THE FLORIDA STATS UNIVERSITY COLLEGE OF ARTS AMD SCIENCES GENERALIZED THREE-tMANIFOLDS WITH ZERO-DLMENSICNAL SINGULAR SET by DUŠAN REPOVS" A Dissertation submitted to the Department of Mathematics and Computer Science in partial fulfillment of the requirements for the degree of Doctor cf Philosophy Approved: April, 1983 GENERALIZED TEREE-iMMIFOLDS WITH ZERO-DIMENSIONAL SINGULAR SET (Publication No. ) Duäan Repovä,Fh.D. The Florida State University, 19Ö3 Major Professor: Robert Christopher Lacher, Ph.D. We study two "disjoint disks properties" in dimension 3 due to H.W.Lambert and R.B.SheiPacific. J .Math. 2k (1968) 511-518), the Dehn lemma property (DLP) and the map separation property (MSP). Theorem 1^. Let G be a cell-like closed O-dimensional upper semi continuous decomposition of a 3-manifold M (possibly with boundary) with N~C int M • Then the following statements are equivalent: (i) M/G has the DLP; (ii) M/G has the MSP; (iii) M/G is a 3-manifold. Theorem 2. Let L be the class of all compact generali zed 3-manifolds X with dimS(X) <_ 0 and let C^C L be the subclass of all X€L with S(X) cz {pt} and X*S . Then the following statements are equivalent: (i) The Poinca-rL conjecture in dimension three is true; (ii) If X€Lhas the DLP or the MSP then S(X) = 0; (iii) If X6C has the DLP or the MSP then S(X) = 0. We also study neighborhoods of peripherally 1-acyclic compacta in nonorientable 3-manifolds. We prove a finiteness and a neighborhood theorem for such ccmpacta and as an application extend a result of J.L.Bryant and R.C.Lacher concerning resolutions of almost TL- -acyclic images of orientable 3-manifolds (Math.Proc.Camb.Phil.Soc. ii 88 (1980) 311-320), to nonorientable 3-manifolds. Theorem 3. Let f be a closed, monotone mapping from a 3-manifold M onto a locally simply connected 2Z~ -homology 3-manifold X. Suppose that there is a 0-dimensional set ZCX such that ftf^fchE ) = 0 for all x € X - Z. Then the set C = {x€X j f~ (x) is not cell-like} is locally finite in X* Moreover X has a resolution. Included is an investigation of the basic properties of generalized 3-manifolds with boundary, a topics on which little study has been done so far, as well as some results on regular neighborhoods of ,ccmpacta in 3-manifolds with applications to homotopic PL embeddings of compact polyhedra into 3-manifolds. Hi ACKNOWLEDGEMENTS This research was completed under the supervision of Professor R.Christopher Lacher. He encouraged me to enroll at FSU. He and his family helped us to settle in Tallahassee and they took care of our well-being during our stay in the States. Professor Lacher's excellent teaching helped me to get the most out of my studies at FSU» He accepted the responsibility of directing my doctoral research and he carried out this task in almost impossible conditions « most of the second half of my graduate studies I was away, in Yugoslavia so we had to communicate by mail and by phone. With his great knowledge, experience, will, patience, and good humor he came to rescue at the most difficult times. Without his help this work would never have been completed. I wish to most gratefully acknowledge him for all his generous support and encouragement through all these years. I wish to thank Professors J.J.Andrews, J.L.Bryant, D.A.Meeter, W.D.Nichols, and J.R.Quine,Jr. for serving on my Ph.D.Committee and for their help. I also had many useful discussions with Professors W.Heil, D.W.Sumners, and T.P.Wright,Jr. Special thanks are due to Professor J.Vrabec from the University of Ljubljana who always took great interest in my progress and devoted a considerable amount of his time to provide many remarks and and improvements of my preliminary manuscript. iv The illustrations were made by Mr. M.štalec from Ljubljana. My friend, Professor J.Rakovec from the University of Ljubljana, read the preliminary and the final version of the manuscript and made many important comments. I wish to thank him and his family for their moral support throughout these years. Finally, I would like to acknowledge the contributions of my wife, Barbara. Even though she had to combine the roles of wife, mother, wage earner and student, she still found time to encourage me in my studies. I also wish to thank my parents for their moral and financial support. The following institutions provided financial support: Cl) Research Council of Slovenia Ca U-year Undergraduate Fellowship, a 12-months Graduate Fellowship, a 12-months Research Grant (l98l), and 2 Travel Grants (1981,1933)); (2) Fulbright Foundation Ca Travel Grant (.1978-80)); (3) FSU Department of Mathematics and Computer Science (a part-time Teaching Assistantship (1978-80)}; (_k) University of Texas at Austin (participation at a conference C198O)); C5) College of Mechanical Engineering (University of Ljubljana) (_2 Graduate Research Grants (1981,1983) and k payed leaves of absence (.1981,1983)); and (6) Mathematisches Forschungsinstitut in Qberwol-fach (West Germany) (participation at a conference (198I)). I wish to gratefully acknowledge these institutions for this support. D.R. Tallahassee, February k9 1983 v TABLE OF CONTENTS Page LIST OF FIGURES ;,,.....................,....... ;¦..............viii LIST OF SYMBOLS AND ABBREVIATIONS.................. ¦.......... x INTRODUCTION........,.......................................... 1 CHAPTER ONE: PRELIMINARIES 1. UV, LC, and Related Properties ..................... 7 2. Upper Semicontinuous Decompositions ..,..¦,¦•¦¦¦¦¦¦¦ 9 3» Generalized n-Manifolds •, •......,................ ¦ • 11 k. Generalized 3-Manifolds .....,..¦•¦.................13 CHAPTER TWO: NEIGHBORHOODS OF COMPACTA IN NONORIENTABLE 3-MANIFOLDS 1. A Finiteness Theorem ¦...............................19 2 • A Neighborhood Theorem.......•............¦•.......22 3. A Resolution Theorem ••••••.......•................. 35 k. Peripheral 1-Acyclicity ,,••,.,,..,••••.............Hi CHAPTER THREE: A DISJOINT DISKS PROPERTY FOR 3-MANIFOLDS 1. Dehn Disks in 3-Manif olds •. • • f..................... ^7 2. Recognizing 3-Manifolds .....• •..................... 59 3. Isolated Singularities ,.,..,,..,.........,,,•••*••, 78 CHAPTER FOUR; GENERALIZED 3-MANIFOLDS WITH BOUNDARY ,.......... 83 CHAPTER FIVE: EPILOGUE .....t,................................. 91 vi Page BIBLIOGRAPHY ...,........................................... ?? APPENDIX: REGULAR NEIGHBORHOODS OF COMPACT POLYHEDRA IN 3-MANIFOLDS......................................105 VITA....................................................... lilt vii LIST OF FIGURES Figure Page 2.1, A 1-Handle Which Generates Several New 1-Handles ..... 28 2.2, Joining the 2-Sphere Components of 3Q..............¦• 30 2.3, The Structure of Neighborhoods N of K ................ 31 2.1*. A Modification of Whitehead's Continuum •••••••.••...• kO 2f5t Surgery on r •....................................... ^3 a 3.1. Main Steps in the Proof of Theorem (3.1) ............. 50 3.2. Special Neighborhoods of the Singular Set ............ 52 3.3» Surgery on f(D) ...,.................................. 55 3*b. Reparametrization of f- ...,,¦..,..................... 56 3.5» Pushing Along a Bicollar Into int M .............,. 57 3,6. Splitting a Handlebody Into 3-cells .................. 60 3.7* Expanding 3-Cell Chambers. .. •. •.......,............... 63 3.8. Cutting Along Compressing Disks . ¦........,..•..••.•.. 63 3.9» Main Steps in the Proof of Assertion 3 .. .......... t.. 65 3.10. Surgery on N., Case 1A; l^if^P .•••••.• t.. t •?••••••• 11 69 3.11. Surgery on N,, Case IB; l^i^P ..........¦•.......... 70 3.12. Surgery on N,, Case 2 ; p+l<^i<^k...........•.•••.•.• 71 3.13. Avoiding S(X)0N., Case 1; l^i^P.....¦...,.......,. 73 3.1H. Avoiding S(X)ONM Case 2: p+l^i^k .¦............... 7^ 3.15. Bypassing Isolated Singularities ...............*••... 82 viii Figure Page k.l. A Wildly Embedded 3-Cell in S3 • •, •............ ¦.;..... 85 1+.2. A Nonresolvable Generalized 3-Manifold With Boundary • ¦ 88 5.1« Extending Resolutions Over a Collar • ••••••••-....... •.. 95 5.2. Shrinking Out a Wild Arc in Bn ,.¦,... •..........•¦•••• 97 A.l. Simple Homotopy Equivalence of Regular Neighborhoods ••"109 ix LIST QF/SYMBOLS AND ABBREVIATIONS [ 1 • •.....•..••• "bibliographical reference 55 -............... the proof (or discussion of it) is ended or omitted *............... a point B............... Euclidean n-space Bn............... (closed) n-ball S............... n-sphere In............... n-cube (Ix ,,fxl, n factors I = [0,1] ) ZZ .............. nonnegative integers E.............. nonnegative reals FrU (CÄ U)....... frontier (resp. closure) of U X (X)............ interior (resp. boundary) of X, a generalized manifold with boundary (See p. 11) int M (3M)....... interior (resp. boundary) of M, a manifold with boundary X -** Y........... surjective map X >—*Y.......... .infective map X >->•*¦ Y ........... bijective map If].............. the greatest integer not exceeding t € JR S .............. singular set of a map f E '.............. the image under f of S PID .............., principal ideal domain ANR ..........••••'. absolute neighborhood retract ENR .............y. euclidean neighborhood retract (See p.11} DDP.......,.... •. .disjoint disks property (See p.13) DLP .¦.....¦.......Dehn's lemma property (See p.^7) MSP ••¦.¦.......... map separation property (See p. U8) KF ..,....,.¦..... Kneser Finiteness (See p.15) dimX.........m,, covering dimension of X, a separable metrizable space g(3M)............. total genus of 3M, M a compact 3-manifold with boundary (See p*2^) k-UVjUV^UV00 .....UV properties (See p.7) k-LC,LC ,LC°° .....LC properties (See p. 8) k k-uv(R),uv (R),uv°° (R) „ uv properties, R a PID (See p.8) k-lc(R),lck(R),lc°° (R) .. lc properties, R a PID (See p.8) 1-LCC............. locally simply-connected (See p.8) g(X,p)............genus of X at p, X a generalized 3-manifold (See p.15) H*(_;R),H#(_jR) .. singular (co)homology with coefficients in R S*(_jR),$*(_jR) • • Sech (co)homology with coefficients in R ^ ,••.,.•••...... homology equivalence a('.s).nMiMMM (resp. simple) homotopy equivalence s % .,.••.••...... homeomorphisja ä.............. PL isomorphism xi INTRODUCTION Generalized manifolds have held an important position in topology ever since they were introduced in the 1930fs. For low dimensions (^.2) their local algebraic properties are strong enough to imply that they are genuine manifolds. In higher dimensions they are interesting for at least two reasons: (i) they arise in many different classes (as quotient spaces of cell-like upper semicon-tinuous decompositions of manifolds, as manifold factors, as quotients of Lie group actions on manifolds, and as suspensions of homology spheres), and (ii) they have the same global algebraic properties possessed by manifolds (local orientability, duality). Recent success in higher dimensions — a remarkably simple characterization of n-manifolds (n> 5) — has stimulated an upsurge in interest in the geometric topology of generalized manifolds, so we first briefly review these results which, in turn, motivated our research in dimension three. The definition of a topological n-manifold (without boundary) is simple — this is a separable metrizable space that locally looks like If1 . However, when working with topological spaces it is quite often difficult to determine whether a certain construction is a topological manifold. Thus it would be desirable to have a short list of topological properties that are reasonably easy to check and that characterize topological manifolds. Such a list 1 2 should not include e.g., horneompr phi sms (since they are usually difficult to construct), or induction on dimension (since nice sub-manifolds are in general hard to find), or homogeneity (since the construction is usually already euclidean at some points so that homogeneity is precisely the problem), etc. His successful solution of J.W.Milnorfs classical problem about the double suspension of homology 3-spheres [17 ;Ch.ll ] , led J.W.Cannon to conjecture that topological n-manifolds are precisely generalized n-manifolds satisfying a "minimal amount of general position" [16;Conjecture. (1.3)] . In higher dimensions (n^5) this conjecture was proved soon thereafter, in two steps: (i) in 1977 R.D.Edwards showed that every resolvable generalized n-manifold (n>^5) with Cannon's disjoint disks property (DDP) [17 ;p.83l is a genuine n-manifold t [23] and [25 ;pp.118-122 ]), and (ii) in 1978 F.Quinn announced a proof that every generalized n-manifold (n^5) is resolvable ([53] and [55 ; Theorem(l.l) ]). Exploring the latest remarkable results of S.Donaldson [21] and M.H.Freedman [27] in dimension four, Quinn proved the resolution conjecture also for dimension four [56 ;Theorem (2.6.1) ] . Therefore in order to get a characterization of U-mani-folds, an analogue of Edwards? shrinking theorem for this dimension should be developed. This dissertation is a study of generalized 3-manifolds with O-dimensional singular set and their.possible role in characterization of 3-manifolds. Before summarizing our results we briefly review the work of others in this area. 3 M.G.Brin and D. R.McMillan, Jr. proved that, modulo the PoincarL conjecture, every compact generalized 3-manifold with O-dimensional singular set has a resolution [12 ;Theorem 5 ] > hence by J.L.Bryant and R.C.Lacher a conservative one [14;Theorem 1] , provided.it satisfies a certain "torsion-free" hypothesis. This extra condition was inherited from Brin1s extension of the Loop theorem and Dehnfs.lemma [10;Theorems 1-3] they used in their proof. T.L.Thickstun removed the "torsion-free" hypothesis from [10] and thus from [12] as well [61]• ^e later proved a positive result [62;Main Theorem] (obtained independently by R.J.Daverman) to the effect that such generalized 3-manifolds are images of "tame" generalized 3-manifolds (whose singular set has genus zero at each point), and consequently disentangled the PoincarS conjecture from the resolution theorem [61] • Another positive result is due to Bryant and Lacher who proved that every locally contractible 2Zp - acyclic image of a 3-manif old has a resolution (and is thus a generalized 3-manifold) [14 ;Theorem 2] . A refinement of their proof enabled them to omit the acyclicity hypothesis over a zero-dimensional set provided that the 3-manifold domain was orientable [14 ;Theorem 3] . In Chapter Two we prove that orientability is not necessary: Theorem 2.7» Let f be a closed, monotone mapping from a 3-manif old M without boundary onto a locally simply connected 7L^ - homology 3-manifold X. Suppose that there is a O-dimensional set ZCX such that S1(f,"1(x);2:2 ) = 0 for all x6 X - Z. Then the set C ={x€X | f (x) is not cell-like } is locally finite in X. Moreover, X has a resolution. M.Starbird introduced two notions of the disjoint disks property (DDP I and DDP II) for decompositions G of IT (rather than for the o quotient spaces B /G ) and he proved that for G a cell-like, upper . semicontinuous O-dimensional decomposition, satisfying either DDP I or DDP II, Ir/Gfe K [60;Theorem (3.1)] . His result is useful for generalized 3-manifolds X which are already known to be a quotient 3 X = E /G . A different approach was taken by Bryant and Lacher who showed that if in a compact generalized 3-manifold X the singular set S(X) lies in a compact, tamely embedded O-dimensional set ZCX (i.e., Z is 1-LCC in X) then X is a topological 3-manifold, provided X contains at most finitely many pairwise disjoint fake 3-cells [14;Theorem k] , (This generalizes previous results of C.H.Edwards, Jr. [22;Theorem 1] and C.T.C.Wall [66;Corollaries 1 and 2] .) However, the condition "S(X)CZ where Z is a closed 1-LCC subset of X" is not suitable since many potential singular sets may be wildly embedded in X. Professor Lacher suggested in July 198O that instead, one should look for a disjoint disks property for generalized 3-manifolds X with O-dimensional singular set such that it would imply