# A CONSTRUCTION OF THE UNIVERSAL COVER AS A FIBER BUNDLE

Save this PDF as:

Size: px
Start display at page:

## Transcription

1 A CONSTRUCTION OF THE UNIVERSAL COVER AS A FIBER BUNDLE DANIEL A. RAMRAS In these notes we present a construction of the universal cover of a path connected, locally path connected, and semi-locally simply connected space. This construction builds the universal cover as a fiber bundle, by gluing together various evenly covered pieces. This process is surprisingly straightforward, and requires one to think less about point-set issues than the standard construction presented in Hatcher [1, Section 1.3]. These notes are based on notes by Greg Brumfiel. 1. Fiber bundles and transition functions We begin with the general definition of a fiber bundle. Note that covering spaces correspond precisely to fiber bundles in which the fibers are discrete spaces. Definition 1.1. A fiber bundle consists of a pair of spaces E, B and a map E p B, subject to the following local triviality condition: For each point b B, there exists an open neighborhood U B (with b B), a space F U, and a homeomorphism τ: p 1 = (U) U F making the diagram p 1 τ (U) U F U p π 1 U commute (here π 1 is the projection onto the first factor). We call E the total space, B the base space, and p the projection. We ll refer to the neighborhoods U appearing in the local triviality condition as locally trivial neighborhoods, and we call the maps τ local trivializations. Note that the space F U is homeomorphic to each fiber p 1 (x), x U, via the restriction of τ, so we refer to F U as the fiber of p. The local triviality condition implies that all fibers in a neighborhood of b are homeomorphic to the fiber over b itself. One can check that if B is connected, then all of the fibers p 1 (b) are homeomorphic to one another (choose a point b B, and use the local triviality condition to show that {x B : p 1 (x) = p 1 (b)} is both open and closed). Hence the spaces F U above will all be homeomorphic to one another. 1

2 2 DANIEL A. RAMRAS If the fibers F U are discrete spaces, then this definition reduces to the definition of a covering space, because then U p 1 (b) is a disjoint union of copies of U. One should think of the local trivializations as giving coordinates for points in E, but these coordinates are valid only locally. Remember that in a covering space, two fibers can be identified with one another via path lifting. If we consider a simply connected neighborhood in the base, then within that neighborhood these identifications won t depend on the specific paths we choose. This is a way of obtaining local trivializations for covering spaces of (semi-)locally simply connected spaces, and forms the basic idea behind our construction of the universal cover Transition functions. Transition functions offer a useful way to think about fiber bundles. Let E p B be a fiber bundle, and imagine that we have a point b B that lies in two different locally trivial neighborhoods U 1 and U 2, with trivializations τ i : p 1 (U i ) U i F i (respectively). We now have two different ways of trivializing p 1 (U 1 U 2 ), and we refer to the map (1) τ 21 = τ 2 τ 1 1 : U 1 U 2 F 1 = U1 U 2 F 2 as a transition function: it tells us how to switch between the two local trivializations of our bundle, or in other words how to change coordinates. For our purposes, the important thing will be to go in the other direction: starting with an open cover {U i } of a (connected) space B, and a collection of transition functions φ ji : U i U j F i = Ui U j F j, how can we build a fiber bundle E B whose fibers are the (homeomorphic) spaces F i? The key observation is that in a fiber bundle, the transition maps always satisfy a relation known as the cocycle condition. Roughly speaking, this condition says that if we have three overlapping locally trivial neighborhoods U i, U j, and U k (meaning that U i U j U k ) with local trivializations τ i, then changing from τ i coordinates to τ j coordinates, and then to τ k coordinates, is actually the same as changing directly from τ i coordinates to τ k coordinates. This idea is summed up by the following equation: τ kj τ ji = τ ki, which is immediate from Definition (1). Note that we ve indexed the transition functions in such a way that the cocycle condition looks like a cancellation between the two adjacent j s. The following result let s us go in the other direction: we can construct a fiber bundle out of locally trivial pieces, together with transition functions satisfying the cocycle condition.

3 CONSTRUCTION OF THE UNIVERSAL COVER 3 Proposition 1.2. Let B be a space with an open covering {U i } i I, and let {F i } i I be a collection of spaces. Consider a collection of continuous maps τ ji : U i U j F i = Ui U j F j, one for each pair (i, j) I I with U i U j, and assume these functions satisfy π 1 τ ji = π 1 (where π 1 denotes the projection to U i U j ) and also satisfy the cocycle condition τ kj τ ji = τ ki whenever U i U j U k. Define a relation of the set i U i F i by (u, f) i (u, τ ji f) j for every u U i U j and every f F i, where the subscripts on the ordered pairs mean that the first is considered as an element of U i F i and the other as an element of U j F j. Then is an equivalence relation, and the quotient space ( ) E = U i F i / is a fiber bundle over B with projection map p ([(u, f)]) = u. i Proof. First, note that taking i = j in the cocycle condition tells us that τ ii : U i F i U i F i is the identity map, and similarly setting k = i tells us that τ ji and τ ij are inverses of one another. In particular, each τ ij is a homeomorphism. With these observations, the cocycle condition translates immediately to the statement that is transitive, reflexive, and symmetric. To check that E p B is a fiber bundle, we will show that the map U i F i ( ) U i F i U i F i / = E, i i which we denote α i, gives rise to a local trivialization over U i. First, note that the image of this map is p 1 (U i ), because any equivalence class of the form [(u, f) j ], with u U i U j, has a representative τ ij (u, f) j U i F i. We will show that α i is a homeomorphism onto its image; the desired local trivialization is then τ i = αi 1. It suffices to check that α i is an open map. Say V U i F i is open. By definition of the quotient topology on E, we need to show that {(u, f) j U j F j : (u, f) j (u, f ) i V } is open in U j F j. But this set is empty when U i U j =, and otherwise it is precisely τ ji (V ), which is open because τ ji is a homeomorphism (note here that the relation induced by the τ ji is already transitive!). Exercise 1.3. Show that the transition functions associated to the local trivializations of E that we just built are none other than the maps τ ji we started with.

