````` The Belt Trick

Size: px
Start display at page:

Download "````` The Belt Trick"

Transcription

1 ````` The Belt Trick

2 Suppose that we keep top end of a belt fixed and rotate the bottom end in a circle around it.

3 Suppose that we keep top end of a belt fixed and rotate the bottom end in a circle around it.

4 We deform the twisted belt keeping the ends `fixed

5 We deform the twisted belt keeping the ends `fixed

6 We deform the twisted belt keeping the ends `fixed

7 We deform the twisted belt keeping the ends `fixed Observe that the twisted belt cannot be untwisted by these transformations

8 Now rotate the bottom end one more time

9 Now rotate the bottom end one more time

10 Now rotate the bottom end one more time

11 Now rotate the bottom end one more time

12 Now rotate the bottom end one more time

13 Now rotate the bottom end one more time Hence the twice rotated belt can be transformed to the untwisted belt while the once rotated belt cannot. This is called THE BELT TRICK

14 In order to understand the Belt trick one needs ` to consider the transformations.

15 In order to understand the Belt trick one needs ` to consider the transformations. These are precisely rigid twists preserving ends.

16 In order to understand the Belt trick one needs ` to consider the transformations. These are precisely rigid twists preserving ends. One needs to construct an appropriate invariant.

17 Along the centre line of the belt consider two perpendicular vectors along ` the plane of the belt

18 Along the centre line of the belt consider two perpendicular vectors along ` the plane of the belt

19 Along the centre line of the belt consider two perpendicular vectors along ` the plane of the belt For every belt configuration we get an interval I (= [0,1]) of pairs of perpendicular vectors in space (R3) which vary continuously.

20 Along the centre line of the belt consider two perpendicular vectors along ` the plane of the belt For every belt configuration we get an interval I (= [0,1]) of pairs of perpendicular vectors in space (R3) which vary continuously. Hence one has a function : Belt configurations

21 Along the centre line of the belt consider two perpendicular vectors along ` the plane of the belt For every belt configuration we get an interval I (= [0,1]) of pairs of perpendicular vectors in space (R3) which vary continuously. Hence one has a function : Belt configurations Maps from I to pairs of perpendicular vectors in R3 with same value at 0 and 1.

22 Consider V1 = { unit vectors in R3 } = S2 ( the sphere )

23 Consider V1 = { unit vectors in R3 } = S2 0

24 Consider V1 = { unit vectors in R3 } = S2 This makes V1 a topological space 0

25 Consider V1 = { unit vectors in R3 } = S2 This makes V1 a topological space 0 Define : V2 = { pairs of perpendicular unit vectors in R } 3 = { (u,v) є R3 x R3 u =1, v =1, u.v = 0 }

26 Consider V1 = { unit vectors in R3 } = S2 This makes V1 a topological space 0 Define : V2 = { pairs of perpendicular unit vectors in R } 3 = { (u,v) є R3 x R3 u =1, v =1, u.v = 0 } This gets a topology as a subset of R x R 3 3

27 Suppose (u,v) є V2. We can uniquely associate to it the vector u x v (cross product) to get an oriented three frame {u, v, u x v} (right-handed orientation)

28 Suppose (u,v) є V2. We can uniquely associate to it the vector u x v (cross product) to get an oriented three frame {u, v, u x v} (right-handed orientation) Therefore, V2 { oriented three frames }

29 Suppose (u,v) є V2. We can uniquely associate to it the vector u x v (cross product) to get an oriented three frame {u, v, u x v} (right-handed orientation) Therefore, V2 { oriented three frames } For each oriented three frame, form the matrix with the corresponding vectors as columns. This is an orthogonal matrix of determinant 1. The set of these matrices is called SO(3).

30 Suppose (u,v) є V2. We can uniquely associate to it the vector u x v (cross product) to get an oriented three frame {u, v, u x v} (right-handed orientation) Therefore, V2 { oriented three frames } For each oriented three frame, form the matrix with the corresponding vectors as columns. This is an orthogonal matrix of determinant 1. The set of these matrices is called SO(3). Therefore, V2 SO(3).

31 Any orthogonal 2x2 matrix of determinant 1 is a rotation. This implies that SO(2) S1 (the circle).

32 Any orthogonal 2x2 matrix of determinant 1 is a rotation. This implies that SO(2) S1 (the circle). Any orthogonal 3x3 matrix of determinant 1 fixes a vector v (one can check that 1 is an eigenvalue).

33 Any orthogonal 2x2 matrix of determinant 1 is a rotation. This implies that SO(2) S1 (the circle). Any orthogonal 3x3 matrix of determinant 1 fixes a vector v (one can check that 1 is an eigenvalue). Consider a ball of radius π ( Bπ ) in R3

34 Any orthogonal 2x2 matrix of determinant 1 is a rotation. This implies that SO(2) S1 (the circle). Any orthogonal 3x3 matrix of determinant 1 fixes a vector v (one can check that 1 is an eigenvalue). Consider a ball of radius π ( Bπ ) in R3 a v Φ Rotation by angle a in the plane normal to v (note Φ(v) = -v)

35 Any orthogonal 2x2 matrix of determinant 1 is a rotation. This implies that SO(2) S1 (the circle). Any orthogonal 3x3 matrix of determinant 1 fixes a vector v (one can check that 1 is an eigenvalue). Consider a ball of radius π ( Bπ ) in R3 a v Φ Rotation by angle a in the plane normal to v (note Φ(v) = -v) Therefore, SO(3) Bπ / (v = -v) RP 3 (3 dim real projective space)

