Gf(qv) = G, (G. is the isotropy group at q). The map f may be taken as strictly. Of: f: pq -F(Z2 E Z2; S4)

Size: px
Start display at page:

Download "Gf(qv) = G, (G. is the isotropy group at q). The map f may be taken as strictly. Of: f: pq -F(Z2 E Z2; S4)"

Transcription

1 AN EXAMPLE FOR SO(3) BY PIERRE CONNER* AND DEANE MONTGOMERY SCHOOL OF MATHEMATICS, INSTITUTE FOR ADVANCED STUDY, PRINCETON, NEW JERSEY Communicated September 7, Introduction.-The purpose of this note is to give an example to prove the following: THEOREM. There exists an action of SO(3) on Euclidean space EX, n >_ 12, which does not have a stationary point. In constructing the example to prove this theorem, we make use of earlier methods;1 see reference 2 for extensions by Kister. We also rely on a result of Floyd which is as yet unpublished. This is mentioned below and we wish to thank Floyd for informing us of this result. It is also necessary to use the fact3' 4that the product of a contractible manifold by a line is an open cell. 2. The map f.-we begin with the irreducible linear action of S0(3) in E6. This action leaves invariant the unit sphere S4. It has been proved by Floyd that there exists a mapf, f: S4 -> S4 which has degree zero and is equivariant; that is, fg = gf, g G G = SO(3). Floyd has used f to construct a four-dimensional continuum with trivial cohomology on which G = SO(3) acts without a stationary point. This map f is used as the basis of our example and a sketch of the construction of f is included for convenience. The group G = SO(3) contains a subgroup Z2. The set of fixed points of this subgroup, that is F(Z2 ED Z2; S4), is a simple closed curve. The orbits of G in S4 are of two types. Two of these orbits are two-dimensional and are projective planes. The remaining orbits are three-dimensional and all of the same type with isotropy group isomorphic to Z2. The simple closed curve F(Z2 (3 Z2; S4) intersects each three-dimensional orbit in 6 points and each two-dimensional orbit in 3 points. Let pq be an arc in F(Z2 Go Z2; S4) with p and q being in distinct two-dimensional orbits and all points in the interior of pq being in three-dimensional orbits. We first choose f as a map f: pq -F(Z2 E Z2; S4) in sueh a way that f(p) = p and f(q) is in the two-dimensional orbit G(p) and so that Gf(qv) = G, (G. is the isotropy group at q). The map f may be taken as strictly increasing. For any t in pq we define fg(t) = gf(t), g E G. This definesf for all of S4, Of: S4--.S4 and it can be seen that f has degree zero and is equivariant. 3. The mapping cylinder.-let X and Y be two copies of S4 each acted upon by G = SO(3) as above and let f be the equivariant map defined above f: X --Y. 1918

2 VOL. 48, 1962 MATHEMATICS: CONNER AND MONTcOMERY 1919 The join X o Y is defined as follows: X 0 Y = {xyt; x C X, Yy E Y. 0 < t < 1} with the usual identifications. Then G acts on X a Y by means of the following definition: g(x,yt) = (gx,gy,t), g G - SO(3). In X o Y there is the mapping cylinder K = {x,f(x),t; 0 < t < 1 and K is invariant under G because of the equivariance of f. A set U in X o Y is defined as follows: U = {x,y,t; d(y,f(x)) < e/2, 0. t < 1i, where e < 1/4. The set U is invariant. LEMMA 1. The set U can be deformed over itself to Y and this can be done in such a way that x X is carried tof(x) in Y. Let x be a point of X and let N(x) be defined as follows: N(x) = {x,y,t; d(y,f(x)) < e/2, 0 < t. 11, and let the points of N(x) where t = 1 be denoted by P(x), that is P(x) = {x,y,1; d(y,f(x)) < e/2}. The sets N(x) and P(x) are subsets of the join X o Y. The set Y was taken as the unit sphere in a Euclidean space and P(x) determines a set Q(x) in this Euclidean space as follows: Q(x) = { points on the intervals from the origin to P(x)}. Of course N(x) is homeomorphic to Q(x), and we may select a definite homeomorphism TX: N(x) -- Q(x) by requiring a point (x,y,t) of N(x) to be mapped to ty. The set Q(x) is a cone over P(x) with vertex at the origin. There is a deformation DX of Q(x) into itself which moves the origin along the radius joining it to f(x) and moves all of Q(x) to P(x). To define this, let z be any point of Q(x). Then z j on a ray parallel to the ray from the origin through f(x). The point z is to be deformed along this ray to P(x), in such a way that at time s, 0 < s < 1, the point has moved the sth part of the length from z to P(x) along the ray. We now define a deformation of N(x) over itself by the formula T-1DxTx; N(x) -- P(x). Or if the parameter s is exhibited for DX in Q(x) in the form Dr(s), then it is exhibited in N(x) by Fx(s) = Tx-'D(s)Tx: N(x) -N(x), T-I1(D (0)) TX = identity,

