A aa-ésxaacwamzma GF camm L3 35*:15W3LF‘S Thesis for the Degree of 'Ph, 9. MlCéfiiGAN STATE UNWERSETY GERHARD WALTER KNUTSON 1968 LIBRARY H Michigan State ' University {3 11—15539. This is to certify that the thesis entitled A CHARACTERIZATI ON OF CERTAIN CLOSED 3-MAN I FOLDS presented by Gerhard W. Knutson has been accepted towards fulfillment of the requirements for Ph.D. degree in Mathematics Date JU1Y 8. 1968 0-169 i . amuse BY ' F ‘uons & sons 9 mwnmtta »‘ ABSTRACT A CHARACTERIZATION OF CERTAIN CLOSED 3—MANIFOLDS by Gerhard Walter Knutson Let M be a closed connected combinatorial 3—manifold. A compact subcomplex A of M is a residual set of M if M is the disjoint union A U U of an open 3-cell U dense in M and a non-separating continuum A of dimension less than 3. The singular set of A, S(A), is the set of points of A that do not have an open 1- or 2-dimensional euclidean neighborhood in A. In this thesis we examine the relationship between A and M. ‘In particular we show that if A does not contain a wild are then we may pick A so that S(A) is a point. Then we prove the following theorem: M has a residual set that contains no wild arc if and only if M is the con— nected sum of closed 3-manifolds each of which is topologi- cally the 3-sphere, real projective 3-space, 51 x 82, or the twisted S2 bundle over 81. We also show that A may be pidked so that A - S(A) is the disjoint union of Open arcs and the interiors of com- pact 2-manifolds with connected boundaries. Under this as- sumption, if S(A) is a simple closed curve, M is the con- nected sum of closed 3-manifolds each of which is topologi- cally 81 x 82, the 3-sphere, real projective 3-space, the twisted 82 bundle over SI, or a lens space. A CHARACTERIZATION OF CERTAIN CLOSED 3-MANIFOLDS BY Gerhard Walter Knutson A THESIS Submitted to Michigan State University in partial fulfillment of the requirements for the degree of DOCTOR OF PHILOSOPHY Department of Mathematics 1968 ACKNOWLEDGMENTS The author wishes to express his gratitude to Professor P. H. Doyle for suggesting the problem and for his helpful suggestions and guidance during the research. ii To My Mother and Father iii CONTENTS CHAPTER Page I. INTRODUCTION . . . . . . . . . . . . . . . 1 1. Homology and Homotopy of Residual Sets 1 2. Local Connectivity of A relative to M o o o o o o o o o o o o o o o o o 4 II. TOROIDAL MANIFOLDS . . . . . . . . . . . . . 7 1. Residual Sets of Connected Sums . . . 7 2. Residual Sets of Disk Sums . . . . . . 8 3. Toroidal 3-Manifolds . . . . . . . . . 9 III. A CHARACTERIZATION OF CLOSED 3-MANIFOLDS WITH RESIDUAL SETS CONTAINING NO WILD ARC . 16 1. A is a One-Point Union . . . . . . . 16 2. The 2-Manifolds of A . . . . . . . . 20 3. Reconstruction of M . . . . . . . . . 24 4. Compact 3-Manifolds with Boundary . . 31 IV. A CHARACTERIZATION OF CERTAIN CLOSED 3- MANIFOLDS WHOSE SINGULAR SET IS A SIMPLE CLOSED CURVE . . . . . . . . . . . . . . . . 34 1. The Singular Set . . . . . . . . . . 34 2. Residual SEts with S(A) a Simple Closed Curve . . . . . . . . . . . . . 44 BIBLIOGRAPHY . . . . . . . . . . . . . . . . 54 iv LIST OF FIGURES FIGURE Page 2.1 . . . . . . . . . . . . . . . . . . . . . . 14 4.1 . . . . . . . . . . . . . . . . . . . . . . 48 CHAPTER I INTRODUCTION In 1962 Doyle and Hocking established a decomposition of a closed n-manifold into an open n-cell and a non—sep- arating continuum of dimension less than n. In this thesis we start with the continuum and under certain conditions reconstruct the manifold. Since our concentration is on connected 3-manifolds, we assume that all our manifolds are combinatorflfl.and connected. Furthermore, all subsets are simplicial and all maps are piecewise linear. In this chapter we establish some elementary relations between the manifold and its decomposition. 1. Homology and Homotopy of Residual Sets Definition 1.1.1: Let M be a compact n—manifold. A com- pact subcomplex A of M is a residual set of M if M is the disjoint union M = A U U of an open n-cell U dense in M and a non-separating continuum A of dimen- sion less than n. A U U is called a decomposition of M. We remark that if M has non-empty boundary, the boundary of M is contained in A. Therefore we must not confuse the residual set with a spine [17]. However, if A is a residual set of M then A is a Spine of M less an Open n-ball of the interior of M. 1 2 Note that we will assume that a residual set does not collapse onto any proper subset of itself. .In [5] Doyle and Hocking prove that every compact n- manifold has a decomposition. The Brown-Casler Theorem [2] asserts the existence of a continuous function f from the closed n-ball Bn onto M such that f|Int Bn is a homeo- morphism, f-1f(Bd Bn) = Bd B“, and dim f(Bd Bn) < n. Thus if M = A U U is a decomposition of M, we will always assume we are given the map f: (Bn,Bd Bn) > (M. A). It will be useful to establish the relationship be- tween the homology and homotopy groups of A and M. Since M is the adjunction Space obtained by attaching Bn to A by means of f, the pair (M,A) is a relative n-cell. n-1) ' e ’ M’A = “ S H nce Hq( ) Hq_1( for 0 < q <-n [10]. Note that we will use equality to mean for all q and Wq(M4A) = 0 group isomorphism or space homeomorphism whenever no con— fusion is likely. n-1 . . ) IS the above isomor- > fi _ q 1(5 fh: ,A I q Hq(M ) h' d f : “’1 p ism an Hq( ) Hq(S ) > Hq(A) is the homomorphism induced by f, we obtain the commutative diagram: Hq(M) > Hq(M,A) > Hq_1(A) > Hq(M) hq Hq_1§f) a n-1 Hq_1(S ) where the unnamed maps are the maps of the exact homology sequence of the pair (MJA). 3 Using these definitions we state two well known theorems and an immediate corollary. Theorem 1.1.2[8]: The following hold; ' = f , -1 1) Hq(M) Hq(A) or q # n n 11) Hn-1(M) = Hn_1(A)/Im Hn_1(f) and > O iii) 0 > Hn(A) > Hn(M) > ker Hn-1(f) is exact. Corollary 1.1.3: If dim A < n-l, M is orientable. Proof: If dim A < n-l, Hn(A) = O = H A) and so n-1( Im Hn_1(f) = 0. Thus ker Hn_1(f) = z and so Hn(M) = 2. Hence M is orientable. Theorem 1.1.4: vq(M) = w A) for 0 :_q < n-1. q( Theorem 1.1.5: Let M be a closed n-manifold. Let A be a residual set of M with dim A < n/2. If n is odd sup- pose that H(n-1)/2 (A) is torsion free. Then M is a homology n-sphere. Egggg: If n = 1 or 2, A is a point and so M is a Sphere. If n = 3, A is a 1-complex and hence A is the homotopy type of an r-leafed rose. By Corollary 1.1.3 M is orientable. Thus ker H2(f) = Z and Im H2(f) = 0. By 'Dheorem 1.1.2, H2(M) = H2(A). By Poincare duality H1(M) = H2 (A) = 0. Thus H1 (A) = 0 = H1(M) and so A is contract- able. Hence M is a 3-sphere. 4 Suppose that n > 3. Then M is orientable and Hn_1(A) = Hn_1(M). By Theorem 1.1.2, we obtain: Z q = O or q = n Hq(M) = 0 n/Z fi.q :.n-1 Hq(A) 1 :.q < n/2. By Poincare duality Hq(M) = Hn-q(M)° Hence Z q 0 and q = n Hq(M) = 0 1.: q < n/2 H (A) n/2 _<_ q :n-1. By Theorem 5.5.3 of {14], Hq(M) = Hom(Hq(M),Z) 0 Tor(Hq+1(M)). Hence Hq(M) = 0 if q # 0 or n and H0(M) = 0 = Hn(M). Corollary to the proof: If M is a closed 3-manifold with a residual set of dimension 1, M is a 3-sphere. 2. Local Connectivity of A Relative to M Let X be a separable metric space and A a subset of X. Let x be an element of X. We say A is locally p- connected in the sense of homotopy at x (p-LC at x) if for every Q > 0 there is a a > 0 such that each map f : s9 —-—-> sx(5) r} A is null homotopic in sx(z;) n A [.7], where SX(§) is an Q-ball centered at x. A iis locally Ap:connected in the sense of homotopy in relation to X (A is p-LC rel X) if A is p-LC at x for each x in X. 5 Lemma 1.2.1: Let M be a closed n-manifold with a decom- position M = A U U. If U is p-LC rel M for O fi.p :.k, then dim A fi.n — (k+2). ‘ggggfi: Let B be a Simplex of A with maximal di- mension m. Let x be an interior point of B. Then x has a neighborhood N in M with (N,N n U) = (Rn,Rn - Rm). Thus N n U contains an (n — (m + 1))-sphere that does not bound in N n U. Hence U is not (n -(m + 1))-LC rel M. Since k < n - (m + 1) by definition of p-LC, and m = dim A, it follows that dim A :,n - (k + 2). Corollary 1.2.2: If U is O-LC rel M, then M is orient- able. Proof: From Lemma 1.2.1, dim A i.n-2 and so, by Corollary 1.1.3, M is orientable. Corolla£y 1.2.3: If M is a closed 2- or 3-manifold and U O-LC rel M, then M is a Sphere. Corollary 1.2.4: Let M be a closed 4-manifold with U O-LC rel M. If 'M is not a 4-sphere dim A = 2. To see that Corollary 1.2.4 cannot be strengthened, consider S2 x 32. This manifold has a residual set that is topologically the one-point union of two 2—Spheres. .Qprollary 1.2.5: Let M be a closed n-manifold and suppose that U is p-LC rel M for 0 j,p :_n-3. Then M is an n—sphere. 