ON THE EMBEDDABILITY OF COMPACTA IN N-BOOKS: INTRINSIC AND EXTRINSIC PROPERTIES Thesis for the Degree of Ph. D. MICHIGAN STATE UNIVERSITY GAIL ADELE ATNEOSEN 196.8 4.5-.va I ....- - mr' Ell-vi! LP} a . v -. .1 A M; 1 Michigm Stats”: Unix"? r813. 57 TthIS' This is to certify that the thesis entitled ON THE EMBEDDABILITY OF COMPACTA IN N-BOOKS: INTRINSIC AND EXTRINSIC PROPERTIES presented by GAIL ADELE ATNEOSEN has been accepted towards fulfillment of the requirements for Ph.D. degree in Mathematics 7~ ' 4 , / I‘ A fir V W Major prolesso Date July 21+ , 1968 0-169 im‘wvf '1— ! BINDING BY noun & sous' ‘ .BI‘II amnm IIIII. Iu'li'l‘: BINDERS “LIICIIAI m._-1 _ Sb ABSTRACT ON THE EMBEDDABILITY OF COMPACTA IN N-BOOKS: INTRINSIC AND EXTRINSIC PROPERTIES by Gail Adele Atneosen An n-book Bn is the union of n closed disks in E3 such that each pair of disks meets precisely on a single arc B on the boundary of each. The disks are called the leaves of Em, and the arc is called its back. The embeddability of compacta in n-books is investigated from two different vieWpoints. In Chapters II and III intrinsic prOperties are considered and in Chapters IV and V extrin- sic prOperties. Chapter II is concerned with the embeddability of certain continua in n-books. It is shown that all compact, connected 2-manifolds with non-void boundary embed in a 3-book. Examples are given of a one-dimensional, locally connected, locally plane continuum that embeds in a 3-book but not in any 2-manifold, of a one-dimensional locally connected continuum that does not embed in any n-book, and of a one-dimensional locally connected continuum that embeds in Bn but not in Bm for 2.3 m < n. In Chapter III the concept of a polyhedron tame in Bn is introduced, and those polyhedrons7tamély;embedded are characterized. Necessary and sufficient conditions are Gail Adele Atneosen given for a polygonal simple closed curve in a 3-book to span a 2-manifold in the 3-bOOk. The monotone Open union of Open n—bOOks is shown tO be an Open n-book. In Chapter IV extrinsic prOperties Of subsets Of n—bOOks in E3 are investigated. Necessary and sufficient conditions are given for a tOpological polyhedron in a tame n-bOOk to be tame in E3. It is shown that every tOpOlogical umbrella in a tame n—book is locally tame at its tangent point and that no disk pierced by an arc lies in an arbitrary 3-bOOk in E3. Next questions of cellularity are considered. The cellular hull of a subset A Of E3 is defined to be a cellular set E containing A such that no prOper cellular subset of B contains A. An are A has a cellular hull that lies in a tame 2-complex in E3 if and only if there is a Space homeomorphism h with the prOperty that the image of A under h lies in a tame 3-book. If A is a cellular arc whose set of wild points is non-empty and does not contain an arc and A lies in an arbitrary n-book in E3, then A has at most one wild point that is not con- tained in the back of the n-bOOk. In the last chapter tamely embedded cones over n-books in E4 are investigated. It is shown that no wild Cantor set lies in a tame cone over an n-bOOk in E4, and that every 1- or 2-cell or 1- or 2-sphere in a tamely ' embedded cone over a l- or 2-bOOk is tame in E4. Examples are given of wild 2- and 3-cells and 2-spheres in tamely embedded cones over n-books, n > 2, in E4. ON THE EMBEDDABILITY OF COMPACTA IN N-BOOKS: INTRINSIC AND EXTRINSIC PROPERTIES By Gail Adele Atneosen A THES IS Submitted to Michigan State University in partial fulfillment of the requirements for the degree of DOCTOR OF PHILOSOPHY Department Of Mathematics 1968 ACKNOWLEDGMENTS I would like to thank Professor Patrick Doyle, my thesis advisor, for being so very helpful, kind, and encouraging. I would like to thank Professor Harold Wolfe, who taught me my first mathematics, for inspiring me to learn more mathematics. And I would like to thank my husband, Dick, for his encouragement and sense of humor. ii CONTENTS Chapter Page_ I. INTRODUCTION . . . . . . . . . . . . . . 1 II. CONTINUA IN N-BOOKS . . . . . . . . . . 11 III. SOME PROPERTIES COMMON TO EUCLIDEAN SPACES AND N-BOOKS . . . . . . . . . . . 25 Iva SUBSETS OF N—BOOKS IN E3 . . . . . . . . 41 v. SUBSETS OF TAMELY EMBEDDED CONES OVER N-BOOKS IN E4. . . . . . . . . . . . . . 58 BIBLIOGRAPHY . . . . .'. . . . . . . . . . . . . . . 71 iii LIST OF FIGURES Figure Page '- 2.1 11 2.2 12 2-3 15 2.4 17 2.5 18 3.1 31 3.2 36 4.1 43 5.1 68 iv CHAPTER I INTRODUCTION An n-bOOk Bn is the union Of n closed disks in E3 such that each pair Of disks meets precisely on a single arc B on the boundary Of each. The disks are call- ed the leaves Of Bn and are denoted by Di’ 1 = l,...,n, and B is its back. In this paper we investigate the embeddability of compacta in n—books from two different vieWpOintS. In Chapters II and III intrinsic properties are consider- ed and in Chapters IV and V extrinsic properties. This investigation of n-books was initiated by P. H. Doyle in [14] when he extended an earlier result [13], and showed that if each Of the leaves Of an n—bOOk tOpologically embedded in E3 is tame then the n-bOOk is tame. C. A. Persinger continued the investigation of extrinsic properties Of subsets Of n-bOOks in [38, 39, 40]. The concept Of n—books has also arisen in a somewhat differ- ent context in the work Of A. H. COpeland, Jr. [10,11] where some results concerning intrinsic properties Of subsets Of n-books are found. Next we give a few comments on notation and some definitions necessary for the reading Of this paper. The following notation will be used. N II the set Of positive integers. En {x I x = (Xl’°"’xn) an n-tuple Of real + numbers, n e Z } . En is assumed to have the topology determined by the Euclidean metric an“ E: = {x I x = (xl,...,xn) Xn.2 O} . n-l s {x I dn(x,0) = 1} A homeomorphic image of S1 is called a Simple closed II 211232- Closed n-cell = {x 5 En | dn(x,O) g_l} . Open n-cell =I{x e En | dn(x,0) < l}.. By a disk will always be meant a space homeomorphic to a closed 2-cell and by an arc a space homeomorphic to a closed 1-cell. If A and B are topological Spaces, a homeomorphism Of A into B is called an embedding. By an n-dimensional manifold M is meant a separable metric space such that each point has a neigh- borhood whose closure is homeomorphic to a closed n-cell. The interior gf_M, Int M, consists Of those points which have neighborhoods homeomorphic to an Open n-cell; the boundary g£_M, Bd M, is defined to be M - Int M. Thus in discussing a disk D, it is clear what is meant by Ed D and Int D. If boundary or interior is: used in the usual topological sense then it will be denoted by Int A and X deA if A is a subset Of X. By the interior of an n-bOOk is meant the set LJE=1 Int DiIJ Int B. Next some terminology from combinatorial tOpology is given; the definitions are essentially those Of Zeeman [45]. By an n-Simplex Ah, 0 3.“: is meant the convex hull Of n+1 linearly independent points (the vertices) {XJ I J = O,...,nI in Ep, n g_p. By a r-face AT Of Ah’ denoted by AT < Ah, is meant the convex hull of r+l distinct points Of [X3 I J = O,...,n} . A simplicial complex, or complex, K Of Ep, p 2_l,is a finite collec- tion of Simplexes of Ep such that: (1) if A.e K, then all the faces Of A.are in K, and (2) if A1, A2 6 K then Allq A2 is a common face Of A1 and A2. L is called a subcomplex Of K if L is a simplicial com- plex and LCIK. If A1, A2 are Simplexes in Ep such that the union of their vertices forms a linearly independent set Of points in Ep, then A1 and A2 are Joinable. If A1 and A2 are Joinable, then the Join Of A1 and A2, denoted by A1*A2, is defined to be the simplex spanned by the union Of their vertices. The subcomplex Of K consisting Of all q-simplexes Of K, where q s m, is called the m—skeleton of K and is denoted by K(m). For A.e K the set st(A,K) = Io e K I A < o} is called the star 93 A in K. The underlying point set IKI Of a simplicial com- plex K is called a Euclidean polyhedron or polyhedron. Sometimes the phrase finite Euclidean polyhedron is used to emphasize the fact that we are only considering simplicial complexes consisting of finitely many Simplexes. By Ist(A,K)I is meant the space |__|{|o-| I {tr e Ist(A,K)I.. If P is a Euclidean polyhedron then a simplicial complex K such that IKI = P is called a triangulation Of P, and P is said to be the carrier Of K. If K is a simplicial complex and and h is a homeomorphism Of IKI onto Q, then the set Ih(|o|) | a e K} is said to be a curved triangulation Of Q. The m-Skeleton Of Q is the set {h(ICI) I 0 e K(m) I. A complex K' is called a subdivision of a complex K if IK'I = IKI and each Simplex of K' is contained in some simplex of K. The dimension Of a simplicial complex K is the largest integer n such that K contains an n-simplex. The carrier Of a 1-dimensional complex is called a graph, note that a graph is not necessarily connected. By an umbrella is meant the Euclidean polyhedron consisting Of a disk D and an arc 0 such that DFIG’=Ix} where x is an interior point Of D and an endpoint Of O. x is called the tangent point Of the umbrella and O the handle. Let I be the interval [O,l] 2. Theorem 1.4 is interesting for it states that 3 wild arcs can lie in very simple subspaces of E . In this connection, we note that the are A Of Example 1.1 Of Fox and Artin [22] is embeddable in a tame 3—bOOk in E3 in such a manner that the images of a set Of generators of W E3 — A) are also contained in the 3~bOOk. (The 1( fundamental group Of the complement Of A in E3, 1rl(E3 - A), is non-trivial.) Another Simple subspace of E3 that contains