3 1920 MATHEMATPICS: CONNER AND MONTGOMERY PROC. N. A. S. Tj-I(Dx(1))Tx: N(x) -* P(x). Of course, points of P(x) remain fixed under this deformation. As x varies Fx(s) defines a deformation on all of U. That is, for u in U, there is an x in X and a y,d(y,f(x))..7/2 such that and then u = (x,y,t), Fx(s)u = T,-'Dx(s)Txu is defined; continuity can be verified. This completes the proof of the Lemma. 4. The neighborhood W.-It will be convenient to consider X and Y as being imbedded as unit spheres in two E5's. Let P and Q be solid balls in these two fivedimensional Euclidean spaces, each of radius 2. The group G = SO(3) acts on P and Q in such a way as to induce the actions on X and Y used above. Extend f radially, f. P-* Q, so f is equivariant and so the restriction is f: X - r. In the join P o Q, define a neighborhood WV of K in P o Q as followers: W= WI U W2 U Wi, where W1 = {x,yt; 1- E < x < 1 + E, d(yf(x)) < e/2}. (Note that a band around the sphere X o Y is the topological product of the sphere X o Y and a disk, actually a square an interval in X and another in Y. It has a product space metric and this metric is used above.) Continuing with the definitions, WH2 = lxyt; 1 - E < x K 1 + E, ye Q, 0 _ t <E}; W3 = {x,y,t; x E P, 1-3E/2 < 1Iy 11 < I + 3E/2, 1 - e < t. 1}. LEMMA 2. The set W is an open invariant set in P o Q and may be deformed in itself into the set U of Lemma 1. In order to see that W is an open set, we may proceed as follows. Note first that W2 and W3 are open sets in P o Q. The set W1 is not open but what we shall prove is that W1 is open at a point {x,y,t} with 0 < t < 1. This will suffice to prove that W is open since W2 and W3 are open. Hence, let (x,y,t), 0 < t < 1 be a point of W1. We must show that if (x',y',t') is a point of P o Q near to (x,y,t) then (x',y',t') is in W1. We know that d(y,f(x)) < E/2 so we have for some (', d(f,f(x)) < El < E2. (1) Given 8' > 0, we may choose a < 0 so that, if d(x,x') < 6, then d(f(x),f(x')) < V'. (2) We may assume 9 d (y,!/') <'. (3)

4 VOL. 48, 1962 MAf.1 T'HEMIA TIC'S: CON.NER AND MONTGOMER Y 1921 Then (1), (2), and (3) prove By choosing 6' properly, we then have d(y',f(x')) < E, + 26'. d(y',f(x')) < e/2. This implies, by definition, that (x',y',t') is in W, as was to be proved. We may deform W2 to t = 0 and WV3 to t = 1, where we leave fixed the set 1/3.. 2/3 and where each x,y,t moves (by varying t) along its own join interval. Then we deform x and y radially to X and Y and this carries W1 through itself to U. 5. The union of cells. The join P o Q is an 11-cell. We consider two of the constructions made above, denoting the first, by PI o Qi and the second by P2 0 Q2. In the second copy, we choose the left-hand band around the unit sphere to extend from 1 -.3E/'2 to 1 + &3E/2. For defining U and the W's, we use e/4 so that the right hand band runs from 1-7E7'4 to 1 + 7(,,/4. In successive copies, we make analogous adjustments so that we are always inside a band from 1-2E to 1 + 2E. In1P1o Q1, the set Bd P, o Qi is a 10-cell on the boundary and it is tamely imbedded. This 10-cell is given by OyIO; x E Bd P1, y E Qi, 0 < t _ 1$, and it is invariant under G. Inside this 10-cell there is a slightly smaller 10-cell Qi* as follows: Qi* =x,y,t; x C Bd P1, 11 y 2 l/2 _1 t 1. Then Qi* is also invariant under G. In P2 0 Q2 there is the 10-cell P2 o Bd Q2, and in it there is the slightly smaller 10-cell defined as follows: -2= t1',8,; x * 1 0 _ t _ 1/,, y E Bd Q2}. Next, we form the union P1' Q1 U P2 0 Q2 and identify Qi* and P2* equivariantly. The union (taken with this identification) is an 11-cell invariant under G. Continuing in this way with P3 0 Q3 and so on, we form Q2* and P3* analogous to the above and identify them to form the union P1 0 Q1 U P2o0 Q2 U P-3 OQQ. Step by step, we identify Qi* and P* and form the union U Xi a Qi with these identifications. In Pi o Qi there is a mapping cylinder Ki, and in the set above we take K2 U K3 U. This set can be seen to have an invariant contractible locally Euclidean neighborhood, which we shall denote by V. Note that we begin with K2 in order to provide

