MTH 507 Midterm Solutions
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Symplectic Group Actions and Covering Spaces
Symplectic group actions and covering spaces Montaldi, James and Ortega, Juan-Pablo 2009 MIMS EPrint: 2007.84 Manchester Institute for Mathematical Sciences School of Mathematics The University of Manchester Reports available from: http://eprints.maths.manchester.ac.uk/ And by contacting: The MIMS Secretary School of Mathematics The University of Manchester Manchester, M13 9PL, UK ISSN 1749-9097 Symplectic Group Actions and Covering Spaces James Montaldi & Juan-Pablo Ortega January 2009 Abstract For symplectic group actions which are not Hamiltonian there are two ways to define reduction. Firstly using the cylinder-valued momentum map and secondly lifting the action to any Hamiltonian cover (such as the universal cover), and then performing symplectic reduction in the usual way. We show that provided the action is free and proper, and the Hamiltonian holonomy associated to the action is closed, the natural projection from the latter to the former is a symplectic cover. At the same time we give a classification of all Hamiltonian covers of a given symplectic group action. The main properties of the lifting of a group action to a cover are studied. Keywords: lifted group action, symplectic reduction, universal cover, Hamiltonian holonomy, momentum map MSC2000: 53D20, 37J15. Introduction There are many instances of symplectic group actions which are not Hamiltonian—ie, for which there is no momentum map. These can occur both in applications [13] as well as in fundamental studies of symplectic geometry [1, 2, 5]. In such cases it is possible to define a “cylinder valued momentum map” [3], and then perform symplectic reduction with respect to this map [16, 17]. -
[Math.AT] 4 Sep 2003 and H Uhri Ebro DE Eerhtann Ewr HPRN-C Network Programme
ON RATIONAL HOMOTOPY OF FOUR-MANIFOLDS S. TERZIC´ Abstract. We give explicit formulas for the ranks of the third and fourth homotopy groups of all oriented closed simply con- nected four-manifolds in terms of their second Betti numbers. We also show that the rational homotopy type of these manifolds is classified by their rank and signature. 1. Introduction In this paper we consider the problem of computation of the rational homotopy groups and the problem of rational homotopy classification of simply connected closed four-manifolds. Our main results could be collected as follows. Theorem 1. Let M be a closed oriented simply connected four-manifold and b2 its second Betti number. Then: (1) If b2 =0 then rk π4(M)=rk π7(M)=1 and πp(M) is finite for p =46 , 7 , (2) If b2 =1 then rk π2(M)=rk π5(M)=1 and πp(M) is finite for p =26 , 5 , (3) If b2 =2 then rk π2(M)=rk π3(M)=2 and πp(M) is finite for p =26 , 3 , (4) If b2 > 2 then dim π∗(M) ⊗ Q = ∞ and b (b + 1) b (b2 − 4) rk π (M)= b , rk π (M)= 2 2 − 1, rk π (M)= 2 2 . 2 2 3 2 4 3 arXiv:math/0309076v1 [math.AT] 4 Sep 2003 When the second Betti number is 3, we can prove a little more. Proposition 2. If b2 =3 then rk π5(M)=10. Regarding rational homotopy type classification of simply connected closed four-manifolds, we obtain the following. Date: November 21, 2018; MSC 53C25, 57R57, 58A14, 57R17. -
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. -
Riemann Mapping Theorem
Riemann surfaces, lecture 8 M. Verbitsky Riemann surfaces lecture 7: Riemann mapping theorem Misha Verbitsky Universit´eLibre de Bruxelles December 8, 2015 1 Riemann surfaces, lecture 8 M. Verbitsky Riemannian manifolds (reminder) DEFINITION: Let h 2 Sym2 T ∗M be a symmetric 2-form on a manifold which satisfies h(x; x) > 0 for any non-zero tangent vector x. Then h is called Riemannian metric, of Riemannian structure, and (M; h) Riemannian manifold. DEFINITION: For any x:y 2 M, and any path γ :[a; b] −! M connecting R dγ dγ x and y, consider the length of γ defined as L(γ) = γ j dt jdt, where j dt j = dγ dγ 1=2 h( dt ; dt ) . Define the geodesic distance as d(x; y) = infγ L(γ), where infimum is taken for all paths connecting x and y. EXERCISE: Prove that the geodesic distance satisfies triangle inequality and defines metric on M. EXERCISE: Prove that this metric induces the standard topology on M. n P 2 EXAMPLE: Let M = R , h = i dxi . Prove that the geodesic distance coincides with d(x; y) = jx − yj. EXERCISE: Using partition of unity, prove that any manifold admits a Riemannian structure. 2 Riemann surfaces, lecture 8 M. Verbitsky Conformal structures and almost complex structures (reminder) REMARK: The following theorem implies that almost complex structures on a 2-dimensional oriented manifold are equivalent to conformal structures. THEOREM: Let M be a 2-dimensional oriented manifold. Given a complex structure I, let ν be the conformal class of its Hermitian metric. Then ν is determined by I, and it determines I uniquely. -
