2-MANIFOLDS IN EUCLIDEAN 4-SPACE I {I I: Thesis for the Degree of Ph. D. MICHIGAN STATE UNIVERSITY JAMES LEE MURPHY 1 9 7 0 t'HUClC LIB REAR 1? Michigan Stan: University This is to certify that the thesis entitled 2-MANIFOLDS IN EUCLIDEAN LI-SPACE presented by JAMES LEE MURPHY has been accepted towards fulfillment of the requirements for M;— degree in Wt i C S Major professor Date 19th. March 1970 0-169 amomo av ‘5 HMS & SUNS' 9995.,E'EQEFXJEQ- ABSTRACT Z-MANIFOLDS IN EUCLIDEAN 4-SPACE by James Lee Murphy This thesis is a study of polyhedral and almost polyhedral 2-manifolds in Euclidean 4-space. In chapter I a special 2-disk, fly is constructed in B such that fi’is locally polyhedral at every point except one on the boundary and is universal in the sense of the: Main Theorem: Given any 2-disk, D, embedded in E4 in such a way that D is locally polyhedral at every point except at one boundary point P, then there exists a space homeomorphism h: E4 4 E4 such that h(D) c fi~and h(P) is the point at which 3 fails to be locally polyhedral. we show that 11(E4-fi) is trivial. Then we extend the results on the disk to 2-manifolds which fail to be locally polyhedral at just one point. In chapter II we discuss a polyhedral 2-disk in E4 which has a flat triangle for boundary but which can not be moved by a space homeomorphism.into a three-dimensional hyperspace. Finally, we construct a polyhedral 2-sphere in E4 which fails to be locally flat at exactly one point. Z-MANIFOLDS IN EUCLIDEAN 4-SPACE BY James Lee Murphy A THESIS Submitted to Michigan State University in partial fulfillment of the requirements for the degree of DOCTOR OF PHILOSOPHY Department of Mathematics 1970 C;-pé;Z/:52.0 ll) “1"‘7‘70 ACKNOWLEDGEMENTS The author wishes to express his gratitude to Professor P. H. Doyle for suggesting the problem and for his helpful suggestions and patient guidance during the research. He also wishes to thank the faculty members and graduate students of the Department of Mathematics with whom.he has had many stimulating discussions during the investigation. ii TABLE OF CONTENTS INTRODUCTION ... ......... . ....................... ...... CHAPTER O. NOTATION AND TERMINOLOGY ... ................ .... . I. ALMOST POLYHEDRAL 2-MANIFOLDS IN E4 .......... .... II. 1. Construction of a Special 2-Disk b in E4 ...... 2. b’is a Universal Disk of its Kind ............. 3. W1(E4-fi) 3 O ................................ .. 4. 2-Manifolds in E4............. ......... . ..... .. POLYHEDRAL 2-MANIFOLDS IN E4 ...................... . 1. Madison Problems ...... ...................... .. 2. A Polyhedral 2-Sphere which Fails to be Locally Flat at Just One Point .. ..... . ..... ... BIBLIOGRAPHY iii 16 19 22 22 23 26 LIST OF FIGURES iv INTRODUCTION In [6] Doyle and Hocking construct a wild 2-disk in E4 by taking cones over an infinite number of polygonal trefoil knots in such a way that the cone points and the trefoil knots approach a point P. They join the cones with polygonal strips and add the point P. Their disk is, by construction, locally polyhedral at all points except P which lies on the boundary. We show that all 2-disks in E4 which are locally polyhedral except at one boundary point arise in this way, i.e.. they are equivalently embedded with a disk of the construction of Doyle and Hocking using perhaps polygonal knot types other than that of the trefoil knot. In particular, we construct in section 1 of Chapter I a type of universal 2-disk, 3) in E4 which is locally poly- hedral at all points except one boundary point P. We then show that for any 2-disk, D, embedded in E4 which is locally polyhedral at all points except a boundary point P' there is a space homeomorphism.h: E4 4 E4 such that h(D) : fl and h(P') = P. In [10] Gugenheim, restricting himself to polyhedral objects and PL maps, shows that the embedding classes for q-disks in 2q-space are in 1-1 correspondence with finite sequences of q-l-spheres in 2q-l-space independent of order. We note that for q = 2, having removed the polyhedral re- quirement at