6 A concept similar to p-LC is obtained using Singular chains and cycles. Using the corresponding definitions we obtain similar results. CHAPTER II TOROIDAL MANIFOLDS In this chapter we investigate the relationship be- tween the residual set of a connected sum and the residual sets of the summands and between the residual set of a disk sum and the residual sets of the summands. Finally we will investigate the residual set of a toroidal manifold. 1. Residual Sets of Connected Sums Definition 2.1.1: Let M and M' be two closed combina- torial n-manifolds. The connected sum M # M' is obtained by removing the interior of a closed n—ball from each mani- fold and matching the resulting boundaries by means of a piecewise linear homeomorphism. If the manifolds are orientable this sum is not always well defined unless the homeomorphism is orientation reversing. >When we write M # M' we will imply that the sum is well defined. In latter chapters we will use the connected sum of 3-manifolds. In the construction it will follow that the homeomorphism will be orientation reversing whenever neces- sary. We note that if M has an orientation reversing self homeomorphism, M is homogeneous in the sense of Brown and Gluck, and so M #’M' is well defined. We remark that S3. 31 x 52, and RP3 (real projective 3-Space) have orienta- tion reversing self homeomorphisms. 7 8 Theorem 2.1.2: If A and A' are residual sets of the closed n—manifolds M and M'. M #'M' has a residual set homeomorphic to the one.point union of A and A' (writ- ten A VA'). Proof: If we pick the n-balls of the connected sum to be n-Simplexes of some triangulation of M, and M' that. meet the respective residual sets at a point, the theorem follows. 2. Residual Sets of Disk Sums Definition 2.2.1: Let M and M' be connected compact n- manifolds with connected non-empty boundaries. The disk sum M A M' is obtained by pasting an (n-1)-ball of Ed M onto an (n-1)—ball of Ed M'. Theorem 2.2.2: If A and A' are residual sets of the compact n-manifolds M and M'. where Bd M and Ed M' are connected and non-empty, -M A M' has a residual set homeomorphic to the Space obtained by removing the interior of an (n-1)-ball from both A and A' and sewing the re- sulting sets together along the boundaries of the removed balls. Proof: If we pick the (n+1)-balls of the disk sum to be (n-1)-Simplexes of Ed M and Ed M'. under some triangu- lation of M and M'. the theorem follows. 9 3. Toroidal 3—Manifolds It is well known that each closed connected orientable 3-manifold M may be obtained by sewing two solid tori of the same genus together by a boundary homeomorphism. We investigate M when we know how M is obtained from two tori. In particular we investigate toroidal manifolds. Theorem 2.3.1: Let M be a closed orientable 3-manifold. Suppose that T1 and T2 are two solid tori of genus n, and h is a homeomorphfimlof Bd T1 onto Bd T2 with M = T1 Uh T2. Let Ti = Ai U Ui be a nice decomposition of Ti' Then there is a 2—cell C in A1 with Int C open in A1 and h(Int C) open in A2 such that M has a decomposi— tion M = A U U, where U = (U1 U Int C) U1‘] (U2 U h(Int C)) and A = (A1 Int c) Uh (A2 - h(Int c)). Proof: A nice decomposition of a solid torus T of genus n, T A U U, is obtained by taking A to be the boundary of T plus n disjoint 2-cells C1. ---. C n where Ci 0 Bd T = Bd C1 and Ed T - U Bd Ci is a sphere i=1 with 2n holes. For example in the torus of genus one, n T = 51 x B2, A would be (S1 x Bd B”) U (p x 8’) where p is a point of 81. We will consider T1 and T2 as submanifolds of M with M=T1UT2 and T10T2=BdT1=BdT2. Then M= A1 U A2 U 01 U U2. Since A1 U A2 is 2-dimensiona1, there is a 2-cell C in T1 n T2, such that C is the carrier of a 2-Simp1ex of some triangulation of M. Then M has a 5‘ 10 decomposition of the desired form, namely M = ((A1 - Int C) U (A2 - Int c)) U (U1 U IntC UU2). Corollary 2.3.2: Any closed orientable 3-manifold has a residual set which is an orientable surface of genus n, less an open 2-cell, to which 2n 2-disks are attached by means of homeomorphisms of their 1-Sphere boundaries. If M is a 3-manifold obtained by attaching two solid tori of genus n by a boundary homeomorphism, we will call M an n-tuple toroidal manifold or an n-TM. Bing [1] has Shown that any l-TM is either the 3—sphere, 81 x S2, or a lens Space. In Chapter IV we will need to know the residual set of a l-TM, so we turn our attention to that goal. Eadh l-TM is obtained by attaching two solid tori T1 and T2 by an appropriate boundary homeomorphism. We now describe such a homeomorphism. Let Mi and Li be meridianal and kmgitudinal Simple closed curves on Bd Ti' for i - 1 or 2. Suppose that n and m are relatively prime positive integers. Let a1, -~-, an be n points on M2, cyclicly ordered by their subscripts. Let J(n,m) be a Simple closed curve on Bd T2 that meets M2 at the n points ai, with the ai cyclicly ordered on J(n,m) as a1, a , ..., a(n-1)m+1° Let h be a homeomorphism of Ed T1 onto Bd T2 such that h(Ml) = J(n,m). Define T(n,m) to be the adjunction Space T1 Uh T2. Set J(1,0) = M2 and J(0,1) = L2. Then S1 x 82 = T(l,0) and S3 = T(0,1). Since isotopic maps 11 yield homeomorphic l-TM's and since each isotopy class of homeomorphisms of Ed T1 onto Bd T2 has a representative that maps M1 onto J(n,m), each 1-TM is a T(n,m) mani- fold. To obtain a decomposition for T(n,m), we consider T1 and T2 as submanifolds of T(n,m) with T1 U T2 - T(n,m) and T1 0 T2 = Bd T1 - Bd T2. In T1, J(n,m) is a meridianal simple closed curve and so bounds a disk D in T1 with Int D C Int T1. Then T1 has a residual set Bd T1 U D. Considering T2 as B2 x 81, where B2 is the closed unit 2—ball, let G be the simple closed curve in T2 corre— 3ponding to (0) x 51. Let B be the singular annulus ob- tained by pushing J(n,m) onto C by a radial projection; that is, B corresponds to the image of the function F : J(n,m) x I > 82 x 81 defined by F(((x,y),s),t) = (((1-t)x,(1-t)y),s), where (x,y) is a point of J(n,m) and s lies on 81. Since T(n,m)-(B U D) is an open 3-cell B U D is a residual set of T(n,m). Notice that B U D is topologically the quotient space of an n-gon ob- tained by identifying each edge with a simple closed curve in an orientation preserving manner. In [3] Casler defines a standard Spine of a 3-manifold with non-empty boundary. Following this definition we will define a standard residual set of a closed 3-manifold. Let K be a 2-complex. A vertex v of K is of type I if v has a 2-cell neighborhood, of type II if v is not of type I and has a 3—book neighborhood, and of type III 12 if v is not of type I or II and has a neighborhood homeo- morphic to the cone over a set consisting of a circle to- gether with three of its radii. K is a standardgg:gomplex if the following hold: i) each vertex of K is of type I, II or III. ii) K less its singular 1-skeleton, K1, is a count- able number of disjoint open disks, and iii) K1 less the Singular 0-skeleton of K1 is the sum of a countable number of pairwise disjoint open arcs. If A is a standard 2-complex and if A is a residual set of a closed 3-manifold M, then A is defined to be a standard residual set. Likewise A is a standard Spine if A is a standard Z-complex and A is also a Spine. The main result of [3] is: Theorem 2.3.3: If K is a standard Spine of a compact 3— manifold M with non-empty boundary and K' is a standard Spine of a compact 3-manifold M' with non-empty boundary, and if K and K' are homeomorphic, then M is homeomor- phic to M'. Recall that we are in the piecewise linear category so that the above homeomorphisms are piecewise linear. Corollary 2.3.4: If two closed 3-manifolds have homeomorphic standard residual sets, then the manifolds are homeomorphic. 13 Proof: We need only note that a standard residual set of a closed 3-manifold is a standard spine of the manifold less an Open 3-Simplex and then apply Theorem 2.3.3. We would now like to find a standard residual set for the manifold T(n,m). If n ? 