Gromov s Proof of Mostow Rigidity
|
|
- Barnard Turner
- 5 years ago
- Views:
Transcription
1 Gromov s Proof of Mostow Rigidity Mostow Rigidity is a theorem about invariants. An invariant assigns a label to whatever mathematical object I am studying that is well-defined up to the appropriate equivalence. For example, the diameter of a metric space is an invariant of metric isometries, since two isometric spaces must have the same diameter. Topological spaces can be assigned homology or homotopy groups. Polyhedra can be assigned their volume. These labels, which may be numbers, groups, or more sophisticated objects, are tremendously useful for distinguishing objects but are rarely exact. Two metric spaces with the same diameter can be very, very far from isometric, for example. A complete invariant is one where the assignment is injective, so that if two objects get assigned the same label they are in fact isomorphic. Of course, complete invariants are very rare. Complicated mathematical structures can almost never be uniquely identified by a simple statistic, even if that statistic is something complex like a group. The problem is, in a sense, that the objects are too flexible to be tied down by any one single restriction. Fix the volume of a polyhedron and you can still wiggle all the angles about, if you do so carefully. What Mostow asserted was that the class of closed, hyperbolic manifolds of dimension n 3 are very rigid, so that there exists a very simple complete invariant. That invariant is, in fact, the fundamental group! Consider how bizarre and incredible this is: if you only tell me what the fundamental group of my manifold looks like I can determine not only its entire topological structure, but even the geometric structure! Underlying every hyperbolic manifold is a simple frame consisting of a set of based loops, and that structure can be completed in a unique way. There are, actually, other excellent invariants for these manifolds: their volume! This has the advantage of being a single number but is not quite complete, although it does determine the hyperbolic manifold up to a finite list. The proof of that is another matter entirely and not the focus of these notes. 1
2 1 Boundaries at infinity Virtually every proof of Mostow s rigidity goes through the same first step, which is outlined in this section. Take M and N to be closed, hyperbolic manifolds of dimension n 3. Their universal covers are the contractible hyperbolic space H n, so these spaces are K(π, 1) s. What that means is that all their higher homotopy groups vanish. K(π, 1) spaces are uniquely determined up to homotopy equivalence, so we can find a pair of maps f : M N and g : N M that are inverses up to homotopy. If we are careful, f and g can be taken C 1. If we lift these maps to the universal covers equivariantly they remain Lipschitz, and are in fact quasi-isometries. Hyperbolic space H n has a boundary at infinity homemorphic to the sphere S n 1. This boundary at infinity can be thought of the space of geodesics escaping H n up to the fellow traveling equivalence. Since quasiisometries send geodesics to quasi-geodesics that fellow travel a unique geodesic, we can in fact extend our maps to the boundary of hyperbolic space. By abuse of notion I refer to these maps as f and g also. Some tinkering with hyperbolic geometry reveals that these extensions are continuous, and in fact the assignment that takes a map on hyperbolic space to the associated boundary map is injective. What is then the strategy of the proof? We wish to show that the map on the boundary must be conformal. If n 3, the conformal maps on S n 1 all extend isometries on H n, and hence, up to isometry, the map on the boundary is the identity, so the map on the interior is the identity, and thus f and g are isometries. This fails for n = 2 because any diffeomorphism of the circle is conformal, and virtually none of these extend to the interior as isometries. 2
3 2 Gromov s Proof Gromov s strategy is the following. To show that the boundary map is, up to isometry, the identity, we take a collection of points on the boundary which are the vertex of an ideal simplex of maximal volume. Gromov then claims that these must also be sent to the vertices of an ideal simplex of maximal volume. Up to isometry, we can assume that these points are fixed. Next, reflect one of these vertices through the face determined by the others and consider where this new vertex is sent. This new vertex, together with the vertices of that face, give a new simplex which also has maximal volume (because it is regular, a separate result). This must again be sent to the vertices of an ideal simplex of maximal volume, that is the vertices of an ideal regular simplex, but because we have fixed all but one of the vertices and because the boundary map is injective the newest vertex must also be fixed. Repeating this we see that a whole mesh of the boundary is fixed, and if we take reflections carefully we can show that a smaller mesh is fixed, and so on, so that a dense mesh if fixed. By continuity, the boundary map is the identity. So how do we justify this claim that the vertices of an ideal simplex of maximal volume must be sent to the vertices of an ideal simplex of maximal volume? Put another way, how does Gromov make arguments about volume simply from the topological information that the maps f and g are homotopy equivalences? He uses a norm on homology that encodes volumetric information. 2.1 The Gromov Norm For any topological space X let C (X) be the real singular chain complex. Any k-chain can be written uniquely as c = i a i σ i We can endow our chain complex with the l 1 norm by setting c = i a i. This descends to homology as a seminorm by taking the infimum of the norms in the homology class. Note that if f : X Y is a continuous map and z is a homology class that 3