one boundary point necessitates changing the finite sequence of l-spheres in 3-space to an infinite sequence of 1-spheres in 3-space, still independent of order. Results appear to be forthcoming which would trivialize this result of Gugenheim for q 2 3, i.e.. for q 2 3 all q-disks in 2q-space may turn out to be equivalently embedded. In Chapter I section 3 we relate our results to almost polyhedral 2-manifolds in E4. Finally, we turn our attention to the polyhedral category in Chapter II to point out that results of Gugenheim in [9] can be used to answer two questions posed at Madison in [12] p. 55. We then construct a polyhedral 2-sphere which fails to be locally flat at exactly one point. CHAPTER 0 NOTATION AND TERMINOLOGY Our entire discussion will take place in E4. By En we mean all n-tuples of real numbers with the topology in- )2. n duced by the euclidean metric d(x,y) = v/izalxi - yi 1.: while 1/2 En = {ern: xn 3 O}. For a and b in En we denote by ab the segment from a to b, i.e., ab = {ta+(1-t)beEn: O S t‘S 1}- A topological n-manifold. MP, is a second countable Hausdorff space with an open covering {U6} and a set of homeomorphisms {ho} such that ha: U 4 1/2 Bn 0' and hc(qd) is open in 1/2 En. The set of all points with En neighborhoods is the interior 9f Mp. written Int Mp, and the bgundary of Mn is Bd M? = M9 = M9 - Int Mp. We will use the piecewise linear terminology of Hudson in [11], Gugenheim in [9], or Zeeman in [15]. Polyhedra will be the spaces in En underlying locally finite rectilinear complexes, i.e., finite unions of convex linear cells. If X and Y are homeomorphic spaces embedded in En we say that they are equivalently embedded if there exists a homeomorphism h: En 4 En such that h(x) = Y. This is clearly an equivalence relation on a particular class of homeomorphic spaces and allows us to speak of the equivalence classes of embeddings. If X is a space embedded in En and P e X we say X is locally polyhedral at P if there exists a neighborhood N of P such that C1(N FIX) is a polyhedron. X is locally tame at P if there exists a neighborhood N and homeomorphism hp: N 4 En such that hp(N) is a polyhedron and hp(N FIX) is also a polyhedron. If M3 is a k-manifold embedded in En and P 6 M3 we say Mk is locally flat at P if there exists a neighborhood U of P and a homeomorphism of the pair (U,U FINE) onto (En, ER) or (En, 1/2 Ek) depending on whether P 6 Int M3 or P 6 Bd Mk. Given a 2-disk D with simple disjoint spanning arcs A and B, the disk bounded by A and B, and pieces of the boundary of D is denoted [A:B]. B(P,e) = {x 6 En: d(x,P) < (S) e] is the open (closed) ball in En centered at P with radius a. When using cube neighborhoods, N(P, e) = [xe En: [xi-Pik e for i = 1, 2....rfl , e is still referred to as the radius of the cube neighborhood N. For any space X‘we denote by id.: X 41X the identity map defined by id.(x) = x for all x c X. When writing i e N, N denotes the natural numbers. CHAPTER I ALMOST POLYHEDRAL 2-MANIFOLDS IN E4 Section 1. Construction of a Special 2-Disk fi-in E4 There are countably many different polygonal knot types in E3 with representatives. say {Ki}. Diagonalize the infinite matrix {aij}. with elements aij = j. to give a sequence {ak} in the order 1. l. 2. l. 2. 3.... Define a new sequence of knots {Ki} where Ki = K;.. note that the Ki occur frequently in the sequence {Ki}? Let Ti denote the trapezoidal cube in E4 determined by the inequalities l/ i+l 5 x1 5 l/i and -x1/2 < xj le/Z for j = 2. 3. 4 and for all i e N. Ti is a PL 3-Sphere. Let ai = (l/i.1/2i.0.0) and bi = (l/i.