36 A belt configuration yields a map γ : I V2 such that γ(0) = γ(1). This is a loop in V2.

37 A belt configuration yields a map γ : I V2 such that γ(0) = γ(1). This is a loop in V2. Therefore, { Belt configurations } { loops in V2 }

38 A belt configuration yields a map γ : I V2 such that γ(0) = γ(1). This is a loop in V2. Therefore, { Belt configurations } { loops in V2 } { loops in SO(3) }

39 A belt configuration yields a map γ : I V2 such that γ(0) = γ(1). This is a loop in V2. Therefore, { Belt configurations } { loops in V2 } { loops in SO(3) } { loops in RP } 3

40 Suppose X is a topological space. A loop in X is a map γ : I X so that γ(0) = γ(1).

41 Suppose X is a topological space. A loop in X is a map γ : I X so that γ(0) = γ(1). X γ

42 Suppose X is a topological space. A loop in X is a map γ : I X so that γ(0) = γ(1). X γ X γ

43 Suppose X is a topological space. A loop in X is a map γ : I X so that γ(0) = γ(1). X γ X γ Two loops γ and γ' are said to be homotopic (γ ~ γ') if there is a continuous deformation from γ to γ'.

44 Suppose X is a topological space. A loop in X is a map γ : I X so that γ(0) = γ(1). X γ γ' X γ γ ~ γ' Two loops γ and γ' are said to be homotopic (γ ~ γ') if there is a continuous deformation from γ to γ'.

45 Suppose X is a topological space. A loop in X is a map γ : I X so that γ(0) = γ(1). X γ γ ~ γ' γ' X γ γ' γ ~ γ' Two loops γ and γ' are said to be homotopic (γ ~ γ') if there is a continuous deformation from γ to γ'.

46 Suppose X is a topological space. A loop in X is a map γ : I X so that γ(0) = γ(1). X γ γ ~ γ' γ' X γ'' γ γ' γ ~ γ' but γ ~ γ'' Two loops γ and γ' are said to be homotopic (γ ~ γ') if there is a continuous deformation from γ to γ'.

47 Suppose X is a topological space and x is a point of X.

48 Suppose X is a topological space and x is a point of X. Define : π1(x,x) = { loops starting and ending at x (γ(0) = γ(1) = x) } (Homotopy fixing x)

49 Suppose X is a topological space and x is a point of X. Define : π1(x,x) = { loops starting and ending at x (γ(0) = γ(1) = x) } (Homotopy fixing x) Elements of π1(x,x) can be multiplied :

50 Suppose X is a topological space and x is a point of X. Define : π1(x,x) = { loops starting and ending at x (γ(0) = γ(1) = x) } (Homotopy fixing x) Elements of π1(x,x) can be multiplied : a x b `

51 Suppose X is a topological space and x is a point of X. Define : π1(x,x) = { loops starting and ending at x (γ(0) = γ(1) = x) } (Homotopy fixing x) Elements of π1(x,x) can be multiplied : a.b = the loop a followed by b a x b `

52 Suppose X is a topological space and x is a point of X. Define : π1(x,x) = { loops starting and ending at x (γ(0) = γ(1) = x) } (Homotopy fixing x) Elements of π1(x,x) can be multiplied : a.b = the loop a followed by b a x b a.b `

53 Suppose X is a topological space and x is a point of X. Define : π1(x,x) = { loops starting and ending at x (γ(0) = γ(1) = x) } (Homotopy fixing x) Elements of π1(x,x) can be multiplied : a.b = the loop a followed by b a x With this multiplication π1(x,x) b a.b becomes a group. `

54 Example : X = the circle S and x = any point in S 1 1

55 Example : X = the circle S and x = any point in S 1 x a 1

56 Example : X = the circle S and x = any point in S 1 x a 1 The loop a is homotopic to the circle thought of as a loop.

57 Example : X = the circle S and x = any point in S 1 x a 1 The loop a is homotopic to the circle thought of as a loop. In fact, two loops are homotopic if and only if they wind around the circle the same number of times in the same direction.

58 Example : X = the circle S and x = any point in S 1 x a 1 The loop a is homotopic to the circle thought of as a loop. In fact, two loops are homotopic if and only if they wind around the circle the same number of times in the same direction. This demonstrates : π1(s1,x) = Z ( the group of integers )

59 We have seen the map { Belt configurations } { loops in RP3 }

60 We have seen the map { Belt configurations } { loops in RP3 } A deformation of belt configurations leads to a homotopy of the corresponding loops.

61 We have seen the map { Belt configurations } { loops in RP3 } A deformation of belt configurations leads to a homotopy of the corresponding loops. Therefore we get an invariant : { Belt configurations } π1(rp3,x) (x is some point of RP ) 3

62 Recall RP3 Br / (v = -v) for some radius r. `

63 Recall RP3 Br / (v = -v) for some radius r. Br `

64 Recall RP3 Br / (v = -v) for some radius r. Any loop in RP can be homotoped to miss an interior point of the ball. Such a loop can be expanded radially to homotope it to the boundary sphere (image in RP3`). 3 Br

65 Recall RP3 Br / (v = -v) for some radius r. Any loop in RP can be homotoped to miss an interior point of the ball. Such a loop can be expanded radially to homotope it to the boundary sphere (image in RP`3). 3 Br S2

66 Recall RP3 Br / (v = -v) for some radius r. Any loop in RP can be homotoped to miss an interior point of the ball. Such a loop can be expanded radially to homotope it to the boundary sphere (image in RP`3). 3 Br S2 Similarly any loop can be made to miss a point and such a loop can be homotoped to the equator.