4 4 DANIEL A. RAMRAS 2. The universal cover With the preliminaries out of the way, let s construct the universal cover. Let X be a path connected, locally path connected, semi-locally simply connected space. We want to build a path connected covering space X p X with π 1 ( X) = 0. By assumption, we can cover X by open neighborhoods U i such that each U i is path connected, and the maps π 1 (U i, u i ) π 1 (X, x 0 ) are trivial. Note that the second condition is true for one choice of basepoints if and only if it s true for all choices of basepoints: either way, the condition amounts to saying that every loop in U i is nullhomotopic in X. Said another way, (2) Any two paths in U i with the same endpoints are homotopic in X, through a homotopy fixing the basepoints. This is the form of the condition that we will use below. These will be the evenly covered (i.e. locally trivial) neighborhoods for the universal cover. We need to decide how to represent the fibers of X. Just as in Hatcher s construction, we ll think of the fiber over x X as the collection of (based) homotopy classes of paths from x 0 to x. Here x 0 X is a basepoint that will be fixed throughout our discussion. The idea behind this is simple: if we had the universal cover X in front of us, we could fix a point x 0 in the fiber over x 0 and connect each point in the fiber over x to x 0. These paths, projected down into X, would give representatives for all the based homotopy classes of paths between x and x 0, because each such path would lift to a path in X ending somewhere in the fiber over x. So we will choose points u i U i define F i to be the set of all based homotopy classes of paths from x 0 to u i, with the discrete topology. (Remember that fiber bundles with discrete fibers are covering spaces.) Condition (2) tells us that within U i we can canonically identify F i with the set of based homotopy classes of paths from x 0 to any other point u U i. This corresponds to the discussion in the paragraph just before Section 1.1. Next, we need to construct transition functions τ ji : U i U j F i U i U j F j. Given (x, [q]) U i F i, we need to define τ ji (x, [q]) = (u, [q ]) where q is a path from x 0 to u j. To figure out what q should be, think of (x, [q]) U i F i as a homotopy class of paths from x 0 to x as follows: choose a path α in U i from u i to x, and now qα is a path to x. Note that [qα] (the based homotopy class of this path in the full space X) is independent of α, by Property (2). We choose q so that (x, [q ]) U F j corresponds to (x, [qα]) under this same convention: choose any path β in U j with β(0) = u j and β(1) = x,

5 CONSTRUCTION OF THE UNIVERSAL COVER 5 and set q = qαβ. Again, this does not depend on the choice of α or β (so long as these paths stay inside U i and U j, respectively). To summarize, we have defined τ ji (x, [q]) i = (x, [qαβ]) j, where α and β are paths in U i and U j (respectively) from u i and to u j to x. Theorem 2.1. There is a fiber bundle X p X associated to the above data, and it is a simply connected covering space of X. Proof. We need to check the cocycle condition, in order to see that X is well-defined. This is simple enough: if α, β, and γ are paths in U i, U j, and U k from u i, u j, and u k to x U i U j U k (respectively), and q is a path from x 0 to u i, then τ kj τ ji (x, [q]) i = τ kj (x, [qαβ]) j = (x, [qαββγ]) k = (x, [qαγ]) k = τ ki (x, [q]) i. This is illustrated by the following picture: By Proposition 1.2 we now have a fiber bundle X p X, and since the fibers are discrete this is actually a covering space. We need to show that X is path connected and that π 1 ( X) = 0. Since q is a covering map, we know that q is injective on fundamental groups, with image the set of (homotopy classes of) loops in X whose lifts to X are loops. So to prove that X is simply connected, it will suffice to show if γ is a loop in X that lifts to a loop in X, then γ is null-homotopic in X. We will use one of the points u i U i as our basepoint, and we ll write it as x 0 U 0. We will show that if q is a path in X from x 0 to x U i, then the lift of q to X starting at x 0 = [(x 0, [c x0 ])] ends at [(x, [qα]) i ], where (as above) α is any path in U i from u i to x. (Note that under the convention set up above, we can think of this point as (x, [q]).) Assuming this lifting property for the moment, we check that X is path connected. If we choose any point [(x, [q]) i ] X, then the lift of q starting at x 0 ends at [(x, [q]) i ]. So this point is connected to x 0 by a path in X. The proof that X is simply connected is similar: if γ is a loop in X (at x 0 ),