2, B U D is a copy of the real projective plane. By a result of Hocking and Kwun [9], T(2,m) is real projective 3-space. Since T(3,1) and T(3,2) are homeomorphic we need only consider T(3,1). For T(3,1), s u D is'a 3-book with its ends identified after a twist of 120 degrees. Let v be a vertex Of B U D Of type II. Let N be a 3-book neighborhood of v in B U D, with pages P1, P2 and P3. Swell up v to a 3-cell C that meets B U D in a disk E contained in P1 U P2 with E contained in the boundary of C. By collapsing C onto a copy of Bing's house with two rooms leaving E fixed, we obtain a standard residual set for T(3,1). If n.Z 4, we do not consider the residual set B U D, but rather start all over. As before T(n,m) - T1 Uh T2, and D is the same nice disk in T2. We now decompose T1 into two Open sets U1 and U2, each topologically the upper half 3-space, and a continuum H. -Then U1 U U2 U Int(T2-D) is an open 3-cell and if H is sufficiently nice H U D is a standard residual set for T(n,m). To construct H, consider T1 as being Obtained as the identification Space of Bzrx I, under the action of a homeomorphism f Of B2 x (0) onto 82 x (1), i9+(2v/m) 1). defined by f(tele,0) - (te Then T1 is 14 homeomorphic to (B2 x I)/R, where R is the equivalence relation x §'f(x). Let p be the composite map p : 32 x I > (B2 x I)/R > T1. where the first map is the quotient map and the second is the above homeomorphism. \L, //11_ Figure 2.1 Let L1, ---, Ln be n arcs in Ed (B2 x I) such that n . p(U Li) 3 J(n,m). Now consider B2 x I as a cube with 1 B2 x (O) as top and B2 x (1) as bottom. Furthermore consider L1, L2, L3 and L4 as the four edges on the Sides of the cube. The remaining arcs, L5, ---, Ln' are on the side that has L1 and L4 as edges. Now collapse B2 X I onto a copy of Bing's house with two rooms as in Figure 2.1. Setting H equal to the image of the house under the map p, it follows that H U D is a residual set Of T(n,m). '5‘ 15 However, H U D was so constructed that, if a little care is taken as to how we collapse onto the house, H U D will be a standard residual set for T(n,m). — _- -—_ Jun-x1 CHAPTER III A CHARACTERIZATION OF CLOSED 3—MANIFOLDS WITH RESIDUAL SETS CONTAINING NO WILD ARCS In this chapter we establish that a closed 3-manifold has a residual set containing no wild arc if and only if it is the connected sum of closed 3-manifolds each of which is homeomorphic to a 3—sphere, real projective 3-space, 81 x 52, or the twisted S2 bundle over 81. Finally we establish a Similar characterization for compact 3-manifolds with boundary. 1. A Is a One-Point Union Definition 3.1.1: An arc B in a complex X is wild if there does not exist a homeomorphism of X onto itself carrying B onto a polyhedral arc of X. Since a trefoil knot may be embedded in a 3-bOOk, a 3- book contains a wild arc [14]. Thus if A is a residual set Of a closed 3-manifold that does not contain a wild arc, then A does not contain a 3-book. Recall that A does not collapse onto any subcomplex of itself. If v is a vertex Of A and N(v,A) the second derived neighborhood of v in A [16], we may classify the vertices of A into three disjoint types: 1) N(v,A) is an arc, ii) N(v,A) is a disk, and iii) N(v,A) is the one-point union of arcs and disks. 16 17 Lemma 3.1.2: Let A be a residual set for the closed 3- manifold M which contains no wild arcs. Then there is another residual set A' for M that is the one-point union of closed 2—manifolds and 1-spheres. 2522:: If the dimension of A is less than two, M is a 3-sphere and SO A is a point. Suppose that A has dimension two and that a is a vertex of A. Let St(x,X) and Lk(x,X) be the star and link of x in the second derived subdivision Of a complex triangulating X [16]. Then St(a,M) is a 3-ball with St(a,A) contained in St(a,M) as the join Of a with Lk(a,A). Since Lk(a,A) = Lk(a,M) n A is the disjoint union Of p.1 0 1-spheres and q 3.0 points, we may associate with a the pair (p,q) and a will be called a (p,q)-point. We will define a series of moves that change A into .N; where A' contains only (1,0)-points, (0,2)-points and one (m,n)-point. MOVE A: Let a be a (p,q)-point of A with pq > 2. Let x be an isolated point of Lk(a,A). In Lk(a,M) there is an arc C with Bd C = A D C - x U y, where y is a point of a 1—sphere Of Lk(a,A). There is a 2-cell B in St(a,M) with A n B = A n Bd B = aox U aoy and Bd B = aox U aoy U C. Here "o" denotes the join Operator. An A-move expands A to A U B and collapses from aox across B onto Cl(A-aox) U C. 18 MOVE B: Let a be a (p,0)—point Of A with p > 1. There is a l-Sphere S of Lk(a,A) that bounds a 2-disk D in (Lk(a,M) - Lk(a,A)) U S. Thus there is a 3-cell C in St(a,M) with A n C = A n Bd C = aoS and Bd C = aOS U D. A B—move expands .A to A U-C and collapses onto (A-aoS) U aoy U D, where y is a point of S. MOVE C: Let a be a (O,q)-point of A with q > 2. Suppose that x is a point Of Lk(a,A) and aox may be extended to an arc B in A with Ed B = a U y, where y is a (p,q)-point of A with pq # 0, such that Int B con- tains only (0,2)-points. Let 2 # x be a point of Lk(a,A). There is a 2-cell C in M with A n C = A n Bd C = B U aoz. Let D = Cl(Bd C - (aoz U B)). A C-move expands A to A U C and collapses from aoz across C onto (A-aOZ) U a U D. MOVE D: Let a be a (1,1)-point of A with x the isolated point of Lk(a,A). Suppose that y is a (p,q)- point of A in the same 2—chainable component of A as a. There is an arc C in A, with Bd C = a U y and Int C containing only (l,O)-points, and a 2-cell B in .M with A n B = A 0 Bd B = C U aox. Let D = Cl(Bd B - (C U aox)). A D-move expands A to A U B and collapses from aox across B onto (A-aox) U a U D. MOVE E: Let a be a (1,1)-point of A with x the point and S the 1—Sphere Of Lk(a,A). Suppose that aox may be extended to an arc B in A with Bd B = a U y, 19 where y is a (p,q)-point of A with pq # O, and Int B containing only (0,2)-points. There is a 3-cell C of M with A n C = B U aoS and A n Bd C = y U aOS, such that C collapses onto B U aOS. ‘Let. D = Cl(Bd C - aos). An E- move expands A to A U C and collapses from aOS across C onto (A — (B U aoS)) U D. We observe that each move transforms A into a resid- ual set Of M. By a finite series of A-moves we may assume that each vertex of A is either a (1,1)-point, a (O,q)- point or a (p,O)-point. By A- and B-moves we may assume that each (p,0)-point of A has p = 1. By A- and C- moves each vertex of A is a (1,1)-, (1,0)— or a (0,2)- point. By D- and E-moves we Obtain the desired form. Lemma 3.1.3: If A is as in the conclusion Of Lemma 3.1.2, where A is the one-point union Of n 1-Spheres and m closed 2-manifolds, n j,m. Egggfi: Since Theorem 1.1.2 holds for arbitrary coef- ficients, it follows that H2(M;Zz) ='H2(A:Z2) and H1(M;Zz) = H1(A;Zz). By Poincare duality and the universal coeffici- ent theorem for cohomology, H1(A;Z2) = H2(A:Z2). Since H2 (A;Zz) = #495 n Z2, and O Z2 C H1(A;Zz), the lemma follows. 1 Remark 3.1.4: Given m Z-n 3.0 there is a closed 3-mani- fold with a residual set that is the one-point union of n 1-spheres and m closed 2-manifOldS. The connected sum Of n copies of 81 x 82 and m-n OOpies Of RP3 has the desired residual set. 20 Remark 3.1.5: By swelling up a principal Simplex of a re- sidual set and collapsing onto a OOpy of Bing's house with two rooms we see that every 3-manifold has a residual set that contains an arc that is wild. If A is as in the conclusion of Lemma 3.1.2, and M is not a 3—Sphere, then m f 0 and so M has a non-trivial second homology group with 22 coefficients. Thus we Ob- tain: Corollary 3.1.6: A residual set of a counter example to the 3-dimensional Poincare conjecture must contain a wild arc. 2. The 2-Manifolds of A If A is a residual set that is the one-point union of n 1-spheres Si and m 2-manifolds Pi we will call A an (n,m)—residual set. Again N(X,M) will be the second derived neighborhood of X in M. We will set N(X) = N(X,M) if the manifold M is understood. Remark 3.2.1: If S is a 1-sphere embedded in a closed 3- manifold M, N(S,M) is either a solid Klein bottle or a solid torus. Notice that if M is non—orientable N(Sl) may be either a solid torus or a solid Klein bottle. -For example in J, the twisted $2 bundle over 81, both types of neighborhoods are easily found. 