67 The image of the equator in RP is of the form S1/(x=-x). The corresponding loop α in RP3 is the boundary of a disc (image of the sphere). This shows that α is homotopic to constant. 3

68 The image of the equator in RP is of the form S1/(x=-x). The corresponding loop α in RP3 is the boundary of a disc (image of the sphere). This shows that α is homotopic to constant. 3 The image of the half circle in RP is a loop β as the end points map to the same point. This cannot be homotoped to a constant. Also note α = β.β. 3

69 The image of the equator in RP is of the form S1/(x=-x). The corresponding loop α in RP3 is the boundary of a disc (image of the sphere). This shows that α is homotopic to constant. 3 The image of the half circle in RP is a loop β as the end points map to the same point. This cannot be homotoped to a constant. Also note α = β.β. 3 Therefore, π1(rp,x) Z2 3 (the group of integers modulo 2)

70 We have constructed an invariant : { Belt configurations } π1(rp,x) 3

71 We have constructed an invariant : { Belt configurations } We can compute this to show π1(rp,x) 3

72 We have constructed an invariant : { Belt configurations } π1(rp,x) 3 We can compute this to show β є π1(rp,x) 3

73 We have constructed an invariant : { Belt configurations } π1(rp,x) 3 We can compute this to show β є π1(rp,x) 3 This is a non-trivial element.

74

75 β.β = α є π1(rp,x) 3

76 β.β = α є π1(rp,x) 3 This is homotopically trivial.

Math 225A: Differential Topology, Final Exam

Math 225A: Differential Topology, Final Exam Math 225A: Differential Topology, Final Exam Ian Coley December 9, 2013 The goal is the following theorem. Theorem (Hopf). Let M be a compact n-manifold without boundary, and let f, g : M S n be two smooth

More information

Algebraic Topology Homework 4 Solutions

Algebraic 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 information

Topological Theory of Defects: An Introduction

Topological Theory of Defects: An Introduction Topological Theory of Defects: An Introduction Shreyas Patankar Chennai Mathematical Institute Order Parameter Spaces Consider an ordered medium in real space R 3. That is, every point in that medium is

More information

Part II. Algebraic Topology. Year

Part 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 information

Handlebody Decomposition of a Manifold

Handlebody Decomposition of a Manifold Handlebody Decomposition of a Manifold Mahuya Datta Statistics and Mathematics Unit Indian Statistical Institute, Kolkata mahuya@isical.ac.in January 12, 2012 contents Introduction What is a handlebody

More information

April 13, We now extend the structure of the horseshoe to more general kinds of invariant. x (v) λ n v.

April 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 information

AN INTRODUCTION TO THE FUNDAMENTAL GROUP

AN INTRODUCTION TO THE FUNDAMENTAL GROUP AN INTRODUCTION TO THE FUNDAMENTAL GROUP DAVID RAN Abstract. This paper seeks to introduce the reader to the fundamental group and then show some of its immediate applications by calculating the fundamental

More information

Algebraic Topology M3P solutions 2

Algebraic Topology M3P solutions 2 Algebraic Topology M3P1 015 solutions AC Imperial College London a.corti@imperial.ac.uk 3 rd February 015 A small disclaimer This document is a bit sketchy and it leaves some to be desired in several other

More information

Def. A topological space X is disconnected if it admits a non-trivial splitting: (We ll abbreviate disjoint union of two subsets A and B meaning A B =

Def. A topological space X is disconnected if it admits a non-trivial splitting: (We ll abbreviate disjoint union of two subsets A and B meaning A B = CONNECTEDNESS-Notes Def. A topological space X is disconnected if it admits a non-trivial splitting: X = A B, A B =, A, B open in X, and non-empty. (We ll abbreviate disjoint union of two subsets A and

More information

Bending deformation of quasi-fuchsian groups

Bending 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 information

THE FUNDAMENTAL GROUP AND BROUWER S FIXED POINT THEOREM AMANDA BOWER

THE FUNDAMENTAL GROUP AND BROUWER S FIXED POINT THEOREM AMANDA BOWER THE FUNDAMENTAL GROUP AND BROUWER S FIXED POINT THEOREM AMANDA BOWER Abstract. The fundamental group is an invariant of topological spaces that measures the contractibility of loops. This project studies

More information

7. Homotopy and the Fundamental Group

7. Homotopy and the Fundamental Group 7. Homotopy and the Fundamental Group The group G will be called the fundamental group of the manifold V. J. Henri Poincaré, 895 The properties of a topological space that we have developed so far have

More information

1. Classifying Spaces. Classifying Spaces

1. Classifying Spaces. Classifying Spaces Classifying Spaces 1. Classifying Spaces. To make our lives much easier, all topological spaces from now on will be homeomorphic to CW complexes. Fact: All smooth manifolds are homeomorphic to CW complexes.

More information

Bredon, Introduction to compact transformation groups, Academic Press

Bredon, 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 information

FFTs in Graphics and Vision. Groups and Representations

FFTs in Graphics and Vision. Groups and Representations FFTs in Graphics and Vision Groups and Representations Outline Groups Representations Schur s Lemma Correlation Groups A group is a set of elements G with a binary operation (often denoted ) such that

More information

Theorem 3.11 (Equidimensional Sard). Let f : M N be a C 1 map of n-manifolds, and let C M be the set of critical points. Then f (C) has measure zero.

Theorem 3.11 (Equidimensional Sard). Let f : M N be a C 1 map of n-manifolds, and let C M be the set of critical points. Then f (C) has measure zero. Now we investigate the measure of the critical values of a map f : M N where dim M = dim N. Of course the set of critical points need not have measure zero, but we shall see that because the values of

More information

Math 637 Topology Paulo Lima-Filho. Problem List I. b. Show that a contractible space is path connected.