6 6 DANIEL A. RAMRAS then its lift it to X, starting at x 0, ends at [(x 0, [γ])] (we may choose the path α to be constant). This will be a loop if and only if [γ] = [c x0 ], so only null-homotopic loops lift to loops. To examine how a path γ in X lift to X, we use the Lebesgue Lemma: there exists a decomposition of γ into a sequence of paths γ 0, γ 1,, γ k such that γ i lies entirely inside of some U i (since we are really just working with paths up to homotopy, we can reparametrize the γ i so that they are defined on [0, 1]). We may assume that U 0 is the piece of our cover containing x 0 that we chose above. Choose paths α i in U i from u i to γ i (1). We will prove, by induction on k, that the lift of γ ends at [(γ k (1), [γ 1 γ 2 γ k α k ]) k ]. When k = 1, the path γ stays inside the locally trivial neighborhood U 0 and hence its lift stays inside p 1 (U 0 ) = U 0 F 0 and hence must be constant in the second coordinate. In other words, the lifted path is simply t [(γ(t), [c x0 ]) 0 ], and ends at [(γ(1), [c x0 ])]. Since loops in U 0 are nullhomotopic in X, we have [γ 0 α 0 ] = [c x0 ], as desired. For the induction step, we assume that the lift of γ 1 γ k 1 ends at [(γ k 1 (1), [γ 1 γ k 1 α k 1 ]) k 1 ]. Applying the transition function τ k k 1, this point is the same as [(γ k 1 (1), [γ 1 γ k 1 α k 1 α k 1 γ k α k ]) k ] = [(γ k 1 (1), [γ 1 γ k 1 γ k α k ]) k ] = [(γ k 1 (1), [γα k ]) k ] as desired. (Note here that we are free to use the path γ k α k from γ k 1 (1) to u k in computing the transition function τ k k 1.) 3. problems Exercise 3.1. Let G be a group, and consider a fiber bundle built using Proposition 1.2, in which F i = G for each i and the transition functions τ ji : U i U j G U i U j G are equivariant, in the sense that τ ji (x, gh) = g τ ji (x, h). Here G acts on U j G by g (x, h) = (x, gh). Prove that there is a well-defined left action of G on the fiber bundle E associated to this data. (In this problem, you may assume that G has the discrete topology if you like, but that won t really be important. Any topological group would do.) Fiber bundles of the sort constructed in Exercise 3.1 are called principal bundles, or more specifically principal G bundles. Exercise 3.2. In this exercise, we ll modify the construction of the universal cover so as to make it a principal π 1 (X, x 0 ) bundle. In our construction above, we viewed the fiber of X over x X in terms of paths from x0 to x. Now we ll make some choices so that each fiber becomes a copy of π 1 (X, x 0 ). Let {U i } i I be a covering of X consisting of path connected open sets such that π 1 U i π 1 X is trivial. Choose points u i U i and paths q i from x 0 to u i. Then we have maps F i π 1 (X, x 0 ) sending [q] to [qq i ].

7 CONSTRUCTION OF THE UNIVERSAL COVER 7 (1) Show that these maps are bijections, so we can use them to identify F i with π 1 (X, x 0 ). (2) After making these identifications, the transition maps τ ji used above can be identified with maps U i U j π 1 (X, x 0 ) U i U j π 1 (X, x 0 ). Calculate these maps, and show that they satisfy the equivariance condition from Problem 3.1. (3) Problem 3.1 now tells us that π 1 (X, x 0 ) acts on X from the left. Show that this action corresponds to the action of Aut( X) on X, under = Aut( X) constructed in Hatcher [1, the isomorphism π 1 (X, x 0 ) Proposition 1.39] (note that this isomorphism depends on a choice of basepoint in X: you ll need to choose the basepoint [(x 0, c x0 )]). (4) Show that after identifying the fiber p 1 (x 0 ) with π 1 (X, x 0 ) by sending [γ] π 1 (X, x 0 ) to γ fx0 (1), the above left action (restricted to this fiber) corresponds to left multiplication in π 1 (X, x 0 ). Exercise 3.3. Let X p X be the universal cover. The fundamental group π 1 (X, x 0 ) acts on X via deck transformations, as in the previous problem. But it also acts on the fiber p 1 (x 0 ) via path lifting: given x in this fiber and [γ] π 1 (X, x 0 ), we can lift γ to a path γ ex starting at x, and we set x [γ] = γ ex (1). (1) Prove that this defines a right action of π 1 X on the fiber p 1 (x 0 ). (2) Show that after identifying the fiber p 1 (x 0 ) with π 1 (X, x 0 ) by sending [γ] π 1 (X, x 0 ) to γ fx0 (1), the above right action corresponds to right multiplication in π 1 (X, x 0 ). The next two exercises are meant to emphasize the fact that the right and left actions of π 1 (X, x 0 ) on p 1 (x 0 ) are really very different actions. Exercise 3.4. Check that these actions agree (in the sense that [γ] x = x [γ]) if and only if π 1 (X, x 0 ) is an abelian group. Exercise 3.5. Any left action G S S of a group G on a set S can be converted into a right action by setting s g = g 1 s. Consider what happens when we convert the left action of π 1 ( X, x 0 ) on p 1 (x 0 ) (from Exercise 3.2) into a right action. Show that this action agrees with the right action from Exercise 3.3 if and only if every element in π 1 (X, x 0 ) has order two. (This exercise could equally well have been phrased in terms of converting the action in Exercise 3.3 into a left action.) References [1] Allen Hatcher. Algebraic topology. Cambridge University Press, Cambridge, Department of Mathematical Sciences, New Mexico State University, Las Cruces, New Mexico , USA address:

### Sets of Fibre Homotopy Classes and Twisted Order Parameter Spaces

Communications in Mathematical Physics Manuscript-Nr. (will be inserted by hand later) Sets of Fibre Homotopy Classes and Twisted Order Parameter Spaces Stefan Bechtluft-Sachs, Marco Hien Naturwissenschaftliche

### The fundamental group of the Hawaiian earring is not free (International Journal of Algebra and Computation Vol. 2, No. 1 (1992), 33 37) Bart de Smit

