Algebraic Topology Homework 4 Solutions
|
|
- Mitchell Elliott
- 5 years ago
- Views:
Transcription
1 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 the identity on A. Then a homotopy relative to A (or just: a homotopy rel. A) from f to g is a map H : X I X satisfying: (1) H(a, t) = a for all a A and all t I, (2) H(x, 0) = f(x) for all x X, and (3) H(x, 1) = g(x) for all x X. Exercise 1. Let C be the cylinder [ π, π] S 1, π 2 : C S 1 the projection. Consider the map f : C C defined by f(s, x) := (s, A s+π x), where ( ) cos θ sin θ A θ := sin θ cos θ is the map S 1 S 1 given by counter-clockwise rotation through the angle θ. Note that f restricts to the identity map on C = {±π} S 1. Define γ : I C by γ(t) := ( π + 2πt, e 1 ). Let T := C/ be the torus obtained from C by identifying the two boundary circles: ( π, x) (π, x) for x S 1. By the universal property of quotients, f determines a map f : T T and γ determines a map γ : S 1 = I/{±1} T. Define b : S 1 T by b(x) := (0, x). Fix a generator e for H 1 (S 1 ) = Z. Our Mayer-Vietoris computation in class shows that H 1 (T ) is freely generated by α := γ e and β := b e. a) Show that f : C C is not homotopic rel. C to the identity. Hint: If it were, then argue that π 2 fγ and π 2 γ would be homotopic as loops based at e 1 S 1. b) Show that, in the ordered basis (α, β), H 1 (f) : H 1 (T ) H 1 (T ) is given by ( ) Hint: First use the explicit basis of H 1 (T ) so check that the projections π 1 : T [ π, π]/{±π} = S 1 and π 2 : T S 1 induce an isomorphism (π 1, π 2 ) : H 1 (T ) H 1 (S 1 ) H 1 (S 1 ). Conclude that f is not homotopic to Id : T T. c) Observe that b gives an alternative proof of a. Solution: For (a): Suppose H : C I C is a homotopy rel. C from f to the identity. Then one sees immediately from the definition of relative homotopy that the composition I I γ Id C I 1 H C π 2 S 1
2 2 would be a homotopy of paths from π 2 fγ to π 2 γ. But π 2 γ is the constant loop at e 1 S 1, while (π 2 fγ)(t) = π 2 (f( π 2 πt, e 1 )) so that π 2 fγ is a generator of π 1 (S 1 ) = Z. = π 2 ( π + 22πt, A 2πt e 1 ) = (cos(2πt), sin(2πt)), For (b): In fact, π 1 π 2 : T S 1 S 1 is a homeomorphism so the fact that π 1 π 2 is an isomorphism is just the general principle that π 1 is compatible with products. Now one just calculates that π 1 α = [π 1 γ] π 2 α = [π 2 γ] = [1 e1 ] = 0 π 1 β = [1 0 ] π 2 β π 1 fα = [π 1 fγ] π 2 fα π 1 fβ = [π 1 fb] = 0 π 2 fβ = [π 2 fb]. For (c): If H : C I C is a homotopy rel. C from f to the identity, then, in particular, we have H( π, x, t) = ( π, x) (π, x) = H(π, x, t), so H would descend to a homotopy H : T I T from f to Id. Exercise 2. Let S n R 2 be the circle of radius 1/(2n) centered at the point (1/(2n), 0) and let X := n=1 be the Hawaiian earring (with the subspace topology from the inclusion X R 2 ). Let C(X) R 3 be the cone on X, v the vertex of the cone. View X as a subspace of C(X) (the top of the cone). Let x 0 X C(X) be the origin. Show that C(X) does not deformation retract onto x 0 even though the inclusion of x 0 into C(X) is a homotopy equivalence. Hint: Suppose H : C(X) I C(X) is a deformation retract onto x 0. Show that for every n {1, 2,... }, there is a point x n S n C(X) and a t n I such that H(x n, t n ) = v. (If not, then you could somehow use H to get a deformation retract of S n onto x 0.) Solution: The tricky part is to establish the assertion in the hint. If this were not true, then for some n {1,... }, H would restrict to a continuous map H : S n I C(X)\{v}. Composing this map with the projection from the vertex p : C(X) \ {v} X, followed by
