1 to 0 on the xaxis. The endpoints of loops limit on (0,0) in the standard embedding, so on the "midpoint" (or, more precisely, some interior point) of [p,q] in any other embedding. Endpoint p of limit segment [p,q] is the limit point of points on R selected from successive troughs, endpoint q is the limit point of points selected from successive peaks. We suppose R to be coordinatized by the function g: [0,) * R so that the endpoint of R is g(0) and for x E (0,1] odd positive integers correspond to t(x,l): sin 1/x = 13, even positive integers correspond to t(x,l): sin 1/x = 13, and fractions with denominator 2 and odd numerator greater than 1 correspond to [(x,0): x=l/(in), for positive integers n]. This embedding, denoted K, and its corrdinatization by g are illustrated in Figure 4.1. An embedding e:X E 2 of a continuum X is a homeomorphism into the plane; however, we shall somewhat loosely suppress reference to an embedding function and refer to the image in the plane of a continuum X as the embedding of X. An embedding of the sin 1/x continuum can be described in terms of how the limit segment and each loop L = [(2nl)/2, n, (2n+l)/2, n+l, (2n+3)/2 ] n for odd positive integers n, is embedded in the plane. We will continue to designate the image of R, [p,q] Ln, or a point x in the sin 1/x continuum as R, [p,q] L or x, 90 suppressing reference to any particular embedding function. A schema will be a set of directions (necessarily infinite, though countable) for embedding (and, ambiguously, the embedding of) the loops of ray R with respect to limit segment [p,q]. A subschema will be directions for embedding some finite number of loops. Subschemata will be linked in sequence to form a schema. We will reference the various subschemata rather informally. An example will indicate our procedure. We assume the limit segment [p,q] is fixed, and describe the embedding of the loops with reference to [p,q] The embedding M0 in Figure 4.2 is the simplest of the uncountably many we will show to exist. For this embedding we require only one subschema So: from (2nl)/2 bend around q at n, then toward, but not around p at n+l, and extend to (2n+3)/2. The schema P0 for embedding M0 is then the infinite sequence of So's linked so that corresponding parts of successive subschemata are closer to [p,q] than their predecessors. 4.2.1 The Prime End Structures of K and MO. Before proceeding to construct our uncountably many embeddings of the sin 1/x continuum, we illustrate some of the concepts involved in prime end theory by applying them to K and MO. Definitions and further references may be found in Brechner [19781 . Prime ends are a way of looking at and classifying the approaches to the boundary of a simply connected domain with nondegenerate boundary. The complement in S of a 91 nonseparating nondegenerate plane continuum X, denoted S2X, is a simply connected domain. While E X is not simply connected, as E2U[,), the onepoint compactification of E2, 2 2 is S we can refer to the prime end structure of E X by associating it with the prime end structure of S X, where the embedding at X misses the point at infinity. A prime end of E X is defined by a chain of crosscuts converging to a point of X, where a crosscut is an open arc in E X whose endpoints lie in X. If Q is a crosscut of 2_ 2 E X, then QUX separates E A sequence of crosscuts (Q.i.i=l is a chain provided that Q. converges to a point, that no two crosscuts have a common endpoint, and that Qi separates Q ii and Qi+l. So, for example, the prime ends E and F of E K in Figure 4.1 are defined by chains of crosscuts (Q..il and (Q.)i=l respectively, while in Figure 4.2, [T.] i= defines prime end H of E2 The impression of a prime end E, denoted I(E), is the intersection of the closures of the domains cut off by the crosscuts in a chain defining E. For example, in Figure 4.1 it can be noted that I(E)=[p,q] =I(F), while in Figure 4.2, I (H) =p,q ] . The set of principal points of a prime end E, denoted P(E), is the collection of all points in X to which some chain of crosscuts defining E converges. For example, in Figure 4.1, P(E)={p), and