Math 637 Topology Paulo Lima-Filho. Problem List I. b. Show that a contractible space is path connected. Problem List I Problem 1. A space X is said to be contractible if the identiy map i X : X X is nullhomotopic. a. Show that any convex subset of R n is contractible. b. Show that a contractible space is

More information

Hairy balls and ham sandwiches

Hairy balls and ham sandwiches Hairy balls and ham sandwiches Graduate Student Seminar, Carnegie Mellon University Thursday 14 th November 2013 Clive Newstead Abstract Point-set topology studies spaces up to homeomorphism. For many

More information

FUNDAMENTAL GROUPS AND THE VAN KAMPEN S THEOREM

FUNDAMENTAL GROUPS AND THE VAN KAMPEN S THEOREM FUNDAMENTAL GROUPS AND THE VAN KAMPEN S THEOREM ANG LI Abstract. In this paper, we start with the definitions and properties of the fundamental group of a topological space, and then proceed to prove Van-

More information

SEIFERT MATRIX MIKE MCCAFFREY

SEIFERT MATRIX MIKE MCCAFFREY SEIFERT MATRIX MIKE MAFFREY. Defining the Seifert Matrix Recall that oriented surfaces have positive and negative sides. We will be looking at embeddings of connected oriented surfaces in R 3. Definition..

More information

Homework 4: Mayer-Vietoris Sequence and CW complexes

Homework 4: Mayer-Vietoris Sequence and CW complexes Homework 4: Mayer-Vietoris Sequence and CW complexes Due date: Friday, October 4th. 0. Goals and Prerequisites The goal of this homework assignment is to begin using the Mayer-Vietoris sequence and cellular

More information

MATH 547 ALGEBRAIC TOPOLOGY HOMEWORK ASSIGNMENT 4

MATH 547 ALGEBRAIC TOPOLOGY HOMEWORK ASSIGNMENT 4 MATH 547 ALGEBRAIC TOPOLOGY HOMEWORK ASSIGNMENT 4 ROI DOCAMPO ÁLVAREZ Chapter 0 Exercise We think of the torus T as the quotient of X = I I by the equivalence relation generated by the conditions (, s)

More information

INVARIANCE OF THE LAPLACE OPERATOR.

INVARIANCE OF THE LAPLACE OPERATOR. INVARIANCE OF THE LAPLACE OPERATOR. The goal of this handout is to give a coordinate-free proof of the invariance of the Laplace operator under orthogonal transformations of R n (and to explain what this

More information

Stratification of 3 3 Matrices

Stratification of 3 3 Matrices Stratification of 3 3 Matrices Meesue Yoo & Clay Shonkwiler March 2, 2006 1 Warmup with 2 2 Matrices { Best matrices of rank 2} = O(2) S 3 ( 2) { Best matrices of rank 1} S 3 (1) 1.1 Viewing O(2) S 3 (

More information

The Fundamental Group of SO(n) Via Quotients of Braid Groups

The Fundamental Group of SO(n) Via Quotients of Braid Groups The Fundamental Group of SO(n) Via Quotients of Braid Groups Ina Hajdini and Orlin Stoytchev July 1, 016 arxiv:1607.05876v1 [math.ho] 0 Jul 016 Abstract We describe an algebraic proof of the well-known

More information

1 Invariant subspaces

1 Invariant subspaces MATH 2040 Linear Algebra II Lecture Notes by Martin Li Lecture 8 Eigenvalues, eigenvectors and invariant subspaces 1 In previous lectures we have studied linear maps T : V W from a vector space V to another

More information

Homology of a Cell Complex

Homology of a Cell Complex M:01 Fall 06 J. Simon Homology of a Cell Complex A finite cell complex X is constructed one cell at a time, working up in dimension. Each time a cell is added, we can analyze the effect on homology, by

More information

Lisbon school July 2017: eversion of the sphere

Lisbon school July 2017: eversion of the sphere Lisbon school July 2017: eversion of the sphere Pascal Lambrechts Pascal Lambrechts Lisbon school July 2017: eversion of the sphere 1 / 20 Lecture s goal We still want

More information

THE POINCARE-HOPF THEOREM

THE POINCARE-HOPF THEOREM THE POINCARE-HOPF THEOREM ALEX WRIGHT AND KAEL DIXON Abstract. Mapping degree, intersection number, and the index of a zero of a vector field are defined. The Poincare-Hopf theorem, which states that under

More information

NATIONAL BOARD FOR HIGHER MATHEMATICS. Research Scholarships Screening Test. Saturday, February 2, Time Allowed: Two Hours Maximum Marks: 40

NATIONAL BOARD FOR HIGHER MATHEMATICS. Research Scholarships Screening Test. Saturday, February 2, Time Allowed: Two Hours Maximum Marks: 40 NATIONAL BOARD FOR HIGHER MATHEMATICS Research Scholarships Screening Test Saturday, February 2, 2008 Time Allowed: Two Hours Maximum Marks: 40 Please read, carefully, the instructions on the following

More information

Solution: 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:

Solution: 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 information

Definition We say that a topological manifold X is C p if there is an atlas such that the transition functions are C p.

Definition 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 information

A friendly introduction to knots in three and four dimensions

A friendly introduction to knots in three and four dimensions A friendly introduction to knots in three and four dimensions Arunima Ray Rice University SUNY Geneseo Mathematics Department Colloquium April 25, 2013 What is a knot? Take a piece of string, tie a knot

More information

MATH540: Algebraic Topology PROBLEM SET 3 STUDENT SOLUTIONS

MATH540: Algebraic Topology PROBLEM SET 3 STUDENT SOLUTIONS Key Problems 1. Compute π 1 of the Mobius strip. Solution (Spencer Gerhardt): MATH540: Algebraic Topology PROBLEM SET 3 STUDENT SOLUTIONS In other words, M = I I/(s, 0) (1 s, 1). Let x 0 = ( 1 2, 0). Now

More information

Hopf Fibrations. Consider a classical magnetization field in R 3 which is longitidinally stiff and transversally soft.