4 f (z) z Because the induced map on homology can collapse certain simplices on to each other. Now, as M and N are orientable they have fundamental classes [M] and [N]. The quantity [M] is called the Gromov norm of the manifold, or the simplical volume. In general, the Gromov norm of a manifold may be zero, but the Gromov norm of hyperbolic manifolds will be positive, and in fact related to their volume. Before we go foreward we will define an operation on simplices that normalizes them. Given a simplex σ in M we lift it to the universal cover (thought of as the hyperboloid model in R n+1 ), build the affine simplex in the ambient Euclidean space, project it back down to the hyperboloid, and then let that descend to M. This is called the straightening of σ, written str(σ) and it induces a well-defined map on homology. Let v n be the supremum of the volumes of straight n-simplices in H n. We claim that this is in fact bounded. To be precise, for n 2, v n π (n 1)! For n = 2 this is Gauss-Bonnet. In higher dimensions one proves that v n v n 1. This is derived via an explicit calculation and some convenient n 1 normalizations. The next stop in the proof is to justify the claim that the Gromov norm is related to the volume. The exact relation is M = Vol(M) v n That is, if we scale the simplical norm by the maximum possible norm for straight simplices we get exactly the volume of M. This is quite the incredible claim, and the proof is a little involved. It turns out that it is convenient to replace simplical homology with an isomorphic, measure-theoretic homology theory. In this new theory, the k-chains are signed Borel measures on C 1 ( k, M) with bounded total variation and compact support. We denote these chains by C k (M). 4
5 What is the boundary map on these chains? Well, on ordinary chains we can take projection onto the ith face. This induces a pushforward on the space of measures, where the ith projection of a measure on k-chain takes a collection of (k 1)-chains, finds the collection of k which i-project to it, and measures those. The alternating sum of these projections gives the differential, and it is not hard to see that we get a chain complex, i.e. the composition of differentials is zero. There is a natural chain map from our original chain complex to this new one. It sends a chain σ to the Dirac measure on σ. This descends to homology and one can check that this gives an isomorphism. The usual DeRham pairing of k-chains and k-forms extends to a pairing with the k- chains in C k (M), via ( ) (µ, α) τdα dµ(τ) τ C 1 ( k,m) k Put in works, integrate the form over all possible k-chains, summing the contributions via the weighting of the measure. We are now ready to prove that vol(m) v n M. The volume of M can be obtained by pairing the fundamental class [M] with the volume form Ω M. Take a generator for the fundamental class composed only of small, straight simplices. Via the isomorphism between our homology theories we can replace the fundamental class with a fundamental measure µ. Now, the straightening operation is also defined on measures, simply by stating that the straightened measure of a simplex is the measure of the straightened simplex. Since we took the fundamental class to be straight we also have that µ is straight, so it only puts mass on straight simplices. Thus we just end up integrating the form against straight simplices. Let τ be such a simplex. The pullback τ Ω M is equal to τ Ω H n for τ a lift of τ. Each of those integrals can have measure at most v n by definition, so a simple ML-estimate tells us that the whole integral is bounded by v n times the mass of µ. Taking the infimum over representaties of [M], one obtains vol(m) v n M. This works because every representative for [M] can be straightened (and this just shrinks the norm). This proves that the norm of M, which can be obtained by integrating a volume form over a fundamental class (taken here to be straight), is bounded by the size of that fundamental 5
6 class times the largest possible volume of each of its pieces. Now to show that the vol(m) v n M, we will explicitly construct a cycle whose norm is Vol(M)/v n. First, recall that Isom + (H n ) is a unimodular Lie group. If we let Γ = π 1 (M) and denote the quotient Isom + (H n )/Γ by G, then since Γ was discrete G is again unimodular, with a Haar measure h G. We can normalize this measure to have norm Vol(M). Fix σ a simplex in H n. We get a map ψ σ from G into C 1 ( n, M) given by hitting σ with an element of G and projecting it down to M. This defines a map smear(σ) from C 1 ( n, H n ) into C n (M) which sends σ to the pushforward measure ψ σ (h G ). This measure gives all of its support to projected G-shifts of σ, and given a collection of such simplices assigns them value that h M would assign the associated shifts in G. The smear map has a number of nice properties. For one thing, σ and gσ have the same smear. If we take the ith face of σ and smear that it is equivalent to smearing σ and taking the ith face of the resulting measure. The norm of smear(σ) is just the volume of M because of our normalization. Lastly, pairing smear(σ) with the volume form Ω M gives Vol(M) Vol(σ). To see this last fact, observe that this pairing only weights the integrals of the volume form over projected G-translated of σ, all of which are equal to n σ Ω H n = Vol(σ) We d like to take this smeared measure as our witness for a cycle that hits the floor of Vol(M)/v n provided by our earlier bound. The only problem is that it is not a cycle. However, if we take σ and reflect it through one of its faces to get σ, and then write ζ(σ) = 1 2 (smear(σ) smear(σ )) then this is indeed a cycle, but it is non trivial. Moreover, this measure has norm equal to Vol(M). Since ζ(σ) is a top-dimensional measure it must be equal to [M] for some λ. We already know that Vol(σ) Vol(M) = ζ(σ), Ω M = λ [M], Ω M = λ Vol(M) Thus λ = Vol(σ). We also know that 6