-l/2i.0.0) for all i e N. Let Ki be in Ti in such a way as to include the in the order a.a. b b and segments aibi and b. 1 1+1 i+1 i 1+lai+l such that Ki FI(hyperp1ane x1 = l/i) = aibi exactly. Let mi = (2i+l/2i(i+l).0.0.0) and form polyhedral disks Di by joining mi to Ki for each i separately. Finally. form 3 a by taking the union ( -L&Di ) L) (0.0.0.0). By construction 1: this disk is locally polyhedral at every point except possibly the origin. That it actually fails to be locally polyhedral at the origin follows from the result of Doyle and Hocking in [6]. Section 2. fl is a Universal Disk of its Kind We show that the disk fi’constructed in section 1 is "Universal" in the sense of the following: MAIN E REM 2 1: Given any 2-disk. D. embedded in E4 in such a way that D is locally polyhedral at all points except at one boundary point P. then there exists a space homeomorphism h: E4 4 B4 such that h(D) C fi'and'h(P) = origin. Proof: The theorem is proved in the course of the following six lemmas. Lemma 2.2: Given a 2-disk. D. embedded in E4 in such a way that D is locally polyhedral at every point except at one boundary point P. then there is a space homeomorphism f: E4 4 E4. fixed outside a compact set. such that f(D) is locally polyhedral at every point except f(P) and locally flat at all but at most a sequence of points lying on a polygonal arc spanning f(D). Prggf; Let D' = [(x.y):x.y e R. x + y S 1. x 2 0.y ECU and let m: D' 4 E4 be an embedding such that ¢(D') = D and cp(0.0) = P. In D' denote segments {(x.y): x+y = 1/n} (I D' =1n; {m(ln)} form a sequence of arcs spanning D. For k 3 1. consider the disks on D between ¢(12k) and ¢(12k+2)° On these disks consider the arcs ¢(12k+1)' For each point x on ¢(12k+l) there eXists a neighborhood Ux such that Ui FID is a polyhedral disk and is equal to UXIW[¢M12k):m(12k+2)]. There exist a finite number of these open Ui which cover ¢(1 ). Order these as ¢(1 ) traverses them say from 2k+l 2k+l m(0.l/2k+1) to m(l/2k+l.0). consecutive disks having a point in common. In the first disk there is a polygonal arc from ¢(0.l/2k+l) to this common point. in the second disk a polyhedral arc from this point to the point in common to the second and third disks etc. to m(l/2k+l.0). Drop any loops that occur leaving a simple polygonal arc Ak from m(0.l/2k+1) to m(l/2k+l.0) and contained in D between > ¢(12k) and ¢(12k+2) for k _ 1. Let AO denote the polygonal arc m({(x.y): x + y = 1]) from.¢(0.l) to m(l.0). Each disk [Ai;Ai+l] is locally polyhedral at every point and thus by the proof of Bing's lemma 1 in [l]. is polyhedral. Compact polyhedra have only a finite number of vertices so these disks [Ai:Ai+1] can fail to be locally flat at at most a finite number of points which. by Theorem 4.2 of Tindell in [13]. must be interior points. Starting at m(l/2.l/2) we pass a simple polygonal arc through all points of [AO;A1] which fail to be locally flat ending at a point interior to a segment of A Starting at 1. this point continue similarly to A and so to Ai so traversing 2 each [AigA Let the arc traversing [AO;A1] be the image 1+1]‘ of the interval from 2 to 1. and the arc traversing [Ai;Ai+l] be the image of the interval from l/i to l/i+1 for i 2 1. Let m(0.0) be the image of 0 giving an arc A which is the image of the interval [0.2] and is locally polyhedral at every point except possibly P. Taking this arc together with ¢(0.y) for 0 S y S l and ¢(x.y) for x + y'= l and 0 f x < 1/2 yields a simple closed curve C which is locally polyhedral except possibly at P. By lemma 2 of Cantrell and Edwards in [3]. given 6 > 0 there exists a homeomorphism f: E4 4 E4 such that: a. f is the identity on E4-B(P.e) b. f is piecewise linear except at P c. f(C) is polyhedral. f(D) is locally polyhedral except perhaps at f(P) and is locally flat except perhaps at a sequence of points which lie on the polygonal arc, f(zx), spanning f(D). f is the homeomorphism required to satisfy the lemma. We modify Doyle's proof of theorem 3.2 in [7] to give 4 a PL homeomorphism f E 4 E4 such that fl(A) is a segment 1: where A is a polygonal arc. Thus we have: Lemma 2, : Given the disk f(D) of the conclusion of 4 lemma 2.2 we can find a PL homeomorphism f E 4 E4 such 1: that f1(f(A)) is a segment. Erggf: Using an inductive argument. we only need to define maps which will reduce the number of segments of the polygonal arc by one. Let v be an end-point of the polygonal arc and let vl be the other end-point of the segment containing v. Find a cube neighborhood of v