Notes on Basic 3-Manifold Topology Chapter 1. Canonical Decomposition
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Part III 3-Manifolds Lecture Notes C Sarah Rasmussen, 2019
Part III 3-manifolds Lecture Notes c Sarah Rasmussen, 2019 Contents Lecture 0 (not lectured): Preliminaries2 Lecture 1: Why not ≥ 5?9 Lecture 2: Why 3-manifolds? + Intro to Knots and Embeddings 13 Lecture 3: Link Diagrams and Alexander Polynomial Skein Relations 17 Lecture 4: Handle Decompositions from Morse critical points 20 Lecture 5: Handles as Cells; Handle-bodies and Heegard splittings 24 Lecture 6: Handle-bodies and Heegaard Diagrams 28 Lecture 7: Fundamental group presentations from handles and Heegaard Diagrams 36 Lecture 8: Alexander Polynomials from Fundamental Groups 39 Lecture 9: Fox Calculus 43 Lecture 10: Dehn presentations and Kauffman states 48 Lecture 11: Mapping tori and Mapping Class Groups 54 Lecture 12: Nielsen-Thurston Classification for Mapping class groups 58 Lecture 13: Dehn filling 61 Lecture 14: Dehn Surgery 64 Lecture 15: 3-manifolds from Dehn Surgery 68 Lecture 16: Seifert fibered spaces 69 Lecture 17: Hyperbolic 3-manifolds and Mostow Rigidity 70 Lecture 18: Dehn's Lemma and Essential/Incompressible Surfaces 71 Lecture 19: Sphere Decompositions and Knot Connected Sum 74 Lecture 20: JSJ Decomposition, Geometrization, Splice Maps, and Satellites 78 Lecture 21: Turaev torsion and Alexander polynomial of unions 81 Lecture 22: Foliations 84 Lecture 23: The Thurston Norm 88 Lecture 24: Taut foliations on Seifert fibered spaces 89 References 92 1 2 Lecture 0 (not lectured): Preliminaries 0. Notation and conventions. Notation. @X { (the manifold given by) the boundary of X, for X a manifold with boundary. th @iX { the i connected component of @X. ν(X) { a tubular (or collared) neighborhood of X in Y , for an embedding X ⊂ Y . -
Ordering Thurston's Geometries by Maps of Non-Zero Degree
ORDERING THURSTON’S GEOMETRIES BY MAPS OF NON-ZERO DEGREE CHRISTOFOROS NEOFYTIDIS ABSTRACT. We obtain an ordering of closed aspherical 4-manifolds that carry a non-hyperbolic Thurston geometry. As application, we derive that the Kodaira dimension of geometric 4-manifolds is monotone with respect to the existence of maps of non-zero degree. 1. INTRODUCTION The existence of a map of non-zero degree defines a transitive relation, called domination rela- tion, on the homotopy types of closed oriented manifolds of the same dimension. Whenever there is a map of non-zero degree M −! N we say that M dominates N and write M ≥ N. In general, the domain of a map of non-zero degree is a more complicated manifold than the target. Gromov suggested studying the domination relation as defining an ordering of compact oriented manifolds of a given dimension; see [3, pg. 1]. In dimension two, this relation is a total order given by the genus. Namely, a surface of genus g dominates another surface of genus h if and only if g ≥ h. However, the domination relation is not generally an order in higher dimensions, e.g. 3 S3 and RP dominate each other but are not homotopy equivalent. Nevertheless, it can be shown that the domination relation is a partial order in certain cases. For instance, 1-domination defines a partial order on the set of closed Hopfian aspherical manifolds of a given dimension (see [18] for 3- manifolds). Other special cases have been studied by several authors; see for example [3, 1, 2, 25]. -
Computing Triangulations of Mapping Tori of Surface Homeomorphisms