wild arcs is an infinite croquet game. By an infinite croquet_game is meant a flat disk D in E3 union countably many disjoint polygonal arcs {01 I i 6 2+} that intersect D only in their endpoints and such that the {01} converge to a point p in the interior of D. If A is any are that lies in a tame 3-bOOk in E3 and has a Single wild point, then it is easy to see that A is equivalent to an arc that lies in an infinite croquet game. A set C in En is said to be cellular if there exists a sequence Of tOpological closed n-cells + on {Ci | iezIsuch thatC CInt c1 and C=I—Ii=1 Ci. 1+1 This notion was defined in 1960 by M. Brown [8]. An arc may be wild and also be cellular as Example 1.2 of [22]. Wild points of cellular subsets Of 2-Spheres in E3 are considered by Loveland in [32]. McMillan in [35] has Obtained results about cellular subsets Of higher dimen- sional Space. A k-cell in En, k‘g n, is said tO be flatly embedded, or $193, if there is a Space homeomorphism Of n E onto itself mapping it onto a k-simplex. A (k-l)- sphere in En is said to be flatly embedded, or flat, if there is a space homeomorphism that maps it onto the boundary Of a k-Simplex in En. Thus a polygonal trefoil 3 knot in E is tame but not flat. An n-book is flatly 3 embedded in if each of its leaves is a Euclidean 2-Simplex. If M is a k—manifold with boundary contained in an n—manifold N, M is locally flat at‘a point p E Int M if there is a neighborhood U Of p in N such that (U, UfWM) is homeomorphic to the pair (En,Ek); M is locally flat a£_a_point E.£.§§.M if there is a neighborhood U Of p in N such that (U, UIIM) is homeomorphic to (En, EE). These notions have been recently studied by Lacher [31] and Kirby [27]. It is known that a locally flat k-cell IO in En is flat in En [31]. One other theorem that is used and Should be mentioned is the Brouwer Theorem on the invariance of domain. Theorem 1.5 If U and V are homeomorphic subsets Of Sr1 and U is Open in Sn, then V is Open in Sn. MMNERII CONTINUA IN N-BOOKS This chapter is concerned with intrinsic pro- perties of n-bOOks rather than, say, positional proper- ties Of n-bOOkS in Euclidean space. In particular the embeddability Of certain continua, that is compact connected sets, in n-bOOks is considered. Theorem 2.1 All compact, connected 2-manifolds with non-void boundary embed in a 3-bOOk. Proof. Figure 2.1 illustrates what is meant by a disk with (a) a Single bridge, (b) a twisted bridge, (c) a double bridge. Figure 2.1 Using scissors—and-paste techniques (see Chapter 1 Of [34]), it can be shown that all compact, connected 2-manifolds With 11 l2 non—void boundary are homeomorphic to: (i) a disk with r _>_ 0 single bridges and h > 0 double bridges, or (ii) a disk with r _>_ 0 single bridges and q 2 O twisted bridges. Thus to prove that all 2-manifolds with non-void boundary can be embedded in a 3-bOOk, it suffices to show that 2-manifolds Of type (i) or (ii) can be embedded in B3. Figure 2.2 indicates that this is, in fact, the case. Figure 2.2 (a) consists of a disk with three double bridges and five single bridges and (b) consists Of a disk with two twisted bridges and four single bridges. Figure 2.2 Corollary 2.2 All compact, connected 2-manifolds With non-void boundary can be embedded in a 3-bOOk so as to carry a subcomplex Of some triangulation Of the 3-bOOk. 13 Corollary 2.3 1. Every prOper compacthubset70f a compact connected 2-manifold embeds in a 3-bOOk. Proof. If C is a prOper compact subset of a compact, connected 2-manifold M, then M - C is a non-empty Open set. Hence there is a closed disk D contained in M that doesrxnzintersect C. M - Int D is a compact, connected 2-manifold with non-void boundary containing C and can be embedded in a 3—book by Theorem 2.1. Corollary 2.4 All graphs embed in a 3-book. Proof. Let G be a graph and let n denote the number of vertices of G. Select n distinct points on a 2-sphere and accomodate each arc Joining two vertices by a "handle" appropriately attached to the 2-Sphere. Thus G can be embedded in a 2-sphere with handles which is a 2-manifold without boundary. Hence by Corollary 2.3 G embeds in a 3-bOOk. Corollary 2.5 All compact, connected 2—manifolds embed in the triple cone over three points. 3 Proof. B is the double cone over three points, so it suffices to show that all 2-manifOldS embed in C(B3). Let M be a compact, connected 2-manifold; if Bd M # C then by Theorem 2.1 M embeds in 133:0(133). If Bd M = D, 14 then M - Int D embeds in B3 where D is a disk in M. But C(B3) contains the cone over the boundary Of D and (M - Int D)LJC(Bd D) is homeomorphic to M. Hence M embeds in C(B3). Corollary2.6 All compact, connected 2-manifolds embed in a 2-dimensional continuum in E4 that fails tO be locally polyhedral at only one point. Proof. There are countably many distinct 2—manifolds with non—void boundary. These can be polyhedrally embedd- ed in B3, by Corollary 2.2, so as to converge to a point p on the back Of B3. Consider B3<: E3 x O, a 3-dimension- a1 hyperplane in B“, such that these countably many distinct 2-manifolds are embedded in B3 as described above. In E4 form the cone over the boundary components of each of these 2-manifolds in such a manner that the cones are disjoint and the vertices Of the cones converge to p. Then B3 union these cones is a 2-dimensional continuum that is locally polyhedral except at p and con- tains all 2—manifolds with or without boundary. In Corollary 2.3 it was noted that every prOper compact subset Of a compact, connected 2-manifold embeds in B3. Thus the question may be asked if there exists a locally plane, locally connected, one-dimensional 15 continuum that embeds in a 3-bOOk but not in any 2-manifold. By locally plane is meant that each-point has a neighbor- hood that embeds in E2. An example due to Borsuk [7] is used to answer this question in the affirmative. His example utilizes one of Kuratowski's primitive skew curves [29] which is the union Of all the edges Of a tetrahedron plus a segment joining two points lying in the interiors Of two Opposite edges. This graph is not embeddable in the plane. Example 2.7 [7] A locally plane, locally connected one-dimensional continuum that is not embeddable in any 2-manifold. Let a b , and d be the vertices of a tetrahedron 1’ 1’ Cl 1 in E3, cn denote the point dividing the segment alcl in the ratio 1: n-1, and dn the point dividing 5131 in the same ratio,.for n e 2+. See Figure 2.3 below. Figure 2.3 16 Let Xn = alElUalcnualdnUEIcnUEldnUcndnUcn+15[n+1 and ' a X n=l Xn . Note that X is not homeomorphic to any subset Of a 2-manifold M. For if g' were such an embeddinthhen there would exist a disk D in M such that g'(§:FI) 0 there exists an n(e) such that for all n > n(e) the diameters of the sets g(L(n,i)) are less than e. let Y = g(Y')LIE:dI, then Y is a locally plane, lOcally connected continuum that contains X and is hence not embeddable in any surface. a; b. I I Ch 7 . .6; a' I 13' :bd; L(|,I) d1 . d? Figure 2.4 Theorem 2.8 There exists a locally plane, locally connected one-dimensional continuum that embeds in a 3-bOOk but not in any 2-manifold. 18 Proof. Example 2.7 is a locally plane, locally connect- ed one-dimensional continuum that embeds in no 2-manifold. To show that it is embeddable in B3 consider Figure 2.5. (‘4 (b) Figure 2.5 19 Figure 2.5 (a) shows the embedding Of xlu xzux3 - —cI+d4. In general,if LJn_l X1 — Cn+1dn+l has been embedded in B3, 3 then xn+ 'c-—T“‘m2 n+2 is embedded in B as in Figure 2.5 (b). Thus continuing in this manner it is clear that X embeds in B3 and also that Y embeds in B3. However, all one—dimensional continua do not necessarily embed in Bn, for some positive integer n, as the argument of the following theorem illustrates. Theorem 2.9 There exists a one-dimensional, locally connected continuum that cannot be embedded in an n-bOOk for any positive integer n. Proof. In order to define such a continuum let: K = the graph which is the union of all the edges Of a tetrahedron plus a segment joining two points lying in the interiors Of two Opposite edges, J(1) =I(x,o,o) eE3 | 05x31}, K(l,2) = a graph in E homeomorphic to K whose diameter is less than 1/2 and such that K(l,2) intersects J(1) only in the point (l/2,0,0), and I J(2) = J(l)UK(1,2). Assume that J(n) has been defined and let: J(n+1) = J(n)\J LJ3=1 K(m,n+l). 