U1. which intersects the 1! are f(A) only in the two segments adjacent to v Let Wv 1. be a conical neighborhood of v such that Wv intersects f(A) in precisely the segment vvl. Let v' be a point of the segment vv1 which is contained in U1. v' ¢ v1. Define PL homeomorphisms h1' h2. and h3 as determined by the simplicial maps indicated in figure 1. These maps are fixed on and outside the boundary of each figure shown. Figure 1 10 Lemma 2.3 could have been proved easily by using the general result of Gugenheim in Theorem 5 of [9]. He shows that for any q-dimensional polyhedra P and Q in En with 2q + 2 S n there exists a PL homeomorphism h: Bn 4 En such that h(P) = Q. i.e.. polygonal arcs are equivalent to . n > . . . segments in E for n _ 4. But it seems instructive to use the above proof for our particular case. Lemma 2.4: The disk of the conclusion of lemma 2.3. call it D. is equivalent under a space homeomorphism to a disk which is locally polyhedral except possibly at a boundary point. P. the only non-locally flat points are on a spanning segment with end-point P. and all interior vertices of the disk lie on this segment. grggi: Assume the segment of lemma 2.3 lies on the x1 axis with fl(f(P)) at the origin and the other end-point at (1.0.0.0). Introduce by starring if necessary a vertex between any two consecutive non-locally flat vertices on our segment and let {vi} denote the vertices on the segment in the order traversed fromvO = (1.0.0.0) to the origin. If a last vK. denote by vK+k the point x1 = IvK] / k+l x2 = x3 = x4 = 0. There are at most countably many vertices in the disk (countable union of finite numbers of vertices in the P1 image of the [Ai:Ai+l] of lemma 2.2). Construct cube neighborhoods Ni centered at the origin with radius I = . I ' ' ri Ivi + vi+l| /2 Replace the Ni Wlth cube neighborhoods Ni of radius ri such that Iri - ril < Ivi - vi+1|/Q and the ll 3-hyperp1anes xj = +ri (j = 1. 2. 3. 4; for all i) miss all the vertices of the disk. These hyperplanes are. of course. the hyperplanes containing the boundary of the Ni' This choice of the Ni assures that the intersection of the disk with the boundary of the Ni are one dimensional. Denote the intersection of the segment vb(0.0.0.0) with Ni by Ri' Then the intersection of Ni with the disk has a component containing Ri which we denote by Qi' Note that Qi is a polygonal arc spanning the disk. Qi is one dimensional as noted above and is polygonal since given ri we can find an Aj in lemma 2.2 such that the image of (D - [Aj;AO]) G B(0.ri). Thus Qi is in fact in the intersection of a polyhedral disk with Ni. Qi cannot be a loop since if it were it would necessarily intersect the spanning segment at least twice but this is impossible as Qi 9 Ni which intersects the segment in precisely Ri' The segment VO(0.0.0.0) divides the boundary of the disk into two arcs. Denote one as "upper" the other as "lower" fixed for the remainder of the discussion. We will work with the disk bounded by the segment and the upper are a similar argument serving the other half of the disk. Given Qi there exists an 6i > 0 with 6i < min [lvi-RiI/Q. |v1+1 the disk to Qi is greater than 61. We can always do this -Ril /2] and such that the distancefrom any vertex of since Qi is compact and contains no vertices of the disk and only finitely many vertices are within Ivi+1-Ri| of Qi. Within ei of Qi we can span the disk with a pair of non- intersecting polygonal arcs on each side of Qi and thus 12 find a polyhedral 4-ball neighborhood Mi of the disk bounded by the inner arcs which does not intersect the disk outside . . 4 the outer arcs. Then With a space homeomorphism.hi: E 4 E4 such that hi|E4_M = id.. hi(D) C D. and the component of A . i hi(D) FIMi containing Ri is at most a pair of segments of length less than 6i. This is possible since upper diskIW Mi is polyhedral and flat and so equivalent by gi: E4 4 E4 to Di . . . . 