first, the existence of a resolution f:M —*X and second, the shrinkability of the associated cell-like decomposition G ={ f (x)j x€X } of M. In Chapter Three we prove that a concept due to H.W, Lambert and R.B.Sher, called the map separation property (MSP) [44 ;p.51^] characterizes the 3-manifold property in certain cases (modulo the Poincare conjecture). We also study a similar concept from [44] called the Dehnfs lemma property (DLP) and show it plays the same role as the MSP: 5 Theorem 3.8. Let G be a cell-like, closed O-dimensional upper semi-continuous decomposition of a 3-manifold M (possibly with boundary) such that N-CintM. Then the following statements are equivalent: (i) M/G has the DLP; (ii) M/G has the MSP; (iii) M/G is a 3-manifold, Theorem 3* 10. Let L be the class of all compact generalized 3-niani- folds X with dimS(X) < 0 and let C C C be the subclass of all X€C — -o — — o with S(X)c {pt} and such that X» S . Then the following statements are equivalent: (i) PoincarŠ conjecture in dimension three is true; (ii) If X€L has the DLP or the MSP then S(X) = 0; (iii) If XeC has the DLP or the MSP then S(X) = 0. —o We conclude this introduction by a description of the organization of the dissertation. In Chapter One we collect most important facts about UV and LC properties, about upper semicontinuous decompositions of manifolds, and about generalized manifolds. In Chapter Two we investigate the nature of the neighborhoods of certain compacta in nonorientable 3-manifolds and prove a finiteness and a neighborhood theorem. We then use these results to prove Theorem (2.7). Also included is a comparative study of various kinds of acy-clicity over Z5p for embeddings of compacta in 3-manifolds. In Chapter Three we introduce the DLP and the MSP, verify that every 3-man-ifold has both properties and then prove Theorems (3.8) and (3.10). We conclude by an application of the DLP/MSP to the study of isola- 6 ted singularities. In Chapter Four we present a study of generalized 3-manifolds with boundary, an area where almost no research has been done yet. We prove several results analogous to those known for generalized 3-manifolds. In Chapter Five we collect some open problems and state some conjectures related to this topics. In the Appendix we study regular neighborhoods of compact polyhedra and prove some results concerning homotopic. .PL embeddings of compact polyhedra into 3-manifolds. We have included these results since they are related to (and were motivated by) those from Chapter Two, I. PRELIMINARIES . In this chapter we collect some important definitions and results from three subjects that underline our dissertation topics: UV, LC, and related properties, upper semicontinuous decompositions, and generalized manifolds. Standard references for other topics are: E.H.Spanier [59]for algebraic topology, C.P.Rourke and B.J.Sanderson [57] for PL topology, J.Hempel [32]for 3-manifolds, K.Borsuk [7] for ANRfs, and R.C.Lacher [40] for cell-like mappings. 1. UV, LC, and Related Properties A continuum is a compact and connected set. A compactum K in an ANR X has property k-UV (resp. UV^jUV00 1 (k6 E ) if for each neighborhood UcX of K there is a neighborhood Vc U of K such that any singular k-sphere in V is null-homotopic in U (resp. any singular j-sphere in V (0 <^j <_k) is null-homotopic in U; V is null-homo-topic in U). An n-manifold is a topological n-manifold without boundary. A compact subset K of an n-manifold M is cellular in M if K is the intersection of a properly nested decreasing sequence of n-cells in M. A space X is cell-like if there exist a manifold N and an embedding f:X*-* N such that f(X) is cellular in N. For finite-dimensional compacta this is known to be equivalent to "X has property UV00 ff [40;p#509l ..A mapping (or a map) is a continuous map fei^t not necessarily also PL. A map defined on a space (resp. an ANR; a manifold) X is monotone (resp. cell-like; cellular) if 7 8 its point-inverses are continue (resp. cell-like sets; cellular sets) in X. A closed map is proper if its point-inverses are compact. A compactum K in a manifold M is point-like if M - K % M -{ptl # A subset Z of a space X is ü-, -negligible if for each open set U in X the inclusion-induced homomorphism II, (U - Z).'•"-* IL (U) is 1-1. A space X is k-LC (resp. LL ; LC00 ) at_ x€ X (kLE+ ) if for every neighborhood U in X of x there exists a neighborhood VcU of x such that any singular k-sphere in V is null-homotöpic in U (resp. any singular J-sphere in V (0<_j<_k) is null-homotopic in U; V is null-homotopic in U). A subset ZcX is 1-LCC (for "locally simply cocon*-nected") if for every x6X and every neighborhood UCX of x there is a neighborhood VCU of x such that the inclusion-induced homomorphism Ji-, (V - Z) —> IL (U - Z) is 0. A compactum K in an ANR X has k-uv(R) (resp. uvk(R); uv" (R)) property (k€ZZ+ , R a PID) if for each neighborhood UCX of K there is a neighborhood VCU of K such that any singular k-cycle in V is null-homologous in U (resp. any singular j-cycle in V (0*Lj5.k) is null-homologous in U; any singular j-cycle in V (jLO) is null-homologous in U), with coefficients in R understood. The uv properties are related to Cech co-homology: if a compactum X has properties j-uv(R) (j = k-l,k; R a PID) then ^(X-^R) = 0 and conversely, if KJ(X;R) = 0 (j = k,k+l; R a PID) then X has property k-uv(R) [40;p.502] . A map defined on an ANR is U^ (resp. uvk(R)) (k€Z+ , R a PID) if its point-inver- k k ses have UV (resp. uv (R)) property. The following two results will often be needed in our proofs: the first one is a consequence of the Vietoris-Smale-Eegle theorems [40 ;pp.505-508] , while the 9 second one is due to R.C.Lacher and D.R.McMillan, Jr. [43;Lemma (U.l)]' : Proposition 1.1, Suppose that f :X -^Y is a.proper UV*"1 (resp, uvk{R)) map (k€Z+, R a PID) and that Y is LCk (resp. lck(R)). Then the inclusion-induced homomorphism II (X,*) —+ U (Y,#) (resp. H (X;R) —*H (Y;R)) is bijective for Q^qf.k-1 and surjective for Proposition 1.2« Let M be a manifold, V a connected open set in M, and suppose that the inclusion-induced homomorphism H. (V;Ep ) —+ H-(M;2Z2 ) is 0. Then V is orientable. ** 2. Upper Semicontinuous Decompositions Let G¦« {gCX } be a decomposition of a space X into compact (and connected) sets and let ir :X—*-*X/G be the corresponding quotient map, IL, the collection of all nondegenerate (i.e., g ž *) elements of G, and N- their union. A set UCX is G-saturated if U = Cj --------:------- fr TT (U). A decomposition G is upper semicontinuous if for each g€G and for each open neighborhood UCX of g there exists a G-saturated open neighborhood VCU of g. Equivalently, tt is a closed map, A decomposition G of a separable metrizable space X is k-dimensional (resp. closed k-dimensional) (k+l€Z ) if dim ir(w ) = k (resp. dimtr (cl(Nn)) = k). A decomposition G of a metric space X is weakly shrinkable if for each z> 0 and each neighborhood UCX of NL there is a homeomorphism h:X—*X such that h | (X - U) = id and for each g€G, diamh(g) 0 and every G-saturated open cover .0 of NG 10 there is a homeomorphism h:X -^ X such that (i) h| (X - ß *) = id, where fl*=U{U6fi} ; (ii) for each g€G, diamh(g) < e ; (iii) for each g€G there is a U6fi such that h(g)UgcU. Theorem 1,3* Let G he a cell-like upper semicontinuous decomposition of an n-manifold M. For n = 3 assume, in addition, that each g 6 G 3 has a neighborhood in M embeddable in E .Then G is shnnkable if and only if M/G% M. Proof. Follows by Bingfs Shrinking criterion [45] and Armentrout-Quinn-Siebenmannfs Approximation theorem ([1]» (56;Corollary (2.6.2)], [58]).** An upper semicontinuous decomposition G of an n-manifold M has a defining sequence if there is a sequence {M. | i^l} of closed subsets M.CM with the following properties: (i) for each iy each component of M. is a compact n-manifold with boundary; (ii) for each i, M. ,CintM. ; * l+l i ' (iii) for each g€G, g€En if and only if g is a nondegener-ate component of A, _ M.. It is well-known (and easy to prove) that an upper semicontinuous decomposition of a PL manifold has a defining sequence if and only if it is closed 0-dimensional. In our studies of decompositions we shall mainly consider those decompositions which are definable by (homology) cubes with handles (cf. a paper of McMillan,Jr. [47]), 11 3. Generalized n-Manifolds. A space X is an euclidean neighborhood retract (ENR) if it is homeomorphic to a retract of an open subset of some B • Equiva-lently, X is a separable, locally compact, finite-dimensional met-rizable ANR. Let R be a PID, A Hausdorff space X is an R-homology n-manifold (n€lf) if for each x€X, 8MX,X-{x} ;R)> 5n~*( {x} ;Rh A Hausdorff space X is an R-homology n-manifold with boundary (n€ "M ) if for each x€X, either 2*(X,X-{x} ;R) = 5nr*({x} ;R) or = 0. The subset X = (x€x| 5*(X,X- (x) ;R) = 0 } is called the boundary of X and jt = X - X the interior of X. Lemma 1.1*. Let X be an ANR and R a PID, (i) If X is an R-homology n-manifold then for each x€X and each q € 7L ( R ; q = n H (X,X-{x};R) ± { q I 0 ; q^ n (ii) If X is an R-homology n-manifold with boundary then for each x € X and each q 6 2Z 0 (R ; q = n and x€ Jt 0 ; otherwise Proof . On the class of ANR's the Cech cohomology agrees with the singular cohomology. The conclusion now follows by the Universal Coefficients theorem. ** Lemma 1.5. Let X be an R-homology n-manifold and an ANR, where R is a PID. Choose any x€X and let i«:H (X-{x};R) —* H (X;R.) be the in- * q q elusion-induced homomorphism (q62Z). Then the following holds: 12 (i) if q = n then i# is 1-1; • (ii) if q s n-1 then i# is onto; (iii) if q ± n-l,n then i^ is bijective. Proof. Consider the homology sequence of the pair (X,X-{x}) over R and apply Lemma (X.U) • $L A generalized n-manifold is an ENR that is also a TL -homology n- manifold. A generalized n-manifold with boundary is an ENR X such • • • that X is a TL -homology n-manifold with boundary and X is a generalized (n-l)-manifold. Let X be a generalized n-manifold (possibly with boundary). The set S(X) = {x€X | x has no neighborhood in X homeomorphic to an open subset of B } is the singular set of X, its complement M(X) = X - S(X) is the manifold set of X. The points of S(X) (resp. M(X)) are called the singularities (resp. manifold points). If X = 0 or S(X)C& then M(X) is a topological n-manifold. Generalized manifolds arise as (i) the finite-dimensional quotient spaces of cell-like upper semicontinuous decompositions of manifolds; (ii) the manifold factors; (iii) the qoutients of the Lie group actions on manifolds; and (iv) the suspensions of TL -homology spheres. A resolution of a generalized n-manifold X is a pair (M,f) where M is an n-manifold and f :M ->-* X is a proper celllike surjection. A resolution (M,f) of X is conservative if for each x€M(X), f-1(x) = * . Theorem 1.6. Let X be a generalized n-manifold. If n = 3 assume the Poincare conjecture and also that dimS(X)<^0. Then X has a conservative resolution. 13 Proof. If ni 2 then S(X) = 0 ([68;Theorems (IX.1.2) and (IX. 2.3)1). If n = 3 then X has a resolution by T.L.Thickstun [61] hence by J. L.Bryant and R.C.Lacher [14;Theorem l], a conservative resolution. For n - k the assertion was recently proved by F.Quinn ([56;Theorem (2.6.1) and Corollary (2.6.2)]). If n>. 5 then X resolves by Quinnfs Resolution theorem [55;Theorem (l.l)] and the assertion then follows by L.C. Si ebenmann1 s Approximation theorem [58] . JjJ A metric space X has the disjoint disks property (DDP) [17] if 2 for every e > 0 and every two maps f.,fp:B -*• X there are disjoint maps g1,g2:B2 —> X such that d(f1>g1)< e >d(fg>g2). Theorem 1.7. Topological n-manifolds (n>^5) are precisely the generalized n-manifolds satisfying the DDP. Proof. Follows by R.D.Edwards1 Shrinking theorem [23] and Theorems (1.3) and (1.6). ** h. Generalized 3-Manifolds Dimension three is in many respects peculiar mostly due to the unresolved status of the PoincarL conjecture. We list some of the most important facts. First, X cannot have "cone" singularities (which are common in higher dimensions) [42;p#8U] and [14;p,3ll] : Proposition 1.8. Let X be a generalized 3-manifold (possibly with boundary). Then no singularity of X can have an open cone neighborhood in X. ** Corollary 1.9. Let X be a PL generalized 3-manifold (possibly with boundary). Then X is a 3-manifold. L$ Another property of generalized 3-manifolds peculiar for this dimension is a kind of algebraic finiteness, as it was observed by Bryant and Lacher [14;pp.312-3131 : Proposition 1.10, Let X be a compact generalized 3-manifold (resp. with a resolution). Then there exists an integer k such that among any k+1 pairwise disjoint 2Zp-homology 3-cells at least one is con-tractible (resp. a 3-cell). L* So far we have made no assumption on the dimension of S(X). The following result of Brin and McMillan [12 ;Theorem 1 ] delineates the O-dimensional singular set case as natural and closely related to the embedding problem for open 3 -man i folds: Proposition 1.11. Let X be a compact generalized 3-manifold with dimS(X) <^0. Then the following statements are equivalent: (i) X has a resolution; (ii) M(X) embeds in a compact 3-manifold; (iii) S(X) has a neighborhood NCX such that NAM(X) embeds in a compact orientable 3-manifold; (iv) S(X) has a neighborhood NCX such that NHM(X) embeds in ]R3 . ** Let X be a generalized 3-manifold with O-dimensional singular set. Then by [12 ;Lemma l] every p€X has arbitrarily small compact generalized 3-manifold-with-boundary neighborhoods NCX such that N is a compact orientable surface in M(X). We say that X has genus en at^ p if p has arbitrarily small such neighborhoods N with N a surface of genus Ln. We say that X has genus n at_ p if X has genus 15 ^n at p and doesn't have genus ^n-1 at p. If X doesn't have genus Ln at p for any n we say that X has genus « at_ p. We shall denote the genus of X at p by g(X,p) [42] • We say that a generalized 3-manifold X satisfies Kneser Finiteness (KF) if for each compact subset X CX there is an integer k such that X contains at most k o o pairwise disjoint fake cubes* A sequence of pairwise disjoint com-pacta (c.) in a metric space X is a null-sequence if for every L> 0 all but finitely many among C.fs have diameter < e. It is not surprising that the Poincare conjecture enters into the picture as soon as we try to resolve generalized 3-manifolds (just recall Proposition (l.ll)). We consider an example which will be used later on in the dissertation. Suppose fake cubes exist and consider in S a null-sequence of pairwise disjoint 3-cells { B.} 3 converging to a point p€S . Replace each B. by a fake cube F. and choose a metric in W = (S3 - U ~ int.B. )U ( U.°\F.) so that F.'s v 1=1 i v 1=1 i i also converge to p. 3 . Proposition 1.12. W is a compact generalized 3-manifold with the following properties: (i) S(W) = { p } ; (We shall call such singularities "soft singularities".) (ii) W doesn't have a resolution; (iii) W Ä S3; (iv) g(W,p) * 0. Proof, (i) Follows by Kneser's Finiteness theorem [32;Lemma (3.1^)]. (ii) Follows by Proposition (1.10). 16 (iii) Let f:W —»>S be the map which shrinks out all F.'s. Then f is cell-like hence by [40 ;Theorem (k,2)] a homotopy equivalence* (iv) Clear. ** Proposition 1,13* Let X be a generalized 3-manifold with S(X)c Z, where ZCX is a closed, 0-dimensional set. Then the following statements are equivalent: (i) Z is 1-LCC in X; (ii) Z is n -negligible; (iii) For every z€Z, g(X,z) - 0. Furthermore, anyone of the statements (i)-(iii) implies that all singularities of X are ffeoftfI i.eMX is obtained from a 3-manifold by replacing null-sequences of pairwise disjoint 3-cells by null-sequences of pairwise disjoint fake cubes. The latter property — that all singularities of X are ffeoft"— is strictly weaker than (i)-(iii) if Poincarž conjecture is false. Proof, (i) =^(iii): See the proof of Theorem k in [14 ;pp.317-318] . (iii) =r>(ii) : Let UCX be an open set and J a loop in U - Z. Since dim Z = 0 and g(X,z) = 0 for all z € Z there is a covering V-, ..., V of Znu with pairwise disjoint compact generalized 3-man-ifolds with boundary V. = S C M(X) for all i, (We may assume that X is compact.) Suppose now that J bounds a (singular) disk in. U. With techniques described in details in the proofs of Theorems (3.1) and (3-9) ve can make this disk locally PL near the surface S =U . -V. , put it in general position with respect to S, and cut it off at S, thus pushing it into U - Z, or just get it off V.n Z for each. i. 17 (ii) ^(i): Let x€X be an arbitrary point and choose a neighborhood UCX of x. Since X is an ANR it is 1-LC. Thus there is a neighborhood VcU of x such that the inclusion-induced homomorphism n,(V) —*¦ n.