The fundamental group of the Hawaiian earring is not free Bart de Smit The fundamental group of the Hawaiian earring is not free (International Journal of Algebra and Computation Vol. 2, No. 1 (1992),

### FIBRATION SEQUENCES AND PULLBACK SQUARES. Contents. 2. Connectivity and fiber sequences. 3

FIRTION SEQUENES ND PULLK SQURES RY MLKIEWIH bstract. We lay out some foundational facts about fibration sequences and pullback squares of topological spaces. We pay careful attention to connectivity ranges

### Chapter 7. Homotopy. 7.1 Basic concepts of homotopy. Example: z dz. z dz = but

Chapter 7 Homotopy 7. Basic concepts of homotopy Example: but γ z dz = γ z dz γ 2 z dz γ 3 z dz. Why? The domain of /z is C 0}. We can deform γ continuously into γ 2 without leaving C 0}. Intuitively,

### A NOTE ON TRIVIAL FIBRATIONS

A NOTE ON TRIVIAL FIBRATIONS Petar Pavešić Fakulteta za Matematiko in Fiziko, Univerza v Ljubljani, Jadranska 19, 1111 Ljubljana, Slovenija petar.pavesic@fmf.uni-lj.si Abstract We study the conditions

### FIBER PRODUCTS AND ZARISKI SHEAVES

FIBER PRODUCTS AND ZARISKI SHEAVES BRIAN OSSERMAN 1. Fiber products and Zariski sheaves We recall the definition of a fiber product: Definition 1.1. Let C be a category, and X, Y, Z objects of C. Fix also

### ORIENTATIONS. Contents

ORIENTATIONS Contents 1. Generators for H n R n, R n p 1 1. Generators for H n R n, R n p We ended last time by constructing explicit generators for H n D n, S n 1 by using an explicit n-simplex which

### Lecture 1-1,2. Definition. Sets A and B have the same cardinality if there is an invertible function f : A B.

Topology Summary This is a summary of the results discussed in lectures. These notes will be gradually growing as the time passes by. Most of the time proofs will not be given. You are supposed to be able

### FIXED POINT SETS OF FIBER-PRESERVING MAPS

FIXED POINT SETS OF FIBER-PRESERVING MAPS Robert F. Brown Department of Mathematics University of California Los Angeles, CA 90095 e-mail: rfb@math.ucla.edu Christina L. Soderlund Department of Mathematics

### Course 421: Algebraic Topology Section 1: Topological Spaces

Course 421: Algebraic Topology Section 1: Topological Spaces David R. Wilkins Copyright c David R. Wilkins 1988 2008 Contents 1 Topological Spaces 1 1.1 Continuity and Topological Spaces...............

### Fiber bundles and non-abelian cohomology

Fiber bundles and non-abelian cohomology Nicolas Addington April 22, 2007 Abstract The transition maps of a fiber bundle are often said to satisfy the cocycle condition. If we take this terminology seriously

### Chapter 7: Products and quotients

Chapter 7: Products and quotients Matthew Macauley Department of Mathematical Sciences Clemson University http://www.math.clemson.edu/~macaule/ Math 42, Spring 24 M. Macauley (Clemson) Chapter 7: Products

### A REMARK ON ALMOST MOORE DIGRAPHS OF DEGREE THREE. 1. Introduction and Preliminaries

Acta Math. Univ. Comenianae Vol. LXVI, 2(1997), pp. 285 291 285 A REMARK ON ALMOST MOORE DIGRAPHS OF DEGREE THREE E. T. BASKORO, M. MILLER and J. ŠIRÁŇ Abstract. It is well known that Moore digraphs do

### Mathematical Research Letters 1, 249 255 (1994) MAPPING CLASS GROUPS ARE AUTOMATIC. Lee Mosher

Mathematical Research Letters 1, 249 255 (1994) MAPPING CLASS GROUPS ARE AUTOMATIC Lee Mosher Let S be a compact surface, possibly with the extra structure of an orientation or a finite set of distinguished

### Effective homotopy of the fiber of a Kan fibration

Effective homotopy of the fiber of a Kan fibration Ana Romero and Francis Sergeraert 1 Introduction Inspired by the fundamental ideas of the effective homology method (see [5] and [4]), which makes it

### FIBER BUNDLES AND UNIVALENCE

FIBER BUNDLES AND UNIVALENCE TALK BY IEKE MOERDIJK; NOTES BY CHRIS KAPULKIN This talk presents a proof that a universal Kan fibration is univalent. The talk was given by Ieke Moerdijk during the conference

### (2) the identity map id : S 1 S 1 to the symmetry S 1 S 1 : x x (here x is considered a complex number because the circle S 1 is {x C : x = 1}),

Chapter VI Fundamental Group 29. Homotopy 29 1. Continuous Deformations of Maps 29.A. Is it possible to deform continuously: (1) the identity map id : R 2 R 2 to the constant map R 2 R 2 : x 0, (2) the

### Assignment 7; Due Friday, November 11

Assignment 7; Due Friday, November 9.8 a The set Q is not connected because we can write it as a union of two nonempty disjoint open sets, for instance U = (, 2) and V = ( 2, ). The connected subsets are

### MAT2400 Analysis I. A brief introduction to proofs, sets, and functions

MAT2400 Analysis I A brief introduction to proofs, sets, and functions In Analysis I there is a lot of manipulations with sets and functions. It is probably also the first course where you have to take

### Quantum manifolds. Manuel Hohmann

Quantum manifolds Manuel Hohmann Zentrum für Mathematische Physik und II. Institut für Theoretische Physik, Universität Hamburg, Luruper Chaussee 149, 22761 Hamburg, Germany Abstract. We propose a mathematical

### Introduction to Topology

Introduction to Topology Tomoo Matsumura November 30, 2010 Contents 1 Topological spaces 3 1.1 Basis of a Topology......................................... 3 1.2 Comparing Topologies.......................................