Hopf Fibrations. Consider a classical magnetization field in R 3 which is longitidinally stiff and transversally soft. Helmut Eschrig Leibniz-Institut für Festkörper- und Werkstofforschung Dresden Leibniz-Institute for Solid State and Materials Research Dresden Hopf Fibrations Consider a classical magnetization field in

More information

Delta link homotopy for two component links, III

Delta link homotopy for two component links, III J. Math. Soc. Japan Vol. 55, No. 3, 2003 elta link homotopy for two component links, III By Yasutaka Nakanishi and Yoshiyuki Ohyama (Received Jul. 11, 2001) (Revised Jan. 7, 2002) Abstract. In this note,

More information

Stable periodic billiard paths in obtuse isosceles triangles

Stable periodic billiard paths in obtuse isosceles triangles Stable periodic billiard paths in obtuse isosceles triangles W. Patrick Hooper March 27, 2006 Can you place a small billiard ball on a frictionless triangular pool table and hit it so that it comes back

More information

Introduction to Braid Groups Joshua Lieber VIGRE REU 2011 University of Chicago

Introduction to Braid Groups Joshua Lieber VIGRE REU 2011 University of Chicago Introduction to Braid Groups Joshua Lieber VIGRE REU 2011 University of Chicago ABSTRACT. This paper is an introduction to the braid groups intended to familiarize the reader with the basic definitions

More information

Chapter Five. Cauchy s Theorem

Chapter Five. Cauchy s Theorem hapter Five auchy s Theorem 5.. Homotopy. Suppose D is a connected subset of the plane such that every point of D is an interior point we call such a set a region and let and 2 be oriented closed curves

More information

The fundamental theorem of calculus for definite integration helped us to compute If has an anti-derivative,

The fundamental theorem of calculus for definite integration helped us to compute If has an anti-derivative, Module 16 : Line Integrals, Conservative fields Green's Theorem and applications Lecture 47 : Fundamental Theorems of Calculus for Line integrals [Section 47.1] Objectives In this section you will learn

More information

+ 2gx + 2fy + c = 0 if S

+ 2gx + 2fy + c = 0 if S CIRCLE DEFINITIONS A circle is the locus of a point which moves in such a way that its distance from a fixed point, called the centre, is always a constant. The distance r from the centre is called the

More information

Math 440 Problem Set 2

Math 440 Problem Set 2 Math 440 Problem Set 2 Problem 4, p. 52. Let X R 3 be the union of n lines through the origin. Compute π 1 (R 3 X). Solution: R 3 X deformation retracts to S 2 with 2n points removed. Choose one of them.

More information

Math 752 Week s 1 1

Math 752 Week s 1 1 Math 752 Week 13 1 Homotopy Groups Definition 1. For n 0 and X a topological space with x 0 X, define π n (X) = {f : (I n, I n ) (X, x 0 )}/ where is the usual homotopy of maps. Then we have the following

More information

AN EXPLICIT FAMILY OF EXOTIC CASSON HANDLES

AN EXPLICIT FAMILY OF EXOTIC CASSON HANDLES proceedings of the american mathematical society Volume 123, Number 4, April 1995 AN EXPLICIT FAMILY OF EXOTIC CASSON HANDLES 2ARKO BIZACA (Communicated by Ronald Stern) Abstract. This paper contains a

More information

The Hurewicz Theorem

The 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 information

Solutions to Problem Set 1

Solutions to Problem Set 1 Solutions to Problem Set 1 18.904 Spring 2011 Problem 1 Statement. Let n 1 be an integer. Let CP n denote the set of all lines in C n+1 passing through the origin. There is a natural map π : C n+1 \ {0}

More information

id = w n = w n (m+1)/2

id = w n = w n (m+1)/2 Samuel Lee Algebraic Topology Homework #4 March 11, 2016 Problem ( 1.2: #1). Show that the free product G H of nontrivial groups G and H has trivial center, and that the only elements of G H of finite

More information

Topology 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 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 information

CW-complexes. Stephen A. Mitchell. November 1997

CW-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 information

MA40040: Algebraic Topology

MA40040: Algebraic Topology MA40040: Algebraic Topology Fran Burstall, based on notes by Ken Deeley Corrections by: Dane Rogers Emily Maw Tim Grange John Stark Martin Prigent Julie Biggs Joseph Martin Dave Colebourn Matt Staniforth

More information

1 )(y 0) {1}. Thus, the total count of points in (F 1 (y)) is equal to deg y0

1 )(y 0) {1}. Thus, the total count of points in (F 1 (y)) is equal to deg y0 1. Classification of 1-manifolds Theorem 1.1. Let M be a connected 1 manifold. Then M is diffeomorphic either to [0, 1], [0, 1), (0, 1), or S 1. We know that none of these four manifolds are not diffeomorphic

More information

Exercises for Algebraic Topology

Exercises for Algebraic Topology Sheet 1, September 13, 2017 Definition. Let A be an abelian group and let M be a set. The A-linearization of M is the set A[M] = {f : M A f 1 (A \ {0}) is finite}. We view A[M] as an abelian group via

More information

Quiz-1 Algebraic Topology. 1. Show that for odd n, the antipodal map and the identity map from S n to S n are homotopic.