1 the Spin Homomorphism SL2(C) → SO1,3(R) a Summary from Multiple Sources Written by Y
1 The spin homomorphism SL2(C) ! SO1;3(R) A summary from multiple sources written by Y. Feng and Katherine E. Stange Abstract We will discuss the spin homomorphism SL2(C) ! SO1;3(R) in three manners. Firstly we interpret SL2(C) as acting on the Minkowski 1;3 spacetime R ; secondly by viewing the quadratic form as a twisted 1 1 P × P ; and finally using Clifford groups. 1.1 Introduction The spin homomorphism SL2(C) ! SO1;3(R) is a homomorphism of classical matrix Lie groups. The lefthand group con- sists of 2 × 2 complex matrices with determinant 1. The righthand group consists of 4 × 4 real matrices with determinant 1 which preserve some fixed real quadratic form Q of signature (1; 3). This map is alternately called the spinor map and variations. The image of this map is the identity component + of SO1;3(R), denoted SO1;3(R). The kernel is {±Ig. Therefore, we obtain an isomorphism + PSL2(C) = SL2(C)= ± I ' SO1;3(R): This is one of a family of isomorphisms of Lie groups called exceptional iso- morphisms. In Section 1.3, we give the spin homomorphism explicitly, al- though these formulae are unenlightening by themselves. In Section 1.4 we describe O1;3(R) in greater detail as the group of Lorentz transformations. This document describes this homomorphism from three distinct per- spectives. The first is very concrete, and constructs, using the language of Minkowski space, Lorentz transformations and Hermitian matrices, an ex- 4 plicit action of SL2(C) on R preserving Q (Section 1.5). -
Rational Homotopy Theory: a Brief Introduction
Contemporary Mathematics Rational Homotopy Theory: A Brief Introduction Kathryn Hess Abstract. These notes contain a brief introduction to rational homotopy theory: its model category foundations, the Sullivan model and interactions with the theory of local commutative rings. Introduction This overview of rational homotopy theory consists of an extended version of lecture notes from a minicourse based primarily on the encyclopedic text [18] of F´elix, Halperin and Thomas. With only three hours to devote to such a broad and rich subject, it was difficult to choose among the numerous possible topics to present. Based on the subjects covered in the first week of this summer school, I decided that the goal of this course should be to establish carefully the founda- tions of rational homotopy theory, then to treat more superficially one of its most important tools, the Sullivan model. Finally, I provided a brief summary of the ex- tremely fruitful interactions between rational homotopy theory and local algebra, in the spirit of the summer school theme “Interactions between Homotopy Theory and Algebra.” I hoped to motivate the students to delve more deeply into the subject themselves, while providing them with a solid enough background to do so with relative ease. As these lecture notes do not constitute a history of rational homotopy theory, I have chosen to refer the reader to [18], instead of to the original papers, for the proofs of almost all of the results cited, at least in Sections 1 and 2. The reader interested in proper attributions will find them in [18] or [24]. The author would like to thank Luchezar Avramov and Srikanth Iyengar, as well as the anonymous referee, for their helpful comments on an earlier version of this article. -
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. -
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. -
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. -
Theory of Covering Spaces