21 Lemma 3.2.2: A regular neighborhood of a compact 2-mani- fold embedded in the interior of an orientable 3-manifold is topologically independent of the 3—manifold. ‘ggggg: Suppose that P and Q are isomorphic, (that is, P and Q are homeomorphic under a simplicial map), compact 2-manifolds simplicially embedded in the interior of two orientable 3-manifolds M and N respectively. Triangulate M and N so that under the induced triangula- tion P is isomorphic to Q. Now N(P,M) is a solid torus H of genus n plus some 3-cells attached to H along annuli. Also N(Q,N) is a solid torus K of genus m plus some 3-cells attached to K along annuli. Since P and Q are isomorphic, we may take H and K as the second derived neighborhood of the respective l-skeletons so that n = m. Moreover the isomorphism of P onto Q extends to an isomorphism of P U H onto Q U K. By collapsing P U H and. Q U K care— fully, we Obtain P' and 0', standard spines of N(P,M) and N(Q,N) respectively. Furthermore, P' and Q' will be isomorphic. By Theorem 2.3.3, N(P,M) is isomorphic to N(Q,N). The above collapse is Obtained by collapsing each neighborhood Of a vertex to a copy of Bing's house with two rooms, so that the house meets the tubes of H and K in disks whose interiors are open in the house. Then collapse the tubes by pushing the disks into the middle of the tubes. Since an orientable 2—manifold embeds in R3, we Obtain: 22 Corollary 3.2.3: A regular neighborhood of a compact orient- able 2-manifold embedded in the interior of an orientable 3-manifold is a product neighborhood. Lemma 3.2.4: Let A be an (n,m)-residual set of a closed orientable 3-manifold M. Then each Pi is either a 2-Sphere or a real projective plane. n Proof: Consider N(A) = ( U N(Si)) U ( U N(Pi)). . .i=1 . .i=1 , Since N(A) is topologically M less an Open 3-cell, Bd N(A) .is a 2-Sphere. If a is the join point of A, N(a,M) = B is a 3-ball with .N(A) - B the disjoint union of the n sets N(Si) - B and the m sets N(Pi) - B. By definition Of the second derived neighborhood, it is clear that Ed (N(Pi) - B) is Bd N(Pi) less two disks. Since Bd N(Pi) is a 2—manifold and. Bd (N(Pi) - B) is contained in Bd N(A), a 2—sphere, Bd N(Pi) is either one or two 2-Spheres. Suppose that P1 is orientable. Then N(Pl) is topologically P1 x I and so Bd N(Pl) is two disjoint OOpies of P1. Hence if P1 is orient- able, P1 is a 2-Sphere. Suppose that P1 is non- orientable. If P1 has an orientable handle, Bd N(Pl) must contain a torus with a hole. .Since a torus with a hole does not embed in a 2-sphere, P1 does not have an orient- able handle. Since the Klein bottle embeds in 81 x S2 with a regular neighborhood having a torus boundary, 23 Lemma 3.2.2 implies that P1 is not a Klein bottle. Hence if P1 is non-orientable, P1 is a real projective plane. Lemma 3.2.5: Let A be an (n,m)-residual set for a closed 3-manifold M. Then each Pi is either a 2-sphere or a real projective plane. 3322;} If .N(P1) is orientable, Lemma 3.2.4 produces the desired result. Suppose that N(Pl) -'N is non- orientable. -AS in Lemma 3.2.4, Bd N is one or two 2- spheres. If Bd N is one 2-Sphere, let E be a 3-ball attached to N by a boundary homeomorphism. Since N col- lapses onto P1 and N U E is non-orientable, we Obtain the Mayer Vietoris sequence: anm n s)‘ > qu) e Hq(E) —> qu U/E) -—-> Hence 0 -—s Z ——> H2(P1) -—> H2(N U E) ——> O is eXact. Thus H2(P1) # 0 and so P1 is orientable. If P1 has genus g, H1(P1) is the direct sum of Zg copies Of Z. Since H1(N n E) - O, H1(N U E) = i; Z. Thus, if x(X) is the Euler characteristic of x, x(N U E) - 1 - Zg + O - O = 0, since N U E is a closed 3-manifold. Since 9 is an integer, we have a contradiction. Thus Rd N is two 2- spheres. Let E and F be two 3-cells attached to N by boundary homeomorphisms. We Obtain the M-V sequence —.> “3““ U E U F) —> H2(N n (F U E)) —>H2(N) enzua: U F)—>. Since N U E U F is non-orientable and N n (E U F) 3 Bd N, 0 >Z$Z—>H2'(P1)'_>H2(NUEUF) > on. 24 is exact. However H2(P1) = O or Z. This contradiction establishes the lemma. Corollary to the proof: If A is an (n,m)-residual set for the closed 3-manifold M, N(Pi,M) is orientable for all i. 3. Reconstruction of AM Suppose that A is an (n,k+m)-residual set Of the closed 3-manifold M, with k of the 2-manifolds of A 2-Spheres and m of the 2-manifolds real projective planes. By the argument of Lemma 3.1.3, -Hz(A;Zz) = H1(A:Zz). But n+h1 k+m H1(A722) = O 22 and H2 (A722) = O 220 ‘Thus k = n. 1 1 Hence A is the one-point union Of n 1-spheres $1, ---, Sn' n 2-Spheres T1, °°-, Tn and m .real projective planes P1, ..., Pn. If A is in this form, we will call A an (n,n,m)-residual set. Lemma 3.3.1: Let A be an (n,n,m)-residual set for the closed 3-manifold M. Then N(A) = N(( 3 Si) v (.3 Ti)) A N(Pl) A ... A N(Pm). i=1 l=1 grggr: Suppose that a is the join point of A. Then the Simple closed curve L = P1 0 Lk(a,N) bounds two disks D and D' in Lk(a,M). If Int(D) n Lk(a,A) is empty, by a B-move we may change A into A' = (A - aOL) U xoL U B, where x is an interior point of aoD and B is the straight line segment from a to x. Let 25 P' = (P1 - aoL) U xoL. Then A' = (A - P1) U P' U B and since N(B) is a 3-cell N(A') is homeomorphic to N((A - P1) U a) A N(P'). Since N(P') is homeomorphic to N(Pl) the lemma will follow by induction if we are able to justify our initial assumption, that Int (D) n Lk(a,A) is empty. Let L1, ..., Lp be the simple closed curves in A n Int D and suppose that x1, ..., xq are the points of A n Int D. Likewise let L be the simple p+1, ..., Ln+m-1 I closed curves Of A n Int D and Xq+1’ ..., xan the points Of A 0 Int D'. By the elementary moves of Lemma 3.1.2, we may change A into a residual set A" with A" P q n+m-1 2n (A-St(a,M))Uxo[( U Li)U( U xi)]UBUyo[( U Li)U( U xi)]UB' i=1 i=1 p+1 q+1 where x is an interior point of aoD, y is an interior point of aoD', B = aox and B' = aoy. Since N(P1,M) is orientable, N(P1,M) is homeomor- phic to N(RP2,RP3). Since RP3 is Obtained by the anti- podal identification of the boundary of the unit 3-ball, we may consider N(RP2,RP3) as the quotient space (S2 x I)/R, where R is the equivalence relation R: {((x,0).(-X.0))|x e 52] U [((X.t).(X.t))|(X.t) 6 52 x I} (For the definition of quotient space see [6].) Suppose that q: 82 x I > N(P1,M) is the composite of the quo- tient map and the obvious homeomorphism. [Without loss of generality we may assume that q((1,0,0),0) = a, 26 q((1,0,0) x I) = B n N(Pl) and q((-1,0,0) x I) = B' n N(Pl). Let E be the 2-cell in 82 x I given by E = {((x,y,0),t)| x2 +y2 = 1, ygo, and o:t:1/2}. Then ‘q(E) is a singular 2-cell in VM. q(E) meets A" on (B n q(E)) U (B' n q(E)) U (91 n q(E). -Let c be the arc of q(E) given by C = q((((x,y,0),1/2)|x2 + y2 = 1 and y.: 0]). To make the desired change of A", expand A" to A" U q(E) and collapse .A" U p(E) ,Onto h(A" +H(B n q(E))) U C U a. Let A"' denote the resulting residual set.' Let F be the arc F = Cl(B - q(E)) U Cl(B' - q(E)) U C. Con- sider the relation R: (F x F) U {(z,z)|z e M}. Let M' = M/R. Since F is point-like, M' is homeomorphic to M. Note that A"'/R is a residual set for M' that has the desired form. Thus M itself has a residual set of the desired form and the lemma is established. Lemma 3.3.2: Suppose that .M is a closed 3-manifold with an (n,n,0)-residual set. Then M has an (n,n,0)-re- sidual set such that for all i, 1) Si pierces Ti and no other 2-sphere, ii) Ti is pierced by Si and no other 1-sphere and iii) N(Si V Ti) is tOpOlogically either 81 X S2 less an Open 3-cell or the twisted $2 bundle over 31 less an open 3-cell. Moreover, N(A) = N(31 V T1) A ... A N(Sn V Tn). 27 .grggg: Suppose that n = 1. If .81 does not pierce T1, then Bd N(Sl v T1) = Bd N(Sl) # Bd|N'('r1):.