Quiz-1 Algebraic Topology. 1. Show that for odd n, the antipodal map and the identity map from S n to S n are homotopic. Quiz-1 Algebraic Topology 1. Show that for odd n, the antipodal map and the identity map from S n to S n are homotopic. 2. Let X be an Euclidean Neighbourhood Retract space and A a closed subspace of X

More information

Polynomial mappings into a Stiefel manifold and immersions

Polynomial mappings into a Stiefel manifold and immersions Polynomial mappings into a Stiefel manifold and immersions Iwona Krzyżanowska Zbigniew Szafraniec November 2011 Abstract For a polynomial mapping from S n k to the Stiefel manifold Ṽk(R n ), where n k

More information

GEOMETRY FINAL CLAY SHONKWILER

GEOMETRY FINAL CLAY SHONKWILER GEOMETRY FINAL CLAY SHONKWILER 1 Let X be the space obtained by adding to a 2-dimensional sphere of radius one, a line on the z-axis going from north pole to south pole. Compute the fundamental group and

More information

Notes 10: Consequences of Eli Cartan s theorem.

Notes 10: Consequences of Eli Cartan s theorem. Notes 10: Consequences of Eli Cartan s theorem. Version 0.00 with misprints, The are a few obvious, but important consequences of the theorem of Eli Cartan on the maximal tori. The first one is the observation

More information

arxiv: v1 [math.gt] 25 Jan 2011

arxiv: v1 [math.gt] 25 Jan 2011 AN INFINITE FAMILY OF CONVEX BRUNNIAN LINKS IN R n BOB DAVIS, HUGH HOWARDS, JONATHAN NEWMAN, JASON PARSLEY arxiv:1101.4863v1 [math.gt] 25 Jan 2011 Abstract. This paper proves that convex Brunnian links

More information

Part IB GEOMETRY (Lent 2016): Example Sheet 1

Part 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 information

Some linear transformations on R 2 Math 130 Linear Algebra D Joyce, Fall 2013

Some linear transformations on R 2 Math 130 Linear Algebra D Joyce, Fall 2013 Some linear transformations on R 2 Math 3 Linear Algebra D Joce, Fall 23 Let s look at some some linear transformations on the plane R 2. We ll look at several kinds of operators on R 2 including reflections,

More information

DIFFERENTIAL TOPOLOGY AND THE POINCARÉ-HOPF THEOREM

DIFFERENTIAL TOPOLOGY AND THE POINCARÉ-HOPF THEOREM DIFFERENTIAL TOPOLOGY AND THE POINCARÉ-HOPF THEOREM ARIEL HAFFTKA 1. Introduction In this paper we approach the topology of smooth manifolds using differential tools, as opposed to algebraic ones such

More information

arxiv: v1 [math.gt] 23 Apr 2014

arxiv: v1 [math.gt] 23 Apr 2014 THE NUMBER OF FRAMINGS OF A KNOT IN A 3-MANIFOLD PATRICIA CAHN, VLADIMIR CHERNOV, AND RUSTAM SADYKOV arxiv:1404.5851v1 [math.gt] 23 Apr 2014 Abstract. In view of the self-linking invariant, the number

More information

43.1 Vector Fields and their properties

43.1 Vector Fields and their properties Module 15 : Vector fields, Gradient, Divergence and Curl Lecture 43 : Vector fields and their properties [Section 43.1] Objectives In this section you will learn the following : Concept of Vector field.

More information

Simply Connected Domains

Simply Connected Domains Simply onnected Domains Generaliing the losed urve Theorem We have shown that if f() is analytic inside and on a closed curve, then f()d = 0. We have also seen examples where f() is analytic on the curve,

More information

Differential Topology Solution Set #3

Differential Topology Solution Set #3 Differential Topology Solution Set #3 Select Solutions 1. Chapter 1, Section 4, #7 2. Chapter 1, Section 4, #8 3. Chapter 1, Section 4, #11(a)-(b) #11(a) The n n matrices with determinant 1 form a group

More information

274 Curves on Surfaces, Lecture 4

274 Curves on Surfaces, Lecture 4 274 Curves on Surfaces, Lecture 4 Dylan Thurston Notes by Qiaochu Yuan Fall 2012 4 Hyperbolic geometry Last time there was an exercise asking for braids giving the torsion elements in PSL 2 (Z). A 3-torsion

More information

ROTATIONS, ROTATION PATHS, AND QUANTUM SPIN

ROTATIONS, ROTATION PATHS, AND QUANTUM SPIN ROTATIONS, ROTATION PATHS, AND QUANTUM SPIN MICHAEL THVEDT 1. ABSTRACT This paper describes the construction of the universal covering group Spin(n), n > 2, as a group of homotopy classes of paths starting

More information

MAS435 Algebraic Topology Part A: Semester 1 Exercises

MAS435 Algebraic Topology Part A: Semester 1 Exercises MAS435 Algebraic Topology 2011-12 Part A: Semester 1 Exercises Dr E. L. G. Cheng Office: J24 E-mail: e.cheng@sheffield.ac.uk http://cheng.staff.shef.ac.uk/mas435/ You should hand in your solutions to Exercises

More information

Assignment 4; Due Friday, February 3

Assignment 4; Due Friday, February 3 Assignment ; Due Friday, February 3 5.6a: The isomorphism u f : π (X, x) π (X, y) is defined by γ f γ f. Remember that we read such expressions from left to right. So follow f backward from y to x, and

More information

Max-Planck-Institut für Mathematik Bonn

Max-Planck-Institut für Mathematik Bonn Max-Planck-Institut für Mathematik Bonn Prime decompositions of knots in T 2 I by Sergey V. Matveev Max-Planck-Institut für Mathematik Preprint Series 2011 (19) Prime decompositions of knots in T 2 I

More information

arxiv:math/ v1 [math.gt] 4 Jan 2007

arxiv:math/ v1 [math.gt] 4 Jan 2007 MAPS TO THE PROJECTIVE PLANE arxiv:math/0701127v1 [math.gt] 4 Jan 2007 JERZY DYDAK AND MICHAEL LEVIN Abstract. We prove the projective plane RP 2 is an absolute extensor of a finite-dimensional metric