5 1922 MATHEMATICS: G. T. WHYBURN PROC. N. A. S. a locally Euclidean neighborhood. We observe that V has vanishing homotopy groups and hence is contractible. See reference (1) for a similar argument. Now V is an open subset of an open cell, as can be seen from the construction of U P, o Qu; hence, V is a differentiable manifold. It is known (ref. 4) that and G has an action on V X El given by V X El = E12 g(v,r) = (g(v),r), g E G. This action has no stationary point, and this completes the proof of the theorem. 6. Concluding remarks.-corollary. The tetrahedral, icosahedral, and octahedral groups have an action on E'2 with no stationary point. This follows from the fact that the only isotropy groups occurring in the above action of G on E12are Z2 (3 Z2, Z2, e, N (N being the normalizer of a circle group T). By adding a point at infinity, we get an action of G = SO(3) on 512 with precisely one stationary point. This action can be seen not to be differentiable in a neighborhood of the stationary point. If it were, it would be equivalent to a linear action locally, and this linear action would be the sum of irreducible actions. However, the known isotropy groups of the irreducible actions do not fit with the isotropy groups listed above. * Alfred P. Sloan Fellow. 1 Conner and Floyd, "On the construction of periodic maps without fixed points," Proc. Am. Math. Soc., 10, (1959). 2 Kister, "Examples of periodic maps on Euclidean spaces without fixed points," Bull. Am. Math. Soc., 67, (1961). 3 McMillan and Zeeman, "On contractible open manifolds," Proc. Camb. Phil. Soc., 58, (1962). 4 Stallings, "The piecewise linear structure of Euclidean space," Proc. Camb. Phil. Soc., 58, (1962). A MEASURE DISTORTION MAPPING* BY G. T. WHYBURN UNIVERSITY OF VIRGINIA Communicated September 6, 1962 If a, b, a, 13, and 1 are real numbers with a < a < <3K b and 0 < 21 < 13 - a, by the central contraction c(x) of ab relative to a 1, and 1 is meant the homeomorphism of ab onto itself, linear on the intervals aa, a13, and 13b, which leaves a, b, and the midpoint 0 of ac3 fixed and maps a and 13 into points 1 units on the left and right of 0 respectively, i.e., if we take 0 as the origin so that a = -13, then c(x) is 1 1l+ a defined byy = x for -.< x <, y-a = (x- a) for a < x <-, b 1 y - b (x - b) for 13 <x < b. In this paper, these contractions are used to b -1

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

Each copy of any part of a JSTOR transmission must contain the same copyright notice that appears on the screen or printed page of such transmission.

Each copy of any part of a JSTOR transmission must contain the same copyright notice that appears on the screen or printed page of such transmission. Uncountably Many Inequivalent Analytic Actions of a Compact Group on Rn Author(s): R. S. Palais and R. W. Richardson, Jr. Source: Proceedings of the American Mathematical Society, Vol. 14, No. 3 (Jun.,

More information

ALGEBRAICALLY TRIVIAL, BUT TOPOLOGICALLY NON-TRIVIAL MAP. Contents 1. Introduction 1

ALGEBRAICALLY 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

Homotopy and homology groups of the n-dimensional Hawaiian earring

Homotopy and homology groups of the n-dimensional Hawaiian earring F U N D A M E N T A MATHEMATICAE 165 (2000) Homotopy and homology groups of the n-dimensional Hawaiian earring by Katsuya E d a (Tokyo) and Kazuhiro K a w a m u r a (Tsukuba) Abstract. For the n-dimensional

More information

Topological properties

Topological properties CHAPTER 4 Topological properties 1. Connectedness Definitions and examples Basic properties Connected components Connected versus path connected, again 2. Compactness Definition and first examples Topological

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

Math 213br HW 3 solutions

Math 213br HW 3 solutions Math 13br HW 3 solutions February 6, 014 Problem 1 Show that for each d 1, there exists a complex torus X = C/Λ and an analytic map f : X X of degree d. Let Λ be the lattice Z Z d. It is stable under multiplication

More information

Lecture 4: Stabilization

Lecture 4: Stabilization Lecture 4: Stabilization There are many stabilization processes in topology, and often matters simplify in a stable limit. As a first example, consider the sequence of inclusions (4.1) S 0 S 1 S 2 S 3

More information

On open 3-manifolds proper homotopy equivalent to geometrically simply-connected polyhedra

On open 3-manifolds proper homotopy equivalent to geometrically simply-connected polyhedra arxiv:math/9810098v2 [math.gt] 29 Jun 1999 On open 3-manifolds proper homotopy equivalent to geometrically simply-connected polyhedra L. Funar T.L. Thickstun Institut Fourier BP 74 University of Grenoble