~ Bthorollary 3.2.3, N(T1) is homeomorphic to S2 X I. and by the re- mark after Lemma 3.2.1, N(Sl) is either a solid torus or a solid Klein bottle. Thus if 31 does not pierce T1 we will contradict the connectivity of Bd N(A). ~Therefore $1 pierces T1. . § Since N(A) S N(S1) U N(Tl), N(A) = (32 x I) Uh (S2 x I), where h is a homeomorphism of. B2 X Ed I onto two disks of Ed (S2 X I). Since Bd N(A) is connected, h must take each Of the disks of B2 X Ed I into distinct 2-Spheres in Ed (S2 X I). If N(Sl) is a solid torus, h is either orientation preserving or orientation reversing on both ends of B2 X I. If N(Sl) is a solid Klein bottle, then h is orientation preserving on one end and orienta- tion reversing on the other. In the first case N(A) is orientable and by attaching a 3-cell to ‘N(A) by a boundary homeomorphism we obtain 81 X 82. [In the second case N(A) is non-orientable and by attaching a 3-cell to N(A) we. obtain the twisted S2 bundle over Sl.'iThuS the Lemma holds if n H 1. Assume the lemma is true for n = 1, 2, ..., k-l. De— k. k fine x by x‘= Cl(Bd[N(a.M) - ( U N(Ti) U U N(Si))]). i=1 . i=1 . where a is the join point Of A. -X is a 2—Sphere with k disjoint open annuli and 2k disjoint Open disks removed. We may Obtain Bd N(A) from X by attaching 2k disks to 28 the boundaries of the annuli and k annuli to the boundaries of the 2k disks. Note that the annuli may be attached with different orientations on each end. Since X has k+1 components and Bd N(A) is a 2-Sphere, the annuli must bridge the components of X. Hence each component has a disk removed. MOVE F: Let a be the join point of A. Suppose x1 and x2 are two isolated points of Lk(a,A) that lie in the same component of Lk(a,M) less the 2-manifolds of A. Since Bd N(A,M) is a 2-Sphere, x1 and x2 belong to two distinct 1-spheres, say 51 and $2, of .A. There is a 2-ball B in M with A n B - A n Bd B = aoxl U aoxz. Let C = Cl(Bd B - aoxl U aoxz). Then there is a 2-ball D in M with A n D = A n Bd.D Cl(Sa - aoxz) U x1 and C. Let E Bd D - (C U Cl(Sz - aox2)). B H D = B 0 Ed D An F-move expands A to A U B U D and collapses from aoxl across B U D onto (A - (aox1)) U a U E. The effect of an F-move is to slide a disk from one component of X along an annulus to another component. Thus we may assume that k of the disks lie in one compo— nent of ~X and that each of the other components contain exactly one of the disks. Since at least one Of the 1-Spheres of Lk(a,A) is nullhomotopic in Lk(a,M) less the other 1-spheres of Lk(a,A), suppose that L = T1 n Lk(a,M) is the 1-Sphere. Then L bounds a disk D in Lk(a,M) with Ti n D empty 29 for i 3,2. By F-moves we may assume that only 81 inter- sects D. Then D n 81 is a point x1. By another series of F-moves, we may assume that each 1-sphere Of A meets the component of Lk(a,M) less all the 2-manifolds that has L as one of its boundary components. Thus 31 pierces T1. Since any other Si lies on one side of T1, Si cannot pierce T1. Likewise $1 lies on one side of the other T and SO cannot pierce them. Let A' = (A - (81 U T1)) Ua and suppose that x is an interior point of aoxl and y is an interior point of ($1 - aoxl) n St(a,M). Let .T' = (T1 - aoL) U xoL. Then by elementary moves change A to A' U aox U T' U 81. -By another series of elementary moves, move aoy along aox to an arc C from y to x. Set 8' = ($1 - ao(x U y)) U C. Thus we may change A into a new residual set A' U S' U T' U aox = A". Now N(A") is homeomorphic to both N(A) and N(S' V T') A N(A'). Clearly A' is an (n-1,n-1,0)-residual set for a closed 3—manifold, for N(A') is a 3-manifold with 2-Sphere boundary. Since N(S' V T') is homeomorphic to N(81 V Ti), the lemma follows by induction. From Lemma 3.3.1 and Lemma 3.3.2, we Obtain: Lemma 3.3.3: Let A be an (n,n,m)-residua1 set of a closed 3-manifold M. Then N(A) = N(81 v T1) A A N(Sn v Tn) A N(P1)A A N(Pm). 30 Theorem 3.3.4: Let A be an (n,n,m)-residual set and let B be a (p,p,q)-residual set for the same closed 3-manifold M. Then A and B are homeomorphic. 2599;: The second homology groups with Z2 coeffici- ents of A, B and M are isomorphic and so n + m = p + q. Since the rank of H1(A), H1(M), and H1(B) is the same, n = p. Thus A and B are homeomorphic (n,n,m)-residual sets for M. Theorem 3.3.5: Suppose that M and M' are two closed 3- manifolds with the same orientability. If M and M' have homeomorphic (n,n,m)-residual sets A and A', then .M is homeomorphic to M'. lgrggg: Suppose that the 1-spheres of A are denoted by Si’ the 2-spheres by Ti' and the real projective planes by Pi' Let Si. Ti, and Pi denote the correSponding parts of A'. Let N(X,M) = N(X) and N(X,M') = N'(X). By Corollary 3.3.3, N(A) = N(51 v T1) A ... A N(sn v Tn) A N(Pl) A ... A N(Pm) and N'(A') = N'(Si v Ti) A ... A N'(Sfi v TA) A N'(Pi) A ... A N'(P$). Suppose that X in a subcomplex Of M with Ed N(X) a 2-Sphere. Then we will denote the closed 3-manifold ob- tained by attaching a 3-cell to N(X) by M(X). In the same manner define M'(X). Then M(A) is homeomorphic to M and M'(A') is homeomorphic to M'. 31 Notice that if N(X U Y) - N(X) A N(Y), it follows that M(x U Y) = M(X) # M(Y), whenever N(X U Y) has a 2- Sphere boundary. Thus we obtain: M = M(81 v T1) # ... #M(sn v Tn) # M(Pl) # ...# M(Pm) and M' - M'(si v Ti) # ...# M'(Sfi v TA) # M'(Pi) #...# M'(Pé). Since Pi is a 2-manifold that is a residual set of the closed 3-manifold M(Pi) it follows from [9] that M(Pi) is a copy of real projective 3-space. Also M'(Pi) - RP3. Since J # J = J # 81 X 82 [13], we may assume that M(Si V Ti) and M'(Si V Ti) are topologically 81 x 82 for i # 1. Since M and M' are either both orientable or both non-orientable, the same is true for M(A) and M'(A'). Hence each term in the connected sum is pairwise homeomorphic. Thus M and M' are homeomorphic. Combining these results we Obtain: Theorem 3.3.6: A Closed 3-manifold has a residual set that contains no wild arc if and only if it is the connected sum of closed 3-manifolds each Of which is homeomoprhic to a 3-sphere, RP3,i 51 X S2 or J. 4. Compact 3-Manifolds with Boundary Lemma 3.4.1: Let M be a compact 3-manifold with or without boundary. Suppose that M = A U U is a decomposition of M such that A contains no wild arc. Let C be a 3-cell in Int M. Then M - Int C has a decomposition M - Int C = A' U U' such that A' contains no wild arc. 32 grggr: Since C is point-like, there is a homeomorphism h of M onto itself such that h(C) C‘U. After a possible subdivision there is a simplicial arc B in U U x, where x is a point of A, with B n h(C) = y and Bd B = x U y. Thus we may expand h(C) U B to a 3-cell D in U U x. Then there is a homeomorphism k of M onto itself such that k(D) = h(C). Thus c - h-1k(D) and M - Int D is a compact 3-manifold with a decomposition M - Int D = (A U Ed D) U (U - D). Since A n Ed D = x, A U Ed D con- tains no wild arc. Thus M --Int C has a decomposition M - Int c = h'1k(A U Ed D) U h'1k(U — D), and h’1k(A U Bd D) contains no wild arc. Lemma 3.4.2: Let M1 and M2 be compact 3-manifolds with boundary. Suppose that B1 and 82 are homeomorphic boundary components of M1 and M2 reSpectively. Let M be obtained by sewing M1 and M2 together along BL and B2. Suppose that Mi - Ai U Ui is a decomposition of M1 such that Ai contains no wild arc. Then M has a residual set that contains no wild arc. Proof: Consider M1 and M2 as submanifolds of M so that M - M1 U M2 and M1 n.M2 = 31 = B2. Set B = B1. Since Ai does not contain a wild arc B n Cl(Ai - B) is a finite point set. By elementary moves we may assume that B n Cl(Ai - B) = x for i = 1 or 2, where x is a (1,2)- point of A1 U A2. Let C be a 2-Simplex of B. Then M has a decomposition 33 M = ((A1 -B) U (B - Int c) U (A2 -B)) U (U1 U Int(C) UU2). By collapsing B - Int C to a Spine K, we Obtain residual set A for M, where A = (A1 - B) U K U (A2 - B). Clearly A contains no wild arc. With the notation of the above lemma, N(KPA) is K with two whiskers. If M is not a closed 3-manifold, take the double of M. It follows that A less some Open disks Of A embeds in a residual set of a closed 3-manifold. Thus Bd N(K,M) less two disks embeds in a 2-sphere. Since K has the homotopy type of an r-leaved rose, Bd N(K,M) is a 2-sphere with r handles, orientable or not. Thus r = O and N(K,M) is a 3-ball. Therefore K is a point and B is a 2—Sphere. Thus we have established: Corollary 3.4.3: If M is a compact 3-manifold with boundary having a residual set containing no wild arc, each boundary component is a 2-sphere. Theorem 3.4.4: Let M be a compact 3—manifold. M has a residual set that contains no wild are if and only if M is obtained from the connected sum of closed 3-manifolds each of which is S3, 51 X 82, RP3 or J, by deleting n.: O disjoint 3-balls. Proof: One way follows from Theorem 3.3.6 and Lemma 3.4.1. The converse follows by sewing n 3-balls onto M and applying Lemma 3.4.3 and Theorem 3.3.6. CHAPTER IV A CHARACTERIZATION OF CERTAIN CLOSED 3*MANIFOLDS WHOSE SINGULAR SET IS A SIMPLE CLOSED CURVE In this chapter we define the singular set Of a re- sidual set A of a closed 3-manifold, denoted by S(A). We Show that A may be chosen so that A - S(A) is the disjoint union of Open arcs and the interiors of compact 2-manifolds with connected boundary. ~We then classify all closed 3-manifolds with S(A) a simple closed curve. 