3 3 a retract r : X S n of the inclusion S n X, we would get a deformation retract from S n onto x 0, which is absurd. Now that this is known, just pick, for each n {1, 2,... } a point x n S n and a time t n I such that H(x n, t n ) = v. Now use the fact that X I is compact to pass to a subsequence of (x n, t n ) that converges, say to (x, t), in X I. We must have x = x 0 because for any ɛ > 0 we can pick an N big enough that S N, S N+1,... are contained in the ɛ-ball around x 0, hence, in particular x N, x N+1,... are contained in this ball. Now continuity of H implies that H(x 0, t) = v x 0, contradicting the assumption that H is a deformation retract of C(X) onto x 0. Exercise 3. Show that the degree map deg : Hom HotTop (S 1, S 1 ) Z yields a bijection between homotopy classes of maps S 1 S 1 and the integers. It might be helpful here to first show that any map f : S 1 S 1 is homotopic to a map that fixes 0 S 1 = R/Z, then note that the Hurewicz isomorphism is natural so that if f : (S 1, 0) (S 1, 0) is a map of pairs, then the Hurewicz isomorphism identifies H 1 (f) and π 1 (f). Exercise 4. For any x R n, let t x : R n R n be the homeomorphism given by translation along x: t x (y) := x+y. By abuse of notation, we also use t x to denote the homeomorphism t x (R n \ {y}) : R n \ {y} R n \ {x + y} for any y R n. Suppose f : R n R n is a homeomorphism. Then f 0 := t f(0) (f (R n \ {0})) is an automorphism of R n \ {0}. This induces an automorphism (f 0 ) of H n 1 (R n \ {0}) = Z, which is necessarily either the identity (in which case we say that f is orientation preserving, or that the degree deg f of f is 1) or multiplication by 1 (in which ase we say that f is orientation reversing, or that deg f = 1). Show that this notion of orientation preserving has the following expected properties: a) Degree is a group homomorphism from Aut(R n ) to the two element group {±1}. b) Every translation of R n is orientation preserving. c) If f : R n R n is the homeomorphism obtained from an invertible matrix A GL n (R), then deg f is the sign of det A R. (This might be a tiny bit tricky, painful, and/or tedious if you don t know anything about the topology of GL n (R), but if worst comes to worst, you can use part a) to reduce to analysing elementary matrices. For this, it will be convenient to use property (e) of degree for maps of spheres on Page 134 in Hatcher, which I did not discuss in class.) Solution: No one really did a good job with the last part. The point is that any invertible matrix is a product of elementary matrices, so by the first part we reduce the last part to the case where A is an elementary matrix E. If the elementary matrix E is a permutation matrix, then det E is the sign of the corresponding permutation to see that this agrees with the degree, just write the permutation as a product of transpositions and apply Property (e) on Page 134 in Hatcher. If E is the elementary matrix given by adding λ times row i to row j (i j), then det E = 1. On the other hand, a path from 0 to λ in R gives an obvious path (through elementary matrices) from Id to E (a homotopy from Id to E), so the degree of E is also 1. If E is given by multiplying row i by λ R, then