4 in Figure 2. Take Di to Di‘in Figure 2 by a PL Fi: E 4 E4 fixed outside a small cube neighborhood of gi(upper disk FIMi) contained in gi(Mi) and disjoint from the remainder of gi(D). Di Di F. \ l ‘91 (oi) A) Figure 2 -1 Let hi = gi ofiogi. Similarly for the lower disk. Clearly Mi FIMj = ¢ for i # j. Thus define a space. homeomorphism h: E4 4 E4 by setting h = hi on Mi and the identity outside the union of the M,., h(D) FINi has as.the. 1 component containing Ri a pair of sements meeting at Ri' denote the end points ui and li for end points in the upper and lower disks respectively. Now in h(upper disk) uiRi and u together with i+lRi+1 bound a polyhedral disk Di. RiRi+l and Ed D from ui to ui+ Join u. to u. 1 1+1 1 by a simple polygonal arc Pi such that .. ' ' ' I I Pi FIBd Di — {ui'ui+l} and Pi is contained in M.i LlMi+l LJCi. Ci is a cylindrical neighborhood of the segment RiRi+l of l3 . , . . radius ei < min [uiRi'ui+lRi+l'6i} where 6i is less than the distance from the segment RiRi+l to the vertices of the disk not on R.R. and less than r.. M! denotes M. intersect 11+ 1 i i l the cylinder about the segment Ri+lRi-l of radius uiRi' We can find a closed neighborhood Bi of Di such that (1(3in U D) 0 Bi = uiRi U RiRi+1 U ui+lRi+l and Bi 0 D = Di. By a PL homeomorphism cpi: E4 4 E4 such that cpiIE4_B is the identity and mi(D) C D take the polygonal boundary from u. to u. in the interior of B. to P.. Let m = m. on B.. i i i i i 1 +1 Similarly for the lower disk find h' and m'. Then ¢'°h'° ¢°h(D) satisfies the lemma. Lemma 2.5: The disk resulting in lemma 2.4. call it D. is equivalent to one with interior vertices on the segment and the intersection of this disk with Ni is a segment lying 3 = 0. x4 = o. and all upper end, points are on the same side of the segment v0(0.0.0.0). in the plane 9. defined by x Prgof: Define two PL homeomorphisms f.g:E4 4 E4. one to correct the intersection with Ni for odd i the other. for eVen i. Let Ti be the trapezoidal neighborhood.from R2i to R2i+2‘Wlth base at R2i of radius 62i° Picking one Slde of the segment in this plane 9.call it upper 9.in N [W 9 let 2i+1 , . . . . u2i+1 be a point in upper 9 at a distance p21+172 from R _ . . . . . pi — min {5i.IRiI] Similarly let 12i+1 be in lower 0 at a 21+1' Define a vertex map which fixes distance p from R 2i+1/2 21+1‘ all vertices of D except u2i+1 and l2i+1 which map to l4 u2i+l and 121+1 respectively and extend this linearly to obtain fi' Define f to be fi on Ti as described for all 1. Define 9i and g similarly resulting in a disk locally polyhedral at all points except the origin and Ni intersects this disk in the segment uili. Note all interior vertices of the disk are on the segment from the origin to (1.0.0.0). Using the segments uili from the proof of lemma 2.5. define trapezoidal cube neighborhoods by Ti = {(xl.x2.x3.x4): < < _ .. _ - _ < < |R1+1I - x1 - IRH'(pi pi+l)/2|Ri Ri+l|(xl lR1" pi/z -’$j‘ pi-pi+l/2|Ri-Ri+l| (xl-IRiI) + pi/Z for j = 2. 3. 4]. Lemma 2, : The disk of the conclusion of lemma“2.5 is equivalently embedded with a disk D with the same intersections with Ni and within the region |Ri+1| 5 x1 5 lRil 6 : i2. 1 Preef: At each v, a pseudo radial projection fixed 1 on and outside the boundary of the rectangular cube defined by IR. | for j = 2. I < 1+1 ' i+l 3.4. takes the boundary of D within the region IRi+IISxIS IRiI x1 5 IRiI and -|Ri+l| 5 xj S |R to Ti. This can be extended to a space homeomorphism fixed, outside the union of these cubes and locally PL except at the origin. The intersection of D and T; is a polygonal S1 in a PL 3-sphere and thus is equivalent to some knot Ki which is trivial if and only if the disk D is locally flat at v, so we have: i 15 Lemma 2.7: Given the disk D of lemma 2.6 there exists a homeomorphism g: E4 4 B4 such that g(D) ; B. Preef: Letegl be fixed in the complement of N Let 1. vk be the first vertex of the sequence [vi] at which D is 1 not locally flat. There is a knot Kk' in the sequence [Ki] 1 such that Kk' is equivalent to D (Iii . Then let g1 send 1 l N - ' _ I _. l Nkl+1 linearly to C(0.l) C(0.l/k1) and send NR1 Nk1+1 to C(O.l/ki) - C(0.l/ki+l). Let vk be the next vertex at 2 which D fails to be locally flat then there exists Kk' in 2 [Ki] such that k2 > k1 and Kk5 is equivalent to D FITkZ Let g1 send Nk +1 - Nk to C(0.l/ki+l) - C(0.l/ki) and l 2 N to C(0.l/ké) - C(0.l/ké+l) and continue in this - N k2 k2+1 way to define g1: E4 4 E4 a locally PL homeomorphism except at the origin. Let 92' 93: E4 4 E4 be PL homeomorphisms defined in such a way as to make the segments gl(uk lk ) i i coincide with the segments ak'bk' in 1» by changing the size i i of the trapezoidal cubes. 