Computing Triangulations of Mapping Tori of Surface Homeomorphisms Peter Brinkmann∗ and Saul Schleimer Abstract We present the mathematical background of a software package that computes triangulations of mapping tori of surface homeomor- phisms, suitable for Jeff Weeks’s program SnapPea. The package is an extension of the software described in [?]. It consists of two programs. jmt computes triangulations and prints them in a human-readable format. jsnap converts this format into SnapPea’s triangulation file format and may be of independent interest because it allows for quick and easy generation of input for SnapPea. As an application, we ob- tain a new solution to the restricted conjugacy problem in the mapping class group. 1 Introduction In [?], the first author described a software package that provides an en- vironment for computer experiments with automorphisms of surfaces with one puncture. The purpose of this paper is to present the mathematical background of an extension of this package that computes triangulations of mapping tori of such homeomorphisms, suitable for further analysis with Jeff Weeks’s program SnapPea [?].1 ∗This research was partially conducted by the first author for the Clay Mathematics Institute. 2000 Mathematics Subject Classification. 57M27, 37E30. Key words and phrases. Mapping tori of surface automorphisms, pseudo-Anosov au- tomorphisms, mapping class group, conjugacy problem. 1Software available at http://thames.northnet.org/weeks/index/SnapPea.html 1 Pseudo-Anosov homeomorphisms are of particular interest because their mapping tori are hyperbolic 3-manifolds of finite volume [?]. The software described in [?] recognizes pseudo-Anosov homeomorphisms. Combining this with the programs discussed here, we obtain a powerful tool for generating and analyzing large numbers of hyperbolic 3-manifolds. -
Lens Spaces We Introduce Some of the Simplest 3-Manifolds, the Lens Spaces
CHAPTER 10 Seifert manifolds In the previous chapter we have proved various general theorems on three- manifolds, and it is now time to construct examples. A rich and important source is a family of manifolds built by Seifert in the 1930s, which generalises circle bundles over surfaces by admitting some “singular”fibres. The three- manifolds that admit such kind offibration are now called Seifert manifolds. In this chapter we introduce and completely classify (up to diffeomor- phisms) the Seifert manifolds. In Chapter 12 we will then show how to ge- ometrise them, by assigning a nice Riemannian metric to each. We will show, for instance, that all the elliptic andflat three-manifolds are in fact particular kinds of Seifert manifolds. 10.1. Lens spaces We introduce some of the simplest 3-manifolds, the lens spaces. These manifolds (and many more) are easily described using an important three- dimensional construction, called Dehnfilling. 10.1.1. Dehnfilling. If a 3-manifoldM has a spherical boundary com- ponent, we can cap it off with a ball. IfM has a toric boundary component, there is no canonical way to cap it off: the simplest object that we can attach to it is a solid torusD S 1, but the resulting manifold depends on the gluing × map. This operation is called a Dehnfilling and we now study it in detail. LetM be a 3-manifold andT ∂M be a boundary torus component. ⊂ Definition 10.1.1. A Dehnfilling ofM alongT is the operation of gluing a solid torusD S 1 toM via a diffeomorphismϕ:∂D S 1 T. -
Singular Hyperbolic Structures on Pseudo-Anosov Mapping Tori
SINGULAR HYPERBOLIC STRUCTURES ON PSEUDO-ANOSOV MAPPING TORI A DISSERTATION SUBMITTED TO THE DEPARTMENT OF MATHEMATICS AND THE COMMITTEE ON GRADUATE STUDIES OF STANFORD UNIVERSITY IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE OF DOCTOR OF PHILOSOPHY Kenji Kozai June 2013 iv Abstract We study three-manifolds that are constructed as mapping tori of surfaces with pseudo-Anosov monodromy. Such three-manifolds are endowed with natural sin- gular Sol structures coming from the stable and unstable foliations of the pseudo- Anosov homeomorphism. We use Danciger’s half-pipe geometry [6] to extend results of Heusener, Porti, and Suarez [13] and Hodgson [15] to construct singular hyperbolic structures when the monodromy has orientable invariant foliations and its induced action on cohomology does not have 1 as an