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Separation of the N-Sphere by an (N - 1)-Sphere University of Tennessee, Knoxville TRACE: Tennessee Research and Creative Exchange Masters Theses Graduate School 12-1961 Separation of the n-Sphere by an (n - 1)-Sphere James C. Cantrell University of Tennessee - Knoxville Follow this and additional works at: https://trace.tennessee.edu/utk_gradthes Part of the Mathematics Commons Recommended Citation Cantrell, James C., "Separation of the n-Sphere by an (n - 1)-Sphere. " Master's Thesis, University of Tennessee, 1961. https://trace.tennessee.edu/utk_gradthes/2948 This Thesis is brought to you for free and open access by the Graduate School at TRACE: Tennessee Research and Creative Exchange. It has been accepted for inclusion in Masters Theses by an authorized administrator of TRACE: Tennessee Research and Creative Exchange. For more information, please contact [email protected]. To the Graduate Council: I am submitting herewith a thesis written by James C. Cantrell entitled "Separation of the n- Sphere by an (n - 1)-Sphere." I have examined the final electronic copy of this thesis for form and content and recommend that it be accepted in partial fulfillment of the equirr ements for the degree of Doctor of Philosophy, with a major in Mathematics. Orville G. Harrold, Major Professor We have read this thesis and recommend its acceptance: E. Cohen, Lee Simmons, & W.S. Makavier Accepted for the Council: Carolyn R. Hodges Vice Provost and Dean of the Graduate School (Original signatures are on file with official studentecor r ds.) December 5, 1961 To the Graduate Council: am submitting herewith a thesis written by James C. Cantrell I entitled 11Separation of the n-sphere by an (n - l)-sphere.11 re­ I commend that it be accepted in partial fulfillment of the requirements for the degree of Doctor of Philosophy, rdth a major in Mathematics. Major0.91/� Professor We have read this thesis and recommend its acceptance: \5?�£� le_;;. � Accepted for the Council: �£�Dean of the Graduate School Copyright by James CecU Cantrell 1962 SEPARATIC!f QF· THE n-SPHERE BY AH (n-1)-SPBERE A Dissertation Presented to the Graduate CouncU of University of Tennessee In Partial Fulfillment of the Requirements for the Degree Doctor of Philosopb7 by ..1 f"... .. James cf'eantrell December 1961 ACKNOWLEDGEMENT The author wishes to acknowledge his indebtedness to Professor Orville G. Harrold, Jr., for his direotion and assistance in the prepa­ ration of this dissertation. This researchwas supported in part through the National Science Foundation Grant 8239. 525255 • TABLE OF CONTENTS PAGE I. DfTRODlJCTION • • • • • • • • • • • • • • • • • • • • • 1 II. ALMOST LOCALLY POLYHEDRAL 2-SPBERES m � • • • • • • 4 n-1 n III. SOME EMBEDDDfGS OF s m s • • • • • • • • • • • • 19 4 s • • • • • • • • • • IV. SOME 3-SPBERES m • • • • • • • 36 4 Three-Spheres in s Obtained by Suspension • • • • • 36 4 Three-Spheres in s Obtained by Rotation • • • • • • 40 Three-Spheres Obtained by CappingA 0,1inder • • • • • 48 BIBI.IOORA.PHI' • • • • • • • • • • • • • • • • • • • • • • • • 53 LIST OF FIGURES FIGURE PAGE 1. A Chambered n-cell • . • • • • • • • • • • 0 • • • • 31 • • • • • • • • 2. A Countable Partition of an n-cell . 33 3. A Modified n-cell . • • • • • • • • • • • • • 0 0 . 34 • . • • . • • • • • • • • 4. A Wild 2-Sphere in s3. • 46 5. A Cross Section of a Wild 3-Sphere in ·s4 • • • . • 47 CHAPTER I INTRODUCTIOO n-l An (n - 1)-sphere is a topological image of s a i5�,x , ••• , 2 2 • • • = x e � +x 2 + +xn 2 , an open n-cell is a topological n) Ff I 2 i} ••• • image of {<�,x ·, , xn) e Ef I x1 2 +:x 2� � ··:+ x.;.2 � Y. and a· closed 2 2 J ••• n n-cell is a topological image of {<x , x , , x ) e E 2 +x 2 + 1 2 n I � 2 • • • 1 +xn 2<- } . In this thesis we consider certain (n - 1) -spheres embe dded in n n S (we will frequently use the fact that S is topological.ly equivalent to the one point compactification of En). The problem is then to estab­ lish the existence or non-existence of certain topological properties of the two domains into which sn is separated by the given (n - 1)-spheres. For the cases n = 1, 2 it is known that each (n - 1)-sphere in Sn separates Sn into two do mains, either of which is an open n-cell and has a closure which is a closed n-cell. That this is not the case * for n a 3 is shown by numerous counter examples (see [2] and [S] ). 