7 Vol(M) = ζ(σ) Vol(σ) M Taking the supremum over all straight simplices gives Vol(M) v n M. Put into words, we built a measure whose norm is exactly the volume of M and which represents the class of Vol(σ)[M]. 3 The Proof So far we ve seen that the Gromov norm of the fundamental class is exactly proportional to the volume. What remains to be shown is our claim about sending vertices of ideal maximal volume simplices to vertices of ideal maximal volume simplices. Take v 0,, v n to be the vertices of an ideal simplex of maximal volume σ. Suppose that the volume of the straightening of h(σ) is not maximal. Then I could wiggle around the v i within open sets U i H n and keep the volume of the straightened image bounded away from v n by 2ɛ, for some ɛ > 0. Within each U i one can take V i, open subsets, such that the set A(G) = g Isom + (H n ) : v i V i gv i U i } has positive measure m A > 0. However, for any δ > 0 we can find some wiggled simplex σ 0. with volume at least v n δ. Let s smear this to obtain a measure on the n-chains of M, push it forward to N and straighten it there. What do we get if we pair this with Ω N? By definition we integrate over all n-chains in N, pairing each chain with the volume form. By definition of the pushforward measure this is just the integral over G with its Haar measure of the volume of the pullback of Ω N through the f-pushforward of gσ 0. Now this integral splits into two pieces. On A(G), which has measure m A, this is the integral of the volume form of H n over the straightened fgσ 0, which by definition is less than v n 2ɛ, and certainly less than v n 2ɛ + δ. When integrating over the rest of G the integral of the volume form over the pushforward is no bigger than v n which is less than Vol(σ 0 ) + δ. This is, all in all m A (Vol(σ 0 ) 2ɛ + δ) + (Vol(M) m A )(Vol(σ 0 ) + δ) 7
8 which simplifies to Vol(M)(Vol(σ 0 ) + δ) 2m A ɛ If we take δ < (ɛm A )/ Vol(M) then this is < Vol(M) Vol(σ 0 ) ɛm A However, we know f : M N is a homotopy equivalence, sending [M] to [N], so M and N must have the same volume. Since ζ(σ 0 ) represents Vol(σ 0 )[M] its straightened pushforward must represent Vol(σ 0 )[N]. However, the above calculation suggests that the pushforward must represent λ[n] with λ < Vol(σ 0 ) ɛm A Vol(M) This gives a contradiction. To summarize this argument, since f is a homotopy equivalence it must send fundamental classes to fundamental classes, preserving volume, sending representatives of λ[m] to representatives of λ[n]. But if h does not preserve collections of maximal volume vertices on the boundary one can find a simplex whose smear represents λ[m] and whose image represents ρ[n] for some ρ < λ. 8
Volume and topology. Marc Culler. June 7, 2009
Volume and topology Marc Culler June 7, 2009 Slogan: Volume is a topological invariant of hyperbolic 3-manifolds. References: W. Thurston, The Geometry and Topology of 3-Manifolds, http://www.msri.org/publications/books/gt3m
More informationMostow Rigidity. W. Dison June 17, (a) semi-simple Lie groups with trivial centre and no compact factors and
Mostow Rigidity W. Dison June 17, 2005 0 Introduction Lie Groups and Symmetric Spaces We will be concerned with (a) semi-simple Lie groups with trivial centre and no compact factors and (b) simply connected,
More informationThe Hurewicz Theorem
The Hurewicz Theorem April 5, 011 1 Introduction The fundamental group and homology groups both give extremely useful information, particularly about path-connected spaces. Both can be considered as functors,
More informationThe Skinning Map. 1 Motivation
The Skinning Map The skinning map σ is a particular self-map of the Teichmüller space T (Σ). It arises in the theory of geometrization of 3-manifolds. Let us start by understanding the geometrization of
More informationRigidity result for certain 3-dimensional singular spaces and their fundamental groups.
Rigidity result for certain 3-dimensional singular spaces and their fundamental groups. Jean-Francois Lafont May 5, 2004 Abstract In this paper, we introduce a particularly nice family of CAT ( 1) spaces,
More informationEILENBERG-ZILBER VIA ACYCLIC MODELS, AND PRODUCTS IN HOMOLOGY AND COHOMOLOGY
EILENBERG-ZILBER VIA ACYCLIC MODELS, AND PRODUCTS IN HOMOLOGY AND COHOMOLOGY CHRIS KOTTKE 1. The Eilenberg-Zilber Theorem 1.1. Tensor products of chain complexes. Let C and D be chain complexes. We define
More informationEXPLICIT l 1 -EFFICIENT CYCLES AND AMENABLE NORMAL SUBGROUPS
EXPLICIT l 1 -EFFICIENT CYCLES AND AMENABLE NORMAL SUBGROUPS CLARA LÖH Abstract. By Gromov s mapping theorem for bounded cohomology, the projection of a group to the quotient by an amenable normal subgroup
More informationDEHN SURGERY AND NEGATIVELY CURVED 3-MANIFOLDS
DEHN SURGERY AND NEGATIVELY CURVED 3-MANIFOLDS DARYL COOPER AND MARC LACKENBY 1. INTRODUCTION Dehn surgery is perhaps the most common way of constructing 3-manifolds, and yet there remain some profound
More informationAlgebraic Topology Homework 4 Solutions
Algebraic Topology Homework 4 Solutions Here are a few solutions to some of the trickier problems... Recall: Let X be a topological space, A X a subspace of X. Suppose f, g : X X are maps restricting to
More information3-manifolds and their groups
3-manifolds and their groups Dale Rolfsen University of British Columbia Marseille, September 2010 Dale Rolfsen (2010) 3-manifolds and their groups Marseille, September 2010 1 / 31 3-manifolds and their
More informationOPEN PROBLEMS IN NON-NEGATIVE SECTIONAL CURVATURE
OPEN PROBLEMS IN NON-NEGATIVE SECTIONAL CURVATURE COMPILED BY M. KERIN Abstract. We compile a list of the open problems and questions which arose during the Workshop on Manifolds with Non-negative Sectional
More informationfor some n i (possibly infinite).
Homology with coefficients: The chain complexes that we have dealt with so far have had elements which are Z-linear combinations of basis elements (which are themselves singular simplices or equivalence
More informationNote: all spaces are assumed to be path connected and locally path connected.