(U) is 0. Since Z is II -negligible, the homomorphisms ni(V - Z) -*• Tl^V) and ni(U - Z) -* H (U) are 1-1. Consider the commutative diagram: n1(v - z)—^-* nx(u - z) I • • F 4 trivial | nx(v) map > n^u) Clearly, i# = 0. Assume now, say the statement (i). By [14 ;Theorem k ] no open subset VCX has the KF unless VCM(X). This implies X has only^oft" singularities. -.¦-¦' The last assertion is demonstrated as follows: take any wild 3 ... Cantor set in S , direct to it a nice null-sequence of pairwise dis- 3 joint 3-balls in S and then replace each of them by a fake 3-cell. Denote the new space by Y. Clearly, Y is a generalized 3-manifold, S(Y) is precisely the chosen wild Cantor set, and by [14 ^Theorem k ] S(Y) cannot be 1-LCC in Y, so Y doesn't satisfy the statement (i) (hence neither (ii) and (iii)). ** Corollary l.lU. Let X be a generalized 3-manifold satisfying KF and suppose that S(X)CZ, where Z is a closed, 0-dimensional set in X. Then X is a 3-manifold if and only if for every x6X, g(X,x) = 0. Proof. Follows by [14 ;Theorem k ] and Proposition (1.13). $$ 18 We conclude by stating two important results due to T.L.Thicks-tun: his extension of the Loop theorem [61](see also [11 ;p.30] ) and his Resolution theorem [62] which considerably improves the n - 3 case of Theorem '(1.6) — most notably, it disentangles the Poincarl conjecture from (1.6). Theorem 1,15. Let X be a compact generalized 3-manifold-with-bound-ary neighborhood of the singular set of a generalized 3-manifold, where X is a 2-manifold. Let C be a component of X, let N be a normal subgroup of II-(C), and let J be a loop in C that shrinks in X but that has [JJ L N . Then in any neighborhood of J in C there is a simple closed curve K such that [K] L N and K shrinks in X. *L Theorem l.l6. Let X be a compact TL -homology 3-manifold with bound-ary such that dimS(X) 1.0, S(X)CX, and X satisfies KF. Then there exist a proper cell-like surjection f: (Y,Y) .—*-*• (X,X) from a compact generalized 3-manifold Y with boundary, with only "soft" singularities. (So, in particular, if the Poincare conjecture is true, X has a resolution.) %% II. NEIGHBORHOODS OF COMPACTA IN NONORIENTABLE 3-MANIFOLDS The main result of this chapter is Theorem (2.7) — a generalization of a theorem of J.L.Bryant and R.C.Lacher [l4;Theorem 3 ] on resolutions of Z5p -homology 3-manifolds which are almost 1-acy-clic (over 2Zp ) images of orientable 3-manifolds, to 5Zp-homology 3-manifolds which are almost 1-acyclic images of nonorientable 3-manifolds. In the first two sections we develop two technical results — a finiteness and a neighborhood theorem (Theorems (2.1) and (2.2)). We then use them in Section Three to prove Theorem (2.7)• In the last section we present a comparative analysis of various kinds of 1-acyclicities for compacta in 3-manifolds. Some related results that were inspired by these findings are collected in the Appendix at the end of the dissertation. 1. A Finiteness Theorem T.E.Knoblauch [35] proved that in a closed orientable 3-man- ifold there can be but a finite number of pairwise disjoint compact 3 sets that do not have a neighborhood embeddable in TR .He also gave an example in [35]to show that this need not hold for nonorientable 3-manifolds. In the theorem below we give an additional condition under which the statement is true also in the nonorientable case. 19 20 Theorem 2.1. For every closed nonorientable 3-manifold M there exists an integer K such that if 3L ,... ,XK _ CM are pairwise disjoint compact sets and each X. has a neighborhood U. C M such that the inclusion-induced homomorphism H. (U,-X. \!ZL2 ) ---¦+¦ H (M;2Zp ) is 0, 3 then at least one X. has a neighborhood which embeds in E . Proof. We work with TL coefficients and we shall supress them from the notation. Let X ,... ,X C M be pairwise disjoint compact sets and suppose that each X. has a neighborhood U. C M such that the inclusion-induced homomorphism IL(U.-X.) ---> H (M) is trivial and if i i j then U.AU4 = 0. Let X = U * X. and U = U * U. and consi-l j 1=1 i 1=1 i der the following commutative diagram ©* IL(U.-X.) ©* -H.(U.) wi=l 1 i i' wi=l V i | 2t | 2t H^U-X) ----i-»- HX(U) ----^ H^U.U-X) -> H^M-X)----f—> H^M) ----—+ H^M^M-X) where the horizontal sequences are exact and pf is. the excision isomorphism. Suppose that for some u. 6 im [ H. (U.) —> H (U) } Zi=l p(ui} = 0# Then p( ri=l ui^ = ° hence f?( Ei=l ui} = (p1^0 Ff.p( S.n_ u.) = 0 so Z.n, u. € kerf1 = imf. Therefore s .n. u. = * 1=1 i 1=1 i 1=1 i f(v) for some v€H (U-X").- Now, v = Z^ v± where v.. €im[ H^IL-X^ —* IL(U-X) ] therefore f (v.) = u. for each i since the U.'s are pairwise disjoint. We conclude that p(u.) = pf (v.) = e(v.) = 0 since by hypothesis e is trivial. Therefore the image of the inclusion-induced homomorphism H (U) —+ H_ (M) is the direct sum of the 21 images of the inclusion-induced homomorphisms H (U.) —>¦ H. (M), lLi<_n. So if we let b. = rank H. (M) then n-b- of the homomorphisms H-(U,) —* ;H.(M) are trivial and so n-b neighborhoods U. axe orientable by Proposition (1.2). Consider the orientable 3-manifold double cover |5:M-—>.M of M. Let k($f) be the Knoblauch number of M [35 ;Theorem 1 ]. Since every orientable neighborhood lifts in R to two homeomorphic copies it follows that if 2(n-b ) >k($f) then some X. has a neighborhood which 3 embeds in B .We can now determine the number K from the equation 2(K-b1)-k(M) = 0 : K = [| H^M;^ )Z2 -S—+ "H^l^ZS )----> Tor(HQ(M;Z ) ,ZZ2 )—^0 Since TorCH^lL-X^ffi );^2 ) = 0 = Tor(H (M;ZZ1 ) ,Z2 ), f and g are isomorphisms. Thus if j^=0 then j#=0. On the other hand Theorem (2.1) is false over 2Z , p any odd 2 1 prime number, as the following example illustrates: let M = P xS , 22 2 1 2 where P denotes the projective plane. For each tLS let X = P *{t}t 2 2 Since M-X, contracts onto P and since IL (P \TL ) - 0 it follows t • • 1 ¦ p that the inclusion-induced homomorphism H. (M-X \7L ) —* H (M;2Z ) . . 3 is trivial. However, no X has a neighborhood embeddable in B 2 ^ since P doesn't embedd in E [28 ;Theorem (27.11) ] . 2. A Neighborhood Theorem Let K be a continuum in a 3-manifold M. How nice a neighborhood can K have? For example, if K is cellular in M then K is the intersection of properly nested 3-cells, while if it is cell-like then K is the intersection of properly nested homotopy 3-cells with 1-handles [47;Theorem 3 1 . We describe below neighborhoods of almost 1-acyclic (over ZL ) continua K. Theorem 2.2, Let K be a continuum in the interior of a 3-manifold M with (possibly empty) boundary. Suppose that K does not separate its connected neighborhoods and that for every neighborhood UC M of K there exists a neighborhood VCU of K such that the inclusion-induced homomorphism E.(V-K;ZZ2 ) -^ H (U;2Z2 ) is trivial. Then K = ft . , N. where each N.CintM is a compact 3-manifold with boundary satisfying the following properties: (i) for each i, N.x,CintN. ; ' l+l i (ii) N. is obtained from a compact 3-manifold Q. with a 2-sphere boundary by adding to 3Q. a finite number of orientable (solid) 1-handles; (iii) for each i, the inclusion-induced homomorphism V 3Ni+l;2Z2 ) "^VV^ } iS trivia1' 23 (iv) there is a homeomorphism h. :N. "*** N. such that h.'| 3N, = id and h.(Q*) = Q. ., where Q*CintQ. is formed by i ^1' l+l .11 J pushing Q. into int Q. along a collar of 3Q.. Remark. Theorem (2.2):(i)-(iii) was proved for orientable 3-man-ifolds by D.R.McMillan, Jr. [49;Theorem 2] . A.H.Wright observed [70;Theorem 2 ] that McMillan's theorem generalizes to nonorientable 3-manifolds but he did not obtain orientable 1-handles. Neither of the two papers [49]and [70] gave details of the proof because it was enough to indicate necessary changes in the proof of an earlier result of McMillan [47 ;Theorem 2 ] . Theorem (2.2): (iv) was proved for orientable 3-manifolds by J.L.Bryant and R.C.Lacher [14 ;Lemma C ]. We have decided to present the details in order to explain the specific situation for nonorientable 3-manifolds. Our proof of (i)-(iii) is modelled after the proof of [49; The or em 2l as outlined in the lecture notes of McMillan [48;pp.^5-^9 ] » from which we also quote the following folklore lemmas we shall need at several points in the dissertation (cf. [48 ;pp.7-8>p.l+9] ). Lemma 2.3« Let K be a compact set in the interior of a 3-manifold M, K i M and let NCM be a neighborhood of K. Then there exists a compact polyhedron UCintN with the following properties: (i) each component of U is a 3-manifold with boundary; (ii) each closed surface in U-K separates U-K; (iii) KCintU . JJ Let M be a compact 3-manifold with boundary and let F.,.. • >F c 3M be its boundary components. Then we define the total genus of 2k 3M to be the sum of the genera of F. (l . p w 1 H (M;R) be the inclusion-induced homomorphism. Then rank-(imi# ) = g(3M). ** Proof of Theorem (2*2). First, we shall prove that K = 0 ' * N. where N. satisfy (i) and (ii). It will follow by hypotheses that we can find a subsequence of {N.} satisfying (iii). By choosing a further subsequence we shall demonstrate (iv). We shall suppress the 7L~ coefficients from the notation. To prove (i)-(iii) it therefore suffices to show that given a neighborhood UCM of K there is a compact 3-manifold neighborhood NCU of K such that N is obtained from a compact 3-manifold Q with 33a 2-sphere, by attaching a finite number of orientable (solid) 1-handles to 3Q. So let UCM be a neighborhood of K. We may assume the following about U; (1) U isanonorientable connected compact 3-manifold with boundary; (2) KCintU; (3) U-K is orientable and connected; (k) each closed surface in U-K separates U-K. The condition (3) follows by Proposition (1.2) since, for sufficiently small Ufs, the inclusion induces trivial homomorphisms H (U-K) * H (M). The condition (iv) is provided by Lemma (2.3). 25 Let n €1 he Haken1 s number of U [29 ;p.-U8 ] .Using the hypothesis we can construct an ordered (nQ+2)-tuple Y = {V ,V ,... ,V - } o of compact 3-manifolds with boundary such that: (5) Vo.» U; (6) Vi+1CintVi; (7) 3V. is an orientable (possibly disconnected) two-sided closed 2-manifold; (8) H^av.^) — H^V.) is trivial; (9) KCintVn +1 . o (Note, that (7) follows by (3) and (k).) n +1 Define the complexity of Y to be the integer c(Y) = L .=Q E _Q 2 (n+l) g. (nowhere g.(n) is the number of components of 3V. with genus n [47;p.l30] , We shall show that in a finite number of steps we can improve Y so that it will still satisfy (5)-(8) (but not necessarily also (9)) and that for some ill, 3V. will be a collection 1 no+1 of 2-spheres. We shall achieve this by compressing 3Y = U ,_Q 3V. in a careful manner to reduce the complexity c(Y) and then we shall apply Haken1s Finiteness theorem [29]. The sequence of compressions that accomplish our goal is a sequence of modifications on Y (McMillan [47] calls them "simple moves}) of two types: if a compression of 3V. takes, place along a disk contained in V. we say that we removed a 1-handle while if the compres- o sing disk lies outside V. we say that we added a 2-handle. So suppose first that there is a disk DCintV such that DH3Y = 3DC3V. 0 1 for some i€{l,...,n +1} and such that 3D bounds no disk in 3V... o i So D either lies outside V, (in intV. . ) or inside V, (in V.-V..J. i l-l li l+l 26 In the first case we add a 2-handle to V. while in the second case we remove a 1-handle from V..- Denote the new V. and Y by V? and Y", respectively. Note that in both cases we did not change any V., J i f j. By [47 ;Lemma k] , 1 < c(Y') < c(Y) so by a finite number of compressions we get Y* = {V*,...,V* -} which cannot be compressed "O' in such a manner anymore, A routine "trading disks" argument now implies that each component of 3Y* which is not a 2-sphere is incompressible. We want to verify that Y* satisfies the conditions (5)-(8). We first note that if F is a boundary of a 3-manifold Z it still bounds after the compression: if we added a 2-handle then the new F will bound the manifold Z plus the "half-open" 3-cell attached via the 2-handle, while if we removed a1-handle from Z then the new F will bound the manifold Z minus the "half-open" 3-cell removed via the 1-handle. Therefore Y* is well-defined. Next, Y* satisfies (5) and (6) by our construction. To prove (7) we show that a compression of an orientable boundary of a 3-man-ifold Z .always yields an orientable boundary: suppose first that Z' = Z+(2-handle) had nonorientable boundary. Then we could find a simple closed curve J C3Z" such that J would reverse the orientation in 3Z'. We could isotope J off the cocore of the 2-handle and hence off the entire handle and into 3Z, thus showing 3Z to be nonorientable. Since removing a 1-handle from Z has the same effect on 3Z as adding a 2-handle to the complementary 3-manifold component bounded by 3Z, the preceeding argument also proves that for Z' = Z -(l-handle), 3Z' stays orientable. Finally, the condition (8) fol- 27 lows by [47 ;Lemma B ] because we" made the simplifications V. —^V"T without disturbing V., i ^ J,'. J We now prove that for some k€{l,...,n +1 } , 3V* is a collection o ju of 2-spheres, If not, then by Haken1s Finiteness theorem [29] for some lf.pU such that f(S. x{s}) = S where s=0;l. Let j. s s X = f(S x [0,1]). We may assume that no surface in (intX)fl3Y* is parallel to S. in X. By [65 ;Corollary (3#2)] each incompressible surface in int X is parallel to S. in X. Therefore (intX)H3Y* consists entirely of 2-spheres. Also, X must be irreducible for if there were a 2-sphere in X which would not bound a 3-cell in X then 2 it would be incompressible hence parallel to S / S . Therefore X minus the interiors of a finite disjoint collection of 3-cells lies in V*. Hence every 1-cycle in S is homologous to a 1-cycle in S thus it bounds in V* by (8). Since by Lemma (2.*0 the image of the inclusion-induced homomorphism H. (3V*) —^H.(V*) has rank (as a vector space over 7L2 ) equal to g(3V*), it"follows by (7) that S is a 2-sphere, a contradiction. Let V be a 3-manifold among V? all of whose boundary components are 2-spheres. Clearly, (9) may no longer be true so we now take care of that. During the compressions, when we attached a 2-handle it may have happened that it passed through the space in U that was previously occupied by a 1-handle which was removed at an earlier. stage. In such cases we require that the boundary of the 2-handle be in general position with respect to the boundary of the 1-handle. 28 In addition, ve shall assume that the annulus removed from 3V? (recall 3V? is orientable so it contains no Mobius bands) in the k-th compression be disjoint from all 1-handles or 2-handles involved in the preceeding k-1 compressions. So if we now add to 3V all 1-handles that were removed from V during the compressions, we get several 1-handles attached to 3V. Note that adding of an old 1-hand-le H to 3V may result in many new smaller 1-handles as H may run through several 2-handles that now occupy portions of its original place, (See Figure (2.1).) Figure 2,1. 