3 A 2-sphere K in s that-is loc� polyhedral except at one, two or three points is considered in Chapter II and the following results are established. ·If K, is locally polyhedral except at one point�·, then the closure of one component of s3 - K is a closed 3-cell and the other component is an open 3-cell. If K is locally polyhedral except at two points, then either the closure of one complementar.r domain is a closed * Numbers square brackets refer to numbers in the bibliopapb7 at the end of this inpape r. 2 3-cell or both camplementar,y domains are open 3-cells. If K is locall y polyhedral except at three points, then one of the complementary domains is an open 3-cell. This domain may or may not have a closure which is a closed 3-cell. Let x ) 2 + 2 2 � A e + + = t<Xi,x2' ••• , n �I xl x2 ••• xn �, = B , x ) x 2 2 + x e + 2 {<x1, x2, •• • n Ef I 1 + x2 • • • n � �} , = C x ) & 2 + + and 2 + 2 � {<�, x2, ••• , n Ef I x1 x2 •• • xn 4} , = n + fl x ) E 2 2 + (x + 1) 2 1. The D 1!�, e + � x2, •• • , n I x1 x2 ••• n 4J • Generalized Schoenflies Theorem states that if h is a homeomorphism of (C . into then the closure of either complementary domain Cl. "B) ff1 , of h(Bd A) is a closed n-cell. A proof of a special case of this theorem by Mazur [13] and a proof of the full theorem by Br own [8) point n-l out that properties of the embedding homeomorphism of S = Bd A in Sn can be used to investigate the properties of the complementar,y do- mains . One is naturally led to the following question, if h is a homeomorphism of Cl (A "B) into ffl , is the closure of the component n of S "h(Bd A) which contains h(Bd B) a closed n-cell? This ques­ tion is answered affirmatively by Theorem 3. 2. In fact the theorem follows from the Schoenflies Theorem and the two are therefore equiva- lent. n Two other embeddings of Bd A in S , n > 3 , are considered in n Chapter IIIa (1) a homeomorphism h of Cl (D "B) into S , and a homeomorphism h of Cl into Sn In the first case it (2) (D "A) • is Shown that if h is semi-linear on each finite polyhedron of (Int A) "B , then the closure of either complementary domain of h(Bd A) 3 is a closed n-cell. In the second case it is shown that if h is semi- linear on each finite polyhedron in a deleted neighborhood of (o,o, •• • ,l) (see Definition 3.7), then the closure of the complementar,r domain of h(Bd A) which intersects h(Bd D) is a closed n-cell. The proofs of n these theorems depend quite heavilY on the fact that an arc in E , n > 3 , which is local..l;y polyhedral except at a single point is tame (see Lemma 3. 3) . 4 In Chapter IV three methods of constructing )-spheres in s from 2-spheres in s3 are considered: (1) suspension of a 2-sphere in s3 , (2) rotation of a 2-cell in s3 about the plane of its boundar,r, and 2 3 (3) capping a cylinder over a -sphere in s • The construction methods in cases (1) and (2) were introduced by Artin [4] and have been used by 4 him and by Andrews and Curtis [ 3] to construct 2-spheres in s from 1- 3 spheres and 1-cells in s • Their techniques are used to establish isomorphism theorems relating the fundamental groups of the complements of the constructed 3-spheres and the fundamental groups of the correspon­ ding complements of the given 2-spheres. In Case (1) it is shown that the second ho motopy groups of the complements of the constructed 3- spheres are trivial. Method (2) is also used to construct a 3-S}?here 4 in s , one complementar,r domain of which is simplY connected bu� is not an open 4-cell. The third construction is considered because it seems to give the simplest scheme (in fact the only scheme of which I am aware) tor showing the existence of a 3-sphere in s4 su�h that one complemen­ tary" domain has a closure which is a closed 4-cell, and the other com­ plementar.y domain is an open 4-cell but its closure is not a closed 4-cell. CHAPTER II 2 ALMOST LOCALLY POLYHEDRAL -SPHERES IN g3 Le t K be a set in a geometric complex C • Definition 2.1. K is local.l.lpolyhedral� ! point p 2£. K if there is an open set U containing p such that Cl U n K is a polyhedron in C • K is said to be locally polyhedra l if it "is local.ly' .p01.ybeClr8.1. at each point of its points. Definiti on 2. 2. K is tamel.y embedde d in C if there is a homeomorphism of C onto itself that carries K onto a polyhedron. Definition 2.3.
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