eigenvalue. We also discuss a combina- torial method for deforming the Sol structure to a singular hyperbolic structure using the veering triangulation construction of Agol [1] when the surface is a punctured torus. v Preface As a result of Perelman’s Geometrization Theorem, every 3-manifold decomposes into pieces so that each piece admits one of eight model geometries. Most 3-manifolds ad- mit hyperbolic structures, and outstanding questions in the field mostly center around these manifolds. Having an effective method for describing the hyperbolic structure would help with understanding topology and geometry in dimension three. The recent resolution of the Virtual Fibering Conjecture also means that every closed, irreducible, atoroidal 3-manifold can be constructed, up to a finite cover, from a homeomorphism of a surface by taking the mapping torus for the homeomorphism [2]. -
A Computation of Knot Floer Homology of Special (1,1)-Knots
A COMPUTATION OF KNOT FLOER HOMOLOGY OF SPECIAL (1,1)-KNOTS Jiangnan Yu Department of Mathematics Central European University Advisor: Andras´ Stipsicz Proposal prepared by Jiangnan Yu in part fulfillment of the degree requirements for the Master of Science in Mathematics. CEU eTD Collection 1 Acknowledgements I would like to thank Professor Andras´ Stipsicz for his guidance on writing this thesis, and also for his teaching and helping during the master program. From him I have learned a lot knowledge in topol- ogy. I would also like to thank Central European University and the De- partment of Mathematics for accepting me to study in Budapest. Finally I want to thank my teachers and friends, from whom I have learned so much in math. CEU eTD Collection 2 Abstract We will introduce Heegaard decompositions and Heegaard diagrams for three-manifolds and for three-manifolds containing a knot. We define (1,1)-knots and explain the method to obtain the Heegaard diagram for some special (1,1)-knots, and prove that torus knots and 2- bridge knots are (1,1)-knots. We also define the knot Floer chain complex by using the theory of holomorphic disks and their moduli space, and give more explanation on the chain complex of genus-1 Heegaard diagram. Finally, we compute the knot Floer homology groups of the trefoil knot and the (-3,4)-torus knot. 1 Introduction Knot Floer homology is a knot invariant defined by P. Ozsvath´ and Z. Szabo´ in [6], using methods of Heegaard diagrams and moduli theory of holomorphic discs, combined with homology theory. -
Inventiones Mathematicae 9 Springer-Verlag 1994
Invent. math. 118, 255 283 (1994) Inventiones mathematicae 9 Springer-Verlag 1994 Bundles and finite foliations D. Cooper 1'*, D.D. Long 1'**, A.W. Reid 2 .... 1 Department of Mathematics, University of California, Santa Barbara CA 93106, USA z Department of Pure Mathematics, University of Cambridge, Cambridge CB2 1SB, UK Oblatum 15-1X-1993 & 30-X11-1993 1 Introduction By a hyperbolic 3-manifold, we shall always mean a complete orientable hyperbolic 3-manifold of finite volume. We recall that if F is a Kleinian group then it is said to be geometrically finite if there is a finite-sided convex fundamental domain for the action of F on hyperbolic space. Otherwise, s is geometrically infinite. If F happens to be a surface group, then we say it is quasi-Fuchsian if the limit set for the group action is a Jordan curve C and F preserves the components of S~\C. The starting point for this work is the following theorem, which is a combination of theorems due to Marden [10], Thurston [14] and Bonahon [1]. Theorem 1.1 Suppose that M is a closed orientable hyperbolic 3-man!fold. If g: Sq-~ M is a lrl-injective map of a closed surface into M then exactly one of the two alternatives happens: 9 The 9eometrically infinite case: there is a finite cover lVl of M to which g l([ts and can be homotoped to be a homeomorphism onto a fiber of some fibration of over the circle. 9 The 9eometricallyfinite case: g,~zl (S) is a quasi-Fuchsian group. -