29 Every resulting 1-handle is orientable. For suppose in reattaching the 1-handles sequentially we have added a nonorientable 1-handle. Then for every subsequent reattachment of the remaining 1-hand-les we have only one isotopy class of attaching maps [57 ;Theorem (3.3*0] so we end up with a nonorientable surface» But this is impossible by (3) and (k). We may also assume that for every resulting 1-handle H both ends of H are attached to the same boundary component for otherwise we add H to V thus reducing the number of boundary components of V by one. The 3-manifold N which we get from V by reattaching all 1-hand-les may be disconnected so we keep only the component which contains K. Thus N is obtained from a compact 3-manifold Q with 3Q a collection of 2-spheres by attaching a finite number of orientable 1-hand-les to 3Q so that every 1-handle has both ends on the same component of SQ. Let_p.CZ. (i=l;2) be arbitrary points on two distinct 2-sphere components L and 2 of 3Q. Since K doesnft separate N there is a polygonal arc A in N-K joining p. and p . Suppose that A passes through a 1-handle H. We may assume that AAH is just one arc meeting 3Q in only two points on Ep. Then AH H can be replaced by another polygonal arc BCN-intH attached to I . So we may assume that A doesn't pass through any of the 1-handles. Therefore by drilling tunnels we can effectively join the components of 3Q thus obtaining the desired neighborhood N. (See Figure (2.2).) We can describe the structure of the neighborhoods N of K as follows: N = Q+(l-handles) where Q captures the "nonorientability" of K while the handles capture the "pathology" of K. (Figure (2.3)) (D ro ro 31 Figure 2.3, 32 00 It remains to prove (iv). So assume K = f] . N. where N. = Q.+ (l-handles), as in (i)-(iii). Let 'K.C.intQ. be a spine of Q. • Let ¦• Q. be the closed 3-manifold we obtain by attaching a 3-cell to 3Q. • For each iL.l, N " = (N /K,) # Q, (the interior connected sum). Recall that N. is a nonorientable compact 3-manifold with boundary so by [32;(3.15),(3.17)] jj admits a unique normal, prime decomposition 2 1 N - M #..f#M , M. ^ S xS . Consider normal, prime decompositions for N /K. and Q. (i L.1). Observe that every N /K. is orientable because (N>-N.)Hk = 0 so N--N. is orientable and N./K. DN./K. » an 1 1 1 1 1 1 1.1 orientable cube with handles. Therefore in a normal, prime decomposition N^K^ AX#...#Ap^B^.-ga of 3^/K. such "that p j 0 factors 2 1 m A. Ä S x S . On the other hand Q. must be nonorientable (since N_ 1 Hi 1 is) so in a normal, prime decomposition Q. = C.# ...#C , every C. 1 1 r 1 2 1 / S * S . By [32;Lemma (3.17) ] we may replace each A. by P = the 2 1 . nonorientable S -bundle over S to get a normal, prime decomposition for H. = P# ...#P#B.# ...#B #Cn# ...#C (p factors P). It föl-1 1 q 1 r lows by the uniqueness of normal, prime decompositions that p+q+r=n and after a suitable permutation of the indices of summands each C. is homeomorphic to some M.. Therefore among n+1 Q.fs at least two have the same prime summands (up to a homeomorphism). By choosing an appropriate subsequence of {Q.} we may henceforth assume that for each i Q,. — lj 1 ^ y We first construct h • Let q:N. —*» N /K be the quotient map where K eint Q is a spine of Ö and let i < j. Then the identity on 3N induces a homeomorphism t! :3(N /K.) —* 3(N./K.) which makes the next diagram commutative: 33 3N1^___-------^L_-----»9^ 3(^/1^)1------------^----------»- ad^/K ) We shall show how to extend t! . to a homeomorphism t. . :N-/K. —> lj xj 1 i vv Take a simple closed curve Jc3(N /K ) such that J is essential on a(H./Kj) and null-homotopic in yK.. Since ^ = (N^K^^C^ we can consider J also as an essential simple closed curve on 3N. which is null-homotopic in N . Therefore q.(J) is a simple closed curve on ^(N./K.) which is essential on 3(N /K.) and null-homotopic in N./K )• By Dehn's lemma,J (resp. t!.(J)) bounds an embedded disk i J *^J (D,3D)c(N1/Ki>3(N1/Ki)) (resp. (Df ,3Df) C (N^Kj ,3 (l^/K ))). By (iij N,/K is a cube with (solid) 1-handles, so in finitely many steps we 1 m can cut N./K. along compressing disks D to get a 3-cell R.. Extend t! over each D by mapping it to the corresponding compressing disk D' in N./K. described above. Finally, we can extend t!, over the interior of R. to get t... Recall again that H^ = (^/K )#^ for all m >1. Let B^CQ^ and C Cint (H,/K.) be open 3-cells, m=i,j. Let f.: (Q.-B.) ----> m j. m ill 3((N./K.)-C.) be an attaching homeomorphism for the connected sum (N /K.)#Q.. If we define the attaching homeomorphism f, for (l^/K.) #Qi by f, = (s. |3)«f71*(t!.) then the diagram on the top of the J J ij i iJ next page will commute. (Note that because Q is nonorientable any two attaching homeomorphisms f. are ambient isotopic [57;Ch.3] .) 3k aUl^/K.)^) *ij , 3((N1/Kj)-BJ) ¦r1 . ...[3Q. I fJ Finally, define h..:N—>N. by h..(x) «. s.,(x) if x€Q. and = tjL (x) if x 6 N^Qj.. Clearly, luj 31^ = id and VjtQj) s QJ* The homeomorphism h is the composition of h. 0 and a homeomorphism of N. that is the identity outside a neighborhood of 3Qp in N. and pushes Q* onto Qp. We can get h., i >2 in a similar way (see the proof of Lemma C in [14;pp.317-318] ). $L Let K be a compact set in the interior of a 3-manifold M. We say that K can be engulfed in_ M if the interior of some punctured 3-ball in M contains K. A sequence (K.) of compact 3-manifolds with boundary is a W-sequence if for every i the following conditions hold: (i) K.CintKi+1; (ii) the inclusion-induced homomorphism. is trivial: An open 3-manifold M is called s Whitehead manifold if it can be ex- CO pressed as M = U . K. for some W-sequence of handlebodies [5Qp.313]. An examination of the proofs in a recent paper of D.R.McMillan, Jr. and T.L.Thickstun [50] shows that the orientability hypothesis can be removed from all results in [50] if one uses Theorem (2.2) in the place of [49 ;Theorem 2 ] : Theorem 2.5. Let M be a compact 3-manifold (possibly with boundary) and KcintM a compact subset. Then K can be engulfed in M if and 35 only if there is an open, connected neighborhood Uc M of K such that U embeds in S3 and H (U;2Z) = 0. ** Theorem 2.6. Let M be a compact 3-manifold (possibly with boundary), Then M contains no fake 3-cells if and only if each Whitehead mani- 3 fold that embeds in int M also embeds in S . $L 3» A Resolution Theorem J.L.Bryant and R.C.Lacher have proved that every locally con-tractible 1-acyclic over 2Zp image X of a 3-manifold M without boundary admits a resolution. In particular, X is a generalized 3-manifold [14;Theorem 2] . A refinement of their proof enabled them to omit the acyclicity hypothesis over a O-dimensional set "provided M was orientable [14;Theorem 3] . We show below that orientability is not necessary. (We are referring to the case p=0 or 2 of [14 ;Theorem 3] only,) Theorem 2.7» Let f be a closed, monotone mapping from a 3-manifold M without boundary onto a locally simply connected 7L~ -homology 3-manifold X, Suppose that there is a O-dimensional set ZCX such that S1(f""1(x);Z;2 ) = 0 for all x€X-Z. Then the set C = ix€X \ f~ (x) is not cell-like } is locally finite in X. Moreover, X has a resolution. Proof. Again we supress the coefficients. Let A = {x€X j H (f (x)) ^ 0 } . By [38 ;Theorem (U.l) ] A is locally finite in X. Let B ={ xLX| —1 3 f (x) has no neighborhood in M embeddable in E } . In order to show that B is locally finite in X it suffices by Theorem (2.1) to prove that for each x€X, f" (x) possesses a neighborhood UCM such 36 that H^U-f'^x)) -^^(M) is trivial, So let x€X. Since A is locally finite in X there is a neighborhood WCX of x such that WAAcCx K By hypothesis,X iš LC so there is a connected neighborhood WfcW of x such that any loop in Wf is null-homotopic in W. Consider the following commutative diagram: ^(^(wM-f^x)) _i*—> ^(f^CwJ-f^Cx)) . H^W'-M) —-------—-* H^W-fx}) Hj.(wf) -------^L____:____> ^(w) where the horizontal homomorphisms are induced by inclusions, f|# are the isomorphisms of Proposition (l.l), and j# and j J sure the isomorphisms from the homology sequence of the pairs (W,W-{x}) and (W'jW'-Cx}), respectively. By hypothesis, i# = 0 hence iL = 0. Thus we may apply Theorem (2,1) — we conclude that B is locally finite in X, By Theorem (2.2), f~ (x) is definable by (orientable) cubes with handles for all x€X-B, so by [49 ;Theorem 3], f"L(x) has the 1-UV property. Since cubes with handles have no higher homotopy, each f" (x) has the UYW property and hence CCB (cf. [40] ). Therefore C is locally finite in X, Note that, in particular, by G.Kozlowski and J.J.Walsh [36], x-C (hence also X) is finite-dimensional. It now remains to find a resolution for X. We construct it by improving f over the points of C. Observe first, that if x€X is an arbitrary point and WCX is any of its neighborhoods then combining 37 the isomorphism H (f"" (W)-f~ (x))^H (W-tx)) given by Proposition (1.1}, with the isomorphism H (W-{x}) ^,HQ(W) given by Lemma (1.5)> we can conclude that f~ (x) doesn't separate its connected neighborhoods in M. It follows by the arguments employed in the diagram on page 3.6 that f (x) satisfies the hypotheses of Theorem (2.2)• So —1 r\ °° if we let cLC and put K = f (c) then K = M . N. where N.'s are the compact 3-manifolds with boundary described in the conclusions of Theorem (2.2). We shall use the notation from that theorem in the rest of the proof (i.e., Q., Q*, and h.). (The following is modelled after [14;pp.316-317] •) Let Mf = M/Q* and let h! :N. —> N. be a homeomorphism such that h!J3N. = id and h!(Q.+.) = Q*+1« Define a map h*;M~* M by letting h*(x) = h!h.(x) for x€N. and = x otherwise. Then hf is a homeomorphism and h*(Q?) = Q? . . Let g :M ~~* Mf be the quotient map. Define inductively gj, = gi-1(h*)"1:M^ Mf, i^l. Then g±\ (M-N.) = g^J (M-I^). Also, the only nondegenerate point-inverse of g. is Q*+1• Indeed, gi = gi-i(hirl = ei-2(hi-irl(hi)"1 = ••• = gjh*)-1...^*)"1, h* are homeomorphisms, the only nondegenerate point-inverse of g is clearly Q*, and g~^o(Q*) = (h*.. •^1)gQ(Q|) = (h*.. .h*)(Qj) = (h?...h*)(Q*) = ... = h»(Q*)= Q*+r Let 1H = «iC^), i>.0, and Kf = H " N!. It follows by Theorem (2.2) that for every i, IL(NJ[) is free on finitely many generators. Also, considering the commutative diagram on the top of the next page we observe that i# is onto and j# = 0 by the choice of N.fs and g.'s. Hence iL = 0, too. It follows by [49;Theorem 3] that K1 is cell-like. 38 y»jk>—-^—>.Wfl) n Define a map g:M-K—> Mf by letting g = g. on M-intN. .Then g(M-K) = Mf-Kf and g is a homeomorphism. Finally, we let ff :Mf "**¦ be given by ff(x) = fg-1(x) for xcM'-K1 and ».c for x€Kf. It is easy to see that ff is a continuous, proper onto map. Since K*is cell-like, fis cell-like over X-(C-{c}): M-K---------g---------> Mf-Kf The proof is now completed by repeating the above surgery over the rest of C (i.e., over C-{c}.) JJ An alternative proof of Theorem (2.7)* Let A = {x6X| 5 (f~ (x) f 0}, By [14 ;Assertion 1 on p.315 ]>A is locally finite in X. Since by [48 proposition 2 ] and Proposition (1.2), every f" (x), xLX-A, has an orientable neighborhood in M, it follows by [14 ;Assertion 3 on p.3l6 ] that C-A is locally finite in X-A. It thus remains to show that no limit point of C-A can belong to A. Let a€A and suppose that for a sequence {x }cX-A ,limx .= a. By [14 ;Assertion 2 on p. & r\-Yoo & 316 ] , every f"\ (x) is strongly E?-acyclic hence by [47 ;Theorem 2 ]t the intersection of a nested sequence of 2Zp -homology 3-cells with 39 handles. Thus for each nf^l there exists an orientable neighborhood U CM of the continuum f~ (x ) and a 22L-homology 3-cell with hand-n n 2 les H CU such that f (x )Cint H • We may also assume that if n n n n i ^ j then U.HU. » 0. It is a well-known corollary of Grushko-Neu- mann theorem [32 ;p.25] that in a compact 3-manifold there is but a finite number of pairwise disjoint TL -homology 3-cells that fail to be genuine 3-cells [69]. Therefore, by [49;Theorem 3] all but a finite number among f" (x ) are 1-UV hence cell-like [AOJ. Thus x ^ n n LC for all but a finite number of indices n. Therefore the set C-A is locally finite in X. Consequently, C is locally finite in X. The construction of a resolution for X is now as in the preceeding proof. $$ The next corollary provides a partial converse in dimension 3 to a well-known fact that cell-like upper-semicontinuous decompositions of topological n-manifolds yield generalized n-manifolds (for n>_k assume also the quotient space is finite-dimensional). Corollary 2.8. Let G be a O-dimensional upper semi continuous decomposition of a closed 3-manifold M such that M/G is a generalized 3-manifold. Then the set C *{g€G g is not cell-like J is finite. Proof. Since G is upper semicontinuous the quotient map q:M -^+M/G is closed and monotone. Let Z = q(lO. Then dim Z < 0. The conclusion now follows immediately by Theorem (2.7). ** Remarks. (l) The Hopf maps or the Bing map [13 ;p.U8 ] show that if q(Nn) is a 1-manifold then all nondegenerate elements g€G may fail ko to be cell-like. Thus the restriction dimG = Q in Corollary (2.8) seems reasonable. (2) Spine maps [13;pA8] show that the set C in Corollary (2.8) may have any finite number of elements even when C = Hß. (3) The following modification of the classical construction of the Whitehead continuum [67] shows that all nondegenerate elements of G may fail to be cellular in M even when q.(Nß) is a Cantor set and G is cell-like. Let {T.} be the defining sequence for the Whitehead continuum. Keep TQ. Replace T. by two smaller solid tori Tnn and Tm as shown in Figure (2.U). Figure 2,k. Ill In "an analogous way replace T by four smaller solid tori T 0 > TooicintToo tod Toio'ToiicintToi* etc- Let Y = Ton(TooUToi)n (T0ooUTooiUToiou Ton^A ,f • and let G be the decomP°sition of s3 into points and the components of Y. U. Peripheral 1-Acyclicity We wish to compare various concepts of 1-acyclicity we employed in the preceeding sections. Let K be a subset of an ANR X, We say that the inclusion KCX is strongly (resp. weakly peripherally, strongly peripherally) 1-acyclic over R (R a PID) if for each neighborhood U C X of K there is a neighborhood V C U of K such that the inclusion-induced homcmorphism H.(V;R) -** H (U;R) (resp, H (V-K;R) -* H^UjR); H^V-^R) -> H^U-K^)) is trivial. It is well-known that strong 1-acyclicity does not depend upon the embedding of K into X and that furthermore, for R a field, it is equivalent to the condition K;R) = 0 [40;p.502] .The following example shows that the other two acyclicities may depend upon the embedding. Let X = S2 x S1 and K = S2 v S1, and let f ;K -> 3R3 and g:E3 -*> X be embed-dings. Then KCX is strongly peripherally 1-acyclic over any PID R (since X-K is an open 3-cell) while (gf)(K)cx is not even weakly peripherally 1-acyclic for any PID.R. (just take U = g(E )). It is not a coincidence that dim K - 2 in this example for we prove in Theorem (2.11) that for dimK.il and X an R-orientable 3-manifold, all three 1-acyclicities are equivalent .so, in particular, independ*-ent of the embedding. It is clear that strong peripheral 1-acyclicity implies weak pe- k2 ripheral 1-acyclicity. We now show that for compacta in 3-manifolds the two concepts are equivalent if R » ZL . Theorem 2.9« Let K be a compact set in the interior of a 3-manifold M. Suppose that KcM is weakly peripherally 1-acyclic over Z^ . Then K CM is strongly peripherally 1-acyclic over 5Zp . Proof, We shall supress ZZp coefficients from the notation. Using the hypothesis we can express K as the intersection of a properly nested sequence of compact 3-manifolds N. Cint M with boundary such that all inclusion-induced homomorphisms IL(N.-K) —> H. (N. .) are trivial• Let a be a simple closed curve in N.-K. Then there is an integer j >i such that aCN.-HL. Let ZC3N... be a component of 3N4J.. . i j j+1 * J+l Since S is a closed 2-manifold it contains a bouquet T of finitely many simple closed curves so that Z -T is an open 2-cell. Let ß C T be one of these loops. Since H.(<)N.+1) —> H-Cn.) is trivial, ß bounds a surface Ta in N,, Also, a bounds a surface r in N. . since VN.