4 4 det E = λ. In this case a path from λ/ λ to λ in R gives a homotopy from a matrix whose degree is λ/ λ (by Property (e)) to E. Exercise 5. Exercise 2 on Page 155 in Hatcher. Exercise 6. Exercise 33 on Page 158 in Hatcher. Exercise 7. (Cf. Exercise 8 on Page 155 in Hatcher) Let f(x) = a 0 +a 1 x+ +a d x d C[x] be a degree d polynomial with complex coefficients (a d 0). We also write f : C C for the map determined by this polynomial. Let f(x, y) := a 0 y d + a 1 xy d a d x d be the homogeneous degree d polynomial in C[x, y] obtained by homogenizing f. (1) Show that [x : y] [ f(x, y) : y d ] is a well-defined map ˆf : CP 1 CP 1. (You have to check that this makes sense on equivalence classes and that ( f(x, y), y d ) (0, 0) when (x, y) (0, 0).) (2) Viewing C as a subset of CP 1 via the embedding x [x : 1], show that ˆf extends f. (3) Note that CP 1 = S 2 so it makes sense to speak of the degree of an endomorphism of CP 1. Show that the degree of ˆf is d and that the local degree of ˆf at a root α C CP 1 of f is the multiplicity of α as a root of f. (4) Observe that all the maps ˆf : CP 1 CP 1 thus-constructed (for d fixed) are homotopic. Solution: The first two parts are straightforward. I would next prove the last part, which is easy: the space of all degree d polynomials is just C C d (look at the coefficients), which is connected the last part is immediate. Now, at least for the first assertion in the third part, it suffices to treat the case f = x d, where ˆf[x : y] = [x d : y d ]. Then ˆf 1 ([0 : 1]) = {[0 : 1]} and, on the standard neighborhood C = U 1 CP 1 of [0 : 1], ˆf coincides with the map x x d from C to C, which certainly has local degree d at the origin. In general, suppose α is a root of f of multiplicity e and let us show that the local degree of f : C C at α is e. We can write f = (x α) e g where g(α) 0. Choose a small closed disc D around α in C such that α is the only root of f in D and let U be the interior of D. By definition of local degree, we want to show that f : H 2 (U, U α) H 2 (C, C ) is multiplication by e. Looking at the LESs of the pairs involved here, we see that it is equivalent to prove that f : H 1 (U α) H 1 (C ) is multiplication by d. Note that g : U α C is homotopic to the constant map 1 because C is homotopy equivalent to S 1 via the argument map z z/ z and g/ g makes sense on the entire disc D. It follows that f : U α C is homotopic to x (x α) e, which is clearly degree e. Exercise 8. Let n be the standard n-simplex, so that Id : n n is a singular n-simplex in n. Write [ n ] for the corresponding element of the complex C( n, n ) of singular chains in n modulo singular chains in n. Obviously [ n ] is a cycle in C( n, n ), so it determines some homology class [ n ] H n ( n, n ).
5 5 On the other hand, we showed in class that H n ( n, n ) = Z. We claim that [ n ] is a generator for H n ( n, n ). This is not so obvious because our computation(s) of H n ( n, n ) involve constructions through which it is difficult to follow the homology class [ n ]. For example, the closely-related statement that the cycle n [ n ] := ( 1) i (Id i n ) i=0 generates H n 1 ( n ) = Z is difficult to see from our Mayer-Vietoris-based calculation of H n 1 ( n ) because the singular chain [ n ] is not actually in the complex C U ( n ) for the usual cover U, V of n = S n 1 used in that argument, so one would have to go into the details of barycentric subdivision to try to follow this cycle through that computation (=painful!). Here is an easy way to prove our claim: (1) Show that if the cycle [ n ] were a boundary in C( n, n ), then any singular n-simplex σ : n n would be a boundary in C( n, n ), hence H n ( n, n ) would be zero, which it is not. (2) Since H n ( n, n ) = Z, we conclude from the previous part that our class [ n ] must be d times a generator of H n ( n, n ) for some positive integer d in particular [ n ] is divisible by d in H n ( n, n ). Show that this implies that any homology class in H n ( n, n ) is divisible by d. Since H n ( n, n ) = Z, we must have d = 1, proving our claim. (3) Conclude, by studying the LES for the pair ( n, n ), that [ n ] is a generator for H n 1 ( n ) = Z.