Determining Hyperbolicity of Compact Orientable 3-Manifolds with Torus
Determining hyperbolicity of compact orientable 3-manifolds with torus boundary Robert C. Haraway, III∗ February 1, 2019 Abstract Thurston’s hyperbolization theorem for Haken manifolds and normal sur- face theory yield an algorithm to determine whether or not a compact ori- entable 3-manifold with nonempty boundary consisting of tori admits a com- plete finite-volume hyperbolic metric on its interior. A conjecture of Gabai, Meyerhoff, and Milley reduces to a computation using this algorithm. 1 Introduction The work of Jørgensen, Thurston, and Gromov in the late ‘70s showed ([18]) that the set of volumes of orientable hyperbolic 3-manifolds has order type ωω. Cao and Meyerhoff in [4] showed that the first limit point is the volume of the figure eight knot complement. Agol in [1] showed that the first limit point of limit points is the volume of the Whitehead link complement. Most significantly for the present paper, Gabai, Meyerhoff, and Milley in [9] identified the smallest, closed, orientable hyperbolic 3-manifold (the Weeks-Matveev-Fomenko manifold). arXiv:1410.7115v5 [math.GT] 31 Jan 2019 The proof of the last result required distinguishing hyperbolic 3-manifolds from non-hyperbolic 3-manifolds in a large list of 3-manifolds; this was carried out in [17]. The method of proof was to see whether the canonize procedure of SnapPy ([5]) succeeded or not; identify the successes as census manifolds; and then examine the fundamental groups of the 66 remaining manifolds by hand. This method made the ∗Research partially supported by NSF grant DMS-1006553. -
Endomorphisms of Mapping Tori
ENDOMORPHISMS OF MAPPING TORI CHRISTOFOROS NEOFYTIDIS Abstract. We classify in terms of Hopf-type properties mapping tori of residually finite Poincar´eDuality groups with non-zero Euler characteristic. This generalises and gives a new proof of the analogous classification for fibered 3-manifolds. Various applications are given. In particular, we deduce that rigidity results for Gromov hyperbolic groups hold for the above mapping tori with trivial center. 1. Introduction We classify in terms of Hopf-type properties mapping tori of residually finite Poincar´eDu- ality (PD) groups K with non-zero Euler characteristic, which satisfy the following finiteness condition: if there exists an integer d > 1 and ξ 2 Aut(K) such that θd = ξθξ−1 mod Inn(K); (∗) then θq 2 Inn(K) for some q > 1: One sample application is that every endomorphism onto a finite index subgroup of the 1 fundamental group of a mapping torus F oh S , where F is a closed aspherical manifold with 1 π1(F ) = K as above, induces a homotopy equivalence (equivalently, π1(F oh S ) is cofinitely 1 1 Hopfian) if and only if the center of π1(F oh S ) is trivial; equivalently, π1(F oh S ) is co- Hopfian. If, in addition, F is topologically rigid, then homotopy equivalence can be replaced by a map homotopic to a homeomorphism, by Waldhausen's rigidity [Wa] in dimension three and Bartels-L¨uck's rigidity [BL] in dimensions greater than four. Condition (∗) is known to be true for every aspherical surface by a theorem of Farb, Lubotzky and Minsky [FLM] on translation lengths. -
Cohomology Determinants of Compact 3-Manifolds