Homework 2 Note: all spaces are assumed to be path connected and locally path connected. 1: Let X be the figure 8 space. Precisely define a space X and a map p : X X which is the universal cover. Check
More informationAn introduction to arithmetic groups. Lizhen Ji CMS, Zhejiang University Hangzhou , China & Dept of Math, Univ of Michigan Ann Arbor, MI 48109
An introduction to arithmetic groups Lizhen Ji CMS, Zhejiang University Hangzhou 310027, China & Dept of Math, Univ of Michigan Ann Arbor, MI 48109 June 27, 2006 Plan. 1. Examples of arithmetic groups
More informationBredon, Introduction to compact transformation groups, Academic Press
1 Introduction Outline Section 3: Topology of 2-orbifolds: Compact group actions Compact group actions Orbit spaces. Tubes and slices. Path-lifting, covering homotopy Locally smooth actions Smooth actions
More informationCALCULUS ON MANIFOLDS. 1. Riemannian manifolds Recall that for any smooth manifold M, dim M = n, the union T M =
CALCULUS ON MANIFOLDS 1. Riemannian manifolds Recall that for any smooth manifold M, dim M = n, the union T M = a M T am, called the tangent bundle, is itself a smooth manifold, dim T M = 2n. Example 1.
More informationWilliam P. Thurston. The Geometry and Topology of Three-Manifolds
William P. Thurston The Geometry and Topology of Three-Manifolds Electronic version 1.1 - March 00 http://www.msri.org/publications/books/gt3m/ This is an electronic edition of the 1980 notes distributed
More informationHidden symmetries and arithmetic manifolds
Hidden symmetries and arithmetic manifolds Benson Farb and Shmuel Weinberger Dedicated to the memory of Robert Brooks May 9, 2004 1 Introduction Let M be a closed, locally symmetric Riemannian manifold
More informationSECTION 5: EILENBERG ZILBER EQUIVALENCES AND THE KÜNNETH THEOREMS
SECTION 5: EILENBERG ZILBER EQUIVALENCES AND THE KÜNNETH THEOREMS In this section we will prove the Künneth theorem which in principle allows us to calculate the (co)homology of product spaces as soon
More informationPart II. Algebraic Topology. Year
Part II Year 2017 2016 2015 2014 2013 2012 2011 2010 2009 2008 2007 2006 2005 2017 Paper 3, Section II 18I The n-torus is the product of n circles: 5 T n = } S 1. {{.. S } 1. n times For all n 1 and 0
More informationδ-hyperbolic SPACES SIDDHARTHA GADGIL
δ-hyperbolic SPACES SIDDHARTHA GADGIL Abstract. These are notes for the Chennai TMGT conference on δ-hyperbolic spaces corresponding to chapter III.H in the book of Bridson and Haefliger. When viewed from
More informationRICCI CURVATURE: SOME RECENT PROGRESS A PROGRESS AND OPEN QUESTIONS
RICCI CURVATURE: SOME RECENT PROGRESS AND OPEN QUESTIONS Courant Institute September 9, 2016 1 / 27 Introduction. We will survey some recent (and less recent) progress on Ricci curvature and mention some
More informationA Dual Interpretation of the Gromov-Thurston Proof of Mostow Rigidity and Volume Rigidity for Representations of Hyperbolic Lattices
A Dual Interpretation of the Gromov-Thurston Proof of Mostow Rigidity and Volume Rigidity for Representations of Hyperbolic Lattices M. Bucher, M. Burger and A. Iozzi REPORT No. 19, 2011/2012, spring ISSN
More informationTHE JORDAN-BROUWER SEPARATION THEOREM
THE JORDAN-BROUWER SEPARATION THEOREM WOLFGANG SCHMALTZ Abstract. The Classical Jordan Curve Theorem says that every simple closed curve in R 2 divides the plane into two pieces, an inside and an outside
More informationBOUNDARY AND LENS RIGIDITY OF FINITE QUOTIENTS
BOUNDARY AND LENS RIGIDITY OF FINITE QUOTIENTS CHRISTOPHER CROKE + Abstract. We consider compact Riemannian manifolds (M, M, g) with boundary M and metric g on which a finite group Γ acts freely. We determine
More informationMA4H4 - GEOMETRIC GROUP THEORY. Contents of the Lectures
MA4H4 - GEOMETRIC GROUP THEORY Contents of the Lectures 1. Week 1 Introduction, free groups, ping-pong, fundamental group and covering spaces. Lecture 1 - Jan. 6 (1) Introduction (2) List of topics: basics,
More informationSPHERES AND PROJECTIONS FOR Out(F n )
SPHERES AND PROJECTIONS FOR Out(F n ) URSULA HAMENSTÄDT AND SEBASTIAN HENSEL Abstract. The outer automorphism group Out(F 2g ) of a free group on 2g generators naturally contains the mapping class group
More informationPart II. Geometry and Groups. Year
Part II Year 2014 2013 2012 2011 2010 2009 2008 2007 2006 2005 2014 Paper 4, Section I 3F 49 Define the limit set Λ(G) of a Kleinian group G. Assuming that G has no finite orbit in H 3 S 2, and that Λ(G),
More informationEilenberg-Steenrod properties. (Hatcher, 2.1, 2.3, 3.1; Conlon, 2.6, 8.1, )
II.3 : Eilenberg-Steenrod properties (Hatcher, 2.1, 2.3, 3.1; Conlon, 2.6, 8.1, 8.3 8.5 Definition. Let U be an open subset of R n for some n. The de Rham cohomology groups (U are the cohomology groups
More informationVolume preserving surgeries on hyperbolic 3-manifolds
Volume preserving surgeries on hyperbolic 3-manifolds Peter Rudzis Advisor: Dr. Rolland Trapp August 19, 2016 Abstract In this paper, we investigate two types of surgeries, the parallel surgery and the
More informationLebesgue Measure on R n
CHAPTER 2 Lebesgue Measure on R n Our goal is to construct a notion of the volume, or Lebesgue measure, of rather general subsets of R n that reduces to the usual volume of elementary geometrical sets
More informationA NOTE ON SPACES OF ASYMPTOTIC DIMENSION ONE
A NOTE ON SPACES OF ASYMPTOTIC DIMENSION ONE KOJI FUJIWARA AND KEVIN WHYTE Abstract. Let X be a geodesic metric space with H 1(X) uniformly generated. If X has asymptotic dimension one then X is quasi-isometric
More informationKleinian groups Background
Kleinian groups Background Frederic Palesi Abstract We introduce basic notions about convex-cocompact actions on real rank one symmetric spaces. We focus mainly on the geometric interpretation as isometries
More informationPatrick Iglesias-Zemmour
Mathematical Surveys and Monographs Volume 185 Diffeology Patrick Iglesias-Zemmour American Mathematical Society Contents Preface xvii Chapter 1. Diffeology and Diffeological Spaces 1 Linguistic Preliminaries
More informationLecture 11 Hyperbolicity.