1. The Singular Set Definition 4.1.1: Let A be a residual set for a closed 3-manifold M. The Special singular set S'(A) of A is the set of all points of A that do not have an Open 2- dimensional euclidean neighborhood in A. The singular set S(A) of A is the set of all points of 1A that do not have an Open 1- or 2-dimensional euclidean neighborhood in A. If A is already lOcally euclidean, we will set S'(A) = S(A) - a where a is an arbitrary point of A. Suppose that M is not a 3-Sphere. Then A is a 2- complex. Since S'(A) is a subcomplex of A, S'(A) is contained in the 1-skeleton of A, A1. Let T be a maxi- mal tree of A1. Mod out T: that is consider the quotient space M/T obtained by identifying T to a point [6]. Since T is contractable, T is point-like and SO M/T 34 35 is homeomorphic to M. .Notice that A'/T is a t-leafed rose and since S(A)/T C?S' (A)/T C Al/T, S(A) is an r- leafed rose and S'(A) is an s-leafed rose. Since T is contained in A1, (M~A)/T is an Open 3-cell and A/T is a non-separating continuum of dimension two. Thus M/T has a decomposition M/T = (A/T) U ((MeA)/T). The Singular set of A/T is clearly S(A)/T and S'(A/T) = S'(A)/T. Since M and M/T are homeomorphic, we may assume that A is already in the above form, that is S(A) is an r-leafed rose and S'(A) is an s-leafed rose. Let Mi, ..., MA be the components of A - S'(A). Then each Mi is an open 2-manifold; in fact, each AM; is the interior of a compact 2-manifold Mi with non-empty boundary. Thus A is Obtained by attaching the Mi's to S'(A) by wrapping each boundary component of the Mi's around S(A). To be more precise, let X be the disjoint union of the Mi's and let Y be the disjoint union Of the boundaries of the Mi's. Then there is a continuous map ¢: Y > S'(A) such that A is topologically the space obtained by attaching X to S'(A) by e. For the definition of the attaching of spaces see [6]. In effect we are sewing the Mi onto S'(A) by the map ¢. Let p: X > A denote the composite of the quotient map and the homeomorphism between X U¢ S'(A) and A. 36 To Obtain a better picture of A let us examine the map ea Now e is a map onto S(A), an r—leafed rose. Let L1, ..., Lr be the leaves Of S(A), where the Li are given a definite orientation. Let fi be the map from the unit interval onto Li given by fi(t) = hi(e2”it), where hi is a homeomorphism Of the unit 1-sphere onto Li with the induced orientation of hi(Sl) agreeing with the orientation of Li and fi(Bd I) is equal to the join point .a Of S(A). Suppose that S is a component of the boundary of one Of the Mi' Let ?|S = ¢'. Since we have modded out a maximal tree in the 1-Skeleton of a re- sidual set of M to obtain A, ¢' will induce a subdivi- sion of S into k segments 81, ..., Sk with the inter- iors of the Si disjoint and ¢' mapping each Sj onto one of the Li in the same way that fi or f;1 maps the unit interval onto Li' If ¢' already maps S homeomorphically onto one of the leaves of S(A), we denote ¢' by fi or f;1 depending on how *¢' Operates. If e' is the con- stant map we denote ¢' by 0. If ¢' is more complex we assume that the Si are cyclicly ordered On S by their subscripts. By disregarding the Obvious homeomorphism be- tween the unit interval and each Sj' we regard fi as a map from Sj onto Li' »We then denote the action of ¢' by setting ¢' = hlhz ... hk where hj is one of the maps .—1 . fi or fi for some 1. 37 Definition 4.1.2: .Let A be a residual set for a closed 3- manifold M. -With A as above, a presentation P of A P : S(A), S'(A), M1, ..., Mn; ¢ is a set consisting Of the Singular set, the special singu- lar set, the compact 2—manifolds Mi and the map ¢. We now establish some properties of A. Let ki be the rank Of H1(Mi) and hi the number of boundary components of Mi' Let V be a second derived neighborhood of S'(A) in A and let W be the closure of A - V. Since A = W U V and .W n V is the disjoint n union of 2 hi simple closed curves, X(A) = x(V) + i=1 ‘ x(W) - x(V n W). Since V collapses onto S'(A), an S- leafed rose, x(V) = 1 - S. Since W is homeomorphic to n the disjoint union Of the Mi’ x(W) = 2 (1 - ki). Since i=1 n V 0 W is 2 hi disjoint simple closed curves, x(W n V) i=1 = 0. If C is a 3-simplex Of M - A, .M - Int C collapses onto A and so x(A) = x(M - Int C) = x(M) + 1. However, M is a closed 3-manifold and so x(M) = 0. Hence n n 1 = x(A) = (1 - S) + Z (1 - k.) = n + 1 - (s + Z k.). . 1 . 1 1:1 1:1 n Lemma 4.1.3: n = s + Z ki' i=1 Lemma 4.1.4: Let M be a closed 3-manifold and A a re- sidual set for M with a presentation P:S(A)( S'(A)IM1I “'0! Mn;¢o 38 Suppose that ¢ restricted to any boundary component S is either the constant map or ¢|S = h1h2... hp, with each hi either hi or h;1, where hl and p depend on S. Then M has a residual set A' with a presentation p' smurs'm'). N1. Nn:¢'. where Ni is homeomorphic to Mi for all i, and ¢' re- stricted to any boundary component 8' is either the con- stant map or O'IS' = hlhl...h1 = h?, for 0 < q j,p. Proof: Let S be a boundary component of M1 and hfl. Then 1 ¢|s = h1...hihi+1...hp, With p > 1 and hi*1 ¢ induces a subdivision of S into p arcs $1, ...,.Sp cyclicly ordered by their subscripts. Let B and C be proper subarcs of Si and Si+1 respectively with B U C connected and p(B) ? p(C). Then .B U.C lies on a 2-cell D in .M1 with B U C = D n Bd M1. Clearly p(D) is a 2-cell of A. -Moreover p(D) lies on a 3-cell E in M as three sides of a 3-simplex. -Let F be the remaining side of ~E. Suppose that E n A = p(D). Swell up A to A U E and collapse onto (A - E) U F U p(B). By modding out the closure Of p(Si) - p(B), we Obtain a new residual set A' with a presentation P': S(A'), S'(A'), N1, ..., Nn; s‘, with N1 homeomorphic to Mi and ¢' restricted to the component 5' of N1 corresponding to S is given by ¢'|s' = hlhz...hi_1hi+g...hp. 39 In the above we assumed that E n A = p(D). In general this is too much to ask. However, if E n A # p(D), there is a finite sequence of disks D1, ..., D - D in the dis- t t joint union of the Mi’ with p( U D.) homeomorphic to the i=1 cone over X with vertex v, where X is the planar set that is the union Of the simple closed curves Ci = [(x,y)|(x-i)2 + y2 = 12], for 1,: 1.: t, and voci is mapped onto Di for all i. In M there is a finite sequence of 3-cells E1, ..., E with E = E and p(Di) t’ t lying on Ei as three Sides of a 3-simplex. ~Moreover the Ei may be so chosen that the above homeomoprhism extends to a homeomorphism.of the cone over the bounded region bounded by C onto E in such a way that the cone over t t the compact plane set bounded by C1 is mapped onto Ei' Then by collapsing first E1 as above and continuing for the other Ei we finally collapse E. -In each step we Obtain a residual set with a presentation closer to the de- sired presentation. Thus the Obvious induction establishes the lemma. Corollary 4.1.5: Each closed 3-manifold has a residual set A with a presentation P: S(A),'S'(A):1M1: ooopMn7¢I such that if S is a boundary component of one Of the Mj’ _ . . . 7 -1 . ¢|S - h1---hp Wlth hi distinct from hi+1 for all 1. Corollary 4.1.6: If the dunce hat is a residual set for a closed 3-manifold M, M is a 3-sphere. 