More information

The moduli space of binary quintics

The moduli space of binary quintics The moduli space of binary quintics A.A.du Plessis and C.T.C.Wall November 10, 2005 1 Invariant theory From classical invariant theory (we refer to the version in [2]), we find that the ring of (SL 2 )invariants

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

SELF-EQUIVALENCES OF DIHEDRAL SPHERES

SELF-EQUIVALENCES OF DIHEDRAL SPHERES SELF-EQUIVALENCES OF DIHEDRAL SPHERES DAVIDE L. FERRARIO Abstract. Let G be a finite group. The group of homotopy self-equivalences E G (X) of an orthogonal G-sphere X is related to the Burnside ring A(G)

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

On the Diffeomorphism Group of S 1 S 2. Allen Hatcher

On the Diffeomorphism Group of S 1 S 2. Allen Hatcher On the Diffeomorphism Group of S 1 S 2 Allen Hatcher This is a revision, written in December 2003, of a paper of the same title that appeared in the Proceedings of the AMS 83 (1981), 427-430. The main

More information

MATH 215B HOMEWORK 4 SOLUTIONS

MATH 215B HOMEWORK 4 SOLUTIONS MATH 215B HOMEWORK 4 SOLUTIONS 1. (8 marks) Compute the homology groups of the space X obtained from n by identifying all faces of the same dimension in the following way: [v 0,..., ˆv j,..., v n ] is

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

PSEUDO-ANOSOV MAPS AND SIMPLE CLOSED CURVES ON SURFACES

PSEUDO-ANOSOV MAPS AND SIMPLE CLOSED CURVES ON SURFACES MATH. PROC. CAMB. PHIL. SOC. Volume 128 (2000), pages 321 326 PSEUDO-ANOSOV MAPS AND SIMPLE CLOSED CURVES ON SURFACES Shicheng Wang 1, Ying-Qing Wu 2 and Qing Zhou 1 Abstract. Suppose C and C are two sets

More information

Course 212: Academic Year Section 1: Metric Spaces

Course 212: Academic Year Section 1: Metric Spaces Course 212: Academic Year 1991-2 Section 1: Metric Spaces D. R. Wilkins Contents 1 Metric Spaces 3 1.1 Distance Functions and Metric Spaces............. 3 1.2 Convergence and Continuity in Metric Spaces.........

More information

CHAPTER 9. Embedding theorems

CHAPTER 9. Embedding theorems CHAPTER 9 Embedding theorems In this chapter we will describe a general method for attacking embedding problems. We will establish several results but, as the main final result, we state here the following:

More information

SOLUTIONS TO THE FINAL EXAM

SOLUTIONS TO THE FINAL EXAM SOLUTIONS TO THE FINAL EXAM Short questions 1 point each) Give a brief definition for each of the following six concepts: 1) normal for topological spaces) 2) path connected 3) homeomorphism 4) covering

More information

HOMOTOPY THEORY ADAM KAYE

HOMOTOPY THEORY ADAM KAYE HOMOTOPY THEORY ADAM KAYE 1. CW Approximation The CW approximation theorem says that every space is weakly equivalent to a CW complex. Theorem 1.1 (CW Approximation). There exists a functor Γ from the

More information

fy (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))

fy (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 information

Morse Theory and Applications to Equivariant Topology

Morse Theory and Applications to Equivariant Topology Morse Theory and Applications to Equivariant Topology Morse Theory: the classical approach Briefly, Morse theory is ubiquitous and indomitable (Bott). It embodies a far reaching idea: the geometry and

More information

B 1 = {B(x, r) x = (x 1, x 2 ) H, 0 < r < x 2 }. (a) Show that B = B 1 B 2 is a basis for a topology on X.

B 1 = {B(x, r) x = (x 1, x 2 ) H, 0 < r < x 2 }. (a) Show that B = B 1 B 2 is a basis for a topology on X. Math 6342/7350: Topology and Geometry Sample Preliminary Exam Questions 1. For each of the following topological spaces X i, determine whether X i and X i X i are homeomorphic. (a) X 1 = [0, 1] (b) X 2

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

Fuchsian groups. 2.1 Definitions and discreteness

Fuchsian groups. 2.1 Definitions and discreteness 2 Fuchsian groups In the previous chapter we introduced and studied the elements of Mob(H), which are the real Moebius transformations. In this chapter we focus the attention of special subgroups of this

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

FUNDAMENTAL GROUPS AND THE VAN KAMPEN S THEOREM. Contents

FUNDAMENTAL GROUPS AND THE VAN KAMPEN S THEOREM. Contents FUNDAMENTAL GROUPS AND THE VAN KAMPEN S THEOREM SAMUEL BLOOM Abstract. In this paper, we define the fundamental group of a topological space and explore its structure, and we proceed to prove Van-Kampen