3 K(m,n+l) is a graph in E homeomorphic to K such that: 2O (1) K(m,n+l){]J(n) = (m/(n+l),0,0) m = l,...,n, (ii) K(m,n+l)tl K(m',n+l) = S if m g m', and (iii) the diameter Of K(m,n+l) is less than 1/2n+1 for m = l,...,n. n=l J(n) with the relative topology of E3. Then Let J = J is compact, connected, and locally connected by con- struction; furthermore, J is one-dimensional Since it is the countable union of closed one—dimensional sets. Next it will be shown that J does not embed in any n-bOOk. Suppose this is not the case, andlthere is an embedding h: J a»B”, then h(J(l))CZB, the back or E“. For if not, then there exists an x e J(l) such that d(h(x), B) = e > 0. Since J is compact, h is a uniform homeomorphism and there exists a 5 > C such that if d(u,v) < 6 then d(h(u),h(v)) < E/3. Thus there exists (m/p,0,0) e J(l) such that d(K(m,p),x) < 6 and hence h(K(m,p))<: Di’ a leaf Of Bn. But this contradicts the fact that K cannot be embedded in the plane so h(J(l))CIB. Let 2 be an interior point Of the interval J(1), and N a neighborhood Of h(z) in Bn such that N - h(J(l)) consists Of n components CiCZD i = l,...,n. Then there i 3 exists integers m and p such that h(K(m,p»CZN. Since h(J(l)) separates N, h(K(m,p) - (m/p,0,0)) CICi for some i, but this again contradicts the fact that K cannot be embedded in the plane. Hence there does not exist an embeddingof J into any n-bOOk. 21 Next it is shown that n~bOOks can be distinguish- ed by the one-dimensional continua that embed in them. A locally connected, one—dimensional continuum A(n) is constructed with the property that if X is a compact set of dimensiOn less than or equal to one in the interior Of an n-bOOthhen XC:A'(n) where A'(n) is homeomorphic to A(n). Theorem 2.10 There exists a locally connected, one- dimensional continuum that embeds in En but not in Bm for 2.3 m < n. Proof. Let {Ei I i 6 2+} be a sequence Of mutually disjoint disks in Br1 that do not intersect the boundary of any Of the leaves of En, such that LJ:=1 E is dense i in Em, and such that the diameters Of theuEi-converge to zero. Define A(n) = Bn - I Int E then A(n) is a i=1 i’ locally connected, one-dimensional continuum that embeds in Bn. Note that if D1 is a leaf Of Br1 then DifIA(n) is homeomorphic to Sierpinski's universal plane curve [42, 44]. AsSume that there exists an embedding h mapping A(n) into Bm for 2.3 m < n and reach a contradiction. By definition of A(n), the back Of Bn is contained in A(n); denote this set by B' in A(n). Then h(B')CIB, the back of Bm. For if not there exists a z e B' such that d(z,B) = 3e > 0. Since h is a uniform homeomorphism there exists a 5 > 0 such that if d(u,v) < DJthen d(h(u),h(v)) < e. 22 Note that A(n)flDi is arcwise connected. Let c(i) , i = l,...,n, be an arc in A(n)f‘I(Di - {z}) such that: (1) d(c(i),z) < o, (2) c(i) intersects B' only in its endpoints a and b, which are the same for all i, and (3) if 5? denotes the line segment in B' joining a to b then 2 e 53. Let a(i) denote an arc in A(n) joining a point on c(i) - B' to 2 such that a(i)fIB' = {z}, (thus the diameter Of a(i) is less than 5). Then h(l 2:1(c(i)Lja(i))LjaE) is a graph that is entirely contained in a leaf Of Bm. This graph contains one of Kuratowski's primitive skew curves which is not embedd— able in the plane [29], thus h(B')CIB. Next,using the fact that h(B')CIB it will be shown that A(n) cannot be embedded in Bm. Let z e B' such that d(h(z), IS- h(B')) is greater than 36 > 0. Choose c(i) and a(i) as before, i = l,...,n. Then since m < n there exists 1 % j such that h(c(i))IJ h(c(j)) is contained in one leaf, say Dk’ Of Bm and [h(c(i))umcunms ={h(aiu{h 2 then the back Of Bm is contained in the back of En. The next corollary states that in some sense A(n) is the "universal curve" for B“. Corollary 2.12 If X is a compact set in the interior Of an n-book Er1 such that dim x _<_ 1, then x c A'(n) where A?(n) is homeomorphic to A(n). Proof. Let X = (Xlei)IJB; then X is a closed 1 i subset Of D1 for each i = l,...,n and dim Xi‘g 1. SO there exists in Int Di — X1’ 1 = l,...,n, a sequence {E(i,k) I k'e Z+I Of mutually disjoint disks such that i=1 E(i,k) is dense in Di and the diameters Of the N E(i,k) converge to zero. Let G1 = Di - k=l Int E(i,k), then by construction G1C: X1. But_G is an S-curve.[44], i that is a plane, locally connected, one-dimensional continuum S such that the boundary Of each complementary domain Of S is a simple closed curve and no two of these complementary domain boundaries intersect. Hence by Theorem 3 Of [44] there exist homeomorphisms hi’ i = l,...,n, which are the identity map on Ed D i and map n — l Gi onto A(n)FIDi. If l—Ji=l Gi — A'(n) then X CIA (n). 24. Furthermore,the map h: A'(n)-a A(n) defined by h(x) = hi(x) for x 6 G1 is a homeomorphism Of A'(n) onto A(n). CHAPTER III SOME PROPERTIES COMMON TO EUCLIDEAN SPACES AND N-BOOKS In this chapter various properties Of Euclidean spaces are generalized tO n—books. In particular the notion Of tameness, polygonal simple closed curves Span- ning 2-manifolds,and monotone unions Of Open n-bOOks are considered. A topological polyhedron in Euclidean n-dimen- sional space is said to be tamely embedded if there is a homeomorphism of En onto itself which transforms the embedded polyhedron into a Euclidean polyhedron. It is this notion which is generalized to n-books. For the remainder Of this chapter Bn will be considered embedded in E3 in such a manner that each leaf is a Euclidean 2-simp1ex Thus Bn inherits a linear structure from E3 and the notion of tameness in Bn can be introduced. A set ACIBn is said to be a pplyhedron ip_B: if A,when con- sidered embedded inlEi is a Euclidean polyhedron. A topological polyhedron embedded in the interior Of an n-bOOk is said to be Eamg.12_BE_iff there is a homeomor- phism Of Bn onto itself which transforms the embedded polyhedron into a polyhedron in Bn. Lemma 3.1 If a Euclidean polyhedron is embedded in Bn in such a manner that the image Of the l-skeleton is the 25 26 carrier Of a subcomplex relative tO some triangulation of B“, then the image of the polyhedron is also the carrier Of a subcomplex relative to this triangulation. Proof. Let h be an embedding of a Euclidean polyhedron IPI, with triangulation P, into Br1 such that h(IP(l)I) is the carrier Of a subcomplex Of the triangulation K of B“. Let J = {0 e K I G'< Cl ,, ” Int IciIr\h(IPI) % C', and Ole K} then J is a subcomplex Of K. It will be shown that IJI = h(IPI). By construction h(IPI)CZIJI. TO prove that IJICTh(IPI) assume not and reach a contradiction. Suppose there exists x e IJI - h(IPI), then there are three cases to consider. Case~ld x e IOI where c is a face, not necess- arily proper, of a 2-Simp1ex A.e J. By definition Of J, there is a y 6 Int IAIFIh(IPI). SO there is an arc t joining x to y such that t -[x‘ICInt IAI. Let 2 be the first point Of tth(IPI) in the direction from x. Then 2 must belong to the image Of the l-skeleton Of P, since it does not lie in the interior Of an Open 2-cell in h(IPI). But 2 6 Int IAI, which contradicts the hypothesis that the image Of the l-skeleton is the carrier Of a subcomplex of K. Case 2. x.e IOI where c'is a l-simplex in J that is not the face Of any 2-simplex in J. But this implies there is a y e h(IPI)f\Int IOI such that y must be the 27 carrier of a O-simplex of K if h(IP(l)I) is the carrier Of a subcomplex of K. Thus this case cannot occur. Case 3. x.e IOI where O is a O-simplex in J and O is not the face of any other simplex in J. But by the definition Of J then 0 E J SO this case cannot occur. The above three cases exhaust all possibilities so IJICIh(IPI) and the lemma follows. The following two lemmas are proved elsewhere but are needed in several arguments SO are stated here. Lemma 3.2 The intersection of two Euclidean polyhedrons is a Euclidean polyhedron. Proof. Corollary 2 to Lemma 1, Chapter 1 of [45]. Lemma 3.3 If IKIZDILI then there exists a subdivision K' Of K and L' Of L such that L' is a subcomplex Of K'. Proof. Lemma 4, Chapter 1 Of [45]. The proof of the following lemma depends upon the plane Schoenflies theorem and is similar to one given by Doyle in [16]. Lemma 3.4 Let T be a finite graph, not necessarily connected, and h an embedding Of T into the closed unit 28 square D in E2 such that h(T)r\Bd D is a finite Euclidean polyhedron. Then there is a homeomorphism g Of D onto itself such that g(h(T)) is the union Of finitely many straight line segments, and g restricted to the boundary Of K is the identity map. Proof. Since h(T)rIBd D is a finite Euclidean poly- hedron, there are only finitely many points in Bd D that are limit points of h(T)rIInt D. Let {x1 I i = l,...,n} denote these points plus the images Of the vertices Of T. Let N(xi) be . closed symmetric neighborhoods of xi, 1 = l,...,n, in D such that any two are disjoint. Let O(i,j), i = l,...,n and j = l,...,k(i), denote the finite- ly many arcs in h(T)F\N(xi) such that: (l) c(i, j) has endpoints Xi and yi,j’ 34,36 Bd N(xi), and (2) O( (,i j)- {yi J} is contained in IntDN(xi).. Let,o‘(i,j) be a straight line segment joining [Xi to yi,jh and j = l,...,k(i). Then by the plane Schoenflies theorem for i = l,...,n there are homeomorphisms g1, i = l,...,n, such that: (i) giID — Int N(x1) = the identity map, and (ii) s1(00(i 3)) = (0’(i,J) J = 1, .-,k(i)- Then h(T) - _1LJkg i,j) is the union of finitely i’ i = l,...,m. Let Ui be a closed neighbor- homeomorphic to a closed disk such that many arcs t hood of ti Uith(T) is an arc.