### Surface bundles over S 1, the Thurston norm, and the Whitehead link

Surface bundles over S 1, the Thurston norm, and the Whitehead link Michael Landry August 16, 2014 The Thurston norm is a powerful tool for studying the ways a 3-manifold can fiber over the circle. In

### Classification of Bundles

CHAPTER 2 Classification of Bundles In this chapter we prove Steenrod s classification theorem of principal G - bundles, and the corresponding classification theorem of vector bundles. This theorem states

### FUNCTIONAL ANALYSIS LECTURE NOTES: QUOTIENT SPACES

FUNCTIONAL ANALYSIS LECTURE NOTES: QUOTIENT SPACES CHRISTOPHER HEIL 1. Cosets and the Quotient Space Any vector space is an abelian group under the operation of vector addition. So, if you are have studied

### TOPOLOGY: THE JOURNEY INTO SEPARATION AXIOMS

TOPOLOGY: THE JOURNEY INTO SEPARATION AXIOMS VIPUL NAIK Abstract. In this journey, we are going to explore the so called separation axioms in greater detail. We shall try to understand how these axioms

### MA651 Topology. Lecture 6. Separation Axioms.

MA651 Topology. Lecture 6. Separation Axioms. This text is based on the following books: Fundamental concepts of topology by Peter O Neil Elements of Mathematics: General Topology by Nicolas Bourbaki Counterexamples

### On The Existence Of Flips

On The Existence Of Flips Hacon and McKernan s paper, arxiv alg-geom/0507597 Brian Lehmann, February 2007 1 Introduction References: Hacon and McKernan s paper, as above. Kollár and Mori, Birational Geometry

### Set theory as a foundation for mathematics

Set theory as a foundation for mathematics Waffle Mathcamp 2011 In school we are taught about numbers, but we never learn what numbers really are. We learn rules of arithmetic, but we never learn why these

### MATH 131 SOLUTION SET, WEEK 12

MATH 131 SOLUTION SET, WEEK 12 ARPON RAKSIT AND ALEKSANDAR MAKELOV 1. Normalisers We first claim H N G (H). Let h H. Since H is a subgroup, for all k H we have hkh 1 H and h 1 kh H. Since h(h 1 kh)h 1

### REAL ANALYSIS I HOMEWORK 2

REAL ANALYSIS I HOMEWORK 2 CİHAN BAHRAN The questions are from Stein and Shakarchi s text, Chapter 1. 1. Prove that the Cantor set C constructed in the text is totally disconnected and perfect. In other

### CHAPTER 5: MODULAR ARITHMETIC

CHAPTER 5: MODULAR ARITHMETIC LECTURE NOTES FOR MATH 378 (CSUSM, SPRING 2009). WAYNE AITKEN 1. Introduction In this chapter we will consider congruence modulo m, and explore the associated arithmetic called

### Hatcher C A: [012] = [013], B : [123] = [023],

Ex 2..2 Hatcher 2. Let S = [2] [23] 3 = [23] be the union of two faces of the 3-simplex 3. Let be the equivalence relation that identifies [] [3] and [2] [23]. The quotient space S/ is the Klein bottle

Algebraic Geometry Keerthi Madapusi Contents Chapter 1. Schemes 5 1. Spec of a Ring 5 2. Schemes 11 3. The Affine Communication Lemma 13 4. A Criterion for Affineness 15 5. Irreducibility and Connectedness

### Fiber bundles. Chapter 23. c Amitabha Lahiri: Lecture Notes on Differential Geometry for Physicists 2011

Chapter 23 Fiber bundles Consider a manifold M with the tangent bundle T M = P M T P M. Let us look at this more closely. T M can be thought of as the original manifold M with a tangent space stuck at

### 3. Equivalence Relations. Discussion

3. EQUIVALENCE RELATIONS 33 3. Equivalence Relations 3.1. Definition of an Equivalence Relations. Definition 3.1.1. A relation R on a set A is an equivalence relation if and only if R is reflexive, symmetric,

### Two fixed points can force positive entropy of a homeomorphism on a hyperbolic surface (Not for publication)

Two fixed points can force positive entropy of a homeomorphism on a hyperbolic surface (Not for publication) Pablo Lessa May 29, 2012 1 Introduction A common theme in low-dimensional dynamical systems

### A LEBESGUE MEASURABLE SET THAT IS NOT BOREL

A LEBESGUE MEASURABLE SET THAT IS NOT BOREL SAM SCHIAVONE Tuesday, 16 October 2012 1. Outline (1 Ternary Expansions (2 The Cantor Set (3 The Cantor Ternary Function (a.k.a. The Devil s Staircase Function

### 6.2 Permutations continued

6.2 Permutations continued Theorem A permutation on a finite set A is either a cycle or can be expressed as a product (composition of disjoint cycles. Proof is by (strong induction on the number, r, of

### Basic Concepts of Point Set Topology Notes for OU course Math 4853 Spring 2011

Basic Concepts of Point Set Topology Notes for OU course Math 4853 Spring 2011 A. Miller 1. Introduction. The definitions of metric space and topological space were developed in the early 1900 s, largely

### Course 221: Analysis Academic year , First Semester

Course 221: Analysis Academic year 2007-08, First Semester David R. Wilkins Copyright c David R. Wilkins 1989 2007 Contents 1 Basic Theorems of Real Analysis 1 1.1 The Least Upper Bound Principle................