More information

Geometry and Topology, Lecture 4 The fundamental group and covering spaces

Geometry and Topology, Lecture 4 The fundamental group and covering spaces 1 Geometry and Topology, Lecture 4 The fundamental group and covering spaces Text: Andrew Ranicki (Edinburgh) Pictures: Julia Collins (Edinburgh) 8th November, 2007 The method of algebraic topology 2 Algebraic

More information

Introduction to Group Theory

Introduction to Group Theory Chapter 10 Introduction to Group Theory Since symmetries described by groups play such an important role in modern physics, we will take a little time to introduce the basic structure (as seen by a physicist)

More information

Applications of Homotopy

Applications of Homotopy Chapter 9 Applications of Homotopy In Section 8.2 we showed that the fundamental group can be used to show that two spaces are not homeomorphic. In this chapter we exhibit other uses of the fundamental

More information

Math 6510 Homework 11

Math 6510 Homework 11 2.2 Problems 40 Problem. From the long exact sequence of homology groups associted to the short exact sequence of chain complexes n 0 C i (X) C i (X) C i (X; Z n ) 0, deduce immediately that there are

More information

1. Simplify the following. Solution: = {0} Hint: glossary: there is for all : such that & and

1. Simplify the following. Solution: = {0} Hint: glossary: there is for all : such that & and Topology MT434P Problems/Homework Recommended Reading: Munkres, J.R. Topology Hatcher, A. Algebraic Topology, http://www.math.cornell.edu/ hatcher/at/atpage.html For those who have a lot of outstanding

More information

A NOTE ON QUANTUM 3-MANIFOLD INVARIANTS AND HYPERBOLIC VOLUME

A NOTE ON QUANTUM 3-MANIFOLD INVARIANTS AND HYPERBOLIC VOLUME A NOTE ON QUANTUM 3-MANIFOLD INVARIANTS AND HYPERBOLIC VOLUME EFSTRATIA KALFAGIANNI Abstract. For a closed, oriented 3-manifold M and an integer r > 0, let τ r(m) denote the SU(2) Reshetikhin-Turaev-Witten

More information

arxiv: v2 [math.gt] 19 Oct 2014

arxiv: v2 [math.gt] 19 Oct 2014 VIRTUAL LEGENDRIAN ISOTOPY VLADIMIR CHERNOV AND RUSTAM SADYKOV arxiv:1406.0875v2 [math.gt] 19 Oct 2014 Abstract. An elementary stabilization of a Legendrian knot L in the spherical cotangent bundle ST

More information

Qualifying Exams I, 2014 Spring

Qualifying Exams I, 2014 Spring Qualifying Exams I, 2014 Spring 1. (Algebra) Let k = F q be a finite field with q elements. Count the number of monic irreducible polynomials of degree 12 over k. 2. (Algebraic Geometry) (a) Show that

More information

Chapter 13: universal gravitation

Chapter 13: universal gravitation Chapter 13: universal gravitation Newton s Law of Gravitation Weight Gravitational Potential Energy The Motion of Satellites Kepler s Laws and the Motion of Planets Spherical Mass Distributions Apparent

More information

7.3 Singular Homology Groups

7.3 Singular Homology Groups 184 CHAPTER 7. HOMOLOGY THEORY 7.3 Singular Homology Groups 7.3.1 Cycles, Boundaries and Homology Groups We can define the singular p-chains with coefficients in a field K. Furthermore, we can define the

More information

Math 215B: Solutions 3

Math 215B: Solutions 3 Math 215B: Solutions 3 (1) For this problem you may assume the classification of smooth one-dimensional manifolds: Any compact smooth one-dimensional manifold is diffeomorphic to a finite disjoint union

More information

(x, y) = d(x, y) = x y.

(x, y) = d(x, y) = x y. 1 Euclidean geometry 1.1 Euclidean space Our story begins with a geometry which will be familiar to all readers, namely the geometry of Euclidean space. In this first chapter we study the Euclidean distance

More information

Fiberwise two-sided multiplications on homogeneous C*-algebras

Fiberwise two-sided multiplications on homogeneous C*-algebras Fiberwise two-sided multiplications on homogeneous C*-algebras Ilja Gogić Department of Mathematics University of Zagreb XX Geometrical Seminar Vrnjačka Banja, Serbia May 20 23, 2018 joint work with Richard

More information

RATIONAL TANGLES AND THE MODULAR GROUP

RATIONAL TANGLES AND THE MODULAR GROUP MOSCOW MATHEMATICAL JOURNAL Volume, Number, April June 0, Pages 9 6 RATIONAL TANGLES AND THE MODULAR GROUP FRANCESCA AICARDI Dedicated to V.I. Arnold, with endless gratitude Abstract. There is a natural

More information

arxiv: v1 [math.dg] 28 Jun 2008

arxiv: v1 [math.dg] 28 Jun 2008 Limit Surfaces of Riemann Examples David Hoffman, Wayne Rossman arxiv:0806.467v [math.dg] 28 Jun 2008 Introduction The only connected minimal surfaces foliated by circles and lines are domains on one of

More information

J-holomorphic curves in symplectic geometry

J-holomorphic curves in symplectic geometry J-holomorphic curves in symplectic geometry Janko Latschev Pleinfeld, September 25 28, 2006 Since their introduction by Gromov [4] in the mid-1980 s J-holomorphic curves have been one of the most widely

More information

Course 212: Academic Year Section 9: Winding Numbers

Course 212: Academic Year Section 9: Winding Numbers Course 212: Academic Year 1991-2 Section 9: Winding Numbers D. R. Wilkins Contents 9 Winding Numbers 71 9.1 Winding Numbers of Closed Curves in the Plane........ 71 9.2 Winding Numbers and Contour Integrals............