-K) -^H,(N. .) is trivial. Put the surfaces T and TQ into i l l-l a p general position. Let p.,,••.,?. LT flß be the points of the intersec- J» w CT tion, ordered in such a way that for each i, p. lies between p. , and p. . on ßt Note also that each p. lies in intT because 3T flß l+l i a a -a H ß = 0, Let ACH. te a regular neighborhood of ß in*N._-¦ and C = 3A. Thus A can either be a solid torus or a solid Klein bottle. For each i, there is a disk D. C int T centered at p. such that 9 i a *i A ^T = U. _ D.. Let C.CC be the annulus determined by the pair ** a i=l li (3D2i-1,3D2i), lli±[|] > i.e., ^„^C.UD^ is the boundary of 1*3 P/ O/ n-t oc Figure 2,5, the 3-cell E which Dp. and Dp. cut off on A and which doesn't contain any D , j i 2i-l,2i. (See Figure (2.5).) 2i-l' * We now do the following surgery on r : replace each pair (D t # Dp.) by the annulus C, lf^iülj] • Denote the new surface by ^a If t is even then PjH 3= 0 while if it is odd then r*n ß= (pt> , Suppose t were odd. Consider X = r*nrband let ACX be the compo- u p nent containing the point p • Then A is a compact 1-manifold hence an arc. Plainly, p 6 3A. Let q€3A be the other endpoint. Wow, qL r* because 3r * nrft»anr s 0 sinceT-cif- and a cN.-N.. Also, q ) A T*= (ßAr*)-{pt> = 0.Since qe^r^U 3T this yields a contradiction to our hypothesis that t was odd. kk We can therefore assume that ot bounds a surface T in N. . such that l-l for every loop 3 CT, TAß = 0. Thus if f hits N . at all, it "enters through open disks in 3N. . and so it can be cut off at 3N. . J^l -J -*• Hence abounds a surface in N.-N.A.cN,-K. $$ 1 j+1 1 ** Theorem, 2,10, Let R be a PID and let K be a compact set in the interior of an R-orientable 3-manifold M, Suppose that K is strongly 1-acyclic over R, Then KCM is strongly peripherally 1-acyclic over R, Proof, We shall supress the coefficients from the notation. Let V cue M be neighborhoods of K such that the inclusion-induced homo-morphism BL.(V) —* H (U) is trivial. Consider the following commutative diagram: -* H2(V,V-K) ----> H^V-K) ----> H^V) u X* > H2(U,U-K) ----> H^U-K)----> H^U) where the horizontal sequences are from the homology sequence of the pairs (V,V-K) and (U,U-K), and f,ff are the Alexander duality isomorphisms. By [40 ;p.502 ] , K) = 0 hence i# is a monomorphism, Since j^ = 0 we can therefore conclude that jL = 0. $$ 2 1 The converse of Theorem (2.10) is false: let M = (S x S )-B 2 1 2 1 . where BCS x S is the interior of a 3-cell. Then K = S v S is a p spine of M so that M-K = S x [0,1), Therefore, K is strongly peripherally 1-acyclic over any PID R. On the other hand K certainly is not strongly 1-acyclic over any PID R. Note that in this example 1*5 dim K = 2. The next theorem asserts that there can he no counterexample with dim K (iii) is clear. We show (iii) =^(i): let VCUCM be neighborhoods of K such that the inclusion-induced homomorphism H (V-K) —* H.(U) is trivial. Let z be a 1-cycle in V. By [39 ;Lemma (2.1)] z is homologous to a 1-cycle z*LZ.(V-K). By hypothesis, z*^0 in U hence z^O in U, as well. ** Theorem 2.12. Let K be a compact set in the interior of a 3-manifold M. Then the following statements are equivalent: (i) KCM is weakly peripherally 1-acyclic over TL^ ; (ii) KCM is strongly peripherally 1-acyclic over 7L^ ; (iii) There exists a neighborhood WCM of K such that each simple closed curve in W-K is 7L -homologous to zero in M-K. Remark. Let W be em open neighborhood of K such as in (iii) above. Then by [49 ;Lemma l] K is strongly 1-acyclic over ZZL if and only if, in addition, each simple closed curve in W is ZZ9-homologous to k6 zero in M, This gives us a good measure of the (possible) difference between the two acyclicities (over 2Zp). Proof of Theorem (2.12). We only need to prove (iii) =>(ii) since (ii)=>(iii) is clear and (i)<=*(ii) follows by Theorem (2.9) • So let UCM be a neighborhood of K. We may assume that UCW f that U is a compact 3-manifold with boundary, and that KcintU. Let Sc 3U be a component of 3U. Then there is a bouquet T CS of simple closed curves such that S- U{J€T} is an open 2-cell. By hypothesis, each curve J 6T bounds a surface S in M-K. Let V = int U - U{Sj|j€T} and let J* be a simple closed curve in V-K. Then J* bounds a surface S* in M-K. Using the same argument as in the proof of Theorem (2.9) we can show that J* bounds a surface Sf in M-(KU( U{SJ|J€T} )) and hence enters S through open disks and can thus be cut off on S. We may therefore assume that S'CU-K. This shows that every 1-cycle in V-K bounds in U-K. ** Corollary 2.13» Let K be a compact set in the interior of a 3-niani-fold M and suppose that IL(M-K;Z2 ) = 0. Then KCM is strongly peripherally 1-acyclic over 2Zp . Proof. Apply Theorem (2.12) with W = M. L$ III. A DISJOINT DISKS PROPERTY FOR 3-MANIFOLDS The main results of this chapter are Theorems (3.8) and (3.10) — we show that the map separation property (MSP), a concept due to H.W.Lambert and R.B.Sher [44] is an appropriate analogue of J.W. Cannon's disjoint disks property (DDP) for the class of compact generalized 3-manifolds with 0-dimensional singular set, modulo the Poincarg conjecture. In the first section we introduce the MSP and a similar concept from [44] , called the Dehn1 s lemma property (DLP) and we prove that 3-manifolds have both properties. In the second section we prove the main results. We conclude the chapter by an application of Thickstun's extension of the Loop theorem (Theorem (1.15)) to the study.of isolated singularities 1. Dehn Disks in 3-Manifolds We recall that a mapping means only a continuous hence not necessary PL map. A mapping f of a disk (resp. disk with holes) D into a space X is called a Dehn disk (resp. Dehn disk with holes) if —1 S A3D = 0, where Sf = cä {x€D|f U(x)) i x } is the singular set of f. Also, define I = f(Sf). A space X is said to have the Dehn1 s lemma property (DLP) [44]if for every Dehn disk f:D—>-X and every neighborhood UCX of I ~ there exists an embedding F;D—*X such that F(D)c f(D) u U and F(3D) = f(3D), A space X is said to have the map separation property (MSP) [44] if given any collection of Dehn ki 1*8 disks f ,....,f :D-—*X such that if i j* J then f^CaD) rt f ,(D) = 0, k and given a neighborhood Uc X of U . f,(D) there exist mappings Fl,,,',Fk:D—*U such that for each ij F' l3D = f- l3D and *f ¦* ^ J then F^DjHF (D) = 0. Lambert and Sher say in [44] that "it is a well-known (and use's « ful) fact that -S"* has the DLP and the MSP" but they give no proof or reference [44;p.5,l^] • We prove below that every 3-manifold (possibly with boundary) has both properties (by (3.2) and (3*7))• This result follows by the following stronger result: Theorem 3*1. Let f:D—*M be a Dehn disk in a 3-manifold M (possibly with boundary) and UCM a neighborhood of Ef. Then there exists an embedding F:D—*M such that (i) F(D)-U = f(D)-U; (ii) F J3D = f |3D. Corollary 3*2. Every 3-manifold (possibly with boundary) has the DLP. ** The proof of Theorem (3.1) relies heavily on two deep results from 3-manifolds topology — R*H.Bingfs Surface Approximation theorem [5] and D.W.Henderson's extension of Dehn lemma [33 ;Theorem (IV.3)] . Theorem 3.3. (H.H.Bing [53 ) Let P be a compact surface in a .3-manifold Mf NCP a closed subset, and let f:P—*IR be an arbitrary map, Assume that at each point x€N, P is locally PL in M at x. Then there exists a surface P* c M and a homeomorphism h:P^-^P* such that: (i) h|N:N •—*?* is the inclusion; k9 (ii.) for'every x6P, d(x,h(x)) 0, P* is locally PL in M at h(x)¦ $$ Theorem 3.k. (D.W.Henderson [331) Let M be a 3-manifold and f :D —* M a PL disk with Sf .^ = 0. Then for every e>0 there exists a PL embedding F:D —> M such that (i) F(D)-Ne( Zf) » f(D)-H ( Zf), where H" is the e-neighbor-hood of E ; (ii) F |2D = f|3D. ** Corollary 3*5«(Bing*s extension of Dehn's lemma [15;Theorem (U,5**0J) Let f :D ^ M be a Dehn disk in a 3-manifold M (possibly with boundary) and UCM an open neighborhood of f(int D) . Then there is a ho-meomorphism F of D into f(3D)uu such that F is locally PL except (possibly) on 3D. Proof. Follows by Theorems (3.1) and (3.3).. Proof of Theorem (3.1). We first consider the case when f(D)cintM. Here is an outline of the proof: Put Sf inside pairwise disjoint PL disks with holes C, ,...fC Cf" (U). Let C = U .m c.. Assume that on 1 * * m 1=1 i some neighborhood of 3C, f is a locally PL embedding. Step 1. Consider the surface H = f(D-intC) , Use Theorem (3.3) to make H PL. Step 2. Consider the singular surface L = f(C). Use Zeeman's Relative Simplicial Approximation theorem [72] to make L polyhedral. Step 3. Now HUL is a desired PL Dehn disk. Apply Theorem 50 © I Step 1 (Bing) t SteP * I (replacements) Figure 3.1. 51 (3.4) to get an embedded disk Tc: M. Step k. Replace the portions of T which lie outside U by corresponding pieces of H, (See Figure (3.1).) In general, the curves from f(aC) are going to be"wildly" embedded in M so additional care must be taken to improve f near 3C. This is achieved by using four concentric families of pairwise disjoint PL disks with holes rather than just one such family (our C). Now, the details. Let U** ¦ f_1(U). By [15 ;Theorem (4.8.3)] , there exist families { A. | 1 5.i <.t } , 1 M with the following properties: (5) f-JtD-V = f ICD-D^^); (6) f |D is locally PL; (7) S = S ; 1 where D = int(B,-B.). Apply Theorem (3.3) again to get a Dehn disk fp.-D—> M such that: 52 Figure 3.2, 53 (8) f2|(D-intL) = ^[(D-intL) ; (9) f2 |int L is locally PL; (10) S. = S. . *2 1 Remark. We could have gotten the map f? from f in just one step rather than going via f . However, we shall need f. in assembling the final map F (See Figure (3.U).) Another application of Theorem (3.3) yields a Dehn disk f^:D —*¦ M such that: (11) f3|D2 = f2|D2; (12) f j(D-D2) is locally PL; (13) S * S ; X3 Z2 where D = KUB . Remark. If for some j€{l,2,3,ty the simple closed curves f(3B )cM and f(3K)CM are nicely embedded in M we can skip f and f^ and just apply Theorem (3.3) to f |(D-int B.) to get f . However, if this isn't the case then we must get f and f first to make certain that f ( 3(D-Dp)) is nicely embedded in M. By Zeemanfs Relative Simplicial Approximation theorem 172] there is a Dehn disk f, :D —>• M such that: (1U) fjj(D-int B2) = f3l (D-int Bg) ; (15) fjJ(D-(int K o 3D) is PL; (i6) s. c v;. By Theorem (3-1*) there is an embedding f„:D _¦. M such that: (17) fJintD is locally PL; (18) f5|K = fulK; (19) f5(D)-V2 = fu(D)-V2. 5h In particular, by (U), (5), (8), (ll), (lU), (l8), and (19): (20>-fu(D^inf B3) c'f5.(D).c'f1|(j))UV2. Note, however, that in general, fY and f,- do not agree point-wisely, not even on D-int B_. We wish to know what regions of D are mapped by f,. onto f> (D-intB ). Let C = f ~ f, (D-int B ).. By (20), C is well-defined and non-empty. There exist pairwise disjoint PL disks with holes {e. |l <: i ^ r} such that (21) D-int B = U^ Ei.. By (l6), f, (D-int B ) is a collection of disks with holes, namely f^E^'s hence by (20) so is C = U' * f^f^E.). Define F:D —* M by f V(fJ+|(D-intB3))"1-f5 (X) JX6C (22) F(x) = < [_ f (x) ; x6D-(int CU3D) The map F is well-defined: each x€C lies in precisely one disk with holes f"1f]|(Ei), so f (x) lies in fj^). Now, by (l6), fJ(D- int B3) is an embedding, therefore f7 is well-defined over f, (D-int B_). Also, by (8), (11), (Ik), and (21): (23) T-lI 3B3 = fu| 3B3 hence for every x 63C- 3D: V (fu |(D_int b^'H^ = V^U/H (x) = id»f (x) = f (x) so F is well-defined. By (3), (7), and (20)-(.23), F is an embedding and by (5), (8), (ll), (lU), (19), and;(20)-(23), F(D)-U = f(D)-U as desired. (See Figure (3.3).) _Remark. The disk F(d) is thus obtained from f,-(D) by glueing together the pieces fj-(D-int C u3d) and f,(D-int B_) using the homeomorphism fi »f^ on 3C-3D. (See Figure (3.k).) 55 f(D)...4fc K, HB,) f(B2) V, f(B3) f(B4) U I I If I I I fi(D).. fJD). f3(D).. ^••4fllF 111 f5(D).. F(D)~. Step 1 (Bing) Step 2 (Bing) Step 3 (Bing) Step t (Zeeman) f(L) IIKJ I0D) I I 1 'm Step 5 (Henderson) Step 6 (piecing together) N ^ S Figure 3.3. 56 Figure 3.h. It remains to consider the case when f(D)fi 3M ? 0. Attach a collar CQ = 3M* [0,1] to 3M and extend the neighborhood U over CQ in the obvious way — let U' = U U((UA3M) x [0,1])- Let M' = M U Cq. Apply the preceeding case to the 3-manifold M' to get an embedding F':D -*M: such that F'(D)-tT* = f(D)-U" and F'|3D = f|3D- The disk F'(D) may now hit M"-M so we wish to push it in M by a nice ambient PL isotopy with support in U". Note that by taking a PL collar h: 3M x [0,1] —> M of 3M in M we get a "product structure" in M' close to 3M, i.e,, C Uh(3Mx[o,l]) is PL homeomorphic to 3M x[~l,l] where we identify 3M with 3Mx[o}. , We can now construct the desired ambient PL isotopy H :M" * [0,1] —*• M' by pushing F'(D) from 57 xwdM v^^IMlli^MflMMB^^;.....h(dMx[Oj V) apply Theorem (3.1) for f(D) inside M' push F'(D) down to M along the fibers Figure 3.5. 58 M"-M down to M by means of stretching down the fibers of the product 3Mx [-1,1] . Finally, let F:D -^f(D) UU be given by F = H^F", (See Figure (3.5).) 55 Theorem 3*6. Let f ,. ..,f,:D —>-M be Dehn disks in a 3-manifold M (possibly with boudary) such that if i ^ j then f. (3D) A f .(D) = 0. k Then for every neighborhood UCM of ^j„n f.(D) there exist embed- dings F ,,.,,F :D--* U such that: (i) for each i, F.j int D :int D —* U is locally PL; (ii) for each i, F,| 3D = f.| 3D; (iii) if i ^ j then F.(D)f>F (D) = 0. i J Corollary 3*7» Every 3-manifold (possibly with boundary) has the MSP. ** Proof of Theorem (3*6). By an argument similar to the one in the preceeding proof we may assume that for each i, f,(D)cintM. We use induction on k. For k=l the assertion follows by Theorems (3.1) and (3.3). Assume now that the assertion is true for all k<_n and consider the case k=n+l. By the inductive hypothesis there are embeddings F ,...,JVtoU-f +1(3D) satisfying (i)-(iii) and f +1 can be replaced by an embedding f* :D —> U-( U.^ F.(3D)) such that f' lint D is locally PL, f* - is in general position with respect to the surface S = U * Fi^> and fn+ll3D = fn+ll8D' Hen°e f (D)ns is a finite collection of pairwise disjoint PL simple closed curves* Starting off with an innermost one (on the surface S) of these curves, we can eventually cut f'(D) off S inside the neighborhood U thus obtaining the desired embedding F ¦ 55 59 2. Recognizing 3-Manifolds The next result is an improvement of a theorem of H.W.Lambert and R.B.Sher [44] who proved our result for the case when G is a point-like, closed O-dimensional upper semi continuous decomposition of S . (Consequently, their conclusion (iii) was S /G%S .) Theorem 3*8. Let G be a cell-like, closed O-dimensional upper semi-continuous decomposition of a 3-manifold M. If 3 M Ž 0 assume that N^ÈintM. Then the following statements eure equivalent: (i) M/G has the DLP; (ii) M/G has the MSP; (iii) M/G is a 3-manifold; Remark. The ideas Lambert and Sher used to prove their result in can easily be adapted to prove Theorem (3.8) for the case when every g€G has a neighborhood in M embeddable in 3R (and (iii) then reads M/G%M). Here's how this would go: By [64;Lemma (2.5)1 it suffices that given e>0 and a neighborhood UCM of N- we find a homeomorph-ism h:M—* M that shrinks all elements of G to a size less than e and stays the identity off U. By [47;Theorem 3 ] there axe pairwise disjoint cubes with handles P. ,... ,F.C U such that NpC U intF.. JL iL \J 1 — JL X Let W.,..,,W CU be pairwise disjoint open neighborhoods of F ,..., F, , respectively/Restrict our attention to F^CW. and let Cn = k 11 1 N-HF, • As far as F_ is concerned it suffices to find a homeomorph- Cj 1 1 ism h :M—* M that shrinks C and stays the identity off W . We get h as the composition of two homeomorphisms f ,g :M *-*M. The first one, f :M~^ M shrinks F towards its 1-dimensional spine so that 60 f-XF.) can be split up into adjacent 3-cell "chambers" of size < e/2 ( Figure (3.6)).Puii this chamber partition up in F . It is now clear that if g€G lies in at most two adjacent chambers it will get shrunk under f to a size less than e/2 + e/2 = e. So it now remains to make each g €G meet at most one wall of these "chambers". By-going to M/G and using the DLP (or the MSP) we can recover new walls in F CM as illustrated in Figure (3.7) on p..63.Pick any homeomorph-ism g of M which maps new walls on the old ones and rests off F . Finally, let h* f^g^ Figure 3.6. 