Lecture 11 Hyperbolicity. 1 C 0 linearization near a hyperbolic point 2 invariant manifolds Hyperbolic linear maps. Let E be a Banach space. A linear map A : E E is called hyperbolic if we can find closed
More informationBending deformation of quasi-fuchsian groups
Bending deformation of quasi-fuchsian groups Yuichi Kabaya (Osaka University) Meiji University, 30 Nov 2013 1 Outline The shape of the set of discrete faithful representations in the character variety
More informationLECTURE Itineraries We start with a simple example of a dynamical system obtained by iterating the quadratic polynomial
LECTURE. Itineraries We start with a simple example of a dynamical system obtained by iterating the quadratic polynomial f λ : R R x λx( x), where λ [, 4). Starting with the critical point x 0 := /2, we
More informationRigid/Flexible Dynamics Problems and Exercises Math 275 Harvard University Fall 2006 C. McMullen
Rigid/Flexible Dynamics Problems and Exercises Math 275 Harvard University Fall 2006 C. McMullen 1. Give necessary and sufficent conditions on T = ( a b c d) GL2 (C) such that T is in U(1,1), i.e. such
More informationMetric spaces and metrizability
1 Motivation Metric spaces and metrizability By this point in the course, this section should not need much in the way of motivation. From the very beginning, we have talked about R n usual and how relatively
More informationExercise: Consider the poset of subsets of {0, 1, 2} ordered under inclusion: Date: July 15, 2015.
07-13-2015 Contents 1. Dimension 1 2. The Mayer-Vietoris Sequence 3 2.1. Suspension and Spheres 4 2.2. Direct Sums 4 2.3. Constuction of the Mayer-Vietoris Sequence 6 2.4. A Sample Calculation 7 As we
More informationGroups up to quasi-isometry
OSU November 29, 2007 1 Introduction 2 3 Topological methods in group theory via the fundamental group. group theory topology group Γ, a topological space X with π 1 (X) = Γ. Γ acts on the universal cover
More informationAN INTEGRAL FORMULA FOR TRIPLE LINKING IN HYPERBOLIC SPACE
AN INTEGRAL FORMULA FOR TRIPLE LINKING IN HYPERBOLIC SPACE PAUL GALLAGHER AND TIANYOU ZHOU Abstract. We provide a geometrically natural formula for the triple linking number of 3 pairwise unlinked curves
More informationON NEARLY SEMIFREE CIRCLE ACTIONS
ON NEARLY SEMIFREE CIRCLE ACTIONS DUSA MCDUFF AND SUSAN TOLMAN Abstract. Recall that an effective circle action is semifree if the stabilizer subgroup of each point is connected. We show that if (M, ω)
More informationAnalysis II: The Implicit and Inverse Function Theorems
Analysis II: The Implicit and Inverse Function Theorems Jesse Ratzkin November 17, 2009 Let f : R n R m be C 1. When is the zero set Z = {x R n : f(x) = 0} the graph of another function? When is Z nicely
More informationThree Descriptions of the Cohomology of Bun G (X) (Lecture 4)
Three Descriptions of the Cohomology of Bun G (X) (Lecture 4) February 5, 2014 Let k be an algebraically closed field, let X be a algebraic curve over k (always assumed to be smooth and complete), and
More informationLarge scale conformal geometry
July 24th, 2018 Goal: perform conformal geometry on discrete groups. Goal: perform conformal geometry on discrete groups. Definition X, X metric spaces. Map f : X X is a coarse embedding if where α +,
More informationCW-complexes. Stephen A. Mitchell. November 1997
CW-complexes Stephen A. Mitchell November 1997 A CW-complex is first of all a Hausdorff space X equipped with a collection of characteristic maps φ n α : D n X. Here n ranges over the nonnegative integers,
More informationTEICHMÜLLER SPACE MATTHEW WOOLF
TEICHMÜLLER SPACE MATTHEW WOOLF Abstract. It is a well-known fact that every Riemann surface with negative Euler characteristic admits a hyperbolic metric. But this metric is by no means unique indeed,
More informationSolution: We can cut the 2-simplex in two, perform the identification and then stitch it back up. The best way to see this is with the picture:
Samuel Lee Algebraic Topology Homework #6 May 11, 2016 Problem 1: ( 2.1: #1). What familiar space is the quotient -complex of a 2-simplex [v 0, v 1, v 2 ] obtained by identifying the edges [v 0, v 1 ]
More informationfy (X(g)) Y (f)x(g) gy (X(f)) Y (g)x(f)) = fx(y (g)) + gx(y (f)) fy (X(g)) gy (X(f))
1. Basic algebra of vector fields Let V be a finite dimensional vector space over R. Recall that V = {L : V R} is defined to be the set of all linear maps to R. V is isomorphic to V, but there is no canonical