‘ Let g" = g1...gn, then g"(Ui)rIg"(h(T) - Int ti) consists Of two polygonal arcs, 29 say a1 and b , which were Obtained by the above application 1 Of the plane Schoenflies theorem. Let t' C Int g"(Ui) i be a polygonal arc joining the endpoints.of g"(ti) such that ti intersects g"(h(T)) only in its endpoints. Then by the plane Schoenflies theorem there are homeomorphisms gI, i = l,...,m, such that: (i) gII(D - Int g"(Ui))U(a1ubi) is the identity map, and I " _ l (11) si(s (t1)) — ti- Then g = gi...gég" is the.desired homeomorphism. That is, g | Bd D = the identity map and gh(T) consists of finitely many straight line segments. The following theorem is a characterization Of those polyhedrons which are tamely embedded in Bn. Theorem 3.5 An embedding Of a polyhedron in Int Bn is tame in Bn iff the polyhedron has a.curved triangulation such that the image of the l-skeleton intersected with the back Of Bn is a polyhedron in Bn. Proof. Let P be a polyhedron and h: P SIB“ an embedding. If P is tamely embedded in B“, then by definition there is a homeomorphism g of En onto itself such that gh(P) is a polyhedron in Bn. Let K be any triangulation of En and L any triangulation Of gh(P), (then K and L are 3O simplicial complexes in E3). By Lemma 3.3 there are subdivisions L' Of L and K' Of K such that L' is a sub— complex Of K'. Then (gh)-l: IL'I-a P is a curved triangu- lation Of P. The back Of Bn is a subcomplex of every triangulation OfIBQ so by Lemma 3.2 IL'(1)IfIB is a polyhedron in Bn. Conversely, suppose that P has a curved triangu- lation such that the image Of the l-skeleton, denoted by R, intersects the back of Bn in a finite polyhedron. Then RflD1 is the tOpological image of a finite graph and Rled D is a finite Euclidean polyhedron, i = l,...,n. 1 Hence by Lemma 3.4 there are homeomorphisms g1: Di-a Di’ i = 1,o--,n, which are the identity map on Bd D1 and such that gi(R) consists Of finitely many points and straight line segments (when considered in E3). By Lemma 3.3 there is a triangulation, say K', Of Bn such that g1...gn(R) is the carrier Of a subcomplex Of K'. By Lemma 3.1 g1...gnh(P) is a Euclidean polyhedron,and h is a tame embedding Of P into Bn. Every polygonal simple closed curve in E3 spans an orientable surface in E3, that is,there exists a compact, connected orientable 2-manifold in E3 such that the mani- fold boundary is precisely the simple closed curve. One method Of Obtaining such an orientable surface is given by R. H. Fox in [21]. By a polygonal simple closed curve 31 in B3 is meant a homeomorphic image of S1 that is a polyhedron in B3. Theorem 3.8 characterizes those poly— 3 gonal simple closed curves in B that Span compact, con- nected 2-manifolds in B3. All such curves do not necessar— ily span a surface as is indicated in Figure 3.1 (a). Even if a polygonal simple closed curve does span a 2—manifold, as in Figure 3.1 (b), the 2-manifold need not be orientable. In this case it is a MObius band. Figure 3.1 One approach tO the desired characterization is to consider the simplicial homology Of B3 with coeffi- cients in 22, the group of integers modulO'tWO. TFOP an exposition Of simplicial homology theory and related termi- nology see [Chapter 6, 25]. By a t-dimensional chain on a simplicial complex K with coefficients in Z2 is meant a function m on the t-simplexes Of K with values in 22. 32 It facilitates notationfto let m also denote the subcomplex Of.K whichzis the simplicial closurefof:alLLtASImplexes of K on.whichfim has non—zero.valuea‘ The geometric realization of this subcomplex will also be denoted by m rather than ImI. NO confusion Should arise since Z2 coefficients are particularly well suited to geometric interpretation, and this notation will be used only in relation to chains. Lemma 3.6 If C is a polygonal simple closed curve 3 3 in B then there exists a mod 2 2-chain, me, on B such that Omc = C and mC has the properties listed below. (1) CCch and m0 is compact and connected. (ii) If x e mC 7 C then x has a neighborhood in mC homeomorphic to an Open disk. (iii) If x e C then x has a neighborhood in me whose closure is homeomorphic to either: (1) a closed disk, (2) B3. ' (3) the union Of two closed disks whose intersection is precisely {k} , x being an interior point Of one disk and a boundary point Of the other disk, or (4) the union Of two closed disks which intersect along an arc O with endpoints x and p such that O is in the boundary 33 Of one disk and O — {p} is in the interior Of the other disk. 3 has a triangulation K such that Proof. By Lemma 3.3,B C is contained in the carrier Of the l-skeleton of K. Consider the simplicial homology Of K with coefficients in Z , and let 0 denote the l-chain that has value 1 on all 2 l l-simplexes of K that are contained in C and 0 on all other l—simplexes Of K. Since B3 is contractible, H1(B3,Z ), 3 the first simplicial homology group Of B with coefficients in Z is trivial. Hence there is a 2-chain mC on K such 2, that amc = o1. It will now be shown that mC has the prop- erties stated in the lemma. (1) Since 01 = C, Bmc = C and chDC. mC is compact because it is the point set union Of finitely many compact 2-simplexes. TO prove that mC is connected assume not and reach a contradiction. Suppose that mC can be expressed as the disjoint union Of two non—empty closed sets, Al and A2. Since C is connected it may be assumed that, say, C(ZAl. Let mé be the 2-chain on K that has non—zero value only on those 2-simplexes Of K contained in A2. Since Omc = C, each l-simplex Of K in A2 is the face of an even number of 2—simplexes in A2. Thus Omé = 0, but H (B3, Z2) = 0, since B3 is contractible, 2 hence there must be a 3-chain on K whose boundary is mé. Since the only 3-chain is the zero 3-chain this implies 34 A is empty and hence m0 is connected. 2 (ii) Let x e mC - C and Show that x has a neighborhood in mc homeomorphic to an Open disk. There are three possible cases to consider. Case 1, x is contained in the interior Of a 2-simplex of me. Then x clearly has a neighborhood in mC homeomorphic to an Open disk. Case 2. x is contained in the interior of a l-simplex in me. Then this l-simplex does not lie in C and hence must be the face Of an even number Of 2-simplexes in me. By Corollary 2.11 if Bm is embedded in B“ then m _<_n, so this 1-simp1ex is a face Of precisely two 2—simplexes in mc - Hence x has a neighborhood in mC homeomorphic to an Open disk. Case 3. lastly, consider the case when x is a O-simplex in me. Then Ist(x,mc)| is the union Of n closed disks which contain x in their interior, Since every l-simplex in mC having x as a face must be the face of precisely two 2-Simplexes in me. SO tO Show that x has a neighborhood in mC homeomorphic to an Open disk, it suffices to prove that n = 1. If x e B3 - B then n = 1. For if n 2 2 then Ist(x,mc)I contains a tOpological umbrella which is embeddable in the plane. If x e B and n‘Z 2,there are two closed disks, say E1 and E2, in Ist(x,mc)| such that ElflE2 = {x3 . Lemma 4.1, of the next chapter, implies 35 that x has a neighborhood Ni in E1, 3 Open 2-cell and is contained in precisely two leaves of B . i = 1,2, which is an But this fact contradicts the disjointness of El - {k} and E2 ‘ {XI (iii) Let x e C, and again consider the various cases. Case 1. If x is not a vertex of me, then x is contained in the interior Of a l-simplex of mo that lies in C. This l-simplex is the face Of an Odd number Of 2-simp1exes in me. By Corollary 2.11 it is the face of one or three 2-simplexes and hence x has a neighborhood in mC homeomorphic to (l) or (2) of part (iii) Of the statement Of this lemma. Case 2. If x is a vertex there are exactly two l-Simplexes, t1 and t2, in mC which lie in C and have x as a face. Thus st(x,mC) contains two l-simplexes that are a face of one or three 2—simp1exes in st(x,mc) and all the other 1-simp1exes are the face of two 2-simplexes in st(x,mc). Using the fact that two closed disks which intersect at a single point interior to each cannot be embedded in B3, as shown above, one obtains the following results. If t1 and t2 are both the face Of only one 2-simplex in st(x,mc) then x has a closed neighborhood homeomorphic tO (1) or (3). If t is a face Of only one 1 2-simplex in st(x,mc) and t is a face Of three 2-simplexes 2 in st(x,mc),then x has a closed neighborhood homeomorphic 36 to (4). If both t1 and t2 are the face Of three 2-simplexes in st(x,mc),then x has a closed neighborhood homeomorphic to (2). This exhausts all the possibilities and the lemma follows. Figure 3.2 shows the four possible closed (1) (2) i7 (3) ( neighborhoods. X 4:- V Figure 3.2 The proof of the next lemma is very similar to that of Lemma 3.1 and SO is omitted. Since the polyhedron being considered is a 2-manifold weaker conditions can be imposed on its l-skeleton. Lemma 3.7 Let M be a compact, connected 2-manifold with non-void boundary, C, embedded in Bn. If Bn has a triangulation K such that C is contained in the carrier Of the l-skeleton Of K, then M is the carrier of-a sub- complex Of K. 37 Theorem 3.8 A polygonal simple closed curve in B3 bounds a compact, connected 2-manifold in B3 iff it is a mod 2 cycle which is the boundary of a mod 2 2-chain whose geometric realization contains no umbrellas. Proof. By Lemma 3.3, if C is a polygonal simple closed 3 curve in B there is a triangulation K Of B3 such that CCZ|K(1)I. By Lemma 3.7 the 2-manifold M which C bounds is the carrier Of a subcomplex of K. Let 22 be the mod 2 2-chain which has non-zero value only on those 2-simplexes of K that lie in M. Then Bz2 = C and the geometric real— ization Of z2 is M and hence contains no umbrellas. Conversely, assume that C is a polygonal simple 3 that is a mod 2 cycle with respect to 3 closed curve in B some triangulation K Of B , and that there is a mod 2 2-chain z such that 822 = C. By Lemma 3.6 there is a 2 mod 2 2-chain me, with the prOperties stated there, such that Omc = C. Hence a(mC - z = 0 which implies, since 2) there is only the zero 3-chain on B3, that mC = Z2. Thus by hypothesis mC contains no umbrellas,which implies that points Of 0 do not have neighborhoods in mC Of type (2), (3), or (4) Of part (iii) Of Lemma 3.6. Hence every point of mo has a closed neighborhood homeomorphic to a closed disk, and C bounds mC which is a compact, connected 2—manifold in B3. 38 Corollapy 3.9 A polygonal simple closed curve spans a disk in B3 iff it is a mod 2 cycle which is the boundary Of a mod 2 2-chain whose geometric realization contains no umbrellas and every simple closed curve in the interior of this 2-complex separates it. Proof. A compact, connected 2-manifold M with non- void boundary is a disk iff every simple closed curve in the interior Of M separates M. A topological space Y is said to be the Open m monotone union pf‘a space H.if Y = l—Ji=l U(i%,and U(i) is open in Y, U(i) is homeomorphic to U for all i, and U(i)CIU(i+1). In [9] Morton Brown proved that the Open monotone union Of Open n-cells is an Open n-cell. Theorem 3.10 states that the same kind of result is true for n-books. By an Open n-bOOk is meant a space homeomorphic to Int En which was defined as l—J?