6l This proof will not work for Theorem (3.8). For, the best we can say about G is that by [47 ;Theorem 3] it is definable by homotopy cubes with handles* So if there are fake cubes, some of the chambers in the partition of F above may fail to be 3-cells hence no homeo-morphism g can be produced« A different approach is called for. Proof of Theorem (3.8). The implications (iii) =>(i) and (iii)=» (ii) follow by Corollaries (3.2) and (3.7)> respectively. We prove (i) =>'(iii) and (ii)=>(iii) simultaneously. So assume M/G has either the DLP or the MSP. 3 Assertion 1. If every g € G has a neighborhood in M embeddable in B then M/G %M. As we have already observed above the proof from [44] vill work except that instead of [52 ;Theorem (2.1) ] one should use an improvement due to R.C.Lacher [40 ;Lemma A on p.506] • Assertion 2. If G - { g € G | g has no neighborhood embeddable in B } then *(G ) is locally finite in M/G, where 7r:M-*-*M/G is the quotient map, If M is orientable apply [35 ; Theorem 1 ] and if it is not apply Theorem (2.1). Assertion 3. For every g€G and every neighborhood UCM of g there is a hcmotopy 3-cell HCU such that gCintH. We may assume that U is G-saturated. By [47 ;Theorem 3 ], G is definable by homotopy cubes with handles hence there is a homotopy cube with handles HCU such that gCint H. By going further 62 in the defining sequence for G we may assume that on some neighborhood NCU of 3H the restriction TT|N:N —> M/G is an embedding. The idea of the proof is to use the DLP or the MSP to cut the handles of H along pairwise disjoint compressing disks which miss g. We detect such disks as follows, Assume first that M/G has the DLP. Let C. and Cp be disjoint simple closed curves on 3H such that they are null-homotopic in H but not on 3H. By Dehnfs lemma there exist embeddings f ,fp;(D,3D) —> (H,3H) such that f.(3D) = C, i=l;2. By running a ribbon in U-int H between slightly expanded disks fn(D) and fp(D) we get an embedding f:D —+ U such that for disjoint subdisks D ,DpCint D , f|Di = f., i=l;2 and f (D-(D U D ) )cU-H. Since by our choice tt|N: N —* M/G is an embedding it follows that 7rf :D —* tt(U) is a Dehn disk and that Z=EuZ. Therefore E -CiT(intH) so using 7TI ITi- TTX^ «I the DLP we can get an embedding F:D-+ irf(D) Utt (int H) such that F(3D) = irf(3D). Let q.:D—> tt(H) be the subdisks of F(D) bounded by 7rf.( D), i=l;2. Note that q (DJnqJD) = 0 so there exist disjoint neighborhoods W^u^u) of q^D). Let V,. = ir"1^)'. By [40 ; Lemma A on p.506j, q. lifts to a Dehn disk Q.:D —* V. OH, i=l;2. By Theorems (3*1) and (3.3) we can assume that Q. is a locally PL embedding. Since V.A V? = 0 one of the disks Q.(D) will miss g hence by cutting along it we get a homotopy cube with one handle less, H*, which contains g in its interior. (See Figure (3»8)f) In continuing this process one must be careful to choose the new pair of simple closed curves C*, C* away from the intersections of Nn with H*. That is because in doing the compressions we may have G 63 new T5 's new T2 wall T, new Ts Figure 3.7. H'=newH Figure 3.8. 6k. hit some elements of HL- {g } so now 3H*ANG may no longer be empty. Since any possible intersections lie inside the two copies of the compressing disk on 3H*, we can always push C* and C* off H-0 3H* if necessary* This way tt :M—>-*M/G '.is an embedding on a neighborhood of C*, i=l;2. (See Figure (3.9)•) If instead of the DLP we have the MSP for M/G the procedure is similar. We do not need to introduce f for it suffices to consider f., i=l;2, Use the MSP to separate the Dehn disks wf , wfp:D—> it (H), The rest of the argument stays the same, We now finish off the proof of the theorem, first for the case when 3M * 0. By Assertion 2, G = G UG where G = G-G and the set tt(G ) is locally finite in M/G. Consider Mo= M/G0 and let tt ;M —> M be the corresponding quotient map. Since the elements of G are cell-like, M is a generalized 3-manif o Id. Clearly, S(M ) C ^ (G ), where S(M ) is the singular set of M . Also, M satisfies KF by Proposition (1.10) since it is resolvable, Assertion k. For every pLM , g(M ,p) = 0. If p L ^q^q) *hen p L S (M ) so the assertion is clear, Let p it (G ), By Assertion 2, there is a neighborhood UCM of p such that UHtt (G ) = {p } . Let V =tt~ (u). By Assertion 3, there is a homotopy cube HCV such that tt ""1(p)c int H and 3HO(U{g6G} ) = 0, Therefore, it (3H) is a 2-sphere so tt (H) is the desired neigh- 65 jif(D) Qf(D) V Figure 3.9• 66 borhood of p, It now follows by Assertion k and Corollary (l.l1*) that S(M ) =" 0 since dimS(M ) < dim * (G ) <0, Thus M is a 3-manifold. Consider o — o o — o G* = G. U ^ (G ) as a decomposition of M . By Assertions 2 and 3% 1 loo o the decomposition G* is cellular, closed O-dimensional, and upper semi continuous. Also, M /G* ^(M/G )/G* %M/G so M /G* has the DLP (the MSP, respectively). By Assertion 1, M /G* is homeomorphic to M so M/G is homeomorphic to M and thus is a 3-manifold. This completes the proof if 3M = 0. In the case when 3M ^ 0 we consider the double DM of M (i,e., we identify two copies of M along 9M using the identity map) and apply the preceeding arguments to the to the decomposition DG, the double of G. The proofs of all assertions go through the same although we are not claiming that in general, the hypothesis !!M/G has the DLP/MSP" implies "DM/DG has the DLP/MSP"¦ The point is that we do not need that much to prove Assertions 1-1+¦ Of course, in our case it eventually turns out that DM/DG has the DLP and the MSP since we prove that DM/DG is a 3-manifold. $$ Theorem 3*9. Let X be a generalized 3-manifold with O-dimensional singular set such that for every x€X, g(X,x) = 0. Then X has the DLP and the MSP. Proof. We first prove the DLP. Let f:D —*X be a Dehn disk. We first show that we may assume f(3D)AS(X) = 0. Indeed, by hypothesis there is a neighborhood NCD of 3D such that SfHN = 0. Thus NHf^CsCx)) is O-dimensional so there is a PL simple closed curve JCN-f (S(X)) 67 such that J is ambient PL isotopic in N to 3D. Let ECD be the sub-disk of D bounded by J and consider the Dehn diskf* = f|E:E—> X. If we can show how to find an embedding F':E—* f(E) UU, where UcX is a neighborhood of lf = Z „ such that F"(J) = f"(J), then by defining F:D .—> X to be f on D-E and F' on E we get the desired embedding. So assume that f(3D)HS(X) = 0 and that f(3D)CX-U. By [12 ;Lemma 1] and by Theorem (3*3) we can find a collection N,..., N, CX of pairwise disjoint compact generalized 3-manifold-with-boundary neighborhoods of S(X)Hf(D) such that: (1) for each i, N. is a locally PL 2-sphere; (2) HflS(X) = 0; (3) S(X)n(f(D)-U)cH1CX-(f(3D)UU); (U) S(X)r\ f(D)AUCH2CU; where Hn = U ,P, N. , H = U .*. N. , and H = 1UL, Then by (2), 1 1=1 i * 2 i=p+l l 12 . f(D)nHCM(X). Here is an outline of the proof. First, we want to make f(D) meet H "transversely". But f may not be (even locally) PL so we must improve it to be (locally) PL near H. We do this as follows: close to H we use the Simplicial Approximation theorem while close to H we use Theorem (3-3) in order to keep f an embedding in that region. By applying general position in M(X) we can make f(D) meet H transversely and then we can either cut it off št ¦H1_ (bystandard "cut and paste" techniques) or "push" f(D)HH into M(X) (replacing annuli of f(D)OH by "nicer" annuli in H OM(X)) while intersections of f(D) with Hp are dealt with in a different manner, again .68 making f(D) lie in M(X). Apply Theorem (3.1) to get an embedding F':D-* M(X) such that F'|3D - f|3D and F'(D)-U = f(D)-U. Finally, replace the portions which H. cuts off F'(D) by f(D)fl H and thus obtain the desired embedding F:D .—> X with F(D) Cf(D) U U and F(3D) = f(3D). (See Figures (3ao), (3.11), and (3.12) •) Now the details. By [15 ;Theorem (U.8,3)1* there exists a collection B , • • • »B.C int D of pairwise disjoint PL disks with- holes such that: (5) for each i, f^fH. )c int B. ; (6) ^cD-fV); (7) A2Cf"1(U-S(X)); where An = U.pn B. and A = U .k ^ B.. Let A = A,UAV Applying 1 1=1 i 2 i=P+l i 1 2 ** * Theorem (3.3) to f|A :A —> X and the Simplicial Approximation theorem to f |Ap:Ap —* X we replace f by a map f :D —* X with the following properties: (8) f |intA is locally PL; (9) ^[(D-intA) =fj(D-intA); (10) S. CU. 1 . Note that by (2), HCM(X) hence we can apply general position in M(X) to get a map fp:D .—* X such that: (11) f is ambient isotopic (in X) to f.; - • (12) fp is in general position with respect to H; (13) f2|(D-intA) = fj (D-int-A) ;¦ (1U) S CU. *2 69 Case 1A 1<>i<>p Step 1: Bing Step 2: general position Step 3: "cut and paste9 a Step 7: ! going back \ Step 6: 1 pushing f Step 5: • glueing u Step 4; Theorem (3.1) and "reparametrization* Figure 3,10, To Case IB: Step 1: Bing Step 2: general position Step 3: replacing annul i ¦ Step 7: • going back t Step 6: 1 pushing t Step 5: * glueing back old annufi Ü Step 4: Theorem (3.1) and "reparametrization" Figure 3.11« Case 2: p+1 ; (17) for each i,j: J^-- 3D^; (18) for each i,j: L^K^ - 3A^V (19) for each i,p,q: D(i)H D(i) * D^flA^ »' p q P 1 In order to get f(D)ON. inside M(X) we perform the following surgery: we replace every disk D* (resp. annulus A. ) by another disk 5^ (resp, annulus A^ ) with the following properties: (20) for each i,j: ßj^cH.nM(X) and J ^ «3D*1'; (21) for each i,J: Ä^'cN.AMlX) and L^'ukJ1' = 3Ä^'; (22)for each i,p,q: D(i)D D(i) = 5(i)n Ä(i) = Ä(i)A Ä(i) = 0. * p . By (l), (2), (U), (6), (8), and (12) we may assume that N.-f(D) 4 0 (for otherwise we may take slightly smaller N. obtained by pushing N. into int N. along a collar C C M(X)fiN. on N. )• Thus by shrinking out the disk N.-{p.}, where p. #.'¦¦¦ • €N,-f(D) is an arbitrary point, we may assume that f(D)rt'N. is just a point. (See Figure (3.31| ),) By performing the replacements described in the preceeding two paragraphs we obtain a map f_:D —*X with the following properties; (23) f (DION, can be recovered from f~(D) by replacing each 73 f(D) Figure 3.13. 7^ Figure 3.1^. T5 SW (orÄ^tyD^ (resp. A^); (21») f3(D)0H2 = {xp+1,.,,,xk}, where X.6N., p+1 <_i <_k; (25) f3(D)CM(X); (26) S CU; 3 (27) f, is a Dehn disk (not necessarily PL); (28) f3(D-intA) Cf2(D)UU; (29) f3|3D = f2|3D; (30) f3(D-(HUU)) = f2(P-(HUU)); because (23) follows by (20) and (21); (2h)-{26) by (lU)-(22); (28) by (13)-(21), and (28)-(30) follow by the construction of f_, By (25) we can apply Theorem (3.1) to get an embedding F :D —*¦ X such that: (31) F1(D)-U = f3(D)-U; (32) fJ^D = fj3D. Next, replace the disks Ik (resp. annuli Ä^ ) by the disks D J J J (resp. annuli Ä^ ). Since whenever F and f_ agree they agree (in general) only pointwisely, we must do some "reparametrization" (in a similar way as it was done in the proof of Theorem (3.1)). For each i,j, let DJi} = F"1(D^i)) and A^ = F~1(ÄJi)). By (26), (28), and (31), there exist PL homeomorphisms u. .,v..:D —*• D such that the diagrams "(i) Fll -(i) *(i) Fl' -(i) commute. 76 Define F :D "X by F2(x) = { < f2Uij(x) ; x^f"1^^) for some i,j f2vi(J(x) ; xCf'^Ä^) for some i,j [ F (x) ; otherwise. Then by (26)-(32), we have the following properties for F_: (33) ST 0; (3^) F2(D)Cf2(D)uU; (35) F2(D)-U = f2(D)-U; (36) F2l3D = f2|3D. By (ll) there is an embient isotopy K.:XxI—»-X from fp to f.. Let F = K^Fg. Then by (33)-(36) we have that: (37) S = 0; F3 (38) F3(D)cf1(D)UU; (39) F3(D)-U = f^Dj-U; (kO) F3|3D = f3|3D. Let g. :D —¦• D be a homeomorphism that makes the diagram F^f^B.)) -» F3(D) B. / commute for each iL{l,...,p}. By (9), the map F:D —> X given by fg.(x) ; x€B. for some lL i<_ p F(x) = F-(x) ; otherwise is well-defined. By (3T)-(i+0) we have that Sp = 0, F(D)Cf(D)uu (in fact, we have more— F(D)-U = f (D)-U), and F 13D = f | 3D. This completes the proof that X has the DLP, 77 We now prove that X has the MSP, too. Let f , ...,f,:D -* X be . k Dehn disks, UCX a neighborhood of U. 'f.(D), and suppose that if i 7* j then f,(3D)Af,(D) - 0. As before, we may assume that for each i, f.(3D)o S(X) = 0. Since X was already shown to have the DLP we may also assume that all f.?s are embeddings. Cover S(X)n k (U. , f.(D)) by a collection of pairwise disjoint generalized 3- manifolds with boundary N,..t,N C U such that for each i, N, is a locally PL 2-sphere and I^flSfä) = 0 ((12;Lemma 1] and Theorem (3.3)). We may also make N.'fs small enough as to be sure that for no j is f.(3D)AN. ± 0. Let P = U N. . As before, we can apply tJ 1 1""*JL J* ¦ ¦ Theorem (3.3) close to P in order to make P meet each f.(D) trans- d versely. Then we can cut each f ..(D) off at P (working within M(X)flU J ••all the time) and thus get a new Dehn disk f':D —> X with f' j 3D = f J3D. Since f'(D)CM(X) we can apply Corollary (3.7) to get f'fs disjoint and yet still inside U and keeping their boundaries fixed Since f. and f' agree on the boundary, this completes the proof. LL Theorem 3.10. Let L be the class of all compact generalized 3-mani-folds X with dimS(X)<0 and let C C C be the subclass of all X€C which have ^ 1 singularity and are also homotopy equivalent to 3 . ¦ - S , Then the following statements are equivalent: (i) PoincarL conjecture in dimension three is true; (ii) If X6Lhas the DLP. or the MSP then S(X> = 0; (iii) If X6C has the DLP or the MSP then S(X) = 0. Proof. (i)=^(ii): If Poincar? conjecture is true then X has a conservative resolution f:M —> X by Theorem (1.6). Let G -{ f"\ (x) |x€X}, • 78 be the associated cell-like,. closed 0-dimensional upper semicon- tinuous decomposition of M. It follows by Theorem (3.8) that S(X) = 0. (ii) =L>(iii): Clear. (iii) =?• (i): Suppose the PoincarL conjecture is false. Consider the construction \r from Proposition (.1,12). Then W6C , —¦o On the other hand, W has the DLP and the MSP since g(W,x) = 0 for all x.6W, by Theorem (3.9). Contradiction, since S(W) ? 0. L* 3. Isolated Singularities In this section we give an application of the DLP (MSP) to studying isolated singularities in generalized 3-manifolds. The proof is an application of Thickstunfs extension of the Loop theorem (Theorem (1.15)) for compact generalized 3-manifolds with 0-dimensional singular set. Theorem 3»11. Let X be a generalized 3-manifold satisfying KF and suppose that X has the DLP or the :MSP (in fact, it suffices to assume the MSP only for pairs of Dehn disks). Then X has no isolated singularities. Remarks. If Poincafe conjecture is true this follows by Theorem (3.10), provided dimS(X) <0 and that a "complete" MSP is assumed. Suppose now that fake cubes exist. Then one cannot drop any of the hypotheses from Theorem (3.11): the example of M.G.Brin [9] (or the example of Brin and D.R.McMillan,Jr.[12] ) has S(X) ={#} and satisfies KF; on the other hand the example W from Proposition (1.12) has S(W) ={*} and also the DLP and the MSP by Theorem (3.9). 