More informationLecture 4: Knot Complements
Lecture 4: Knot Complements Notes by Zach Haney January 26, 2016 1 Introduction Here we discuss properties of the knot complement, S 3 \ K, for a knot K. Definition 1.1. A tubular neighborhood V k S 3
More informationTopology Hmwk 6 All problems are from Allen Hatcher Algebraic Topology (online) ch 2
Topology Hmwk 6 All problems are from Allen Hatcher Algebraic Topology (online) ch 2 Andrew Ma August 25, 214 2.1.4 Proof. Please refer to the attached picture. We have the following chain complex δ 3
More informationCOUNTEREXAMPLES TO THE COARSE BAUM-CONNES CONJECTURE. Nigel Higson. Unpublished Note, 1999
COUNTEREXAMPLES TO THE COARSE BAUM-CONNES CONJECTURE Nigel Higson Unpublished Note, 1999 1. Introduction Let X be a discrete, bounded geometry metric space. 1 Associated to X is a C -algebra C (X) which
More informationHEAT FLOWS ON HYPERBOLIC SPACES
HEAT FLOWS ON HYPERBOLIC SPACES MARIUS LEMM AND VLADIMIR MARKOVIC Abstract. In this paper we develop new methods for studying the convergence problem for the heat flow on negatively curved spaces and prove
More information6. Constructing hyperbolic manifolds
6. Constructing hyperbolic manifolds Recall that a hyperbolic manifold is a (G, X)-manifold, where X is H n and G = Isom(H n ). Since G is a group of Riemannian isometries, G acts rigidly. Hence, the pseudogroup
More informationContents. 1. Introduction
DIASTOLIC INEQUALITIES AND ISOPERIMETRIC INEQUALITIES ON SURFACES FLORENT BALACHEFF AND STÉPHANE SABOURAU Abstract. We prove a new type of universal inequality between the diastole, defined using a minimax
More informationOn the exponential map on Riemannian polyhedra by Monica Alice Aprodu. Abstract
Bull. Math. Soc. Sci. Math. Roumanie Tome 60 (108) No. 3, 2017, 233 238 On the exponential map on Riemannian polyhedra by Monica Alice Aprodu Abstract We prove that Riemannian polyhedra admit explicit
More informationNOTES FOR MATH 5520, SPRING Outline
NOTES FOR MATH 5520, SPRING 2011 DOMINGO TOLEDO 1. Outline This will be a course on the topology and geometry of surfaces. This is a continuation of Math 4510, and we will often refer to the notes for
More informationThe Eells-Salamon twistor correspondence
The Eells-Salamon twistor correspondence Jonny Evans May 15, 2012 Jonny Evans () The Eells-Salamon twistor correspondence May 15, 2012 1 / 11 The Eells-Salamon twistor correspondence is a dictionary for
More informationConsequences of Continuity
Consequences of Continuity James K. Peterson Department of Biological Sciences and Department of Mathematical Sciences Clemson University October 4, 2017 Outline 1 Domains of Continuous Functions 2 The
More informationAn introduction to cobordism
An introduction to cobordism Martin Vito Cruz 30 April 2004 1 Introduction Cobordism theory is the study of manifolds modulo the cobordism relation: two manifolds are considered the same if their disjoint
More informationIntroduction to Teichmüller Spaces
Introduction to Teichmüller Spaces Jing Tao Notes by Serena Yuan 1. Riemann Surfaces Definition 1.1. A conformal structure is an atlas on a manifold such that the differentials of the transition maps lie
More informationThe Symmetric Space for SL n (R)
The Symmetric Space for SL n (R) Rich Schwartz November 27, 2013 The purpose of these notes is to discuss the symmetric space X on which SL n (R) acts. Here, as usual, SL n (R) denotes the group of n n
More informationCommensurability between once-punctured torus groups and once-punctured Klein bottle groups
Hiroshima Math. J. 00 (0000), 1 34 Commensurability between once-punctured torus groups and once-punctured Klein bottle groups Mikio Furokawa (Received Xxx 00, 0000) Abstract. The once-punctured torus
More information121B: ALGEBRAIC TOPOLOGY. Contents. 6. Poincaré Duality
121B: ALGEBRAIC TOPOLOGY Contents 6. Poincaré Duality 1 6.1. Manifolds 2 6.2. Orientation 3 6.3. Orientation sheaf 9 6.4. Cap product 11 6.5. Proof for good coverings 15 6.6. Direct limit 18 6.7. Proof
More informationMostow s Rigidity Theorem
Department of Mathematics and Statistics The Univeristy of Melbourne Mostow s Rigidity Theorem James Saunderson Supervisor: A/Prof. Craig Hodgson November 7, 2008 Abstract Mostow s Rigidity Theorem is
More informationApril 13, We now extend the structure of the horseshoe to more general kinds of invariant. x (v) λ n v.