=1 Int DiIJ Int B. The space A x [0,») with A x 0 identified to a point v is called the Open cone, OC(A), over A, If X is a topological Space, a point x e X is said to have an Open cone neighborhood U if there is a homeomorphism f Of some OC(A) onto the Open set U Of X such that f(v) = x. Theorem 3 Of [30] states that if U1CIU2CZ... is a se- quence of Open cone neighborhoods Of x in a locally compact Hausdorff Space,then U = :=1 U1 is also an 39 Open cone neighborhood Of x homeomorphic to each U1. We will use this theorem in the proof of the following theorem. Theorem 3.10 If a tOp61ogical Space Y is the Open monotone union of Open n-bOOks then Y is an Open n-bOOk. Proof. let ‘{U1 I i e 2+} be a sequence Of Open 1+1 1 n-bOOks such that UiC U , and U1 is Open in LJ:=1 U = Y. Next it is established that Y is a locally compact Hausdorff space. Let x and y be two distinct points Of Y, then there exists an integer n such that x, y e Un. Un is a Hausdorff space so there exist disjoint neighbor- hoods, Vx and Vy, Of x and y, reSpectively, such that Vi and Vy are Open in Un. Since Un is Open in Y, Vi and Vy are Open in Y, and Y is a Hausdorff space. Y is also locally compact for let x e Y,then x e Un for some n 6 2+. x has a neighborhood V in Un such that the closure of V in Un is a compact set, denoted by ClUn V. Then Un Open in Y implies V is Open in Y; since Y is a Hausdorff space ClUnV is also closed in Y. SO V is a neighborhood of x in Y whose closure in Y is.compact. Let A be the suspension Of n distinct points, then OC(A) is an Open n-bOOk. If x is a point on the back Of an Open n¥bOOk U1, one readily sees that U1 is an Open cone neighborhood Of x. By an application Of Corollary 2.11, 1+1 1 since UiCIU , the Open back Of U must be contained in 40 the open back Of Ui+1. Thus choose a point x in the back Of U1, by these remarks x is contained in the back Of i l 2 U for i e Z+. SO U CLU (I... is a sequence Of Open cone neighborhoods of x in a locally compact Hausdorff space; by Theorem 3 Of [30],Y is homeomorphic to an Open n-book. CHAPTER IV SUBSETS 0F N-BOOKS IN E3 This chapter is concerned with extrinsic prOperties Of compacta in n-bOOks, that is positional properties Of subsets of n-bOOkS embedded in E3. Euclid- ean polyhedrons tOpologically embedded in tame n-bOOks are investigated, and a characterization is given of those polyhedrons tame in E3 by considering where they can fail to be locally tame. Next, questions concerning cellularity and n-bOOks are examined. The first lemma is, however, only concerned with embeddings into n—books; the result will be useful in characterizing wild points of polyhedrons embedded in n-books in E3, Lemma 4.1 Let h be an embedding Of a disk D into B? and let x be an interior point Of D. Then x has a closed 2-ce11 neighborhood U in D such that h(U) is contained in the union of two leaves Of Bn. Proof. There are two cases to consider. Case 1, h(x) A B, the back of B“. Then there is a neighborhood V Of x such that h(V) is entirely contained in the interior of some leaf of En. By invariance of domain, h(V) is Open in this leaf, hence there is an e > 0 41 42 such that the symmetric neighborhood W c h(V). Then U = h_l(S:T§T) is the desired neighborhood Of x. Case 2. h(x) e B, the back of En. Suppose there does not exist a neighborhood V Of x in D such that h(V) is contained in the union Of two leaves Of Bn. Then there is a sequence of points ‘ka I k e Z+I converging to h(x) such that yk e B - h(D). Since h(D)¢fB, there is also a sequence of points {xk I k e Z+} converging to h(x) such that x e (Bn - B)f\h(D). Let a be an arc in Bn k k joining xk tO yk such that akFIB = {y£I.c Moreover, we may assume that the.diameter Of ak~<.l/2Kst“S¢hcéjh(D)FFak is a compact set there is a first element zk Of ak, in the direction from yk to Xk’ such that 2k 6 h(D). Furthermore, Since yk f h(D), zk t B. SO zk e h(Bd D) for otherwise an umbrella could be embedded in the plane. Because Of the manner in which the a were chosen, the sequence k {Zk I k 6 2+} converges to h(x). But this implies {h-1(zk) I k e ZII is a sequence of points on the bound- ary Of D converging to an interior point x of D which is a contradiction. Thus there is a neighborhood V Of x such that h(V) is contained in the union Of two leaves Of Bn. Then using invariance Of domain and preceding as in Case 1, a neighborhood U is Obtained with the desired prOperties. Lemma 4.1 is not necessarily true for x 6 Ed D as is indicated in Figure 4.1.” ~5 unun; 43 Figure 4.1 Theorem 4.2 Let P be a Euclidean polyhedron embedded in a tame n—bOOk, and let Q be the set Of points Of P that do not have Open 2-cell neighborhoods in P. Then the set Of points at which P fails to be locally tame is contained in QFIB and is a compact, totally disconnected set. Proof. Since Br1 is tame, we may assume that Br1 has planar leaves. If x e P - Q then x lies in the interior Of a tOpological disk D in P. By Lemma 4.1 we may assume that D lies in the union Of two leaves Of Bn. Let N be a 3 suchlthat closed polyhedral neighborhood of x in E erP CID. Then erP is a Euclidean polyhedron and P is locally tame at x. If x 6 Qt“ B then there is a closed polyhedral 44 neighborhood V Of x in Br1 such that V is homeomorphic to a closed disk and such that Q intersects the boundary of V in a finite Euclidean polyhedron. Then Qr1V is the homeomorphic image Of a finite graph, SO by Lemma 3.4 there is a homeomorphism g mapping V onto V which is fix- ed On the boundary Of V and such that g(Vle) is the union Of finitely many straight line segments and points. This homeomorphism can be extended to a closed polyhedral neighborhood N of x in E3 such that NTWP = V and gI Bd N is the identity map. By Lemma 3.1 g(NflP) is a Euclidean polyhedron . Hence P is locally tame at x. From the definition Of local tameness it follows that the set Of points at which P fails to be locally tame is closed and hence compact. Since the back Of Bn is tame, the set Of wild pOintS Of P is'alsovtotally disconnected. Corollary 4.3 If A is an arc in a tame n-bOOk, then A = ELJT where T is the countable union Of tame arcs and E is a compact, totally disconnected set contained in the back of the n-book. Proof. By Theorem 4.2 the set Of points, E, where A fails to be locally tame is compact and totally disconnected and contained in the back Of the n-bOOk. A - E is an Open subset Of A and hence can be expressed as the countable union Of Open arcs. These Open arcs are locally tame and have tame closures. 45 The set of wild points Of an arc in a tame n—bOOk may be uncountable as the next example shows. By an almost tame arc is meant an arc.such that every point lies on a tame subarc of the original arc [l8].‘ Example 4.4 An example of a cellular arc with uncount- ably many wild points that is not almost tame but is embeddable in a tame 3-bOOk in E3. Let B3 be a 3-bOOk in E3 with planar leaves such that the back B Of the 3-bOOk is the unit interval [0,1] 3 on the x-axis of E . Let {61 I i 6 2+} be the sequence Of Open intervals deleted from the unit interval to Obtain the ‘ i=1 51' Replace each closed interval 5: CH?EL i e Z+, with a [23] embedded in B3 usual Cantor ternary set, and let 0 = B Wilder arc J so that the endpoints i of J concide with the endpoints Of 5—, the diameters Of i the J tend to zero, and Jir‘Jk = C for 1 % k. Then i _ w 3 A —- LJi=l JiLJ C is an arc in E . If x e C, then every neighborhood of x in E3 contains a wild arc Ji for suffi- ciently large 1; hence A fails to be locally tame on C plus the set of points where LJ:=1 J fails to be locally 1 tame. If x e C but is not an endpoint of E; for any i, then x does not lie on a tame subarc Of A. Thus, A fails tO be locally tame at uncountably many points and is not almost tame. 46 TO see that A is in fact cellular we use a ./ definition and theorem due to Doyle [15]. If A is an arc in S? we say that A is peshrinkable if A has an endpoint q and in each Open set U containing q in Sn, there is a closed n-cell FCZU such that q lies in Int F while Bd F meets A in exactly one point. If A is an arc in SD such that for each subarc A' Of A, A' is p-shrinkable, then every arc in A is cellular [15]. Since Wilder arcs were inserted, the constructed arc satisfies the necessary conditions.and is therefore cellular. Of course, there are wild arcs with uncountably many wild points in tame 3-bOOks that are not cellular. One example could be Obtained by inserting Example 1.1 Of [22] instead Of the Wilder arcs in the above construc- tion. The next lemma is due to Persinger [40] and is used in the proof Of Theorem 4.6. Lemma 4.5 Let D be a closed disk in a tame n—bOOk. Then D is tame iff Ed D is tame. Theorem 4.6 A topological polyhedron P in a tame 3 n-bOOk in E is tame iff it has a triangulation such that the image Of the l-skeleton is locally tame at each point where it meets the back Of B“. 47 Proof. By Theorem 4.2 and the hypothesis Of this theorem the image Of the l—skeleton is locally tame, and SO Theorem 1.2 implies the 1-skeleton is tame. By Lemma 4.5 the image Of each 2-simplex in P is tame. Theorem 3.1 Of Doyle [14] states that a tOpological polyhedron P in E3 is tame iff each 2-simplex in P is tame and the l—skeleton is tame. Using the notion of tame in Bn, which was defined in Chapter III, one Obtains the following theorem. Theorem 4.7 ' Let Bn be a tame n-book and P a poly- hedron tame in Bn, then P is tame in E3. Proof. Let hl: E3 aIE3 be a homeomorphism such that the leaves of h B“) are 2-simp1exes. Since P is tame in 1( B , there is a homeomorphism h2: hl(Bn)-a hl(Bn) such that h2(hl(P)) is a Euclidean polyhedron. h2 can be extended to a homeomorphism Of E3 onto itself, also called . 