### Mathematics Course 111: Algebra I Part IV: Vector Spaces

Mathematics Course 111: Algebra I Part IV: Vector Spaces D. R. Wilkins Academic Year 1996-7 9 Vector Spaces A vector space over some field K is an algebraic structure consisting of a set V on which are

### p 2 1 (mod 6) Adding 2 to both sides gives p (mod 6)

.9. Problems P10 Try small prime numbers first. p p + 6 3 11 5 7 7 51 11 13 Among the primes in this table, only the prime 3 has the property that (p + ) is also a prime. We try to prove that no other

### Math 225A, Differential Topology: Homework 3

Math 225A, Differential Topology: Homework 3 Ian Coley October 17, 2013 Problem 1.4.7. Suppose that y is a regular value of f : X Y, where X is compact and dim X = dim Y. Show that f 1 (y) is a finite

### Graphs without proper subgraphs of minimum degree 3 and short cycles

Graphs without proper subgraphs of minimum degree 3 and short cycles Lothar Narins, Alexey Pokrovskiy, Tibor Szabó Department of Mathematics, Freie Universität, Berlin, Germany. August 22, 2014 Abstract

University of Oslo MAT2 Project The Banach-Tarski Paradox Author: Fredrik Meyer Supervisor: Nadia S. Larsen Abstract In its weak form, the Banach-Tarski paradox states that for any ball in R, it is possible

### SEMINAR NOTES: G-BUNDLES, STACKS AND BG (SEPT. 15, 2009)

SEMINAR NOTES: G-BUNDLES, STACKS AND (SEPT. 15, 2009) DENNIS GAITSGORY 1. G-bundles 1.1. Let G be an affine algebraic group over a ground field k. We assume that k is algebraically closed. Definition 1.2.

### Degree Hypergroupoids Associated with Hypergraphs

Filomat 8:1 (014), 119 19 DOI 10.98/FIL1401119F Published by Faculty of Sciences and Mathematics, University of Niš, Serbia Available at: http://www.pmf.ni.ac.rs/filomat Degree Hypergroupoids Associated

### Complete graphs whose topological symmetry groups are polyhedral

Algebraic & Geometric Topology 11 (2011) 1405 1433 1405 Complete graphs whose topological symmetry groups are polyhedral ERICA FLAPAN BLAKE MELLOR RAMIN NAIMI We determine for which m the complete graph

### Math 231b Lecture 35. G. Quick

Math 231b Lecture 35 G. Quick 35. Lecture 35: Sphere bundles and the Adams conjecture 35.1. Sphere bundles. Let X be a connected finite cell complex. We saw that the J-homomorphism could be defined by

### TOPOLOGY OF SINGULAR FIBERS OF GENERIC MAPS

TOPOLOGY OF SINGULAR FIBERS OF GENERIC MAPS OSAMU SAEKI Dedicated to Professor Yukio Matsumoto on the occasion of his 60th birthday Abstract. We classify singular fibers of C stable maps of orientable

### Infinite product spaces

Chapter 7 Infinite product spaces So far we have talked a lot about infinite sequences of independent random variables but we have never shown how to construct such sequences within the framework of probability/measure

### DEGREES OF ORDERS ON TORSION-FREE ABELIAN GROUPS

DEGREES OF ORDERS ON TORSION-FREE ABELIAN GROUPS ASHER M. KACH, KAREN LANGE, AND REED SOLOMON Abstract. We construct two computable presentations of computable torsion-free abelian groups, one of isomorphism

### MODULAR ARITHMETIC. a smallest member. It is equivalent to the Principle of Mathematical Induction.

MODULAR ARITHMETIC 1 Working With Integers The usual arithmetic operations of addition, subtraction and multiplication can be performed on integers, and the result is always another integer Division, on

### Introduction to Characteristic Classes

UNIVERSITY OF COPENHAGEN Faculty of Science Department of Mathematical Sciences Mauricio Esteban Gómez López Introduction to Characteristic Classes Supervisors: Jesper Michael Møller, Ryszard Nest 1 Abstract

### CHAPTER 1 BASIC TOPOLOGY

CHAPTER 1 BASIC TOPOLOGY Topology, sometimes referred to as the mathematics of continuity, or rubber sheet geometry, or the theory of abstract topological spaces, is all of these, but, above all, it is

### Sets and Cardinality Notes for C. F. Miller

Sets and Cardinality Notes for 620-111 C. F. Miller Semester 1, 2000 Abstract These lecture notes were compiled in the Department of Mathematics and Statistics in the University of Melbourne for the use

### Math 563 Measure Theory Project 1 (Funky Functions Group) Luis Zerón, Sergey Dyachenko

Math 563 Measure Theory Project (Funky Functions Group) Luis Zerón, Sergey Dyachenko 34 Let C and C be any two Cantor sets (constructed in Exercise 3) Show that there exists a function F: [,] [,] with

### Point Set Topology. A. Topological Spaces and Continuous Maps

Point Set Topology A. Topological Spaces and Continuous Maps Definition 1.1 A topology on a set X is a collection T of subsets of X satisfying the following axioms: T 1.,X T. T2. {O α α I} T = α IO α T.