40 Proof: Let D be the dunce hat. Then D has a pre- sentation p: 51, s1, 32; hhh-l. By Lemma 4.1.4, M has a residual set A with a presenta- tion P': SI, 81, B2; h. Hence A is a disk and SO M is a 3-sphere. Lemma 4.1.7:, Each Mi in a presentation for A is either a disk with holes or a Moebius band with holes. grggfi: Suppose that each Mi has connected boundary. Since each p(Mi) contains a topological OOpy Pi Of Mi in P(Mi) - S(A), Bd(N(Pi,M)*N(Bd Pi,M)) embeds in Ed N(A,M), a 2-Sphere. Let N = N(P1,M). If ‘N is orientable, it is unique by Lemma 3.2.2. Since any 2—manifold with non- empty boundary embeds in R3, N is homeomorphic to N(P1,R3). If P1 is orientable with positive genus, or if P1 is non-orientable with genus greater than two, N' ='N - N(Bd P1,M) has a boundary that contains a torus with a disk removed. If P1 is a Klein bottle less an Open disk, Bd N' is two open annuli attached in such a way that Bd N' does not embed in a 2-sphere. Thus if N(P1,M) is orientable, P1 is either a disk or'a Moebius band. Suppose that N is non-orientable. Since N col- lapses onto P1, H2(N) = O and H1(N) is free. Since N is non-orientable, H3(N,Bd N) = 0. Thus the exact homology sequence of the pair (N,Bd N) yields an exact sequence 41 0 > H2(Bd N) > 0. Since Bd N is a closed 2-manifold, each component of Ed N is noneorientable.i Let» Q = N(Bdfipi,M), and: N' =‘N -NQ; N Then Bd N' embeds in Bd N(A,M) and so each component of Ed N' is a disk with holes. Since Bd N Bd N' U (Bd N.n Bd Q» Bd N 0 Bd Q is non-orientable. Thus Q is a solid Klein bottle and Bd Q is a Klein bottle. Since P1 n Bd Q - C is a Simple closed curve, -N(C,Bd Q) is either an annulus or a Moebius band. In the first case Bd Q - N(C,Bd Q) is a pair of Moebius bands and in the second case it is a single Moebius band. Since Bd N = Bd N' U (Bd Q - N(C, Bd Q)) and Bd N is a closed 2—manifold, the first case implies that Bd N is either two projective planes or a Klein bottle and the second case implies that Ed N is a projective plane. If Bd N is a projective plane, 2X(N) - x(Bd N) = 1. Thus x(N) = 1/2. This is impossible. .If Bd N is two projective planes, we Obtain from the homology sequence of the pair (N,Bd N) the exact sequence 0 -—-w H2(N,Bd N) > 22 0‘22 > H1(N) > ... This is impossible since H1(N) has no torsion and Tor(H2(N,Bd,N))='Zz. [Thus Bd N is a Klein bottle. The exact homology sequence for the pair (N,Bd N) becomes 0 > H2(N,Bd N) -> Z O 22 -> H1(N) -> H1(N,Bd N) -—e 0. Thus H1(N) =‘Z -‘N1(P1). Since P1 has connected boundary P1 is a Moebius band. ’TO complete the proof Observe that if M1 is a sur- face with holes, M1 contains a surface of the same genus 42 with only one hole. In this surface find a P1 as before. The above argument then implies that .M1 is either a disk with holes or a Moebius band with holes. Theorem 4.1.8: If .M is a closed 3-manifold, M has a residual set A with a presentation P, P: S(A).~S'(A), M1, .... Mn:¢. where each Mi has connected boundary. Proof: Suppose that A is a residual set for M with a presentation where at least one of the Mi' say M1, does not have connected boundary. Let S and T be two com- ponents of Ed M1. Let B be an arc in M1 connecting .S and T such that Int B CZInt M1 and p maps one of the end points of B onto a, the join point of S'(A). Either p(B) is an are or a simple closed curve in A. (If p(B) is an arc, -M/p(B) has a residual set A/p(B). ~Since- M is homeomorphic to M/p(B) we need only ShOw that A' = A/p(B) has a presentation that Simplifies the presentation of A in the sense that the number Of boundary components is reduced. A' has a presentation P', P': S(A').-S'(A‘). N1, ..., Nn7 ¢', where Ni is homeomorphic to Mi for 2.: 1,: n and Int N1 = (Int.M1)/p(B) - p(B)/p(B). ~We may think of N1 as being derived from M1 by expanding B to N(B,M1) and removing the Open star Of B in M1. 9 If p(B) is a simple closed curve for all choices of B, ¢|T U S is the constant map. »Let S' be a simple closed 43 curve in the interior of M1 with S U S' bounding an annulus D in M1. Then p(D) is a disk meeting S(A) only at a. .Either S'(A) pierces p(D) or itdoeS not. Suppose that S'(A) does not pierce p(D). By swelling p(D) up to a 3-cell and collapsing as in a B-move and then pushing the resulting are along p(M1 - D) as in an E—move, so as to form a Simple closed curve, we obtain a new re— sidual set A' for .M. A' has a presentation P': S(A'), S'(A'), N1, ...,,Nn: ¢'. Again it is clear that Ni is homeomorphic to Mi for i # 1. N1 may be Obtained from M1 by cutting .M1 along 8' and sewing in a 2-cell. Note that we then must add one more 1—sphere to S'(A) in order to obtain S'(A'). Sup- pose that .S'(A) pierces p(D). ~Without loss of generality, assume Corollary 4.1.5 has been applied. -Swell up p(D) to a 3-cell E as before. Let F = Cl(Bd E - p(D)). A 0 Int E - C is an arc in a l-Sphere of SJ(A). ~Expand A to A U E and collapse A U E onto (A - E) U F U C. There are two arcs of (A - E) U F U C that meet Int F, C and C'. By E-moves push the end points of C and C' along F U p(M1 - D) to the join point a. Let A' be the resulting residual set. A' has a presentation P': S(A'), S'(A'). N1, ...-Nn:¢, with Ni homeomoprhic to Mi for i # 1. -N1 is Obtained from M1 by cutting M1 along 8' and sewing in a 2-cell. Note that we have added a 1-sphere to S'KA) in order to obtain S'(A'). Thus no matter what p(B) is we are able 44 to find a new residual set with a presentation that has re- duced the number of boundary components of the 2-manifolds. Therefore, by an inductive argument, the lemma is established. Remark 4.1.9: Lemma 4.1.8 enables us to assume that each 'Mi has connected boundary. However, we may have to sacri- fice a little, for by reducing the number of boundary com- ponents we may increase the number Of leaves of -S(A). As an example, let .M =‘RP2 X 81. RP2 has a residual set RP1, a 1-Sphere. Let p be a point Of ~31. .M has a residual set A = (RP2 X p) U (RP1 X 81), with a presentation P, 9: RP1 x p,.RP1,x p, M1,.M2;,¢, where _M1 is a disk, .M2 is an annulus, ¢|Bd M1 = ff and ¢ restricted to either component of Bd M2 is f. By changing .A as in Lemma 4.1.8, we Obtain a residual set A' with a presentation P'L P': S(A'), S'(A'), N1, N2: ¢'. where N1 and N2 are disks,' S(A') = S'(A') is a 2—leafed . ' ...—.1 rose, ¢'|Bd N1 =1f1f2f1f2 and ¢'|Bd N2 = flfzfllfz . 2. Residual Sets with S(A) §_§imple Closed Curve In this Section we assume that each 2-manifold in a presentation has connected boundary. Theorem 4.2.1: Let A be a residual set for a closed 3- manifold M. (Suppose that A has a presentation that has only one 2-manifold. -Then »M is either the twisted S2 bundle over 81 or a toroidal manifold. 45 .grggg: By Lemma 4.1.3, 1 = s + k1. If S = 0, S'(A) is a point and so ¢ is the constant map. Thus A is homeomorphic to RP2 and hence M = RP3 = T(2,1). If S = 1 and r = 0, S'(A) is a l-Sphere and S(A) is a point. Thus A is the one-point union of a 1—sphere and a 2-sphere. By Theorem 3.3.6, M is either 81 x S2 = T(0,1) or J. Thus we may assume that S(A) = S'(A) is a 1-sphere. By Lemma 4.1.4, A has a presentation P: SI, 81, B2; hk, k > 0. If k = 1, A is a disk and so M = S3 = T(1,1). If k = 2, A is RP2 and so M = RP3. Thus we assume that k.2.3. Suppose that M is orientable. Since S(A) is a 1-Sphere, it follows that the singular points Of A lie in n-books. Thus A n N is an n-book with its ends identified after a twist of ZW/m degrees for some integer M. Hence A n Bd N is a J(n,m) curve on the boundary of a solid torus. We now proceed as in Chapter II to construct a standard residual set for M. Since the argument goes through exactly as in Chapter II, we find that M and T(n,m) have homeomorphic standard residual sets. By Corollary 2.3.4, M and T(n,m) are homeomorphic if M is orientable. We now Show that M is orientable. As in the proof of Lemma 4.1.3, we Obtain the exact sequence: O >1Hz(A) >2 >Z >H1(A) >0. We will Show that H1(A) = Z for some k. Thus H2(A) = O k and so M is orientable. 46 Let B = N(S(A),A). Since S(A) is a Simple closed curve, W1(B) = (b: ). Since A - S(A) Vis an open disk F1(A - S(A)) is trivial. —Since (A - S(A)) n B is a half open annulus, 7T1((A - S (A)) n B) = (a: ). By the van k Kampen Theorem and the Observation that a ::b in B and a :10 in A - S(A), we Obtain r1(A) - (b: bk = O) - Zk‘ Thus H1(A) : Zk. Suppose that S(A) is a 1-sphere and that S'(A) is an s-leafed rose. By Lemma 4.1.6 and our assumption that each Mi has connected boundary, Mi is either a disk or a Moebius band. Let the M1 be arranged so that the first q are disks and the last n - q are Moebius bands. By Lemma 4.1.4, we may assume that ¢|Bd Mi : fk(i). If k(i) I 0. p(Mi) is either a 2-sphere or a copy of RP2 attached to S(A) at a, the join point of S'(A). Suppose that p(Mi) = RP2. -Let N = N(p(Mi),M) and B = Bd N. If N is non-orientable, consider the exact homology sequence of the pair (N,B). 