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

CHAPTER 7. Connectedness

CHAPTER 7. Connectedness CHAPTER 7 Connectedness 7.1. Connected topological spaces Definition 7.1. A topological space (X, T X ) is said to be connected if there is no continuous surjection f : X {0, 1} where the two point set

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

On the Asphericity of One-Point Unions of Cones

On the Asphericity of One-Point Unions of Cones Volume 36, 2010 Pages 63 75 http://topology.auburn.edu/tp/ On the Asphericity of One-Point Unions of Cones by Katsuya Eda and Kazuhiro Kawamura Electronically published on January 25, 2010 Topology Proceedings

More information

TOPOLOGY HW 2. x x ± y

TOPOLOGY HW 2. x x ± y TOPOLOGY HW 2 CLAY SHONKWILER 20.9 Show that the euclidean metric d on R n is a metric, as follows: If x, y R n and c R, define x + y = (x 1 + y 1,..., x n + y n ), cx = (cx 1,..., cx n ), x y = x 1 y

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

Topology Proceedings. COPYRIGHT c by Topology Proceedings. All rights reserved.

Topology Proceedings. COPYRIGHT c by Topology Proceedings. All rights reserved. Topology Proceedings Web: http://topology.auburn.edu/tp/ Mail: Topology Proceedings Department of Mathematics & Statistics Auburn University, Alabama 36849, USA E-mail: topolog@auburn.edu ISSN: 0146-4124

More information

121B: ALGEBRAIC TOPOLOGY. Contents. 6. Poincaré Duality

121B: 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 information

Math 530 Lecture Notes. Xi Chen

Math 530 Lecture Notes. Xi Chen Math 530 Lecture Notes Xi Chen 632 Central Academic Building, University of Alberta, Edmonton, Alberta T6G 2G1, CANADA E-mail address: xichen@math.ualberta.ca 1991 Mathematics Subject Classification. Primary

More information

LECTURE 15-16: PROPER ACTIONS AND ORBIT SPACES

LECTURE 15-16: PROPER ACTIONS AND ORBIT SPACES LECTURE 15-16: PROPER ACTIONS AND ORBIT SPACES 1. Proper actions Suppose G acts on M smoothly, and m M. Then the orbit of G through m is G m = {g m g G}. If m, m lies in the same orbit, i.e. m = g m for

More information

3-manifolds and their groups

3-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 information

CELL-LIKE SPACES R. C. LACHER1

CELL-LIKE SPACES R. C. LACHER1 CELL-LIKE SPACES R. C. LACHER1 In this note we give a characterization of those compacta which can be embedded in manifolds as cellular sets (the cell-like spaces). There are three conditions equivalent

More information

MATH8808: ALGEBRAIC TOPOLOGY

MATH8808: ALGEBRAIC TOPOLOGY MATH8808: ALGEBRAIC TOPOLOGY DAWEI CHEN Contents 1. Underlying Geometric Notions 2 1.1. Homotopy 2 1.2. Cell Complexes 3 1.3. Operations on Cell Complexes 3 1.4. Criteria for Homotopy Equivalence 4 1.5.

More information

APPLICATIONS OF ALMOST ONE-TO-ONE MAPS

APPLICATIONS OF ALMOST ONE-TO-ONE MAPS APPLICATIONS OF ALMOST ONE-TO-ONE MAPS ALEXANDER BLOKH, LEX OVERSTEEGEN, AND E. D. TYMCHATYN Abstract. A continuous map f : X Y of topological spaces X, Y is said to be almost 1-to-1 if the set of the

More information

MTG 5316/4302 FALL 2018 REVIEW FINAL

MTG 5316/4302 FALL 2018 REVIEW FINAL MTG 5316/4302 FALL 2018 REVIEW FINAL JAMES KEESLING Problem 1. Define open set in a metric space X. Define what it means for a set A X to be connected in a metric space X. Problem 2. Show that if a set

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

Math General Topology Fall 2012 Homework 8 Solutions

Math General Topology Fall 2012 Homework 8 Solutions Math 535 - General Topology Fall 2012 Homework 8 Solutions Problem 1. (Willard Exercise 19B.1) Show that the one-point compactification of R n is homeomorphic to the n-dimensional sphere S n. Note that

More information

Topological dynamics: basic notions and examples

Topological dynamics: basic notions and examples CHAPTER 9 Topological dynamics: basic notions and examples We introduce the notion of a dynamical system, over a given semigroup S. This is a (compact Hausdorff) topological space on which the semigroup

More information

HYPERBOLIC SETS WITH NONEMPTY INTERIOR

HYPERBOLIC SETS WITH NONEMPTY INTERIOR HYPERBOLIC SETS WITH NONEMPTY INTERIOR TODD FISHER, UNIVERSITY OF MARYLAND Abstract. In this paper we study hyperbolic sets with nonempty interior. We prove the folklore theorem that every transitive hyperbolic