P(F)=[q], while in Figure 4.2, P(H)=[p3. A prime end E is of the first kind if I(E)=P(E), both degenerate, of the second kind if I(E)#P(E), only P(E) 92 degenerate, of the third kind if I(E)=P(E), both nondegenerate, and of the fourth kind if I(E)#P(E), both nondegenerate. It can be shown that P(E)CI(E) in any case, and that both are continue in X. Thus prime ends E, F, and H of Figures 4.1 and 4.2 are all of the second kind. Any other prime end G, of either E K or E M0, will be of the first kind, or trivial. Thus we can say that the prime end structure of E2K consists of two prime ends of the second kind and all other prime ends trivial. A more precise description of prime end structure is afforded by the notion of a Cmap (see Brechner [1978].) A Cmap p is a homeomorphism of S X onto Ext B, where Ext B is the complementary domain of the unit disk B in S which contains the point at infinity, which (map) satisfies the conditions: (1) if Q is a crosscut of S2X, then O(Q) is a crosscut of Ext B, and (2) the endpoints of images of crosscuts are dense in Bd B, the boundary of B. If we require, as we may, that 0 take the point at infinity in S 2X to the point at infinity in Ext B, then we can regard the restriction of 1 to S2t_}]=E2 as a Cmap also. Suppose a chain of crosscuts defines a prime end E of E X. Then the images of the crosscuts under a Cmap 0 will converge to a single point e in Bd B. We say e corresponds to E. In fact, there is a onetoone correspondence between the prime ends of E2X and the points of Bd B. For example, 93 in Figure 4.1, points e and f in Bd B correspond to prime ends E and F, respectively. In Figure 4.2, h corresponds to H. No homeomorphism of Bd B in Figure 4.1 to Bd B in Figure 4.2 can carry points corresponding to prime ends of a given kind onto points corresponding to prime ends of the same kind. Hence the prime end structures of K and M are not identical.(See Brechner and Mayer [1980a], Definition 2.10.) Consequently, K and M0 are inequivalently embedded. (See Brechner and Mayer [1980a], Theorem 2.11.) In the general result which follows, each embedding of the sin 1/x continuum will have the same prime end structure as M Note that the accessible points of M0 are the ray R and point p of [p,q] A point of a plane continuum X is accessible if it can be reached by a half open arc in the complement whose closure adds exactly that point. Such a half open arc is called an endcut. Each of our uncountably many embeddings of the sin 1/x continuum will have the same set of accessible points as Mo. The significance of this result is the contrast it provides to the usual procedure for producing inequivalent embeddings of a plane continuum: produce embeddings with different points accessible. Such a procedure is a sufficient, but not a necessary, condition 2 for producing inequivalent embeddings of a continuum in E. In order for subschemata to be constituents of a schema for embedding the sin 1/x continuum, the subschemata must be linkable into a ray that converges to limit segment [p,q ] . Only certain subschemata are so linkable, and the following Lemma identifies countably many of them. 4.2.2 Lemma. There exist a countable infinity of subschemata linkable pairwise in either order. Proof. In the following, all bends in loops are either around q, or toward, but not around p, or away from p. We will describe the first few subschemata fully, and then supress reference to qbends and designate pbends as either in or out with a superscript for the number of bends. Let n be a positive odd integer. The subschemata are: So: from (2nl)/2 bend around q at n, then toward, but not around p at n+l, and extend to (2n+3)/2. (1 loop: in .) S1: from (2nl)/2 bend around q at n, then away from p at n+1, extending to (2n+3)/2; then bend around q at n+2, away from p at n+3, to (2n+7)/2; then bend around q at n+4, toward, but not around p at n+5, to (2n+ll)/2 which will lie between [p,q] and 2 1 previous loops. (3 loops: out in .) 4 1 S$:5 loops: out ,in. Sk: 2k+l loops: out2k, in1 Sk : 2k+l loops: out ,in Subschemata S1 and S2 are illustrated in Figure 4.3. 