April 3, 005 - Hyperbolic Sets We now extend the structure of the horseshoe to more general kinds of invariant sets. Let r, and let f D r (M) where M is a Riemannian manifold. A compact f invariant set
More informationarxiv: v4 [math.dg] 18 Jun 2015
SMOOTHING 3-DIMENSIONAL POLYHEDRAL SPACES NINA LEBEDEVA, VLADIMIR MATVEEV, ANTON PETRUNIN, AND VSEVOLOD SHEVCHISHIN arxiv:1411.0307v4 [math.dg] 18 Jun 2015 Abstract. We show that 3-dimensional polyhedral
More informationINTEGRAL FOLIATED SIMPLICIAL VOLUME OF ASPHERICAL MANIFOLDS
INTEGRAL FOLIATED SIMPLICIAL VOLUME OF ASPHERICAL MANIFOLDS ROBERTO FRIGERIO, CLARA LÖH, CRISTINA PAGLIANTINI, AND ROMAN SAUER Abstract. Simplicial volumes measure the complexity of fundamental cycles
More informationLECTURE 15: COMPLETENESS AND CONVEXITY
LECTURE 15: COMPLETENESS AND CONVEXITY 1. The Hopf-Rinow Theorem Recall that a Riemannian manifold (M, g) is called geodesically complete if the maximal defining interval of any geodesic is R. On the other
More informationDIFFERENTIAL FORMS, SPINORS AND BOUNDED CURVATURE COLLAPSE. John Lott. University of Michigan. November, 2000
DIFFERENTIAL FORMS, SPINORS AND BOUNDED CURVATURE COLLAPSE John Lott University of Michigan November, 2000 From preprints Collapsing and the Differential Form Laplacian On the Spectrum of a Finite-Volume
More informationCOARSE HYPERBOLICITY AND CLOSED ORBITS FOR QUASIGEODESIC FLOWS
COARSE HYPERBOLICITY AND CLOSED ORBITS FOR QUASIGEODESIC FLOWS STEVEN FRANKEL Abstract. We prove Calegari s conjecture that every quasigeodesic flow on a closed hyperbolic 3-manifold has closed orbits.
More informationA FUCHSIAN GROUP PROOF OF THE HYPERELLIPTICITY OF RIEMANN SURFACES OF GENUS 2
Annales Academiæ Scientiarum Fennicæ Mathematica Volumen 28, 2003, 69 74 A FUCHSIAN GROUP PROOF OF THE HYPERELLIPTICITY OF RIEMANN SURFACES OF GENUS 2 Yolanda Fuertes and Gabino González-Diez Universidad
More informationABSOLUTE CONTINUITY OF FOLIATIONS
ABSOLUTE CONTINUITY OF FOLIATIONS C. PUGH, M. VIANA, A. WILKINSON 1. Introduction In what follows, U is an open neighborhood in a compact Riemannian manifold M, and F is a local foliation of U. By this
More informationHYPERBOLIC DYNAMICAL SYSTEMS AND THE NONCOMMUTATIVE INTEGRATION THEORY OF CONNES ABSTRACT
HYPERBOLIC DYNAMICAL SYSTEMS AND THE NONCOMMUTATIVE INTEGRATION THEORY OF CONNES Jan Segert ABSTRACT We examine hyperbolic differentiable dynamical systems in the context of Connes noncommutative integration
More informationThe Geometrization Theorem
The Geometrization Theorem Matthew D. Brown Wednesday, December 19, 2012 In this paper, we discuss the Geometrization Theorem, formerly Thurston s Geometrization Conjecture, which is essentially the statement
More informationMATH 215B. SOLUTIONS TO HOMEWORK (6 marks) Construct a path connected space X such that π 1 (X, x 0 ) = D 4, the dihedral group with 8 elements.
MATH 215B. SOLUTIONS TO HOMEWORK 2 1. (6 marks) Construct a path connected space X such that π 1 (X, x 0 ) = D 4, the dihedral group with 8 elements. Solution A presentation of D 4 is a, b a 4 = b 2 =
More informationDiffeomorphism Groups of Reducible 3-Manifolds. Allen Hatcher
Diffeomorphism Groups of Reducible 3-Manifolds Allen Hatcher In a 1979 announcement by César de Sá and Rourke [CR] there is a sketch of an intuitively appealing approach to measuring the difference between
More informationIntroduction to Poincare Conjecture and the Hamilton-Perelman program
Introduction to Poincare Conjecture and the Hamilton-Perelman program David Glickenstein Math 538, Spring 2009 January 20, 2009 1 Introduction This lecture is mostly taken from Tao s lecture 2. In this
More informationHyperbolic Geometry on Geometric Surfaces
Mathematics Seminar, 15 September 2010 Outline Introduction Hyperbolic geometry Abstract surfaces The hemisphere model as a geometric surface The Poincaré disk model as a geometric surface Conclusion Introduction
More informationBlack Holes and Thermodynamics I: Classical Black Holes
Black Holes and Thermodynamics I: Classical Black Holes Robert M. Wald General references: R.M. Wald General Relativity University of Chicago Press (Chicago, 1984); R.M. Wald Living Rev. Rel. 4, 6 (2001).