19 Corollary 3.12¦ Let M be an open 3-manifold with finitely many ends and M its Freudenthal compactification, Then the following statements are equivalent: (i) M is a 3-manifold; (ii) M is an LC Z -homology 3-manifold, satisfies KP and has either the DLP or the MSP, Proof, (i)=^(ii): Follows by Corollaries (3.2) and (3.7) and by Kneser?s Finiteness theorem. (ii)=^(i): M is clearly finite-dimensional hence it is an ENR as soon as it is LC °° at the points p ,.. ¦ ,p of compactifica-tions (assume that M has t ends). Since for each i, M is always 0-LC at p. and since M deforms onto a Freudenthal compactification of a locally finite 2-dimensional polyhedron with t ends, it suffi- 2 ces to show that M is LC at each p.. The assertion now follows by Theorem (3.11). JJ Proof of Theorem (3.11). Here is the idea of the proof: by Corollary (l.lU) it suffices to show that every point p6X which has a neighborhood UCX such that Ufl S(X)c{ p}, satisfies the condition that g(X,p) = 0. This is done using standard disk-trading techniques from 3-manifolds topology except that instead of the classical Loop-theorem we must invoke Theorem (1,15) and the classical Dehn lemma is replaced here by the DLP (resp, MSP) combined with Theorem (3.3). The latter is done as follows: whenever we want to perform a cut along a compressing disk D which hits p we may use the DLP (or the MSP) on two "close" copies of D to make one of them miss p so that 8o the cut can be performed in M(X). Now the details. Let p€X and let UcX be an open neighborhood of p such that U HS(X)c{p}. By [12 ;Lemma 1] there is a compact orientable connected generalized 3-manifold NcU with boundary a compact orientable 2~manifold such that p€intN. Since X is an ENR it is locally contractible [7 ;Theorem ( V.10.3)] so we may 2 assume that N is null-homotopic in U. Let c = I * (n+l) g(n) where g(n) is the number of components of K with genus n [47;p.l30] . Choose N so that c is minimal. We shall show that c = 0. So suppose that c >0. Then there is a boundary component CCN with positive ge- nus ;C is a 2-sphere with k >0 handles since N is orientable. Let 2 L:3B —* C be an essential simple closed curve. By our choice of N . the inclusion-induced homomcrphism II. (N) —* II-(U) is trivial hence 2 2 there is an extension f:B —* U of L over B . Using methods similar to those employed in the proof of Theorem (3.9) we can assume that f is locally PL near C and that it is in general position with respect to C, because CCM(X). Thus we may assume f (C) is a finite 2 collection of pairwise disjoint PL simple closed curves in B , one 2 2 . of them being 3B . Let JcmtB be an innermost such curve and let 2 ECf(B ) be the (singular) subdisk bounded by f(J). There are three possibilities, Case 1. f(J) is inessential on C. Then f(J) bounds a (singular) disk E'CC. Exchanging E with E' we can go to the next innermost curve. Case 2, f(J) is essential onC and ECU-intN. Since U-int N CM(X) we can use Dehn's lemma to attach a 2-handle to N after we 81 have made E locally PL by Theorem (3.3). This reduces c which, in turn, contradicts the minimality of c. Hence this case "cannot occur, Case 3. F(J) is essential on C and ECN. By Theorem (1.15), f(j) can he replaced by a simple "closed curve J'CC such that J"" is nontrivial on C but bounds a Dehn disk in N, Let RCC be a regular neighborhood of J" in C and let J and Jp be two simple closed curves boundary components of R, Then J. bounds a Dehn disk D. in N for each i = 1;2. Assume first, that X has the MSP, Then we can get D. and Dp disjoint in N — denote them by D* and D*, respectively. Thus one of them will miss p, say p è D*. By Theorem (3.3) we can make D* locally PL — denote it by D**. Then by cutting N along D** we reduce the complexity c which, in turn, again contradicts its minimality. Hence this case cannot occur either. (See Figure (3.15)-) If instead of the MSP we have the DLP the argument is similar — join the Dehn disks D. and D by a ribbon in U-int N to get a Dehn disk D. Apply the DLP to get an embedded disk D* such that D*-int N = D-int N and that 3D* = 3D, This replaces D1 and Dp by embedded disjoint subdisks D*, D*CD*HN so one of them, say D*, misses p. The rest of the argument is now as before: apply Theorem (3.3) and cut N along D** to reduce the complexity c. We conclude that indeed c = 0 hence g(N,p) » 0, Since N satisfies KF and since S(N)c{p}it follows by Corollary (1.1 k) that N is a 3-manifold. In particular, p 6M(X). ** 82 use Theorem (3.1) and Bing | cut N along 0** Figure 3.15« Remark. Suppose that X is a compact generalized 3-manifold with dimS(X) L 0, satisfying KF and having the DLP or the MSP* If S(X) i> 0 then X has the following properties: (i) X admits no resolution ([14 ;Theorem 1 ] and Theorem (3.8)); (ii) S(X) is wildly embedded in X (Proposition (1.13) and Corollary (1.1*0); (iii) S(X) has no isolated points (Theorem (3.11))* IV. GENERALIZED 3-MANIFQLDS WITH BOUNDARY Little investigation has been done concerning generalized manifolds with boundary. In this chapter we present some results which are most of the time analogues of those known for generalized manifolds. Let R be a PID and consider an R-homology n-manifold X with boundary. We first observe that X need not be an R-homology (n-l)-manifold (as it would be the case with topological manifolds with boundary). A simple example is the interior of any n-manifold with boundary together with just one point of its boundary (n>l). It may also happen that X is an R-homology (n-l)-manifold with boundary. The next proposition gives a criterion for determing the boundary points of X: Proposition k.l. Let X be an ANR and an R-homology n-manifold with boundary, R a PID. Suppose that p€X and that H#(X-{p};R) 2ž= H#(X;R). Then p6(X)'. Proof. We supress the coefficients. Consider the homology sequence of the triple (X,X,X-{p}) over R: ... J^+1Cx,x-{p}) JUq+1(x,x) —Hq(x,x-{p}) -->yxi^p}) —> J* ----¦. H (X-,X) —*... Since H*(X) 2tH#(X- {p }) it follows by [59 ;Lemma 6 on p.202] that 83 Qk H*(X,X-{p})iH*(X,X). Hence im A# = 0 = ker i* so H*(X,X-(p}) % ker j#= 0 thus by Lemma (l,*0, p €(X) . $$ By Corollary (1.9) true generalized 3-manifolds with boundary cannot have a PL structure. The next result is from [59;p.2771 • Proposition k<2. Let X be a PL R-homology n-manifold with boundary, R a PID. Let K be a triangulation of X. Then X is a subpolyhedron of X, |K |= X, K is a pseudo n-manifold with boundary and Lefshetz duality holds, i.e., H*(X,X;R)^H (X) and H^(X,X;R) ^^^(X). ** It is not difficult to manufacture examples of generalized manifolds with boundary as the next proposition shows: Proposition U.3* Let X be a compact generalized n-manifold with S(X)CZ, where ZCX is a closed, O-dimensional set. Then there exists an n-cell BCX for which ZC3B, S(Y) = Z, and Y is a compact generalized n-manifold with boundary, where Y = X-int B. Proof. Let B CX-Z be any tamely embedded n-cell. We get B from B by pushing out from B wildly embedded (in X) "feelers" towards the points of Z. $L Example h.k. Let X = S and B = thickened one half of the Fox-Artin wild arc [26 ;Example (3.1)] . Then S(Y) =:{p }= the wild point of the arc. (See Figure (U.l),) Lemma U.5. Let X and Y be generalized n-manifolds with boundary and suppose that there exists a homeomorphism h:X —* Y, Then X ^ Y is a generalized n-manifold. 85 Figure Utl, Proof of Lemma (k.5). Since X is an ENR so is X U Y by [7 jTheorem (lV\6.l)]. It therefore suffices to show that X U y is a Z-homology n-manifold. The argument we give below is valid over any PID R, We shall supress the coefficients. Consider the Mayer-Vietoris sequence for the pairs (X,X-{p}) and (Y,Y-{h(p)}), (By the Excision theorem it suffices to consider only the case when pLX, cf. also Lemma (l.U),}; ----->H (X,X-{p})0H (Y.Y-fotpM^yX Uh Y,(X Ufa Y)-.{p}) _^ H ,(X,X-{p})__^ H. (X,X-{p})®H CY,Y-{h(p)})_^ ... Since p€X and h(p)€Y it follows by Lemma (l.U) that H#(X,X-{p}) ^ 0 =H#(Y,Y-{h(p)}). Also, X is a generalized (n-l)-manifold hence: 86 H (X.. hY,(X hY)-{p})^HH(X,X.{p})^Raf q = n and «O if q * n. The assertion now follows by Lemma (l.U)..- \\ We now turn to dimension three. First, we prove an analogue of the Finiteness theorem of J.L.Bryant and R.C.Lacher (Proposition (1.10) )> Propositions (U.6) and .(U• 8>• Proposition k.6. For every compact generalized 3-manifold with boundary X there is an integer k such that among any k +1 pairwise disjoint 2Zp -homology 3-cells in X at least one is contractible, Proof. By Lemma (U.5) the double DX of X is a generalized 3-manifold so there exists the Bryant-Lacher number n for DX (Proposition (1.10)). Let kQ = [|and Kneser Finiteness in M. $L Proposition U.9» Suppose that X is a compact generalized n-manifold with boundary. If n = 3 assume that dimS(X)<^0 and that the Poinca-rš conjecture in this dimension is true. Then X has a resolution. Proof. Let Y = X + C, .where C = Xxl is a collar on X. By the arguments employed in the proof of Lemma (U.5) and by [7 ;Theorem (IV.6.1)], Y is a generalized n-manifold with boundary. Also, S(Y) CS(X). That Y resolves now follows by [62;Main Theorem] if n=3, by [56;Theorem (2.6.1)] if n=l+, and by [55 ;Theorem (l.l)] if n>^5. (If n<.2*8'(X.) = 0 by [68 ;Theorems (IX.1.2) and (IX.2.3)1.) So there is a proper cell-like surjection f: (M,3M) —*-> (Y,Y) from an n-manifold with boundary. Let g:Y—>-*X be the collapse of Y onto X along the fibers {x} *I (x€X) of the collar C. Then g is clearly cell-like so by [40;p.5H], gf: (M,3M)-*-> (X,X) is a resolution of X. ** Remark. Suppose that the Poincarl conjecture is false. Let X*be the example described in Proposition (l.ll) and let BCX* be a nicely embedded PL 3-cell in X* such that the limit point p lies on 3B. Let X = X*-int B . (See Figure (k.2).) Then X is a compact generalized '3-manifold with boundary, p €X, S(X) ={p} , and X doesn't admit a 88 Figure U.2. resolution, by (U.8). This example together with (k.9) yields: Corollary 1». 10« Let L be the class of all compact generalized 3-mani-¦ folds with boundary such that dimS(X) <_0 and let C^CC^be the subclass of all X€L such that S(X)c * and X is a homotopy 3-cell. Then the following statements are equivalent: (i) Poincarš conjecture in dimension three is true; (ii) If X€L then X has a resolution; (iii) If X6C then X has a resolution, $L Let X be a compact generalized 3-manifold with boundary and suppose that the double DX of X is a 3-manifold. Then X need not be a 3-manifold: e,g., R.H.Bing proved that the double of the Alexander 89 3 solid horned sphere yields S [4] , But we can prove something: Proposition U.ll» Let X be a compact generalized 3-manifoid -with boundary such that DX is a 3-manifold, Then X has no isolated singularities, Proof, Since X is a closed surface in a closed 3-manifold DX it follows by a results due to O.G.Harrold and E.E.Moise [31] that X can be wild at each point at most from one side in DX, But in DX the two sides are "symmetric". Hence XCDX is 1-LCC.so the assertion now follows by Proposition (1.1.3) and Theorem (3.9). ** The following is a generalization to generalized 3-manifolds with boundary of a results due to Bryant and Lacher [14;Theorem k] : Proposition U.12. Let X be a compact generalized 3-manifold with boundary and suppose that S(X)CZ, where ZCX is a closed, 0-dimen-sional set in X, Z is 1-LCC in X. Then X is a 3-manifold if and only if it has the KF. Proof, The "if" direction is Kneser finiteness theorem. So assume now that X has the KF, Cover Z with pairwise disjoint 2-cells in X (note, that by [14 ;Theorem k] we may assume that S(X)CX) so that the boundaries of these 2-cells lie in M(X), Because of the 1-LCC condition these boundaries bound some Dehn disks in X hence real PL disks (apply Dehn1s lemma in M(X)). The methods of the proof of Theorem k in [14] now yield the desired conclusion, JJ 90 Proposition U.12. Let X be a generalized n-manifold with boundary and Y- X+C, where C is a collar on X, Suppose that Y is an n-manifold with boundary. Then the double DX of X is also an n-manifold. Proof. Let G be the decomposition of DY, the double of Y, into points and fibers of the two adjacent copies of the collar C. Then G is a shrinkable, cell-like upper semicontinuous decomposition. By hypothesis, DY is an n-manifold so by Theorem (1.3), DYfy DY/G. Since DXfyDY/G, the assertion follows* '5J V. EPILOGUE In the last chapter we review some open problems which are related to results presented in the dissertation. First, we consider the resolution problem for 22 p-homology 3-manifolds. Theorem (2*7) implies (so does already [14 ;Theorem 3 ] ) that a locally simply connected 2Zp-acyclic image X of a 3-manifold M is a generalized 3-manifold. R.J« Dave rman and J.J.Walsh [20] constructed an example of 3 an upper semicontinuous decomposition G of S with the following properties: (i) each g6G is strongly 2Z -acyclic but not cell-like; (ii) S /G is a Z -homology 3-manifold; (iii) S3/G is lc°°(Z) ; (iv) S /G is not an MR. o Note, that HQ = G. By Theorem (2-7), (i) implies that S /G is not even 1-LC, since if it were 1-LC on just an open set UcS /G then almost all g€G which are mapped into U would have to be cell-like. Thus one cannot drop the 1-LC condition from Theorem (2.7). However, one can try to weaken the hypothesis, on M: let's assume that M is only a generalized 3-manifold. By Proposition (l,l), X is still a Sp-homplogy 3-manif old. What is not clear is whether X is also a TL -homology 3-manifold, i.e. is every ENR which is a 2Zp -homology 3-manifold necessarily also L & -homplQgy 3?manifQld [42]? 91 92 If instead of E? ve have TL ,p any odd prime, then the answer to this question is negative. For, let X be the suspension of the 2 projective plane, X = I P t Then X is locally contractible and since it1s a finite dimensional simplicial complex it follows by . [7; Theorem (V, 10,3)] that X is an ENR. Next, we show that X is a TL •-¦ homology 3-manifold. We shall supress the coefficients TL * By Lern-ma (l.U) and by the Excision theorem it suffices to show that H (EP2,lP2-{a.»2tE if q = 3 and 2i 0 if q f 3, where a ,a 6 t?2 are the suspension points. From the Mayer-Vietoris sequence for the triple (ZP2,ZP2-{a1>,IP2-{a2}) we have that H +1(ZP2)^H (P2) for 2 2 every q . From the homology sequence of the pair (EP ,ZP -{a.}) we have that H (EP ) 2lH (ZP ,ZP -{a.}) for all q>0 hence we can con-q — q i — elude that H (ZP2,ZP2-{a. }) ^H ,(P2) for all q. Since p was an odd q l = q-1 ^ r prime, the assertion follows. Now, the only two singularities are a. and ap. Since each has an open cone neighborhood in X it follows by Proposition (1.8) that X can't be a generalized 3-manifold hence not a TL -homology 3-manifold. Note that this particular example doesn't work for p = 2 since H2(ZP2,ZP -{a. )\TL ) ~ TL + 0. In fact, we can prove the following more general observation: Proposition g,l. Let X be the suspension of a closed surface and suppose that X is a TL -homology 3-manifold, Then X is a 3-manifold. Proof. Clearly, X is an ENR. By the Excision theorem, it has the lo- 3 cal TL- homology of 3R everywhere except maybe at the suspension points a and a ¦ By hypothesis X is a Z -homology 3-manifold so by Lemma (l.U), H^(X,X-{ai }\TL^ % H^CS3;^) hence by the Universal 93 Coefficients theorem H*(X,X-{a.} ;2Z)^ H#(-S ;ZZ).. Thus X is a generalized 3-manifold so by Proposition (1.8),a.€ M(X)f i=l;2. This proves that S(X) = 0. ** We remark here that a negative answer to Lacher1s question qou-ted above would yield an example of a 2Z -homology 3-manifold that is not a 7L.-acyclic image of any 3-manifold, For if it were then by Theorem (2.7)> X would be a generalized 3-manifold, Therefore such a negative example would yield a counterexample to the resolution conjecture for Zp-homology 3-manifolds, Next, we wish to state two problems concerning the DLP and the MSP. Let X be a compact generalized 3-manifold with dimS(X)<0 and satisfying the KF, By Theorem (l.l6) there is a compact generalized 3-manifold Y and a cell-like map f from Y onto X, Also, dimS(Y)f.O, g(Y,y) = 0 for all y€Y, and all singularities of Y are "soft" (in the sense of Proposition (1,13)). Question 5,2, Let f;Y —> X be as above and suppose that X has either the DLP or the MSP. Does Y satisfy the KF? If the answer is "yes" then first, by Corollary (1.13), Y is a 3-manifold so it X has a resolution and is thus, by Theorem (3*8), itself a 3-manifold, This would answer in the affirmative the next question, which tries to disentangle the Poincare conjecture from Theorem (3,10) (as Theorem (l.l6) does with the n = 3 case of Theorem (1.6)): Question 5.3, Suppose that X is a compact generalized 3-manifold with dimS(X)10, satisfying the KF, and with the DLP or the MSP, Is S(X) =0? 9k. Note, that by Theorem (3.8) it would suffice to only show that X has a resolution. We also wish to discuss some open problems concerning generalized 3-manifolds with boundary. The following obstruction arises at once: Question 5«U« Let M be a compact 3-manifold with boundary and f ;M —>->X a cell-like mapping onto an MR X. Let DX = XUX/R, where R —1 is the equivalence relation on X*X given by: xRy f (x)A3M= f" (y)A3M ? 