### Week 5: Binary Relations

1 Binary Relations Week 5: Binary Relations The concept of relation is common in daily life and seems intuitively clear. For instance, let X be the set of all living human females and Y the set of all

### Sets, Relations and Functions

Sets, Relations and Functions Eric Pacuit Department of Philosophy University of Maryland, College Park pacuit.org epacuit@umd.edu ugust 26, 2014 These notes provide a very brief background in discrete

### Relations Graphical View

Relations Slides by Christopher M. Bourke Instructor: Berthe Y. Choueiry Introduction Recall that a relation between elements of two sets is a subset of their Cartesian product (of ordered pairs). A binary

### Structure of Measurable Sets

Structure of Measurable Sets In these notes we discuss the structure of Lebesgue measurable subsets of R from several different points of view. Along the way, we will see several alternative characterizations

### Chapter 13: Basic ring theory

Chapter 3: Basic ring theory Matthew Macauley Department of Mathematical Sciences Clemson University http://www.math.clemson.edu/~macaule/ Math 42, Spring 24 M. Macauley (Clemson) Chapter 3: Basic ring

### Fixed Point Theorems in Topology and Geometry

Fixed Point Theorems in Topology and Geometry A Senior Thesis Submitted to the Department of Mathematics In Partial Fulfillment of the Requirements for the Departmental Honors Baccalaureate By Morgan Schreffler

### Lecture 3: Group completion and the definition of K-theory

Lecture 3: Group completion and the definition of K-theory The goal of this lecture is to give the basic definition of K-theory. The process of group completion, which completes a commutative monoid M

### Classical Analysis I

Classical Analysis I 1 Sets, relations, functions A set is considered to be a collection of objects. The objects of a set A are called elements of A. If x is an element of a set A, we write x A, and if

### 11 Ideals. 11.1 Revisiting Z

11 Ideals The presentation here is somewhat different than the text. In particular, the sections do not match up. We have seen issues with the failure of unique factorization already, e.g., Z[ 5] = O Q(

### Problem Set. Problem Set #2. Math 5322, Fall December 3, 2001 ANSWERS

Problem Set Problem Set #2 Math 5322, Fall 2001 December 3, 2001 ANSWERS i Problem 1. [Problem 18, page 32] Let A P(X) be an algebra, A σ the collection of countable unions of sets in A, and A σδ the collection

### EVALUATION FIBER SEQUENCE AND HOMOTOPY COMMUTATIVITY MITSUNOBU TSUTAYA

EVALUATION FIBER SEQUENCE AND HOMOTOPY COMMUTATIVITY MITSUNOBU TSUTAYA Abstract. We review the relation between the evaluation fiber sequence of mapping spaces and the various higher homotopy commutativities.

### SOLUTIONS TO ASSIGNMENT 1 MATH 576

SOLUTIONS TO ASSIGNMENT 1 MATH 576 SOLUTIONS BY OLIVIER MARTIN 13 #5. Let T be the topology generated by A on X. We want to show T = J B J where B is the set of all topologies J on X with A J. This amounts

### G = G 0 > G 1 > > G k = {e}

Proposition 49. 1. A group G is nilpotent if and only if G appears as an element of its upper central series. 2. If G is nilpotent, then the upper central series and the lower central series have the same

### Reidemeister coincidence invariants of fiberwise maps

Reidemeister coincidence invariants of fiberwise maps Ulrich Koschorke In memory of Jakob Nielsen (15.10.1890-3.8.1959). Abstract. Given two fiberwise maps f 1, f 2 between smooth fiber bundles over a

### Discrete Mathematics & Mathematical Reasoning Chapter 10: Graphs

Discrete Mathematics & Mathematical Reasoning Chapter 10: Graphs Kousha Etessami U. of Edinburgh, UK Kousha Etessami (U. of Edinburgh, UK) Discrete Mathematics (Chapter 6) 1 / 13 Overview Graphs and Graph

### 9. Quotient Groups Given a group G and a subgroup H, under what circumstances can we find a homomorphism φ: G G ', such that H is the kernel of φ?

9. Quotient Groups Given a group G and a subgroup H, under what circumstances can we find a homomorphism φ: G G ', such that H is the kernel of φ? Clearly a necessary condition is that H is normal in G.

### Sets and functions. {x R : x > 0}.

Sets and functions 1 Sets The language of sets and functions pervades mathematics, and most of the important operations in mathematics turn out to be functions or to be expressible in terms of functions.

### CROSS-EFFECTS OF HOMOTOPY FUNCTORS AND SPACES OF TREES. WARNING: This paper is a draft and has not been fully checked for errors.

CROSS-EFFECTS OF HOMOTOPY FUNCTORS AND SPACES OF TREES MICHAEL CHING WARNING: This paper is a draft and has not been full checked for errors. 1. Total homotop fibres of cubes of based spaces Definition

### SOLUTIONS TO EXERCISES FOR. MATHEMATICS 205A Part 3. Spaces with special properties

SOLUTIONS TO EXERCISES FOR MATHEMATICS 205A Part 3 Fall 2008 III. Spaces with special properties III.1 : Compact spaces I Problems from Munkres, 26, pp. 170 172 3. Show that a finite union of compact subspaces

### 4. CLASSES OF RINGS 4.1. Classes of Rings class operator A-closed Example 1: product Example 2:

4. CLASSES OF RINGS 4.1. Classes of Rings Normally we associate, with any property, a set of objects that satisfy that property. But problems can arise when we allow sets to be elements of larger sets