3 3 h2. Then h2hl. E .9 E such that h2hl polyhedron, and hence P is tame in E3. (P) is a Euclidean Corollary 4.8 Every topological umbrella in a tame n—bOOk is locally tame at its tangent point. Proof. The tangent point x Of the tOpological umbrella T must lie in the back B of Bn, since an umbrella cannot be embedded in the plane. By Lemma 4.1 there is a 48 closed neighborhood UJOf x in the 24cell of T that lies in precisely two leaves of Bn. U may be chosen homeomorphic to a closed 2-cell and such that Bd UTIB consists Of two points. Thus there is a subarc a Of the handle of T with endpoint x and such that a - {x} lies in the interior Of a leaf Of Bn. Since (ULJd)rIB is homeomorphic to an arc, Theorem 3.5 implies ULla is tame in Bn. Corollary 4.7 implies ULJd is tame in E3 and hence T is locally tame at its tangent point. In the next theorem arbitrary 3-books in E3 are considered. Let D be a disk in Euclidean 3-space. Let e be an arc such that Drle is a point p which is an interior point both Of D and of e. If for each suffi- ciently small Open neighborhood U Of p, U - D is the sum Of two disjoint Open sets each Of which intersects the component Of Ufle that contains p, then 3 pierces P.§E.E° Theorem 4.9 "NO.dISk pierced by an arc lies in an arbitrary 3-bOOk in E3. Proof. Suppose there did exist a disk D pierced by an 3 3 arc e such that DIJeCZB where B is an arbitrary 3—book in E3. Then Lemma 4.1 implies there is a closed neighbor- hood D' of p,{pI= Dn e, in D that is homeomorphic to a 3 2-cell and D' is contained in precisely two leaves Of B , 49 say D1 and D2. Let the diameter Of D' equal 6’) 0. Without loss Of generality, it may be assumed that e is sufficiently small so that if U is a spherical neighbor- hood Of p in E3 Of diameter e,then U — D equals U — D', and U —LD is the union Of two disjoint Open sets, V1 and V2, each of which intersects the component e' Of Urle that contains p. Let x1 e Vlrle' and x2 5 V2FIe'. Since D'CfDlUD2 it follows that x1, x 5 D3 and there is an 2 3FIU'with endpoints x1 and x2 such that AFID = O. But this contradicts the fact that x1 and x2 joint components of U - D. Hence a disk pierced by an 3 are does not lie in an arbitrary 3-bOOk in E . arc ACD are in dis- Next questions concerning cellularity and n-books are considered. Recall that a set C in E3 is said to be cellular if there is a sequence Of closed 3—cells {Ci | i 6 2+} such that C (2 Int C and C = Fl" C i+l 1 i=1 i' If A is a subset of E3, then the cellular hull 2£:fl’ denoted by?+(A), is a cellular set containing A such that no prOper cellular set BC7‘HA) contains A [16]. Thus the cellular hull Of a cellular subset Of E3 is the set itself. If A- and A1 are two arcs in E3, then A is said to be egpivalent tO Al if there is a homeomorphism h 3 mapping E onto itself such that ’h(A) = A 1' Lemma 4.10 Let A be an arc in E3 and WCZA be the set 50 Of points at which A fails to be locally tame. If W is O—dimensional, then A is equivalent to an arc in a flat 3-bOOk iff W lies in a tame set that embeds in E2. Proof. This result follows easily from a theorem Of Posey [41]. Theorem 4.11 An arc A in E3 has a cellular hull that lies in a tame 2—complex iff A is equivalent to an arc in a flat 3-bOOk. Proof. Assume A has a cellular hull that lies in a tame 2-complex. Without loss Of generality, it may be assumed that A lies in the carrier of a simplicial complex K Of dimension two. If Wc:A is the set Of points at which A fails to be locally tame, then by the argument Of ‘nlsr Theorem 4.2,W is contained in the carrier of the l-skele- ton of K. Furthermore W is a closed, totally disconnected set, so W is contained in a polygonal tree in IK(1)I. Hence the conditions Of Lemma 4.10 are satisfied and there is a homeomorphism h mapping E3 onto itself such that h(A) is contained in a flat 3-book. Conversely, assume A is equivalent tO an arc A1 in a flat 3-b00k B3 under a space homeomorphism h. The intersection Of a maximal chain (ordered by inclusion) of cellular sets containing A1 is a cellular hull Of A1. 3 But B3:)Al and B is a cellular set. Consider a maximal 51 3 chain Of cellular sets (as above) containing B 1’ then h'1(I+(A1)) is a cellular hull Of A and lies in the tame 2-complex h'1(B3). which gives rise to a cellular hu117M(Al) Of A The following corollary follows from the proof Of Theorem 4.11. Corollary 4.12 If A is an are that lies in a tame 2-complex in E3, then A is equivalent to an arc in a flat 3-book. Theorem 4.13 There is an arc A in E3 with the prOperty that:if 2¥(A) is any cellular hull Of A, 3%(A) does not lie in a tame 2-complex. Proof. By Theorem 4.11 it suffices to exhibit an arc that does not embed in a tame 3-bOOk. Let A be an arc through a wild Cantor set in E3, for example an Antoine's necklace [2]. From Theorem 1.3 it follows that no wild Cantor set lies in a tame 3-bOOk in E3 and so 1+(A) does not lie in a tame 2-comp1ex. Theorem 4.14utilizes the following result Of McMillan [36]: Suppose that K is a finite complex, L is a subcomplex Of K, and that K collapses to L. Let h: K-a Mn be a homeomorphism where Mn is a piecewise— 52 linear n-manifold. If n % 4 and if h(K) is cellular in n M , then h(L) is cellular. (For a definition Of collapsing see [45].) Theorem 4.14 If Bn is a cellular book in E3 then each leaf is cellular and the back is cellular, but not conversely. Proof. Since an n-book collapses to any leaf Of the back, it follows immediately from [36] that if En is a cellular book in E3 then each leaf is cellular and the back is cellular. However, the converse is not true. That is,‘ there are n—bOOks in E3 such that each leaf plus the back is cellular but the n-bOOk is not cellular. One such example, for n = 2, is Obtained from the non-cellular arc A Of Example 1.1 of [22]. A can be expressed as the union of two arcs A and A such that A10 A2 = {x} , and x is a 1 2 point in the interior of A. Then A1 and A2 are both cell- ular since they are locally tame except at their endpoints. The arc A can be swollen into a disk D such that D = DlLlDQ, D i = 1,2, is a cellular disk Obtained by swelling A1, 1, and D10 D2 is a straight line segment. Thus D is a 2-bOOk with cellular back Dln D2_ and cellular leaves D1 and D2, that is not cellular. 53 In Theorem 4.18, we describe the wild points Of cellular arcs in arbitrary n—books in E3. In the proof of this theorem we use Theorem 10 Of C. D. Sikkema ["A duality between certain Spheres and arcs in 83: Trans. Amer. Math. Soc. 22 (1966) 339-415]. P. H. Doyle has recently given an alternate proof Of this result which we include. The proofs of Theorems 4.15-4.17 are due to Doyle. Theorem 4.17 is Theorem N) Of Sikkema in the above paper. The space X is either the 3-Sphere or Euclidean 3-space. If A is a compact set in X, let Z be the Space Obtained by identifying A with a point while n: X.» Z is the natural map. Theorem 4.15 Let Ac:X be a wild arc that is locally tame at all points except an endpoint a and let b be the other endpoint. If C is a flat 3-cell in X, A - {OECZInt C, suppose b lies on Bd C SO that ALJBd C is locally tame at b. Then n(Bd C) is wild in Z. Proof. Let D be a 3-cell that is locally tame except at a, Bd DTle C is a disk E on the boundary of each cell while A — ({a} u {b])c Int D and b lies in Int E. D is Obtained by "swelling A". By construction C—:_D is not a 3-cell, but it has a wild 2-sphere boundary R. Note that nI'C_:ID is a homeomorphism and so n(R) is wild in z. If n(Bd C) were tame in Z, then n(R) would have the point n(b) accessible by a tame arc from the side having the 3—cell 54 closure. But by [26] this is impossible. Theorem 4.16 Let A be as in Theorem 4.15. If B is an arbitrary arc in X, BrlA = {b}, then n(B) is wild in Z. Proof. There is a disk D that lies on a tame 2—sphere 82 such that 82 bounds a 3-cell C, A {b} CInt C, b lies on 82 and AL182 is locally tame at b; one may obtain D by swelling A near b and 82 is Obtained by the tameness Of D and an application of [5]. By Theorem 4.15 n(Se) is wild in Z. Then by the same argument n(B) contains no tame arc in X with n(b) as endpoint. SO n(B) is wild in Z. and A Theorem 4.17 Let A be disjoint arcs in X that 1 2 are each wild and fail to be locally tame at just one end— point each, a and a respectively. If A3 is any arc in l 2’ X containing AlUA2 and having al and a2 as endpoints, then A3 is not cellular. Proof. Suppose A3 were cellular. Then each subarc Of A3 must be cellular [35]. In Int A3 select a subarc Q such that Ag—T—Q_ is locally tame except at al and a2. Note that X modulo Q is tOpologically X again. SO for A3 one may select an arc A that fails to be locally tame at its endpoints and by [33] exactly one interior point. But by [5] and Theorem 4.16 this is impossible. 