THEORY OF COVERING SPACES BY SAUL LUBKIN Introduction. In this paper I study covering spaces in which the base space is an arbitrary topological space. No use is made of arcs and no assumptions of a global or local nature are required. In consequence, the fundamental group ir(X, Xo) which we obtain fre- quently reflects local properties which are missed by the usual arcwise group iri(X, Xo). We obtain a perfect Galois theory of covering spaces over an arbitrary topological space. One runs into difficulties in the non-locally connected case unless one introduces the concept of space [§l], a common generalization of topological and uniform spaces. The theory of covering spaces (as well as other branches of algebraic topology) is better done in the more general do- main of spaces than in that of topological spaces. The Poincaré (or deck translation) group plays the role in the theory of covering spaces analogous to the role of the Galois group in Galois theory. However, in order to obtain a perfect Galois theory in the non-locally con- nected case one must define the Poincaré filtered group [§§6, 7, 8] of a cover- ing space. In §10 we prove that every filtered group is isomorphic to the Poincaré filtered group of some regular covering space of a connected topo- logical space [Theorem 2]. We use this result to translate topological ques- tions on covering spaces into purely algebraic questions on filtered groups. In consequence we construct examples and counterexamples answering various questions about covering spaces [§11 ]. In §11 we also -
On Digital Simply Connected Spaces and Manifolds: a Digital Simply Connected 3- Manifold Is the Digital 3-Sphere
On digital simply connected spaces and manifolds: a digital simply connected 3- manifold is the digital 3-sphere Alexander V. Evako Npk Novotek, Laboratory of Digital Technologies. Moscow, Russia e-mail: [email protected]. Abstract In the framework of digital topology, we study structural and topological properties of digital n- dimensional manifolds. We introduce the notion of simple connectedness of a digital space and prove that if M and N are homotopy equivalent digital spaces and M is simply connected, then so is N. We show that a simply connected digital 2-manifold is the digital 2-sphere and a simply connected digital 3-manifold is the digital 3-sphere. This property can be considered as a digital form of the Poincaré conjecture for continuous three-manifolds. Key words: Graph; Dimension; Digital manifold; Simply connected space; Sphere 1. Introduction A digital approach to geometry and topology plays an important role in analyzing n-dimensional digitized images arising in computer graphics as well as in many areas of science including neuroscience, medical imaging, industrial inspection, geoscience and fluid dynamics. Concepts and results of the digital approach are used to specify and justify some important low-level image processing algorithms, including algorithms for thinning, boundary extraction, object counting, and contour filling. Usually, a digital object is equipped with a graph structure based on the local adjacency relations of digital points [5]. In papers [6-7], a digital n-surface was defined as a simple undirected graph and basic properties of n-surfaces were studied. Paper [6] analyzes a local structure of the digital space Zn. -
MATH 215B MIDTERM SOLUTIONS 1. (A) (6 Marks) Show That a Finitely
MATH 215B MIDTERM SOLUTIONS 1. (a) (6 marks) Show that a finitely generated group has only a finite number of subgroups of a given finite index. (Hint: Do it for a free group first.) (b) (6 marks) Show that an index n subgroup H of a group G has at most n conjugate subgroups gHg−1 in G. Apply this to show that there exists a normal subgroup K ⊂ G of finite index in G with K ⊂ H. Solution (a) Suppose we have a finitely-generated free group ∗nZ, and a subgroup H of finite index k. We can build a finite graph X having fundamental group ∗nZ; it is a wedge of n circles, having 1 vertex and n edges. The subgroup p H determines a connected based covering space Xe −→ X, which is also a graph. The number of sheets is the index of p∗π1(Xe) in π1(X), which is k. So Xe is a based graph with k vertices and nk edges. Up to based isomorphism, there are only finitely many based graphs with k vertices and nk edges. So by the classification of covering maps, there are only finitely many subgroups H of index k. Now suppose we have a finitely-generated group. Every finitely-generated group is a quotient of a finitely-generated free group, so we can assume our group is ∗nZ/K for some normal subgroup K ⊂ ∗nZ. An index k subgroup of ∗nZ/K is of the form H/K where H is an index k subgroup of ∗nZ that contains K.