0 (x,yLX). Is then the associated mapping Df:DM—*+DX also cell-like? If the answer is affirmative then we get an analogue of Proposition (l.ll):(i)=*(ii). Proposition 5*5« Let X be a resolvable compact generalized 3-mani-fold with boundary and suppose that dim S(X)L.0. If the answer to Question (5«*0 is affirmative then M(X) embedds in some closed 3-manifold. Proof. Let f: (M,3M)-*+(X,X) be a resolution of X. Then Df:EM—>"*DX is a resolution of DX, by (5-^0• It follows by Proposition (l.ll) that M(DX) embedds in the interior of some compact 3-manifold hence so does M(X)CM(DX). ** Another useful consequence of a possible affirmative answer to Question (5• U) would be the following analogue of Theorem (2,7): Proposition 5.6. Let X be a locally simply connected 2Z -homology 3-mahifold with boundary. Suppose that there exist a 3-manifold M with boundary and a closed, monotone map f:M~~*"*X such that for 95 every x^X-Z, $ (f~ (x);2Z_) = 0, where ZcX is some 0-dimensional set. If the answer to Question (5»M is affirmative then X has a resolution* Proof¦ Let Y = X+C, where C is a collar on X, Then Y is a generalized 3-manifold with boundary, Y is 1-LC, and S(Y)CS(X), Assuming that (5#U) is true we can extend f over N = M+D, where D is a collar on 9M, fiberwise to get a map g:N -_>^Y satisfying the hypotheses we required initially for M,X, and f. By similar methods sis those applied in the proof of Theorem (2,7) one can now show that Y has a resolution h-.P —*+Y, Let h^:Y —**X be the collapse of C onto X (along the fibers of C), Then h = hJa-iP —>-*X is a resolution of X. (See Figure (5.1).') $$ Figure 5fl. 96 Question 5*7« Let X be a compact generalized 3-manifold with boundary and suppose that dimS(X)<0. Are the following statements equivalent: (i) X embeds in the interior of a compact 3-manifod with boundary; (ii) X plus a collar on X is a 3-manifold with boundary. Question 5/ff« Let X be a compact generalized 3-manifold with boundary and suppose that dimS(X)<_0, Suppose also that M(X) embeds in the interior of a compact 3-manifold with boundary. Does then X have a resolution? Example 5*9« Let X be a compact generalized n-manifold with boundary and let Y be X plus a collar C on X, Suppose that Y is an n-manifold with boundary. Then (i) S(X)CX; (ii) X has a resolution (just collapse Y onto X along the fibers of C). The converse need not be true. E.g., let A be a noncellular arc with one wild point p* ACBn, A 0 3Bn = {p} [26] . Then X = Bn/A is a compact generalized n-manifold with boundary and satisfies both properties (i) and (ii). If Y = X+C were an n-manifold with boundary then A would necessary have to be cellular.One can thus only conclude that Y is a compact generalized n-manifold with boundary, S(Y)CS(X), and that Y resolves if and only if X has a resolution. (See Figure (5.2) on p.97«) 91 Figure 5.2. Question 5*10* Suppose that X is a resolvable generalized n-manifold with boundary. Does X have a conservative resolution? Question 5.lit Suppose that X is a compact generalized 3-manifold with boundary and that there is a proper cell-like onto map f/(M.f3M) —*-*(X5X) from a compact 3-manifold M with boundary onto X.. Suppose furthermore that f |3M: 3M—>->-X is proper and cell-like, t~ (X) = 3M, and that S(X)CX. Is then X a 3-manifold with boundary? Question 5«12. Let X be a generalized 3-manifold with boundary with • dimS(X)<0 and satisfying the KF. Assume that X has either the DLP or the MSP, Is then X resolvable? If "yes", is then X a 3-manifold with boundary? 98 For resolvable compact generalized 3-manifolds X with dimS(X)l. except [14 ;Theorem l], Question 5*1^* Let X be a compact generalized 3-manifold with dim S(X) = 1. Suppose that M(X) embedds in the interior of a compact 3-manifold with boundary. Does X have a resolution? Question 5.15. Let X be a compact generalized 3-manifold satisfying the KF and with S(X)cA, where ACX is a locally homotopically unknotted arc in X, Is X a 3-manifold? BIBLIOGRAPHY [l] S.Armentrout, Concerning cellular decompositions of 3-manifolds that yield 3-mahif6lds, Trans, Amer, Math* Soc. 133 (1968) 307 -332, [2] S.Armentrout, Concerning cellular decompositions of 3-manifolds with boundary, Trans. Amer. Math. Soc. 137 (1969) 231-236. [3] S.Armentrout,. 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Soc. 167 (1972) ^79^95. [7Ü E.CZeeman, On the dunce hat, Topology 2 (1963) 3^1-358. [72] E.C.Zeeman, Relative simplicial approximation theorem, Proc. Camb. Phil, Soc, 60 (196U) 39-^3. APPENDIX: REGULAR NEIGHBORHOODS OF COMPACT POLYHEDRA IN 3-MANIFQLDS Strong peripheral 1-acyclicity is a homology version of McMillan's cellularity criterion (CC) [46]while weak peripheral 1-acyclicity is a homology version of his weak cellularity crite-rion (WCC)[46]. It is therefore interesting to observe that by Theorem (2.9) the two acyclicities are equivalent over Ep while CC is clearly a stronger property than WCC (just consider any nun's 2 1 2 1 cellular arc in SJ). The example M = S *S , K - S v S from [39] shows that WCC is not a topological property of K (i.e.,.it may depend on the embedding, as might CC). Same example confirms this for peripheral acyclicities (p.Ul). On the other hand we proved in Theorem (2.1l) that the peripheral 1-acyclicities do not depend on the embedding (if K is compact and M is a 3-manifold) provided dim K L1. This is an analogue of the result of Lacher [39] to the effect that if K is a codimension >.2 compact subset of a PL n-mani-fold M, n ^ h, then WCC is equivalent to the 1-UV property and thus independent of embedding [39 ;p.^99] • In [39] Lacher asked the following (to our knowledge still open) question: Suppose that f is an embedding of a compact set K into an n-manifold N, homotopic to the inclusion KCN. If KCN has WCC does f(K)cN have the same property? In this appendix we answer in the affirmative Lacherfs question for PL embeddings of polyhedrain 3-manifolds. 1Q5. iq6 Theorem A,l. Suppose that f:K—-> M is a PL embedding of a compact polyhedron in the interior of a 3-manifold. Suppose that f is homo-topic to the inclusion KCM,and that KCM has WCC (resp. is peripherally 1-acyclic over ZZp ). Then f(K) C M has WCC (resp, is peripherally 1-acyclic over 2Zp). Lemma A. 2, Let f ,fp:K—^ int M be homotopic PL embeddings of a compact polyhedron K in a PL m-manifold M. Let N.C int M be a regular neighborhood of f.(K) in M. Then xC^n) = x(3Np). Proof. Let r f 0 be any even natural number satisfying r >_2k-m+3, r where k •=* dim K and m = dim M . Choose a triangulation of M^I con- sistent with the one on M and define PL embeddings F.;K —* M* 3R by F. = f. xo, i=l;2. Since f.'s are homotopic there is a homotopy H: K x I -^MxSr from F to Fp. Define a map H*:'K x I —> M x ]Rr * I by H*(x,t) = (H (x),t) for each (x,t)€K*I. By [72]we may assume that H* is PL. Since 2k-m+2 H^U;^) 108 where hjhf are the Hurewicz epimorphisms. Since i „ = 0 it follows that i^ = 0, too/ $$ Proof of Theorem (A.l) .Let N C int M be a regular neighborhood of K and N*CintM a regular neighborhood of f(K). By hypothesis and by Lemma (1.2), 3N is orientable, and by Lemma (A.3)> 3N is a collection of 2-spheres. By Lemma (A.2) so is then 3N*. Another application of Lemma (A.30 now yields the conclusion. $$ We continue with a result concerning pairs of polyhedra in 3-manifolds. So let (K,L)be a compact polyhedral pair in the interior of a 3-manifold M. By Theorem (2.9) and Lemma (A.3), the properties "KCM has WCC", "KCM is strongly peripherally 1-acyclic over 2Z " , and "KM is weakly peripherally 1-acyclic over Z " are equivalent. In the next result "X has property Pff will mean "X has any of these three properties (hence all three)": Theorem A.U. Suppose that K and L are of the same simple homotopy type. Then K has property P if and only if L has property P. Lemma A.5. Let N be a neighborhood of a compact polyhedron KCintM, where M is a PL n-manifold. Then N is a regular neighborhood of K if and only if N is a compact n-manifold with boundary and N is of the same simple homotopy type as K. Proof. The only if part is well-known [57;Corollary (3.30)] . So we prove the other implication. There is a sequence of expansions and collapses (in M) that transform N into K. We may do the expansions first [18 ;Exercise (U.D)l . It suffices to give the proof for the case when there is just one expansion. Let N^^N be N after the ex- 1Q9 pans ion and let N*cintN be N pushed in int N along a collar on 9N (keeping K fixed). Since N* collapses onto K, N" is a regular neighborhood of K in M [57 ;Corollary (3.30)] . Also, N' collapses onto N* and N collapses onto N* hence both N" and N are regular neighborhoods of N* in Mt It follows by the uniqueness theorem [57 ;Theorem (3,2^)]" that there is an ambient PL isotopy of M carrying N onto N' with support in M-N*. Hence N is a regular neighborhood of K in M. (See Figure (A.l).) ** Figure A.l, 110 Proof of Theorem (A.U). We only prove the necessity. The other implication is proved similarily. Let (A,B) be a regular'neighborhood of (K,L) in M. By Lemma (A.5), A is a regular neighborhood of L and thus homeomorphic to B [57 ;Theorem (3.2U) ] . By Lemma (A,3), 3A is a collection of 2-spheres hence so is 3B, The conclusion now follows by another application of Lemma (A.3). ** In the second part of the appendix we wish to discuss a related question concerning regular neighborhoods of homotopically PL embedded compact polyhedra in 3-manifolds. Question A.6. Let K be a compact polyhedron with Hp(K; Zp) = 0 and let f ,f ;K —* M be homotopic PL embeddings of K into a 3-manifold M. Let N. C M be a regular neighborhood of f.(K) in M (i=l,2). Is then N == N ? 3 12 1 Consider the following example: let M=S,K = SvSvS, and let f. be the two standard PL embeddings ~f (K) has both S *s attached at the 2 l 2 same side of S , while fp(K) has one S on each of the two sides of S , Clearly, f «f . However, N^N because 3N = S U (double torus) while 3N2 = (S1x S1) UCS1* S1). This example shows why condition on H2(K) is k necessary. It is known that Dunce hat t7ll can be PL embedded in S in such a manner that the boundary of the corresponding regular neighborhood is not even simply connected. Hence (A.6) has a negative answer in dim U. Proposition A.7. Suppose the answer to Question (A.6) is affirmative. Then the following statements are equivalent: (i) The Poincar! conjecture in dimension three is true; 3 (ii) The spine of any homotopy 3-cell PL embeds in 1R . Ill Proof . (i) =^ (ii);This implication is independent of (A.6) and is obvious, (ii) =^(i):Suppose Kc int F is a spine of a homotopy 3-cell F. Let BCint F be a nicely embedded 3-cell. By hypothesis there is a PL embedding f ;K~* int B . Let C C int B be a regular neighborhood of f(K) in intB. Since F is contractible, f is homotopic to the inclusion K F. Also, F is a regular neighborhood of K [57 corollary (3*30)] . If the answer to Question '(A..6) is affirmative then 3 F = C so C is a homotopy 3-cell• Since C Cint B = E , C is a real 3-cell hence so is F. JJ Theorem A.8. The answer to Question (A.6) is affirmative if K satisfies any of the following conditions: (i) dim K 11; (ii) K is a compact surface with boundary; (iii) K is a closed surface and 3N. is a 2-sphere. Proof. We may assume that K is connected. First, assume (i). By [32 ;Theorem (2.U)], N. is a 3-cell with n. (possibly nonorientable) 1-handles, n.6]N. Since f is homotopic to f-, we have that ^(N,) ^ MNp) hence ni s np# SuPPose now that, say N is orientable and Np nonorientable. Then there is an orientation-reversing loop J in N . Since f - fp and N. collapses onto f.(K), there is a loop J* in N with [J*] s '[J]€ll (m) hence J* also reverses the orientation. This yields a contradiction to the assumption that N is orientable. It now follows by [32;Theorem (2.2)], that NJjj? Ng. 112 Assume next that K satisfies (ii). Then there exists a bouquet T of simple closed curves on K such that K collapses onto T. The conclusion now follows by case (i) above and [57 ^Corollary (3•29)3. Finally, assume (iii). Then N ' is an I-bundle over f (K) so 3N 2 2 is a double covering of 'f.(K) hence f'(K) is either S or P , the 2 2 projective plane. Thus K is either S or P , 2 If K = S then f.(K) is necessarily two-sided since there are 3 no one-sided 2-spheres in 3-manifolds¦ (indeed, suppose SCM is a 2-sphere in a 3-manifold M and consider a regular neighborhood NC M of S. Then N is a product I-bundle since S is simply connected and hence no loop on S can reverse the orientation in M. Thus S must be two-sided.) Consequently, N. = f.(K) xl hence N ~ N . 2 If K = P we first consider the case when M is orientable. Then both embeddings f.(K) are one-sided since otherwise f.(K) xl would represent a nonorientable 3-submanifold of M, an impossibility. 2 2 Hence N. is a twisted I-bundle over P . It is known that over P i there is but one twisted I-bundle (up to a PL homeomorphism). Consequently, ^ =? Hg. 2 Finally, assume that M is nonorientable and that K = P . If both embeddings f.(-K) are one-sided apply the preceeding argument. If both embeddings f,(K) are two-sided then N. = f,(K)x I hence Nn CT No« So it remains to consider the case when, say f -(K) is one -sided and fp(K) is two-sided. Consider the orientable double cover p:M~^M of M. Then f, (K) lifts in M to two disjoint homeomorphic copies while $"" fp(K) is connected (and double covers f (K)). Since f -f the number of components of p~ f. in S agree. Contradiction.^ 113 We conclude the appendix with some further comments regarding Question (A.6). Note, first, that by Lemma (A.2), the boundaries of the regular neighborhoods are always homeomorphic, since they are both connected (because H (K: Z ) = 0 implies that N.-f.(K) is connected) and they are either both orientable or both nonorientable (because Professor W.Heil suggested the following question which — if the answer is affirmative — would*yield a negative answer to Questionr(A.6): Question A.9» Let K - the square knot, K- = the granny knot, and let M. be the corresponding knot space, i.e. M. = S - (open tubular neighborhood of K. in S ). It is well-known that M- ^ M . However, do M. and Mp collapse to homeomorphic spines? There are certain grounds for a belief that the answer to (A.9) is negative: although the fundamental groups of M. are isomorphic, the peripheral systems are different, i.e. the diagram below can never be completed to a commutative one -- and that obstruction could determine the spine of M.. nl(aiL) ^------> nl(M1) •i ni(3H2) >----—> ni(.M2) VITA Dušan RepovŽ was born on November 30, 195^ in Ljubljana, Yugoslavia. He graduated from the University of Ljubljana in the Fall of 1977 with Mathematics as a major and Mechanics as a minor. He wrote his B.Sc.Thesis on Borsuk's shape theory for compacta, under Professor J.Vrabec. In September 1978 he entered the FSU Graduate School. He passed the Preliminary Doctoral Examination in September 1979• In June and July of I98O he attended the Topology Summer Conference at the University of Texas at Austin.. In September 1980 he accepted an assistantship at the College of Mechanical Engineering (University of Ljubljana). In January 198l he attended the Shape.Theory and Geometric Topology Winter Postgraduate School in Dubrovnik, Yugoslavia. The Summer of 198l he spent back at FSU. In September 198l he was invited to present his joint research with Professor R.C.Lacher at a topology conference in Oberwolfach, West Germany. Next month he was drafted for a 12-months military service in his home country, after which he resumed his position at the College. He is a member of the American Mathematical Society, Pi Mu Epsilon, and the Society of Mathematicians, Physicists and Astronomers of Yugoslavia. He is married to the former Barbara Hvala from Ljubljana, a 1982 graduate of the University of Ljubljana (Philosophy/Sociology). They have a 5 years old son Uroš Peter and they live in Ljubljana. Ill*