55 The proof Of the following theorem could have also been Obtaining by using the techniques Of Sikkema in the paper mentioned on page 53. Theorem 4.18 Let A be a cellular arc in the interior Of an arbitrary n-bOOk in E3. If the set of wild points Of A is non-empty and does not contain an arc, then A has at most one wild point that is not contained in the back Of the n-book. Proof. Let Bn be an arbitrary n-bOOk that contains a cellular arc A. Assume that A has two wild points al and a2 that are not contained in the back of BH and reach a contradiction. There are three cases to consider. Case 1. a and a are both interior points Of A. 1 2 Let Al and A2 be disjoint subarcs Of A such that a1 6 Int A1 and A is contained in the union of two leaves Of Bn, 1 i = 1,2. Then both Al and A2 are cellular arcs by [35]. The argument Of Theorem 5 of [5] establishes the existence of a subdisk DI of the two leaves Of Br1 containing Al such that: (l) Alis cOntained in the interior Of D', and (2) D' lies on a 2-sphere Sl in E3. Theorem 1 of [32] states that a cellular are on a 2—Sphere in E3 has a set of wild points that is empty, contains an arc, or consists of a single point. Hence by the hypothesis Of this theorem, al 1' Let T1 and T2 be two subarcs and whose intersection is al. We is the only wild point Of A of Al whose union is Al next Show that both T1 and T2 fail to be locally tame at al. 56 By the Bing approximation theorem [4],it may be assumed that S1 is locally polyhedral except on A argument, A 1. By the above is locally tame everywhere except at a l Of [17] implies that S 1 ; Theorem 1 It 1 is locally tame except at al. then follows from Theorem 9 Of [3] that there is a space homeomorphism h mapping S onto a 2—sphere that is locally 1 polyhedral except at h(a Then Theorem 5 Of [17] implies i)- that h(Tl) and h(T2)’ and hence T and T2,are equivalently 1 embedded in E3. If T1 and T2 are both locally tame at al, then by an application Of Theorem 1 Of [17], Al would be a tame arc. Hence T1 and T both fail to be locally 2 tame at al. The same argument establishes subarcs U1 and U2 of A2 such that U1U U2 = A2 and Uln U2 = {a2} , and such that U and U both fail to be locally tame at a Let A3 be a l 2 subarc Of A with endpoints a1 and a2, which is a cellular arc by [35]. But this 2. then these are isolated wild points Of A3 contradicts Theorem 4.17 and hence this case cannot occur. is an endpoint of A and a2 is an be a subarc Of A with endpoints Case 2. al interior point Of A. Let A3 al and a2. Then by the same argument as in Case 1, a2 is an isolated wild point Of A Let a e TCZA3, where T is 3' 1 an arc contained in the union of two leaves of Bn. By an argument as in Case l,T may be assumed to lie on a 2-sphere in E3. SO by [32], a1 is an isolated wild point of T. Thus A is a cellular arc by [35] such that its endpoints are 3 isolated wild points Of the arc. This contradicts Theorem 4.17 and SO this case cannot occur. 57 Case 3. al and a2 are the endpoints of A. As in Case 2, it follows that al and a2 are isolated wild points Of A. SO by Theorem 4.17 this case cannot occur. Since the above three cases cannot occur, it follows that A has at most one wild point that does not lie on the back Of B”. Next we give an example to show that this is the best possible result for an n-book, n > 2. Example 1.2 of Fox and Artin [22] can be swollen into a 3-ce11 that contains Example 1.2 on its boundary. Let D be a 2-cell contained in the 2-sphere boundary Of the 3-cell, such that D contains Example 1.2 in its interior and is locally polyhedral except at the wild point Of Example 1.2. Let B3 be a 3-bOOk in E3 such that two of the leaves are 2—simplexes and the other leaf is D, and let A be an arc in D that is equivalent to Example 1.2 of [22] such that A intersects the back of B3 only in its endpoint z. Then Example 4.4 Of this chapter 3 can be embedded in B in such a manner that it has z as one Of its endpoints, and it intersects A only in this point. The union Of Example 4.4 and A is a cellular arc, by the p-shrinkable criterion, that has uncountably many wild points on the back Of B3 and precisely one wild point con- tained in the interior Of a leaf of B3. CHAPTER V SUBSETS OF TAMELY EMBEDDED CONES OVER N-BOOKS IN E4 In [40] Persinger considered wild and tame subsets Of tamely embedded n-bOOks in E3. As was re— marked in Chapter II, an n-bOOk may be considered as the double cone over n points. In this chapter wild and tame subsets Of tamely embedded triple cones over n points in En are considered, that is,subsets Of tamely embedded cones over n-bOOks in E4. A cone over an n—bOOk will be denoted by c(B“). The argument Of the first theorem establishes that there exist no wild Cantor sets in tame cones over n-bOOks in E4, just as in [40] it is established that 3 there exist no wild Cantor sets in tame n-bOOks in E . Theorem 5.1 NO wild Cantor set lies in a tame cone over an n-bOOk in E4. Proof. Let C(Bn) be a tamely embedded cone over an n—book in E4. Let h be a homeomorphism Of E)4 onto itself such that h(C(Bn)) is a Euclidean polyhedron with triangu-- lation K. Let s1,...,sr denote the 3-simplexes Of K. . Suppose C is a Cantor set embedded in C(Bn). Then for each i, 1 3’1 3 r, C(\h-1(IsiI) is contained in a Cantor set C -l i such that CiCIh (I31I)- h(Ci)(:IsiI,and IsiI is a 58 59 Euclidean 3-simp1ex in E4 and so is contained in a 3- dimensional hyperplane Of E4. From Klee [28] it follows that h(Ci), hence 01’ is a tame Cantor set in E4. Theorem 8 Of Osborne [37] states that the countable union Of tame Cantor sets in En is a tame Cantor set. Hence C CILJ§=10i and LJ§=1 C1 is a tame Cantor set in E4, and SO C is a tame Cantor set. The above proof is valid for any tamely embedded (n-l)-complex in Euclidean n-space, hence the following corollary. Corollary 5.2 A tamely embedded (n—l)—complex in En contains no wild Cantor sets. Next l-cells and l-Spheres in tamely embedded C(Bn) are considered. The fact that all such l—cells and 1-spheres are tame in El1L follows from Theorem 2 Of Dancis in [12]. This theorem states: A necessary and sufficient condition that a k-complex K, which is a closed subset Of a combinatorial n-manifold (without boundary) n 2.2k + 2, be tame in M is that K lie in the union of a countable number Of locally tame (n-k)-simplexes in M. Theorem 5;; There exist no wild arcs or wild Simple 4 closed curves in tame cones over n-bOOks in E . 60 In the case n = l or 2, this result can be Obtained in another manner which is indicated in the proof Of Theorem 5.6. All arcs, simple closed curves, and disks in a tame l- or 2—book in E3 are tame in E3 [40]. The next two theorems are an analogous kind Of result for cones over 1— or 2-bOOkS tamely embedded in E4. These theorems depend strongly on some recent results Of Kirby [27], and results on embeddings Of subsets in 3-dimensional hyperplanes in E4 Of Klee [28], Bing and Klee [6], and Gillman [24]. A finite sequence Of distinct 3-simplexes in "Sr’ is called a circuit if: (1) LI?=1 ISJI is homeomorphic to a closed 3-cell for l S i‘g r, and (2) (LIZ;=1 ISJI){IISi+lI for i < r is a 2-cell on the boundary of LJ3=1 ISJI and on the boundary Of I31+1I' A circuit which is a sequence with r members is said to have length r. Lemma 5.4 Let B be a Euclidean polyhedron in E4 homeomorphic to a closed 3-cell, K a triangulation Of B, and O a l-simplex Of K. Then the collection Of 3-simp1exes Of K that have 0 as a face can be ordered in such a manner, say 81,...,S 130 that this sequence is a circuit. I.) 61 Proof. The collection Of all circuits, such that the members Of the sequence are 3-simplexes Of K having 0 as a face, is non-empty and contains a finite number Of ele- ments. Thus there is a circuit of maximal length, 81,...,Sk. To prove the lemma it is necessary to show that k = r, where r is the number Of 3-simplexes Of K having O'as a face. Suppose k # r and reach a contra- diction. Let G = Isllu ulskl, then G is homeomorphic to a closed 3-cell. There are two cases to consider. Case 1. There exists x e IOIFIInt G. Then x has an Open 3-cell neighborhood N in G which, by invari- ance Of domain, is also a neighborhood Of x in B. If there exists s'e K, where s' is a 3-simplex with 0 as a face and s' # s1,i = l,...,k, then Int ISII-interseCts every neighborhood Of x in B. But Int IS'IFIN = ¢,hence there does not exist such an s'. Case 2. There does not exist x e IOIfIInt G. Then IOIc:Bd G and there are two 2-simplexes, say Oi and O*-) Next it will be shown that both O* and 0*(X2> lie in the boundary Of B. Suppose not, and that Int IO*|c:Int B. Then O* is the face Of two 3- Simplexes in K, one Of which lies in G and the other S' = O* which is not a member of the circuit 62 determining G. Then IS'IrIG = IO*I or I°’*| u I0*| . for all the 3—simplexes in G have C as a face. Hence Is'I intersects G in a 2-cell on the boundary Of each, and IS'ILJG is homeomorphic to a closed 3-ce11. But then s .,sk,s' is a circuit which contradicts the l". maximal length Of s Hence both G* and l""’Sk' O* lie in the boundary Of B. Since both O* and O* lie in the boundary Of B, if x 6 Int IOI then x has a neighborhood N in G, which by invariance of domain, is also a neighborhood Of x in B. SO by the same reasoning as in Case;l, there does not exist a 3—Simp1ex s' Of k with O < s' and s' # si for i = l,...,k. The lemma followsufrom the above argument. Theorem 5.5 A tame 3-cell B in E4 is flat. Proof. Since B is tame it may be assumed that B is a Euclidean polyhedron in E4 with triangulation K. In the appendix Of [31] Lacher gives a proof that locally flat cells in En are flat. SO it suffices to prove that B is locally flat. In [27] Kirby also proves that if B1 and B2 are two locally flat (n-l)-Cells in En with n-2 BlrlB2 = Bd BlrIBd B2 = B“.2 where B is an (n-2)- cell which is locally flat in Bd B1 and Ed B2, then BlLlB2 63 is a flat (n—1)-ce11 in En. This theorem along with Lemma 5.4 implies that B is locally flat except possibly at its vertices and hence by the previous remarks B is flat. Theorem 5.6 There