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

Smith theory. Andrew Putman. Abstract

Smith theory. Andrew Putman. Abstract Smith theory Andrew Putman Abstract We discuss theorems of P. Smith and Floyd connecting the cohomology of a simplicial complex equipped with an action of a finite p-group to the cohomology of its fixed

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

J. HORNE, JR. A LOCALLY COMPACT CONNECTED GROUP ACTING ON THE PLANE HAS A CLOSED ORBIT

J. HORNE, JR. A LOCALLY COMPACT CONNECTED GROUP ACTING ON THE PLANE HAS A CLOSED ORBIT A LOCALLY COMPACT CONNECTED GROUP ACTING ON THE PLANE HAS A. CLOSED ORBIT BY J. HORNE, JR. The theorem of the title has its origin in a question concerning topological semigroups: Suppose S is a topological

More information

LECTURE 15: COMPLETENESS AND CONVEXITY

LECTURE 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 information

function provided the associated graph function g:x -) X X Y defined

function provided the associated graph function g:x -) X X Y defined QUASI-CLOSED SETS AND FIXED POINTS BY GORDON T. WHYBURN UNIVERSITY OF VIRGINIA Communicated December 29, 1966 1. Introduction.-In this paper we develop new separation and intersection properties of certain

More information

1 Spaces and operations Continuity and metric spaces Topological spaces Compactness... 3

1 Spaces and operations Continuity and metric spaces Topological spaces Compactness... 3 Compact course notes Topology I Fall 2011 Professor: A. Penskoi transcribed by: J. Lazovskis Independent University of Moscow December 23, 2011 Contents 1 Spaces and operations 2 1.1 Continuity and metric

More information

Eilenberg-Steenrod properties. (Hatcher, 2.1, 2.3, 3.1; Conlon, 2.6, 8.1, )

Eilenberg-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 information

MH 7500 THEOREMS. (iii) A = A; (iv) A B = A B. Theorem 5. If {A α : α Λ} is any collection of subsets of a space X, then

MH 7500 THEOREMS. (iii) A = A; (iv) A B = A B. Theorem 5. If {A α : α Λ} is any collection of subsets of a space X, then MH 7500 THEOREMS Definition. A topological space is an ordered pair (X, T ), where X is a set and T is a collection of subsets of X such that (i) T and X T ; (ii) U V T whenever U, V T ; (iii) U T whenever

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

A NOTE ON SPACES OF ASYMPTOTIC DIMENSION ONE

A 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 information

MATH 614 Dynamical Systems and Chaos Lecture 6: Symbolic dynamics.

MATH 614 Dynamical Systems and Chaos Lecture 6: Symbolic dynamics. MATH 614 Dynamical Systems and Chaos Lecture 6: Symbolic dynamics. Metric space Definition. Given a nonempty set X, a metric (or distance function) on X is a function d : X X R that satisfies the following

More information

AN INTEGRAL FORMULA FOR TRIPLE LINKING IN HYPERBOLIC SPACE

AN 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 information

Lifting Smooth Homotopies of Orbit Spaces of Proper Lie Group Actions

Lifting Smooth Homotopies of Orbit Spaces of Proper Lie Group Actions Journal of Lie Theory Volume 15 (2005) 447 456 c 2005 Heldermann Verlag Lifting Smooth Homotopies of Orbit Spaces of Proper Lie Group Actions Marja Kankaanrinta Communicated by J. D. Lawson Abstract. By

More information

THE COMPLEX OF FREE FACTORS OF A FREE GROUP Allen Hatcher* and Karen Vogtmann*

THE COMPLEX OF FREE FACTORS OF A FREE GROUP Allen Hatcher* and Karen Vogtmann* THE COMPLEX OF FREE FACTORS OF A FREE GROUP Allen Hatcher* and Karen Vogtmann* ABSTRACT. We show that the geometric realization of the partially ordered set of proper free factors in a finitely generated

More information

Math General Topology Fall 2012 Homework 6 Solutions

Math General Topology Fall 2012 Homework 6 Solutions Math 535 - General Topology Fall 202 Homework 6 Solutions Problem. Let F be the field R or C of real or complex numbers. Let n and denote by F[x, x 2,..., x n ] the set of all polynomials in n variables

More information

Rigidity result for certain 3-dimensional singular spaces and their fundamental groups.

