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Contractible N-Manifolds and the Double N-Space Property Pete Sparks University of Wisconsin-Milwaukee

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Contractible N-Manifolds and the Double N-Space Property Pete Sparks University of Wisconsin-Milwaukee University of Wisconsin Milwaukee UWM Digital Commons Theses and Dissertations December 2014 Contractible n-Manifolds and the Double n-Space Property Pete Sparks University of Wisconsin-Milwaukee Follow this and additional works at: https://dc.uwm.edu/etd Part of the Mathematics Commons Recommended Citation Sparks, Pete, "Contractible n-Manifolds and the Double n-Space Property" (2014). Theses and Dissertations. 643. https://dc.uwm.edu/etd/643 This Dissertation is brought to you for free and open access by UWM Digital Commons. It has been accepted for inclusion in Theses and Dissertations by an authorized administrator of UWM Digital Commons. For more information, please contact [email protected]. CONTRACTIBLE n-MANIFOLDS AND THE DOUBLE n-SPACE PROPERTY by Pete Sparks A Thesis Submitted in Partial Fulfillment of the Requirements for the Degree of Doctorate of Philosophy in Mathematics at The University of Wisconsin-Milwaukee December, 2014 ABSTRACT CONTRACTIBLE n-MANIFOLDS AND THE DOUBLE n-SPACE PROPERTY by Pete Sparks The University of Wisconsin-Milwaukee, 2014 Under the Supervision of Professor Craig Guilbault We are interested in contractible manifolds M n which decompose or split as n n n M = A ∪C B where A, B, C ≈ R or A, B, C ≈ B . We introduce a 4-manifold M containing a spine which can be written as A ∪C B with A, B, and C all collapsible 4 4 which in turn implies M splits as B ∪B4 B . From M we obtain a countably infinite 4 4 collection of distinct 4-manifolds all of which split as B ∪B4 B . Connected sums at infinity of interiors of manifolds from sequences contained in this collection constitute 4 4 an uncountable set of open 4-manifolds each of which splits as R ∪R4 R . ii Table of Contents 1 IntroductionToManifoldSplitting 1 1.1 Definitions, Motivation, and Summary of Results . 1 1.2 ElementaryResults............................ 2 1.3 HistoryandCurrentWork ........................ 3 2 TheMazurandJester’sManifolds 6 2.1 TheMazurManifold ........................... 6 2.2 TheJester’sManifolds . 10 3 Spines 12 3.1 Collapses.................................. 12 3.2 TheDunceHat .............................. 14 3.3 TheJester’sHat ............................. 16 4MoreJester’sManifolds 22 4.1 Pseudo2-handles ............................. 22 4.2 ATheoremofWright........................... 23 5SumsofSplitters 26 5.1 Some Manifold Sums and the Fundamental Group at Infinity . 26 5.2 CSI’sofSemistableManifolds . 33 5.3 Some Combinatorial Group Theory and Uncountable Jester’s Mani- foldSums ................................. 38 iii 5.4 SumsofSplittersSplit ... .. .. .. .. .. ... .. .. .. .. .. 44 Bibliography 50 CurriculumVitae 53 iv List of Figures 2.1 Γ ⊂ ∂(S1 × B3) ⊂ theMazurManifold ................. 6 2.2 WirtingerdiagramoftheMazurlink . 7 2.3 ThickenedMeridionalDisc . 8 2.4 Triangle in H2 ............................... 9 2.5 C ⊂ ∂(S1 × B3) ⊂ aJester’sManifold. 11 3.1 Elementary Collapse (simplicial) K ցe L, A isafreeface. 12 3.2 Elementary Collapse (CW) X ց Y ................... 13 3.3 TheDunceHat .............................. 15 3.4 DunceHatTriangulated . 15 3.5 TheDunceHatAttachingMap . 15 3.6 TheJester’sHat ............................. 16 3.7 J Triangulated .............................. 17 3.8 Attaching map for J ........................... 17 3.9 J “splits”intoCollapsibles. 18 3.10 Collapses of J’s“Splittands”....................... 18 3.11 Intervals of S1 andtheirclasps...................... 19 3.12 M(g) the Mapping Cylinder of g .................... 19 3.13 M(g) ≈ S1 × B3 .............................. 20 3.14 The Mapping Cylinder of h ....................... 20 4.1 S1 × B3 unionadegree2pseudo2-handle . 23 2 0 4.2 µ λ ∈ π1(∂T )............................... 25 v 5.1 h and h¯ .................................. 27 5.2 Construction of a Rn neighborhood of r ................. 29 5.3 (X, αX )♮(Y, αY ).............................. 30 ∞ 5.4 ♮i=1Mi ................................... 31 5.5 N ′ ⊂ N .................................. 34 5.6 HomotopyBetweenRays. 35 5.7 Dividing H’sDomain........................... 36 5.8 Strip S1 .................................. 37 5.9 Neighborhoods of ∞ ........................... 41 5.10 W1 ⊃ W2 ⊃ W3 in X1♮X2♮... ....................... 42 5.11 |Σ ∩ M n−1| iseven ............................ 45 5.12 Tapered Product Neighborhood of α .................. 45 5.13 N ′ ⊃ S ⊔ T ................................ 46 5.14 M1♮M2 Splits ............................... 47 5.15 Cˇ3 ..................................... 