exist no wild l- or 2—cells or l— or 2-spheres in a tamely embedded cone over a l- or 2-bOOk in E4. Proof. The cone over a l- or 2-bOOk is homeomorphic to a closed 3-cell. Hence by Theorem 5.5,we may assume that C(Bn) n = 1,2 is contained in the hyperplane E3 x O in E4. By a theorem Of Klee [28],any l-cell embedded in a 3-dimensional hyperplane Of'El‘l is tame in E4. In [6] Bing and Klee prove that every simple closed curve in E3 is unknotted in E4. By Theorem 3 of Gillman in [24] every 2-Sphere or 2-cell in a 3-dimensional hyperplane Of E4 4 is tame in E . Thus every 1— or 2-cell Or 1- or 2-sphere 4 in a tamely embedded cone over a l- or 2-bOOk in E is tame in E4. The questionl of whether or not there exist wild 3-cells in tame CCBnL n = 1,2,has not been answered. However, there do exist wild 2-.and 3-ce1ksin tamely embedded cones over n-bOOkS, n > 2, in E“. To Show that this is the case it is necessary to introduce some defini- tions from [5]. Let D be a disk. We say that §_p§p 93 Bd D into a set Y can be shrunk to a constant in Y if 64 the map can be extended to take D into Y. YCIX is locally simply connected at a point p Of'Y if for each neighborhood U Of p in X there is a neighborhood V of p in X such that each map Of Bd D into VFIY can be shrunk to a constant in UFIY. Theorem 5.7 If A is an arc in En whose complement fails to be locally simply connected at an endpoint p of A, then A x [O,l]c:Er1 x E1 = En+1 is a wild disk in En+l, Proof. Assume A x [0,1] is tame in En+1 and reach a contradiction. Since A x [0,1] is tame there exists t e (0,1) such that if p' = p x t then A x [0,1] is locally flat at p'. Hence En+1 — (A x [0,1]) is locally simply connected at p'. A x t is embedded in En x t as A is embedded in En. A contradiction will be reached by proving that En x t — A x t is locally simply connected at p', and hence that En - A is locally simply connected at p. Let U be any neighborhood of p' in Enix t, then 1 U' = U x (0,1) is a neighborhood Of p' in En+ . Since En+1 — (A x [0,1]) is locally Simply connected at p', there exists a neighborhood V' Of p' in En+1 such that each map Of Bd D into V'r\(En+1 ~ (A x [O,l])) can be Shrunk to a constant in U'r\(En+l - (A x [O,l])). Let V be a neighbor- hood Of p' in En x t such that VCIV', and prove that each map Of Bd D into V{'\(En x t - A x t) can be shrunk to a 65 constant in Ufl(En x t - A x t). Let f be any mapping Of Bd D into Vn(Er1 x t - A x t), then f(Bd D) isfcontained in V'f\(En+1 -(A x [0,1]». Hence f can be extended to ‘a map, denoted by f', where f' maps D into U'fl(En+1 - 1 onto En x t ; @ x.[O,l]». Let a be the projection: Of Er1+ a is a continuous map. Then af' maps D into Ufl(En x t - A x t) and is an extension of f. Thus Er1 x t - A x t is locally simply connected at p'. Hence Er1 - A is locally simply connected at p which is a contradiction to the hypothesis Of the theorem. SO A x [0,1] is a wild disk in En+l. If A is the arc of example 1.1 Of Fox and Artin [22],then E3 - A is not locally simply connected at an endpoint Of A. The proof Of this fact may be readily Obtained by considering thepmesentation Of the fundamental group Of the complement of the arc given in [22]. Theorem 5.8 There exists a wild disk in a tamely 4 embedded cone over an n-book, n > 2, in E . Proof. Example 1.1 Of [22] can be embedded in a poly- hedral 3-bOOk, B3CIE3; Then B3 x [-2,2]C'_'E4 is homeomorphic to the cone over B3 and is tamely embedded in E4. By the remarks preceding this theorem and Theorem 5.7, it follows that A x [0,1] is a wild disk contained in a tamely embeddr. ed cone over a 3-bOOk. 66 Next it will be established that there exist wild 3-cells in tamely embedded cones over n—bOOkS, n.) 2. Theorem 5.9 If D is a 2—cell in E3 whose complement fails to be locally simply connected at a point p of Bd D, then D x [0,1] is a wild 3-cell in E4. Proof. Assume D x [0,1] is tamely embedded and reach a contradiction. Let h be a homeomorphism of E4 onto itself such that h(D x [0,1]) is a Euclidean polyhedron with triangulation K. If there exists t e (0,1) such that h(p x t) lies in the interior Of a 2-simplex Of K, then ELl — (D x [0,1]) is locally simply connected at p x t. If, however, h(p x [0,1]) is contained in the l-skeleton of K then there exists t e (0,1) such that h(p x t) is contained in the interior of a l-simplex of K. The point- set realization of all those 3-simplexes of K that have this l-simplex as a face is a closed 3-cell by Lemma 5.4. Furthermore,by [27] and Lemma 5.4,it is a flat 3-cell. Hence E4 - (D x [0,1]) is locally Simply connected at p x t. Using the same argument as in the proof of Theorem 5.7, one Obtains that D is locally Simply connect- ed at p. Thus a contradiction is reached SO D x [0,1] is wildly embedded in E4. 67 Theorem 5.10 There exists a wild 3-ce11 in a tamely embedded cone over an n-bOOk in E4, n > 2. Proof. Example 1.1 of [22] can be swollen inTE3 into a 2—ce11 D whose complement in E3 fails to be locally Simply connected at an endpoint p Of Example 1.1, which by the "swelling construction" lies on the boundary of the 2-cell. Furthermore, D can be embedded in a polyhedral 3-bOOk in E3. Then D x [0,1] lies in a tamely embedded cone over an n-bOOk (as in the proof Of Theorem 5.8). By Theorem 5.9 D x [0,1] is a wild 3-cell in Eu. TO Obtain an example Of a wild 2-sphere in a tamely embedded cone over an n-book,n > 2, that is constructed in a somewhat different manner then the wild 2- and 3-cells above, we utilize an example of Doyle and Hocking [19]. Let B3 be a flat 3—bOOk in E3. Let RE, i e z+, 3 be a sequence Of disjoint 3-bOOkS embedded in B such 3 1 when considered embedded in E3, (2) the books Bf converge that: (1) the leaves Of B are Euclidean 2—simplexes to a point p on the back Of B3, and (3) the diameter Of B3 is less than 1/21. Denote the back of B3 by B i e Z+. 1 1 1’ By [40] a trefoil knot can be polyhedrally embedded in a 3-bOOk. Let T1, 1 e Z+, be a polygonal trefoil knot 3 in such a manner that T is contained in embedded in B1 i 68 the interior of Bi except for two straight line segments ti and si on the boundary Of a leaf Of Bi and in such a manner that tiLlsichl, a leaf of B3, for all i e Z+. Let E i e 2*, be a polygonal disk in B3 such that: i) + (l) E1CD1 for all i e z , (2)E1n T1 = t1 and L a 3 _ Ein T1+1 " Si+l’ (3) Ein L—Ij=l Bj ‘ tiU Si+l’ (4) EiFIEJ = C for i f j, and (5) the diameters Of the E1 tend to zero. See Figure 5.1. P . I33 . E; <’ 3 BO. l—- ’ J'1IE, \ \ 'fi \" 3 ‘75. 8. Figure 5.1 Let B3 be embedded in the 3-dimensional hyper- 4 plane E3 x OCIE . Let 2'(B3) denote the suspension Of 4 B3 in E with suspension points u and v, and let Cu(Bi) denote the cone over Bi which is obtained by joining points in B denote the cone over Bi which is i to u,anduCV(Bi) 69 Obtained by joining points on Bi to v. Let vi be at point in CV(B - (B1L1[v}) and let ui denote a point in 1) C (Bi) — (BiLJ{u]). Choose the points v u and ui, i e Z+, i SO that the sequences {ui I i e Z+}and {vi I i e Zf}converge to p. Let Z'(B§) denote the suspension of B? in ELl with the same suspension points as B3, then Z'(B§)CZZ'(B3). 4 If 2(T denotes the suspension of T in E with suspen- 1) i Sion points ui and Vi’ then 2(T1)CIZI(B§). This fact can be verified using the convexity of the cones over the leaves of B? in Z'(B§) and the fact that ui 3 chosen in the suspension Of the back of Bi Hence and vi were {2(T1) I i e Z+I consists Of a sequence of disjoint 2-spheres converging to the point p. These 2-spheres are now joined together in such a manner that a wild 2—Sphere is Obtained. The E1 can be swollen into a polyhedral 2- spheres EI? in E3 containing E1 such that: (l) EICCZ'(D1), (2) EIFIE3 = O for i # j, (3) EIr\Z(T1) is a polyhedral 2-cell containing t and EII‘] 2(T ) is a polyhedral 1 1+1 2-cell.containing s1, (4) the diameters of the E' converge 1 to zero, and (5) S =I_I:=1 (2(T1)LIEI) is locally poly- hedral except at p. Then S is homeomorphic to a 2-sphere and is the example of Doyle and Hocking [19]. Furthermore by con- struction SCIZ'(B3),Land.2'(B3) is homeomorphic to the cone over B3. Z'(B3) is tamely embedded in E“. A tamely embedded 2-sphere in El‘l can fail to be locally flat at 70 only finitely many points, the vertices Of a triangulation of the 2-Sphere. S fails to be locally flat at the se- quences {vi I i e Z+I and {ui I i e Z+} which converge to p. Hence S is wildly embedded in Eu. The above dis- cussion yields the following theorem. Theorem 5.12 There exists a wild 2—Sphere in a tamely 4 embedded cone over an n-bOOk, n > 2, in E . It is interesting to note that the wild 2-Sphere S constructed above is locally tame everywhere except at p and that S fails to be locally flat on a sequence Of points converging to p. Furthermore,every arc embedded in S is tamely embedded in E4. Also S can be expressed as the union Of either two wild 2-cells or two tame 2—cells. The wild 2- and 3-cells in tame cones over n-bOOkS,n > 2, constructed in this chapter were both products Of cells; therefore,they are both cellular. 10. 11. 12. 13. l4. BIBLIOGRAPHY J. W. Alexander, "An example Of a simply connected surface bounding a region which is not Simply con- nected", Proc. Nat. Acad. Sci. U. S. A. 10 (1924) 8-10. L. Antoine, "Sur l'homeomorphisme de deux figures et de leurs voisinages" J. Math. Pures Appl. 4 (1921) 221- 325. R. H. Bin , "Locally tame sets are tame" , Ann. of Math. 59 I1954) 145 158. R. H" Bing, "Approximating surfaces with polyhedral ones" , Ann. Of Math. 65 (1957) 456- 483. R. H. 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