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 informationPart II. Algebraic Topology. Year
Part II Year 2017 2016 2015 2014 2013 2012 2011 2010 2009 2008 2007 2006 2005 2017 Paper 3, Section II 18I The n-torus is the product of n circles: 5 T n = } S 1. {{.. S } 1. n times For all n 1 and 0
More informationALGEBRAICALLY TRIVIAL, BUT TOPOLOGICALLY NON-TRIVIAL MAP. Contents 1. Introduction 1
ALGEBRAICALLY TRIVIAL, BUT TOPOLOGICALLY NON-TRIVIAL MAP HONG GYUN KIM Abstract. I studied the construction of an algebraically trivial, but topologically non-trivial map by Hopf map p : S 3 S 2 and a
More informationSolution: We can cut the 2-simplex in two, perform the identification and then stitch it back up. The best way to see this is with the picture:
Samuel Lee Algebraic Topology Homework #6 May 11, 2016 Problem 1: ( 2.1: #1). What familiar space is the quotient -complex of a 2-simplex [v 0, v 1, v 2 ] obtained by identifying the edges [v 0, v 1 ]
More informationHomework 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 informationThe 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 informationExercises 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 information1. 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 informationMATH 215B HOMEWORK 5 SOLUTIONS
MATH 25B HOMEWORK 5 SOLUTIONS. ( marks) Show that the quotient map S S S 2 collapsing the subspace S S to a point is not nullhomotopic by showing that it induces an isomorphism on H 2. On the other hand,
More information10 Excision and applications
22 CHAPTER 1. SINGULAR HOMOLOGY be a map of short exact sequences of chain complexes. If two of the three maps induced in homology by f, g, and h are isomorphisms, then so is the third. Here s an application.
More informationApplications 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 informationMath 147, Homework 6 Solutions Due: May 22, 2012
Math 147, Homework 6 Solutions Due: May 22, 2012 1. Let T = S 1 S 1 be the torus. Is it possible to find a finite set S = {P 1,..., P n } of points in T and an embedding of the complement T \ S into R
More informationMTH 428/528. Introduction to Topology II. Elements of Algebraic Topology. Bernard Badzioch
MTH 428/528 Introduction to Topology II Elements of Algebraic Topology Bernard Badzioch 2016.12.12 Contents 1. Some Motivation.......................................................... 3 2. Categories
More informationMath 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 informationAlgebraic Topology exam
Instituto Superior Técnico Departamento de Matemática Algebraic Topology exam June 12th 2017 1. Let X be a square with the edges cyclically identified: X = [0, 1] 2 / with (a) Compute π 1 (X). (x, 0) (1,
More informationAlgebraic 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 information7. 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 informationTopological 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 informationSOLUTIONS TO HOMEWORK PROBLEMS
SOLUTIONS TO HOMEWORK PROBLEMS Contents 1. Homework Assignment # 1 1 2. Homework Assignment # 2 6 3. Homework Assignment # 3 8 4. Homework Assignment # 4 12 5. Homework Assignment # 5 16 6. Homework Assignment
More informationAlgebraic Topology. Oscar Randal-Williams. or257/teaching/notes/at.pdf
Algebraic Topology Oscar Randal-Williams https://www.dpmms.cam.ac.uk/ or257/teaching/notes/at.pdf 1 Introduction 1 1.1 Some recollections and conventions...................... 2 1.2 Cell complexes.................................