Rigidity 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 information

DEVELOPMENT OF MORSE THEORY

DEVELOPMENT OF MORSE THEORY DEVELOPMENT OF MORSE THEORY MATTHEW STEED Abstract. In this paper, we develop Morse theory, which allows us to determine topological information about manifolds using certain real-valued functions defined

More information

Algebraic Topology M3P solutions 1

Algebraic Topology M3P solutions 1 Algebraic Topology M3P21 2015 solutions 1 AC Imperial College London a.corti@imperial.ac.uk 9 th February 2015 (1) (a) Quotient maps are continuous, so preimages of closed sets are closed (preimages of

More information

Math 6510 Homework 10

Math 6510 Homework 10 2.2 Problems 9 Problem. Compute the homology group of the following 2-complexes X: a) The quotient of S 2 obtained by identifying north and south poles to a point b) S 1 (S 1 S 1 ) c) The space obtained

More information

FREUDENTHAL SUSPENSION THEOREM

FREUDENTHAL SUSPENSION THEOREM FREUDENTHAL SUSPENSION THEOREM TENGREN ZHANG Abstract. In this paper, I will prove the Freudenthal suspension theorem, and use that to explain what stable homotopy groups are. All the results stated in

More information

Algebraic Topology European Mathematical Society Zürich 2008 Tammo tom Dieck Georg-August-Universität

Algebraic Topology European Mathematical Society Zürich 2008 Tammo tom Dieck Georg-August-Universität 1 Algebraic Topology European Mathematical Society Zürich 2008 Tammo tom Dieck Georg-August-Universität Corrections for the first printing Page 7 +6: j is already assumed to be an inclusion. But the assertion

More information

On the K-category of 3-manifolds for K a wedge of spheres or projective planes

On the K-category of 3-manifolds for K a wedge of spheres or projective planes On the K-category of 3-manifolds for K a wedge of spheres or projective planes J. C. Gómez-Larrañaga F. González-Acuña Wolfgang Heil July 27, 2012 Abstract For a complex K, a closed 3-manifold M is of

More information

TREE AND GRID FACTORS FOR GENERAL POINT PROCESSES

TREE AND GRID FACTORS FOR GENERAL POINT PROCESSES Elect. Comm. in Probab. 9 (2004) 53 59 ELECTRONIC COMMUNICATIONS in PROBABILITY TREE AND GRID FACTORS FOR GENERAL POINT PROCESSES ÁDÁM TIMÁR1 Department of Mathematics, Indiana University, Bloomington,

More information

Corrections to Introduction to Topological Manifolds (First edition) by John M. Lee December 7, 2015

Corrections to Introduction to Topological Manifolds (First edition) by John M. Lee December 7, 2015 Corrections to Introduction to Topological Manifolds (First edition) by John M. Lee December 7, 2015 Changes or additions made in the past twelve months are dated. Page 29, statement of Lemma 2.11: The

More information

PICARD S THEOREM STEFAN FRIEDL

PICARD S THEOREM STEFAN FRIEDL PICARD S THEOREM STEFAN FRIEDL Abstract. We give a summary for the proof of Picard s Theorem. The proof is for the most part an excerpt of [F]. 1. Introduction Definition. Let U C be an open subset. A

More information

ON NEARLY SEMIFREE CIRCLE ACTIONS

ON 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 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

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

Peak Point Theorems for Uniform Algebras on Smooth Manifolds

Peak Point Theorems for Uniform Algebras on Smooth Manifolds Peak Point Theorems for Uniform Algebras on Smooth Manifolds John T. Anderson and Alexander J. Izzo Abstract: It was once conjectured that if A is a uniform algebra on its maximal ideal space X, and if

More information

DIFFERENTIAL GEOMETRY 1 PROBLEM SET 1 SOLUTIONS

DIFFERENTIAL GEOMETRY 1 PROBLEM SET 1 SOLUTIONS DIFFERENTIAL GEOMETRY PROBLEM SET SOLUTIONS Lee: -4,--5,-6,-7 Problem -4: If k is an integer between 0 and min m, n, show that the set of m n matrices whose rank is at least k is an open submanifold of

More information

On small homotopies of loops

On small homotopies of loops Topology and its Applications 155 (2008) 1089 1097 www.elsevier.com/locate/topol On small homotopies of loops G. Conner a,, M. Meilstrup a,d.repovš b,a.zastrow c, M. Željko b a Department of Mathematics,

More information

arxiv: v1 [math.co] 25 Jun 2014

arxiv: v1 [math.co] 25 Jun 2014 THE NON-PURE VERSION OF THE SIMPLEX AND THE BOUNDARY OF THE SIMPLEX NICOLÁS A. CAPITELLI arxiv:1406.6434v1 [math.co] 25 Jun 2014 Abstract. We introduce the non-pure versions of simplicial balls and spheres

More information

LECTURE 10: THE ATIYAH-GUILLEMIN-STERNBERG CONVEXITY THEOREM

LECTURE 10: THE ATIYAH-GUILLEMIN-STERNBERG CONVEXITY THEOREM LECTURE 10: THE ATIYAH-GUILLEMIN-STERNBERG CONVEXITY THEOREM Contents 1. The Atiyah-Guillemin-Sternberg Convexity Theorem 1 2. Proof of the Atiyah-Guillemin-Sternberg Convexity theorem 3 3. Morse theory

More information

Test 2 Review Math 1111 College Algebra

Test 2 Review Math 1111 College Algebra Test 2 Review Math 1111 College Algebra 1. Begin by graphing the standard quadratic function f(x) = x 2. Then use transformations of this graph to graph the given function. g(x) = x 2 + 2 *a. b. c. d.