92 for the k2i and 93 for the , . k21+1. Now for the vki which are not locally flat 939291(Tki) = Tki. By a PL homeomorphism g4 fixed on and o tsid . , ' u e Ti! take g3ngl(vk.) to mk! There eXists a PL 1 . i i homeomorphism of T , onto itself which keeps the segments i aki‘bki fixed and takes Tkirlg3gzgl(D) to T irIb. Define 16 f : E4 4 E4 by extending over T = mk T , and over the v , k! 2 k. ki i i 1 region G defined by l/k!+1 S x < l/k! such that f is ki i 1 - i vk. i the identity on the boundary of this region. Define gs: E4 4 E4 to be f on G and the identity outside these Vk: ki 1 regions. For vertices between vk and VR at which D is i i+1 locally flat define a map 96 which takes the flat disk 1‘1 ki+l_l the upper boundary of .8 between . and , _ . The aki+l aki+1 1 bounded by alebk,+1 and ak, b to a ribbon along i i i+ composition of these maps 9 = 969594939291 satisfies the lemma. Lemmas 2.2 to 2.7 prove our main theorem 2.1. Section 3. wl(E4-fi) 3 0. In [8]. Fox and Artin give several examples of arcs or l-disks which are wildly embedded in S3. some whose complements fail to be simply connected. some whose com- plements are simply connected but fail to be E3. and some whose complements are E3. We show that any 2-disk D in S4. locally polyhedral at every point except at one boundary point P. has a come plement S4-D which is E4. Let S4 be the boundary of the 5-ball {(xlx ....xs) 6 E5: 0 S x S 2. -2 S xj S 2 for 2 5 j = 1' 2' 3' 4}. 17 Lemma 3.1: If D is a polyhedral 2-disk in S4 and C a compact set in S4-D then there exists a PL 4-ball neighborhood N such that D c N and.N’FIC = ¢. ggeefz By induction on the number of triangles in D. Trivial for n = l where D is a triangle and is at a distance 6 from C. Fatten D to a 4-ball with thickness 6/2 in two orthogonal directions. Assume true for disks of n-l triangles and let D have n triangles. Let T'be a triangle of D with at least one edge on the boundary of D and such that D - T is a pothedral disk D' of n~l triangles. Then by the inductive hypothesis D' has a 4-ball neighborhood N and T has a 4-ball neighborhood NT of the type above and N'FIC = ¢ = fiqu C. TPFID' is a segment or pair of segments and in either case the distance from T FID' to N is greater than some 9 > 0. Then there is a PL homeomorphism.h fixed outside N 'and fixed on D' such that h(D) G N. Then h-1(N) is a PL T , _ _ 4-ball such that D C h 1(N) and h 1(N) FIC = ¢. OR g NT A v h ) D. Figure 3 18 Qerellagy 3.2: Given a polyhedral 2-disk D in S4 and compact C : S4-D there exists a topological 4-cell neighborhood N of C such that N C S4-D. That is. polyhedral 2-disks are cellular in S4 and have complements E4. Theerem 3. : Given a 2-disk D in S4 which is locally polyhedral at every point except at one boundary point P and compact set C in 84-D. then there is a PL 4-ball N containing D such that N FIC = ¢. ggeef: Let D be in the position of the conclusion of lemma 2.6. There exists a cube neighborhood N'(P.r) such that N'(P.r) FIC = ¢. There exists a natural number i such that Ri e N'(P.r) and N(P. IRiI ) me = (I. D flN(P. IR“) is the segment aibi while aibi divides D into two disks D' which is polyhedral and D" Which contains P. By lemma 3.1 there is a PL 4-ball B containing D' which is disjoint from C and B is at a distance from C greater than some 6 > 0. Then there is a 4-cube neighborhood U C N(P. |Ri| ) such that D" c U. D" FIU = aibi' N(P. IRiI )IW U = aibi. and a PL homeomorphism h fixed outside N(P. lRil ) such that h(U) C B. Then h—1(B) is a PL 4-ball containing D and disjoint from C. Then 54-h-1(B) is topologically a 4-cell containing C and itself contained as an open set in 54-D. If every compact set of the countably compact space S4-D is contained in an open 4-cell. then S4-D is a monotone union of open 4-cells and by the result of Morton Brown in [2] 'we have: l9 Corollagy 3,4: If D is a disk as in theorem 3.3. then . 