48 vi ACKNOWLEDGMENTS First and most important, I would like to express my gratitude to Craig Guil- bault, my Ph.D advisor, teacher, UWM Math Department Graduate Coordinator, and mentor for his patient sharing of his knowledge with me. I am also indebted to him for his wise guidance, belief in my abilities and encouragement in general. I would like to acknowledge the work of the many professors and other teachers who imparted their wisdom and knowledge to me. In particular, I would like to thank Boris Okun, Ric Ancel, Chris Hruska, and Jeb Willenbring for their helpful suggestions, friendly encouragement, and sense of humor. From Portland State I want to mention professors Gavin Bjork and Len Swanson whose instruction and advice helped me to find my way to the mathematical sciences. Many thanks to the numerous mathematical authors I have read and even more thanks to those from whom I have borrowed. Particularly (but in no particular order), thank you Rourke, Sanderson [RoSa], Cohen [Coh], James Munkres, Hatcher [Hat], Glaser [Gla65], [Gla66],[Gla72], Curtis, Kwun [CuKw], Wright [Wri], Ancel, and Guilbauilt [AG95], [Gui]. Thank you to my many peers and study buddies in my studies of math especially those in the UW-Milwaukee toplogy research group and other topology students whose acedemic paths have crossed with mine. Mike Howen has my thanks for spending his valuable time illustrating and teach- ing me to illustrate this document. Thank you to my wife Irina for her patience and my son Daniel for giving me perspective and needed cheering up. Much thanks is due also to my many students. vii 1 Chapter 1 Introduction To Manifold Splitting 1.1 Definitions, Motivation, and Summary of Re- sults Our results will generally be in the topological category but because of the niceness of the spaces involved we are able to work in both the piecewise linear and smooth categories in our effort to obtain them. We will primarily be working in the PL category. We may choose to construct manifolds (and other objects) to be piecewise- linear or smooth. Unless stated otherwise the reader should view such constructions as PL. By a PL manifold we mean a simplicial complex in which the link of every vertex is a sphere. Definition 1.1.1. We will write A ∪C B to indicate a union A ∪ B with intersec- n n tion C = A ∩ B. We say a manifold M splits if M = A ∪C B with A, B, and C = A ∩ B ≈ Bn or A, B, and C = A ∩ B ≈ Rn. In the former case we say M n n n “splits into closed balls” or M is a “closed splitter” and write M = B ∪Bn B . In the latter case we say M “splits into open balls” or M is an “open splitter” and n n n write M = R ∪Rn R . We are interested in contractible manifolds M n which are open or closed splitters. We introduce a 4-manifold M containing a spine, which we call a Jester’s Hat, that can be written as A ∪C B with A, B, and C all collapsible. We’ll show that this implies M is a closed splitter. From M we obtain a countably infinite collection 2 of distinct 4-manifolds all of which are closed splitters. Connected sums at infinity of interiors of manifolds from sequences contained in this collection constitute an 4 4 uncountable set of open 4-manifolds each of which splits as R ∪R4 R . These last two statements constitute our two main theorems. Our motivation comes from David Gabai’s result that the Whitehead 3-manifold, W h3, splits into open 3-balls 3 3 3 W h = R ∪R3 R [Gab]. Other terminology in use which is synonomous with open splitting includes double n-space property and Gabai splitting. 1.2 Elementary Results It is clear that the unit ball Bn splits into two “subballs” overlapping in a n-ball. Likewise, Euclidean space itself splits into two Euclidean spaces meeting in a Eu- clidean space. More generally, we have the following. n n n n n Proposition 1.2.1. If a manifold M splits as M = B ∪Bn B then intM splits n n n as intM = R ∪Rn R . n n Proof. Suppose M = A ∪C B with A, B, C ≈ B . We will show that intM = intA ∪intC intB. In order to do this we show (1) intA ∩ intB = intC and (2) intM = intA ∪ intB. For (1), suppose x ∈ intC. Then, as C ≈ Bn, there exists N ⊂ C an open (Euclidean) n-ball neighborhood of x. Then N is an open ball neighborhood of x in both A and B and thus x ∈ intA ∩ intB. For the reverse inclusion, let x ∈ intA ∩ intB and NA and NB be neighborhoods n of x in A and B each homeomorphic to an open ball of R . Then NA ∩ NB is a neighborhood of x in C and it contains a neighborhood of x homeomorphic to an 3 open Rn ball as it is a neighborhood of x in M. Thus, x is an interior point of C and we have shown intA ∩ intB ⊂ intC. For (2), it is clear that intA ∪ intB ⊂ intM. To see the reverse inclusion suppose n for contradiction that there exists x ∈ intM ∩∂A∩∂B so we can choose UA,VB ≈ R+ neighborhoods of x in A and B, respectively. Then UA = A ∩ U and VB = B ∩ V for some open sets U, V ⊂ M n. Let W ≈ Rn be a neighborhood of x in M n contained ′ ′ ′ ′ ′ in U ∩ V, and let UA = A ∩ W and VB = B ∩ W. Then UA ∪ VB = W, with UA ′ n ′ ′ and VB each homeomorphic to an open subset of R+.
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