More information

MATH 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 (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 information

MATH 722, COMPLEX ANALYSIS, SPRING 2009 PART 5

MATH 722, COMPLEX ANALYSIS, SPRING 2009 PART 5 MATH 722, COMPLEX ANALYSIS, SPRING 2009 PART 5.. The Arzela-Ascoli Theorem.. The Riemann mapping theorem Let X be a metric space, and let F be a family of continuous complex-valued functions on X. We have

More information

NON-TRIVIALITY OF GENERALIZED ALTERNATING KNOTS

NON-TRIVIALITY OF GENERALIZED ALTERNATING KNOTS Journal of Knot Theory and Its Ramifications c World Scientific Publishing Company NON-TRIVIALITY OF GENERALIZED ALTERNATING KNOTS MAKOTO OZAWA Natural Science Faculty, Faculty of Letters, Komazawa University,

More information

The Fundamental Group and Covering Spaces

The Fundamental Group and Covering Spaces Chapter 8 The Fundamental Group and Covering Spaces In the first seven chapters we have dealt with point-set topology. This chapter provides an introduction to algebraic topology. Algebraic topology may

More information

From continua to R trees

From continua to R trees 1759 1784 1759 arxiv version: fonts, pagination and layout may vary from AGT published version From continua to R trees PANOS PAPASOGLU ERIC SWENSON We show how to associate an R tree to the set of cut

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

CONSTRUCTING PIECEWISE LINEAR 2-KNOT COMPLEMENTS

CONSTRUCTING PIECEWISE LINEAR 2-KNOT COMPLEMENTS CONSTRUCTING PIECEWISE LINEAR 2-KNOT COMPLEMENTS JONATHAN DENT, JOHN ENGBERS, AND GERARD VENEMA Introduction The groups of high dimensional knots have been characteried by Kervaire [7], but there is still

More information

arxiv: v1 [math.mg] 28 Dec 2018

arxiv: v1 [math.mg] 28 Dec 2018 NEIGHBORING MAPPING POINTS THEOREM ANDREI V. MALYUTIN AND OLEG R. MUSIN arxiv:1812.10895v1 [math.mg] 28 Dec 2018 Abstract. Let f: X M be a continuous map of metric spaces. We say that points in a subset

More information

Algebraic Topology I Homework Spring 2014

Algebraic Topology I Homework Spring 2014 Algebraic Topology I Homework Spring 2014 Homework solutions will be available http://faculty.tcu.edu/gfriedman/algtop/algtop-hw-solns.pdf Due 5/1 A Do Hatcher 2.2.4 B Do Hatcher 2.2.9b (Find a cell structure)

More information

Chapter 12. The cross ratio Klein s Erlanger program The projective line. Math 4520, Fall 2017

Chapter 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 information

On bisectors in Minkowski normed space.

On bisectors in Minkowski normed space. On bisectors in Minkowski normed space. Á.G.Horváth Department of Geometry, Technical University of Budapest, H-1521 Budapest, Hungary November 6, 1997 Abstract In this paper we discuss the concept of

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

Some non-trivial PL knots whose complements are homotopy circles

Some non-trivial PL knots whose complements are homotopy circles Some non-trivial PL knots whose complements are homotopy circles Greg Friedman Vanderbilt University May 16, 2006 Dedicated to the memory of Jerry Levine (May 4, 1937 - April 8, 2006) 2000 Mathematics

More information

Lecture 4: Knot Complements

Lecture 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 information

INVERSE LIMITS WITH SET VALUED FUNCTIONS. 1. Introduction

INVERSE LIMITS WITH SET VALUED FUNCTIONS. 1. Introduction INVERSE LIMITS WITH SET VALUED FUNCTIONS VAN NALL Abstract. We begin to answer the question of which continua in the Hilbert cube can be the inverse limit with a single upper semi-continuous bonding map

More information

Kevin James. MTHSC 206 Section 16.4 Green s Theorem

Kevin James. MTHSC 206 Section 16.4 Green s Theorem MTHSC 206 Section 16.4 Green s Theorem Theorem Let C be a positively oriented, piecewise smooth, simple closed curve in R 2. Let D be the region bounded by C. If P(x, y)( and Q(x, y) have continuous partial

More information

Systoles of hyperbolic 3-manifolds

Systoles 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 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