4 S4-D is topologically E . And then since E4-D is topologically E4 with a point removed we have: Corollagy 3.5: wl(E4-fi) a 0. _ection 4: Z-Meeifolds in E4 The 2-disk is the building block of the 2-manifold so we make a few remarks relating our results to 2-manifolds in E4. Let M? be a compact 2-manifold with boundary M2 embedded in E4 in such a way that M2 is locally polyhedral at every point except at one boundary point P. Using the technique of the proof of lemma 2.2 we can find a sequence of polygonal arcs {Ai] with end points on the boundary of M2 on each Side of P converging to P and such that each arc Ai divides M into a polyhedral 2-manifold and a 2-disk locally polyhedral except at P. Let AO divide M2 into M: and D0° M: will fail to be locally flat at at most a finite number of points. v all in the Int M2. In a finite number of steps 13...,Vk, O we can add these vi to DO by forming a new disk by adding a finite chain of triangles in Int M; reaching from A0 to say v thus forming a new disk D_ with complement in M2 l l a polyhedral 2-manifold M3 1 divided by a polygonal arc A_1. Continue in this way to form the disk D_n and polyhedral 2-manifold Min divided by polygonal arc A-n' Min is also locally flat at every point. By lemma 2.3 there is a PL homeomorphism h: E4 4 E4 such that h(A-n) is a segment giving the following: 20 Theorem 4,1: Given a compact 2-manifold M2 in E4 which fails to be locally polyhedral at just one point P 6 M2. there exists a PL space homeomorphism h: E4 4 E4 and a segment. S. on h(Mz) such that S divides h(MZ) into a locally flat. polyhedral 2-manifold M2' and a 2-disk D' which is locally polyhedral except at the boundary point h(P). If D" is any other 2-disk which arises in this way from M2. there is a space homeomorphism g: E4 4 E4. locally PL except at h(P) and such that g(D') = D". since these embeddings depend only on the knot types at the vertices at which M2 fails to be locally flat. Now suppose the point P at which M2 fails to be locally polyhedral is in Int M2. Using an argument similar to that used above. we can find a polygonal S1 in Int M; which divides M2 into a locally flat polyhedral 2-manifold M2' and a 2-disk D' which fails to be locally polyhedral at just one interior point P. On this disk there is a sequence {Ci} of non intersecting polygonal simple closed curves converging to P. with all points at which M2 fails to be locally flat in the interior of the annuli [Ci:C We can start at . 1 . . i+1]. . points a and b on S With a pair of arcs A and B which on each [Ci:Ci+1] are polygonal and bound a 2-disk which misses the finite number of points of [Ci:Ci+l] at which M2 fails to be locally flat adding in the point P. A and B together with the appropriate half of S1 between a and b form a simple closed curve C which is locally polyhedral except at P. Then as in lemma 2.2. there is an almost PL space homeomorphism f: E4 4 E4 such that f(C) is polygonal. f(C) divides f(Mz) into a 2-disk D which is locally polyhedral except at one boundary point f(P) and a 2-manifold M? which is locally 1 21 polyhedral except at one boundary point f(P) and which is locally flat at every point. This last since it is locally flat except possibly at the boundary point f(P); but then by theorem 4.2 of Tindell in [13]. Mi is locally flat at f(P) also. Note that M2 is not necessarily locally flat at f(P). This gives us: Iheerem 4,2: Given a compact 2-manifold M2 in E4 which fails to be locally polyhedral at just one point P 6 Int M2. there exists a space homeomorphism f: E4 4 E4 and a polygonal simple closed curve f(C) in Int f(M2) which divides f(Mz) into a locally flat 2-manifold Mi which is locally polyhedral except