More informationResults from MathSciNet: Mathematical Reviews on the Web c Copyright American Mathematical Society 2000
2000k:53038 53C23 20F65 53C70 57M07 Bridson, Martin R. (4-OX); Haefliger, André (CH-GENV-SM) Metric spaces of non-positive curvature. Grundlehren der Mathematischen Wissenschaften [Fundamental Principles
More informationLecture 9 Metric spaces. The contraction fixed point theorem. The implicit function theorem. The existence of solutions to differenti. equations.
Lecture 9 Metric spaces. The contraction fixed point theorem. The implicit function theorem. The existence of solutions to differential equations. 1 Metric spaces 2 Completeness and completion. 3 The contraction
More informationEquivalence of the Combinatorial Definition (Lecture 11)
Equivalence of the Combinatorial Definition (Lecture 11) September 26, 2014 Our goal in this lecture is to complete the proof of our first main theorem by proving the following: Theorem 1. The map of simplicial
More informationDiscrete groups and the thick thin decomposition
CHAPTER 5 Discrete groups and the thick thin decomposition Suppose we have a complete hyperbolic structure on an orientable 3-manifold. Then the developing map D : M H 3 is a covering map, by theorem 3.19.
More informationMoment Measures. Bo az Klartag. Tel Aviv University. Talk at the asymptotic geometric analysis seminar. Tel Aviv, May 2013
Tel Aviv University Talk at the asymptotic geometric analysis seminar Tel Aviv, May 2013 Joint work with Dario Cordero-Erausquin. A bijection We present a correspondence between convex functions and Borel
More informationManifolds with complete metrics of positive scalar curvature
Manifolds with complete metrics of positive scalar curvature Shmuel Weinberger Joint work with Stanley Chang and Guoliang Yu May 5 14, 2008 Classical background. Fact If S p is the scalar curvature at
More information612 CLASS LECTURE: HYPERBOLIC GEOMETRY
612 CLASS LECTURE: HYPERBOLIC GEOMETRY JOSHUA P. BOWMAN 1. Conformal metrics As a vector space, C has a canonical norm, the same as the standard R 2 norm. Denote this dz one should think of dz as the identity
More informationA Crash Course in Topological Groups
A Crash Course in Topological Groups Iian B. Smythe Department of Mathematics Cornell University Olivetti Club November 8, 2011 Iian B. Smythe (Cornell) Topological Groups Nov. 8, 2011 1 / 28 Outline 1
More information0. Introduction 1 0. INTRODUCTION
0. Introduction 1 0. INTRODUCTION In a very rough sketch we explain what algebraic geometry is about and what it can be used for. We stress the many correlations with other fields of research, such as
More informationDecomposition Methods for Representations of Quivers appearing in Topological Data Analysis
Decomposition Methods for Representations of Quivers appearing in Topological Data Analysis Erik Lindell elindel@kth.se SA114X Degree Project in Engineering Physics, First Level Supervisor: Wojtek Chacholski
More information(df (ξ ))( v ) = v F : O ξ R. with F : O ξ O
Math 396. Derivative maps, parametric curves, and velocity vectors Let (X, O ) and (X, O) be two C p premanifolds with corners, 1 p, and let F : X X be a C p mapping. Let ξ X be a point and let ξ = F (ξ
More informationPart IB GEOMETRY (Lent 2016): Example Sheet 1
Part IB GEOMETRY (Lent 2016): Example Sheet 1 (a.g.kovalev@dpmms.cam.ac.uk) 1. Suppose that H is a hyperplane in Euclidean n-space R n defined by u x = c for some unit vector u and constant c. The reflection
More informationThe 3-Dimensional Geometrisation Conjecture Of W. P. Thurston
The 3-Dimensional Geometrisation Conjecture Of W. P. Thurston May 14, 2002 Contents 1 Introduction 2 2 G-manifolds 3 2.1 Definitions and basic examples................... 3 2.2 Foliations...............................
More informationChapter 12. The cross ratio Klein s Erlanger program The projective line. Math 4520, Fall 2017
Chapter 12 The cross ratio Math 4520, Fall 2017 We have studied the collineations of a projective plane, the automorphisms of the underlying field, the linear functions of Affine geometry, etc. We have
More informationSystoles of hyperbolic 3-manifolds
Math. Proc. Camb. Phil. Soc. (2000), 128, 103 Printed in the United Kingdom 2000 Cambridge Philosophical Society 103 Systoles of hyperbolic 3-manifolds BY COLIN C. ADAMS Department of Mathematics, Williams
More informationPlane hyperbolic geometry
2 Plane hyperbolic geometry In this chapter we will see that the unit disc D has a natural geometry, known as plane hyperbolic geometry or plane Lobachevski geometry. It is the local model for the hyperbolic
More informationDefinition We say that a topological manifold X is C p if there is an atlas such that the transition functions are C p.
13. Riemann surfaces Definition 13.1. Let X be a topological space. We say that X is a topological manifold, if (1) X is Hausdorff, (2) X is 2nd countable (that is, there is a base for the topology which
More informationWe have the following immediate corollary. 1
1. Thom Spaces and Transversality Definition 1.1. Let π : E B be a real k vector bundle with a Euclidean metric and let E 1 be the set of elements of norm 1. The Thom space T (E) of E is the quotient E/E
More informationALGEBRAICALLY TRIVIAL, BUT TOPOLOGICALLY NON-TRIVIAL MAP. Contents 1. Introduction 1
ALGEBRAICALLY TRIVIAL, BUT TOPOLOGICALLY NON-TRIVIAL MAP HONG GYUN KIM Abstract. I studied the construction of an algebraically trivial, but topologically non-trivial map by Hopf map p : S 3 S 2 and a
More information