### o-minimality and Uniformity in n 1 Graphs

o-minimality and Uniformity in n 1 Graphs Reid Dale July 10, 2013 Contents 1 Introduction 2 2 Languages and Structures 2 3 Definability and Tame Geometry 4 4 Applications to n 1 Graphs 6 5 Further Directions

### Lecture 17 : Equivalence and Order Relations DRAFT

CS/Math 240: Introduction to Discrete Mathematics 3/31/2011 Lecture 17 : Equivalence and Order Relations Instructor: Dieter van Melkebeek Scribe: Dalibor Zelený DRAFT Last lecture we introduced the notion

### INTRODUCTION TO ALGEBRAIC GEOMETRY, CLASS 16

INTRODUCTION TO ALGEBRAIC GEOMETRY, CLASS 16 RAVI VAKIL Contents 1. Valuation rings (and non-singular points of curves) 1 1.1. Completions 2 1.2. A big result from commutative algebra 3 Problem sets back.

### Table of Contents. Introduction... 1 Chapter 1. Vector Bundles... 4. Chapter 2. K Theory... 38. Chapter 3. Characteristic Classes...

Table of Contents Introduction.............................. 1 Chapter 1. Vector Bundles.................... 4 1.1. Basic Definitions and Constructions............ 6 Sections 7. Direct Sums 9. Inner Products

### Finite dimensional topological vector spaces

Chapter 3 Finite dimensional topological vector spaces 3.1 Finite dimensional Hausdorff t.v.s. Let X be a vector space over the field K of real or complex numbers. We know from linear algebra that the

### Algebraic K-Theory of Ring Spectra (Lecture 19)

Algebraic K-Theory of ing Spectra (Lecture 19) October 17, 2014 Let be an associative ring spectrum (here by associative we mean A or associative up to coherent homotopy; homotopy associativity is not

### 6. Metric spaces. In this section we review the basic facts about metric spaces. d : X X [0, )

6. Metric spaces In this section we review the basic facts about metric spaces. Definitions. A metric on a non-empty set X is a map with the following properties: d : X X [0, ) (i) If x, y X are points

### 1 Local Brouwer degree

1 Local Brouwer degree Let D R n be an open set and f : S R n be continuous, D S and c R n. Suppose that the set f 1 (c) D is compact. (1) Then the local Brouwer degree of f at c in the set D is defined.

### Chapter 4. Trees. 4.1 Basics

Chapter 4 Trees 4.1 Basics A tree is a connected graph with no cycles. A forest is a collection of trees. A vertex of degree one, particularly in a tree, is called a leaf. Trees arise in a variety of applications.

### SOME PROPERTIES OF FIBER PRODUCT PRESERVING BUNDLE FUNCTORS

SOME PROPERTIES OF FIBER PRODUCT PRESERVING BUNDLE FUNCTORS Ivan Kolář Abstract. Let F be a fiber product preserving bundle functor on the category FM m of the proper base order r. We deduce that the r-th

### Diagonal, Symmetric and Triangular Matrices

Contents 1 Diagonal, Symmetric Triangular Matrices 2 Diagonal Matrices 2.1 Products, Powers Inverses of Diagonal Matrices 2.1.1 Theorem (Powers of Matrices) 2.2 Multiplying Matrices on the Left Right by

### it is easy to see that α = a

21. Polynomial rings Let us now turn out attention to determining the prime elements of a polynomial ring, where the coefficient ring is a field. We already know that such a polynomial ring is a UF. Therefore

### Math 3000 Running Glossary

Math 3000 Running Glossary Last Updated on: July 15, 2014 The definition of items marked with a must be known precisely. Chapter 1: 1. A set: A collection of objects called elements. 2. The empty set (

### arxiv:math/ v1 [math.nt] 31 Mar 2002

arxiv:math/0204006v1 [math.nt] 31 Mar 2002 Additive number theory and the ring of quantum integers Melvyn B. Nathanson Department of Mathematics Lehman College (CUNY) Bronx, New York 10468 Email: nathansn@alpha.lehman.cuny.edu

### Continuous first-order model theory for metric structures Lecture 2 (of 3)

Continuous first-order model theory for metric structures Lecture 2 (of 3) C. Ward Henson University of Illinois Visiting Scholar at UC Berkeley October 21, 2013 Hausdorff Institute for Mathematics, Bonn

### 6.3 Conditional Probability and Independence

222 CHAPTER 6. PROBABILITY 6.3 Conditional Probability and Independence Conditional Probability Two cubical dice each have a triangle painted on one side, a circle painted on two sides and a square painted

### Geometry 2: Remedial topology

Geometry 2: Remedial topology Rules: You may choose to solve only hard exercises (marked with!, * and **) or ordinary ones (marked with! or unmarked), or both, if you want to have extra problems. To have

### Zachary Monaco Georgia College Olympic Coloring: Go For The Gold

Zachary Monaco Georgia College Olympic Coloring: Go For The Gold Coloring the vertices or edges of a graph leads to a variety of interesting applications in graph theory These applications include various

### You know from calculus that functions play a fundamental role in mathematics.

CHPTER 12 Functions You know from calculus that functions play a fundamental role in mathematics. You likely view a function as a kind of formula that describes a relationship between two (or more) quantities.

### 136 CHAPTER 4. INDUCTION, GRAPHS AND TREES

136 TER 4. INDUCTION, GRHS ND TREES 4.3 Graphs In this chapter we introduce a fundamental structural idea of discrete mathematics, that of a graph. Many situations in the applications of discrete mathematics