at the boundary point f(P) and a 2-disk D which is locally polyhedral except at the boundary point f(P). CHAPTER II POLYHEDRAL 2-MANIFOLDS IN E4 Section 1: Madisog Ezoblems Using the results of Gugenheim throughout the investi- gations for Chapter I stimulated interest in the polyhedral category. It was noticed that two problems which were posed in Madison in [12] could be answered negatively using results obtained by Gugenheim. The first of these problems appeared in [12] as number 5 on page 56 as: If C is a polygonal simple closed curve in E2 and D a polyhedral disk in E4 of which C is the boundary. does there exist a PL space homeomorphism h: E4 4 E4 such that h(C) = C and h(D) C E3? The answer is no. Pgeefi: Let K‘be a polygonal trefoil knot in E3 and let P be a point of E4 - E3. The join P.K is a polyhedral 2-disk with boundary K. call it D'. By theorem 5 of [8] there is a PL homeomorphism f: E4 4 E4 such that f(K) is a polygonal simple closed curve in E2. Let the D of the question be f(D') with polygonal simple closed curve boundary in E2. C = f(K). If there exists a PL homeomorphism h: E4 4 E4 such that h(C) = C and h(D) C E3 (even Without the restriction h(C) = C) then h(D) would be a 2-disk in E3. But all polyhedral 2-disks in E3 are equivalently embedded and hence h(D) would be flat. In particular. h(D) would be locally flat at h(f(P)) and D' would be locally flat at P which it is not. 22 23 The second problem appeared as number 6 on page 56 of [12] as: If K is a polyhedral 2-sphere in E4 does there exist a polyhedral 3-ball in E4 of which K is the boundary? No. The suspension of a polygonal trefoil knot in E3 forms a polyhedral 2-sphere in E4 which fails to be locally flat at its suspension points. But every PL 3-ball in E4 is equivalently embedded by 7.33 of [10] and is therefore flat and thus its boundary is locally flat at every point. Thus no PL 3-ball has the suspension of the trefoil knot as its boundary. SQQLiQE 2: A Pelyhedral 2-sphere which Eeils to be Lecally Flat at Jest Qee Point In [4]. Cantrell and Edwards constructed a 2-sphere in E4 which failed to be locally tame at just one point P and was locally flat at every point except possibly P. In [14]. Tindell showed that the example constructed by Cantrell and Edwards did in fact fail to be locally flat at P. The construction of polyhedral 2-spheres in E4 which fail to be locally flat at two points by suspending polygonal knots in E3. together with the facts of the above paragraph concerning a wild sphere in E4 which fails to be locally flat at just one point. brings us to the following question. Does there exist a polyhedral 2-sphere in E4 which fails to be locally flat at just one point? We answer in the affirmative by constructing an example. Example 2,1: Let K be a polygonal trefoil knot in the 3-dimensional hyperplane E3 in E4 defined by x4 = 0. Let K 24 be so situated that of its vertices. v0.....vn. v0 = (0.0.0.0) and the other n vertices have first coordinate greater than 1. Let a = (0.0.0.1) and b = (0.0.0.-l) and form a polyhedral 2-sphere. S. by joining a to K and b to K. This sphere is locally flat except at a and b. Let H be the 3-dimensional hyperplane defined by x1 = 1. We want to see first that S FIH is a non-trivial polygonal knot in H. Let r: E3 4 E3 be defined by r(xl.x2.x3.0) = (‘X +2.x .x3.0). i.e.. reflection through the plane x = l. l 2 l = = ' = " x4 0. Let H FIva1 v and H FIvan v and let A be the polygonal arc on K from v' through V1 to v". Denote by K'. r(A) LIA. the connected sum of a right and left handed trefoil knot. which is non-trivial since knots under con- nected sums form a semigroup Without inverses (see page 164 of [5]). 3 . . : H denote the restriction of Let p Exlzl. 4 OSX4<]. the projection from E3 into H through a to the points of E3 - > . 3 _ With x 1. Let q. Ex >]_ denote the re - H> - l 0-x4>l striction of the projection from E3 into H through b to the points of E3 with x1 2 l. p and q are homeomorphisms and on 3 H FIE they are both fixed pointwise. Define h: E3 4 H -l