COHOMOLOGY DETERMINANTS OF COMPACT 3–MANIFOLDS CHRISTOPHER TRUMAN Abstract. We give definitions of cohomology determinants for compact, connected, orientable 3–manifolds. We also give formu- lae relating cohomology determinants before and after gluing a solid torus along a torus boundary component. Cohomology de- terminants are related to Turaev torsion, though the author hopes that they have other uses as well. 1. Introduction Cohomology determinants are an invariant of compact, connected, orientable 3-manifolds. The author first encountered these invariants in [Tur02], when Turaev gave the definition for closed 3–manifolds, and used cohomology determinants to obtain a leading order term of Tu- raev torsion. In [Tru], the author gives a definition for 3–manifolds with boundary, and derives a similar relationship to Turaev torsion. Here, we repeat the definitions, and give formulae relating the cohomology determinants before and after gluing a solid torus along a boundary component. One can use these formulae, and gluing formulae for Tu- raev torsion from [Tur02] Chapter VII, to re-derive the results of [Tru] from the results of [Tur02] Chapter III, or vice-versa. 2. Integral Cohomology Determinants arXiv:math/0611248v1 [math.GT] 8 Nov 2006 2.1. Closed 3–manifolds. We will simply state the relevant result from [Tur02] Section III.1; the proof is similar to the one below. Let R be a commutative ring with unit, and let N be a free R–module of rank n ≥ 3. Let S = S(N ∗) be the graded symmetric algebra on ∗ ℓ N = HomR(N, R), with grading S = S . Let f : N ×N ×N −→ R ℓL≥0 be an alternate trilinear form, and let g : N × N → N ∗ be induced by f. -
MTH 507 Midterm Solutions
MTH 507 Midterm Solutions 1. Let X be a path connected, locally path connected, and semilocally simply connected space. Let H0 and H1 be subgroups of π1(X; x0) (for some x0 2 X) such that H0 ≤ H1. Let pi : XHi ! X (for i = 0; 1) be covering spaces corresponding to the subgroups Hi. Prove that there is a covering space f : XH0 ! XH1 such that p1 ◦ f = p0. Solution. Choose x ; x 2 p−1(x ) so that p :(X ; x ) ! (X; x ) and e0 e1 0 i Hi ei 0 p (X ; x ) = H for i = 0; 1. Since H ≤ H , by the Lifting Criterion, i∗ Hi ei i 0 1 there exists a lift f : XH0 ! XH1 of p0 such that p1 ◦f = p0. It remains to show that f : XH0 ! XH1 is a covering space. Let y 2 XH1 , and let U be a path-connected neighbourhood of x = p1(y) that is evenly covered by both p0 and p1. Let V ⊂ XH1 be −1 −1 the slice of p1 (U) that contains y. Denote the slices of p0 (U) by 0 −1 0 −1 fVz : z 2 p0 (x)g. Let C denote the subcollection fVz : z 2 f (y)g −1 −1 of p0 (U). Every slice Vz is mapped by f into a single slice of p1 (U), −1 0 as these are path-connected. Also, since f(z) 2 p1 (y), f(Vz ) ⊂ V iff f(z) = y. Hence f −1(V ) is the union of the slices in C. 0 −1 0 0 0 Finally, we have that for Vz 2 C,(p1jV ) ◦ (p0jVz ) = fjVz . -
On Open Book Embedding of Contact Manifolds in the Standard Contact Sphere
ON OPEN BOOK EMBEDDING OF CONTACT MANIFOLDS IN THE STANDARD CONTACT SPHERE KULDEEP SAHA Abstract. We prove some open book embedding results in the contact category with a constructive ap- proach. As a consequence, we give an alternative proof of a Theorem of Etnyre and Lekili that produces a large class of contact 3-manifolds admitting contact open book embeddings in the standard contact 5-sphere. We also show that all the Ustilovsky (4m + 1)-spheres contact open book embed in the standard contact (4m + 3)-sphere. 1. Introduction An open book decomposition of a manifold M m is a pair (V m−1; φ), such that M m is diffeomorphic to m−1 m−1 2 m−1 MT (V ; φ)[id @V ×D . Here, V , the page, is a manifold with boundary, and φ, the monodromy, is a diffeomorphism of V m−1 that restricts to identity in a neighborhood of the boundary @V . MT (V m−1; φ) denotes the mapping torus of φ. We denote an open book, with page V m−1 and monodromy φ, by Aob(V; φ). The existence of open book decompositions, for a fairly large class of manifolds, is now known due the works of Alexander [Al], Winkelnkemper [Wi], Lawson [La], Quinn [Qu] and Tamura [Ta]. In particular, every closed, orientable, odd dimensional manifold admits an open book decomposition. Thurston and Winkelnkemper [TW] have shown that starting from an exact symplectic manifold (Σ2m;!) 2m as page and a boundary preserving symplectomorphism φs of (Σ ;!) as monodromy, one can produce a 2m+1 2m contact 1-form α on N = Aob(Σ ; φs).