0 > H2(B) > H2(N) > H2(N:B) > H1(B) > H1(N) > H1(N,B) > HO(B)--—> 0. Since H2(N) - H2(p(Mi)) = O, H2(B) = 0. However, B less two disks embeds in a 2-Sphere. Thus B is either one or two 2-spheres and so H2(B) # 0. Thus N is orientable. By the proof of Lemma 3.3.1, we may change A into a resid- ual set topologically (A - p(Int Mi)) U B U RPZ, where B is an are from a point of (A - p(Int Mi)) to a point of 47 RP2 such that Int B does not meet (A - p(Int Mi)) U RPz. We will say that such an RP2 has been "put on a stick". In the same way we put each p(Mi) on a stick if q + 1.: i j.n and k(i) = 0. If p(Mi) is a 2-Sphere, the con- nectivity of Bd N(A,M) implies that p(Mi) is pierced by a 1-sphere of -S'(A). By an argument similar to that of Lemma 3.3.2, we may assume that one and only one Of the 1- spheres of S'(A) pierces p(Mi) and that p(Mi) and that l-sphere may be put on a stick. -Hence we Obtain: Lemma 4.2.2: Let A be a residual set for the closed 3- manifold M. Suppose that A has a presentation P: 31, S'(A), M1, ..., Mn;¢, with M1, ..., M disks and M q q+1' 'Suppose that ¢|Bd Mi = 0 for p+1 j.i :.q and t+1.: i.: n. ...,Mn Moebius bands. Then M ='M' # M" where M' has a residual set A' with a presentation I. 1 I I . P o S I S (A )I N1: °°°Iinl Nq+1r 00-: Ntl with N1 homeomorphic to Mi and M" is the connected 4"; sum of q — p copies of 81 x 52 or J and n - t copies of RP3. With the above notation, if k(i) = 1 for some i be- tween 1 and p, p(Ni) is a disk. By modding it out we Obtain a 3-manifold homeomorphic to M' that has a residual set whose Singular set is a point. Since the main theorem of Chapter III classifies all closed 3-manifolds with this prOperty we assume that k(i) # 1 for 1.: 1.: p. 48 Suppose that N = N(S(A'),M') is a solid Klein bottle. Let Ci = p(Ni) n Bd N. Clearly Ci is a simple closed curve for all 1. By [12], there are exactly four isotopy classes of simple, closed, orientation preserving paths and exactly four isotopy classes of Simple, closed, orientation reversing paths on —Bd N. Let 0, a, -a and b be repre- sentatives of the orientation preserving classes and p1, p2, -p1 and -p2 be representatives of the orientation re- versing classes. These may be pictured as in Figure 4.1. \f .\ I \- \ N'°K. ’\ n.~ ‘\ 1 ‘L -“~“ b \-&- -“‘ \ \ ‘1-‘ " \~ (1 ‘~_)“ V \. ‘P2‘\ \‘ \\ ~ -- \‘C‘gr N s~i -~\)“ ‘\ 3 >Figure 4.1. If Ci 2:0 or b, k(i) = 0, (a contradiction. If Ci 211 p1 or 1 pa for 1 j_i E.p, k(i) I 1, a contra— diction. :Thus Ci 2:: a for 1 fi.i :.b. By reversing the orientation of Ni’ we may assume that Ci 22a for 1.: 1.: p. Likewise. Ci 2:: a, 1 p1 or 1 p2 for q+1 : 1.: t. Sup- pose that ~Ct 2:1 p1. -By an isotopy we may assume that Ct = r p1. Since Bd N - C is a Moebius band, no other t 49 Ci 3' 1 p1. In the same way, if .Ct 21 1 p1 and ct_1 : 1 p2, no other 'Ci may be isotopic to 1 p1 or 1 p2. Thus we Obtain three cases: CASE 1: Ci :1: a for all 1, CASE 2: Ci :;i:a for i # t and Ct :11 p1 and CASE 3: ci ~ i a for i 75 t, t-l, ct Zr p1 and C t-1 1: p2. In any case there are two Ci’ say C and C', that bound an annulus E on Bd N, with Ci C’E if Ci 2:: a. If there is only one C. 2!: a, set C1 - E. Notice that 1 E' = N(E,Bd N) is an annulus. CASE 1: Since Bd.N - E' is two Open Moebius bands and Ed N - E' less some disks embeds in a 2-sphere, we have a contradiction. It is necessary to remove the disks since an arc of S'(A') - S(A') may intersect Bd N - E'. CASE 2: A similar argument excludes this case. CASE 3: Let P = Cl(p(Nt) — N). P is a Moebius band. Clearly N(P,M') is either a solid torus or a solid Klein bottle. -Since Bd N(Ct,N) -N(Ct,Bd N) is a Moebius band embedded in Bd N(P,M'), N(P,M') is a solid Klein bottle. However, Bd N(P,M') - Bd N(Ct,N) is a Moebius band that embeds in a 2-Sphere, excluding case 3. ppmma 4.2.3: Let A be a residual set Of a closed.3-mani- fold M with AS(A) a simple closed curve. Then N(S(A),M) is orientable. 50 grgpr; As in Lemma 4.2.2, we write M = M' # M". No- tice that N(S(A),M) is homeomorphic to N(S(A'),M'). How- ever, the above argument implies that N(A(S'),M) is orient- able. Remark 4.2.4: Our assumption that the Mi have connected boundary is essential in Lemma 4.2.3. Again M - RP2 X 81 gives the counter example, for N(S(A),M) is a solid Klein bottle. Lemma 4.2.5: Suppose that A is a residual set of the closed 3-manifold M and that A has a presentation P P: 51, 31. M1. M2; e. Then M = RP3 # T(n,m) or M = RP3 # J. grpgr: Since 2 = 1 + k1 + k2 by Lemma 4.1.3, we may assume that M1 is a disk and M2 is a Moebius band. If ¢|Bd M1 is the constant map, p(Ml) is a 2-Sphere and so Bd N(A,M) is not connected. If ¢in M2 is the constant map, p(Mz) may be put on a stick. Then M has a residual set RP2 v p(M1) and so M = RP3 # T(n,m) or M - RP3 # J by Theorem 4.2.1. Thus we assume that ¢ is not the con- stant map on either boundary component. By Lemma 4.2.4, we h with k r o r h. may assume that ¢|Bd M1 = fk and ¢|Bd M2 = f By Lemma 4.2.3, N(S(A),M) is a solid torus T. Let C = (Bd T) n p(Ml) and D - (Bd T) n p(Mz). Now C is a J(k,m) curve and D is a J(h,m') curve. Since C does not meet D, after changing the orientation Of M1 if 51 necessary, k = h and m - m'. Thus C U D bounds an annulus E on Bd T. Swell up S(A) to a singular solid torus T' with boundary ((p(M1) U p(M2)) n T) U E. Thus all the Singularities of T' lie on. S(A). »Collapse *T' from p(Mz) n T onto (p(Ml) n T) U E. .Mod out an arc J on p(Ml) n T, with one end point x on p(Ml) n Bd T and the other end point 'a on S(A), such that Int J C: p(Int M1) 0 Int T. ALet A' be the resulting residual set. A' has a presentation P': 31 v 31, 51 v31, 131,132, N; 4>', where the Bi are disks and N is a Moebius band. -More- over, ¢'|Bd 31 = f, ¢'|Bd B2 = gkf and ¢'|Bd N = f, where f is the map around one of the leaves and g is the map around the other leaf. Thus p(B1) is a disk. Mod out p(Bl), Obtaining a residual set A" with a pre- sentation P": sl,.sl, N1, N2; ¢". where N1 is a disk with ¢"|Bd N1 = 9k and N2 is a Moebius band with ¢"]Bd N2 = 0. Hence A" is the one- point union of RP2 and the residual set for a T(k,m) manifold. Thus the Lemma is established. Theorem 4.2.6: Let A be a residual set for a closed 3- manifold M. Suppose that A has a presentation P: 81, S'(A); M1, ...,-Mn;-¢’ then either Q-I n—q i) M = ( f 81 X 32) # ( # RP3) # (T(n,m)) or 1 . 52 (1&1 n-q ii) M = ('# J) # ( # RP3) # T(n,m). 1 1 [Proof: By Theorem 4.1.4, ¢|Bd M1 = k(l). Since N(S(A),M) is a solid torus, the argument Of Lemma 4.2.5 implies that either k(i) = i k or ¢|Bd Mi is the COD? stant map. By Lemma 4.2.2, we may write M =‘M' # M", where M' and »M" are as in the lemma. Let A' be the residual set for M'. As in Lemma 4.2.2, A' has a pre- sentation I, 1. I l . . I P .S I S (A)IN1! 0-01Npqu+11 00-: Ntl ¢ 0 ik . Note that ¢'|Bd Ni- f for all 1. Set ci = p(Ni) n Bd.N(S(AF)}M?)L- Then there are two of the ‘Ci, say C1 and C2, that bound an annulus E on Bd N(S(A)',M') with Ci CiE for all i. As in Lemma 4.2.5, there is a singular torus T' in N(S(A'), M') bounded by ((P(N1) U p(N2)) fl N(S(A'), M')) U E. Collapse T' from p(Nl) n N(S(A'),M') onto E U ( p(Nz) n N(S(A'),M')). Let J be as in Lemma 4.2.5. MOd out J.. If there are any arcs in the resulting residual set that do not form a simple closed curve move their end points, by E-moves, to the image of the join point. 'We thus obtain a residual set A" with a presentation ll. 1 1 I II o n P . s vs , s (A ), Q, Ql, Qp, Qqfl. Qt. <1>, where Q is a disk with ¢"|Bd Q = gkf and Qi is homeo- morphic to Ni for all i with ¢"|Bd Qi = f. Since Q1 is a disk, p(Ql) is a disk. By modding out p(Ql) we obtain a residual set A"' with a presentation 53 III. 1 I III , III p ‘3 IS (A )I DI D2: 00-: Dpl Dq+11 000! Dtl ¢ I with D homeomorphic to Q and Di homeomorphic to 01' and ¢"'|Bd D 3 gk amd ¢"'|Bd Di = O. The theorem follows. [1] [2] [3] [4] [5] [5] [7] [3] [9] [10] [11] [12] [13] BIBLIOGRAPHY R. H. Bing, "Some aSpects of the topology of 3-mani- folds related to the Poincare conjecture", Lectures on Modern Mathematics, ~John Wiley and Sons (1964), Vol. 2, 93-128. M. Brown, "A mapping theorem for untriangulated mani- folds", Topology of 3-manifolds and Related Topics, Prentice- Hall (1962), 92-94. B. G. Casler, "An imbedding theorem for connected 3- manifglds with connected boundary", Proc. Amer. Math. Soc. 16 (1965), 559-566. P. H. Doyle, "Two weak Poincare theorems", Proc. Camb. Phil. SOC. 62 (1966), 23. P. H. Doyle and J. G. HoOking, "A decomposition theorem for n-dimensional manifolds", Proc. Amer. Math. Soc. 13 (1962), 469-471. J. Dugnundji, Topology, Allyn and Bacon (1966). S. Eilenberg and R. L. Wilder, "Uniform local con- nectedness and contractibility", Amer. J. 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