More information

Bordism and the Pontryagin-Thom Theorem

Bordism and the Pontryagin-Thom Theorem Bordism and the Pontryagin-Thom Theorem Richard Wong Differential Topology Term Paper December 2, 2016 1 Introduction Given the classification of low dimensional manifolds up to equivalence relations such

More information

MATH 23b, SPRING 2005 THEORETICAL LINEAR ALGEBRA AND MULTIVARIABLE CALCULUS Midterm (part 1) Solutions March 21, 2005

MATH 23b, SPRING 2005 THEORETICAL LINEAR ALGEBRA AND MULTIVARIABLE CALCULUS Midterm (part 1) Solutions March 21, 2005 MATH 23b, SPRING 2005 THEORETICAL LINEAR ALGEBRA AND MULTIVARIABLE CALCULUS Midterm (part 1) Solutions March 21, 2005 1. True or False (22 points, 2 each) T or F Every set in R n is either open or closed

More information

TOEPLITZ OPERATORS. Toeplitz studied infinite matrices with NW-SE diagonals constant. f e C :

TOEPLITZ OPERATORS. Toeplitz studied infinite matrices with NW-SE diagonals constant. f e C : TOEPLITZ OPERATORS EFTON PARK 1. Introduction to Toeplitz Operators Otto Toeplitz lived from 1881-1940 in Goettingen, and it was pretty rough there, so he eventually went to Palestine and eventually contracted

More information

MATH730 NOTES WEEK 8

MATH730 NOTES WEEK 8 MATH730 NOTES WEEK 8 1. Van Kampen s Theorem The main idea of this section is to compute fundamental groups by decomposing a space X into smaller pieces X = U V where the fundamental groups of U, V, and

More information

Universität Regensburg Mathematik

Universität Regensburg Mathematik Universität Regensburg Mathematik Harmonic spinors and local deformations of the metric Bernd Ammann, Mattias Dahl, and Emmanuel Humbert Preprint Nr. 03/2010 HARMONIC SPINORS AND LOCAL DEFORMATIONS OF

More information

The Borsuk-Ulam Theorem

The Borsuk-Ulam Theorem The Borsuk-Ulam Theorem Artur Bicalho Saturnino June 2018 Abstract I am going to present the Borsuk-Ulam theorem in its historical context. After that I will give a proof using differential topology and

More information

Lecture 2. x if x X B n f(x) = α(x) if x S n 1 D n

Lecture 2. x if x X B n f(x) = α(x) if x S n 1 D n Lecture 2 1.10 Cell attachments Let X be a topological space and α : S n 1 X be a map. Consider the space X D n with the disjoint union topology. Consider further the set X B n and a function f : X D n

More information

MATH 205 HOMEWORK #3 OFFICIAL SOLUTION. Problem 1: Find all eigenvalues and eigenvectors of the following linear transformations. (a) F = R, V = R 3,

MATH 205 HOMEWORK #3 OFFICIAL SOLUTION. Problem 1: Find all eigenvalues and eigenvectors of the following linear transformations. (a) F = R, V = R 3, MATH 205 HOMEWORK #3 OFFICIAL SOLUTION Problem 1: Find all eigenvalues and eigenvectors of the following linear transformations. a F = R, V = R 3, b F = R or C, V = F 2, T = T = 9 4 4 8 3 4 16 8 7 0 1

More information

Math Subject GRE Questions

Math Subject GRE Questions Math Subject GRE Questions Calculus and Differential Equations 1. If f() = e e, then [f ()] 2 [f()] 2 equals (a) 4 (b) 4e 2 (c) 2e (d) 2 (e) 2e 2. An integrating factor for the ordinary differential equation

More information

30 Surfaces and nondegenerate symmetric bilinear forms

30 Surfaces and nondegenerate symmetric bilinear forms 80 CHAPTER 3. COHOMOLOGY AND DUALITY This calculation is useful! Corollary 29.4. Let p, q > 0. Any map S p+q S p S q induces the zero map in H p+q ( ). Proof. Let f : S p+q S p S q be such a map. It induces

More information

Some K-theory examples

Some K-theory examples Some K-theory examples The purpose of these notes is to compute K-groups of various spaces and outline some useful methods for Ma448: K-theory and Solitons, given by Dr Sergey Cherkis in 2008-09. Throughout

More information

SMSTC Geometry & Topology 1 Assignment 1 Matt Booth

SMSTC Geometry & Topology 1 Assignment 1 Matt Booth SMSTC Geometry & Topology 1 Assignment 1 Matt Booth Question 1 i) Let be the space with one point. Suppose X is contractible. Then by definition we have maps f : X and g : X such that gf id X and fg id.

More information

Scharlemann s manifold is standard

Scharlemann s manifold is standard Annals of Mathematics, 149 (1999), 497 510 Scharlemann s manifold is standard By Selman Akbulut* Dedicated to Robion Kirby on the occasion of his 60 th birthday Abstract In his 1974 thesis, Martin Scharlemann

More information