More informationConformal Mappings. Chapter Schwarz Lemma
Chapter 5 Conformal Mappings In this chapter we study analytic isomorphisms. An analytic isomorphism is also called a conformal map. We say that f is an analytic isomorphism of U with V if f is an analytic
More informationFUNDAMENTAL 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 informationAlgebraic 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 informationAn Outline of Homology Theory
An Outline of Homology Theory Stephen A. Mitchell June 1997, revised October 2001 Note: These notes contain few examples and even fewer proofs. They are intended only as an outline, to be supplemented
More informationTHE FUNDAMENTAL GROUP AND CW COMPLEXES
THE FUNDAMENTAL GROUP AND CW COMPLEXES JAE HYUNG SIM Abstract. This paper is a quick introduction to some basic concepts in Algebraic Topology. We start by defining homotopy and delving into the Fundamental
More informationManifolds and Poincaré duality
226 CHAPTER 11 Manifolds and Poincaré duality 1. Manifolds The homology H (M) of a manifold M often exhibits an interesting symmetry. Here are some examples. M = S 1 S 1 S 1 : M = S 2 S 3 : H 0 = Z, H
More informationMATH 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 informationMath 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 informationMATH8808: 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 informationMath 215a Homework #1 Solutions. π 1 (X, x 1 ) β h
Math 215a Homework #1 Solutions 1. (a) Let g and h be two paths from x 0 to x 1. Then the composition sends π 1 (X, x 0 ) β g π 1 (X, x 1 ) β h π 1 (X, x 0 ) [f] [h g f g h] = [h g][f][h g] 1. So β g =
More informationMath 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 informationMath 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 informationSolutions to homework problems
Solutions to homework problems November 25, 2015 Contents 1 Homework Assignment # 1 1 2 Homework assignment #2 6 3 Homework Assignment # 3 9 4 Homework Assignment # 4 14 5 Homework Assignment # 5 20 6
More informationIntroduction to Topology
Introduction to Topology Randall R. Holmes Auburn University Typeset by AMS-TEX Chapter 1. Metric Spaces 1. Definition and Examples. As the course progresses we will need to review some basic notions about
More informationAssignment 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 informationB 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 informationLisbon 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 informationFUNDAMENTAL 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 informationCW-complexes. Stephen A. Mitchell. November 1997
CW-complexes Stephen A. Mitchell November 1997 A CW-complex is first of all a Hausdorff space X equipped with a collection of characteristic maps φ n α : D n X. Here n ranges over the nonnegative integers,
More informationMath 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 informationNOTES ON THE FUNDAMENTAL GROUP
NOTES ON THE FUNDAMENTAL GROUP AARON LANDESMAN CONTENTS 1. Introduction to the fundamental group 2 2. Preliminaries: spaces and homotopies 3 2.1. Spaces 3 2.2. Maps of spaces 3 2.3. Homotopies and Loops
More informationMAS435 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 informationSolutions 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 informationHomework 3: Relative homology and excision
Homework 3: Relative homology and excision 0. Pre-requisites. The main theorem you ll have to assume is the excision theorem, but only for Problem 6. Recall what this says: Let A V X, where the interior
More informationMATH 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 informationAN 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 information1. Let r, s, t, v be the homogeneous relations defined on the set M = {2, 3, 4, 5, 6} by
Seminar 1 1. Which ones of the usual symbols of addition, subtraction, multiplication and division define an operation (composition law) on the numerical sets N, Z, Q, R, C? 2. Let A = {a 1, a 2, a 3 }.
More informationSECTION 2: THE COMPACT-OPEN TOPOLOGY AND LOOP SPACES
SECTION 2: THE COMPACT-OPEN TOPOLOGY AND LOOP SPACES In this section we will give the important constructions of loop spaces and reduced suspensions associated to pointed spaces. For this purpose there
More informationFREUDENTHAL 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 informationChapter 2 Linear Transformations
Chapter 2 Linear Transformations Linear Transformations Loosely speaking, a linear transformation is a function from one vector space to another that preserves the vector space operations. Let us be more
More informationEXAMPLES AND EXERCISES IN BASIC CATEGORY THEORY
EXAMPLES AND EXERCISES IN BASIC CATEGORY THEORY 1. Categories 1.1. Generalities. I ve tried to be as consistent as possible. In particular, throughout the text below, categories will be denoted by capital
More informationHairy 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 informationTopology Hmwk 6 All problems are from Allen Hatcher Algebraic Topology (online) ch 2
Topology Hmwk 6 All problems are from Allen Hatcher Algebraic Topology (online) ch 2 Andrew Ma August 25, 214 2.1.4 Proof. Please refer to the attached picture. We have the following chain complex δ 3
More informationMath 147, Homework 1 Solutions Due: April 10, 2012
1. For what values of a is the set: Math 147, Homework 1 Solutions Due: April 10, 2012 M a = { (x, y, z) : x 2 + y 2 z 2 = a } a smooth manifold? Give explicit parametrizations for open sets covering M
More informationEquivalence of the Combinatorial Definition (Lecture 11)
Equivalence of the Combinatorial Definition (Lecture 11) September 26, 2014 Our goal in this lecture is to complete the proof of our first main theorem by proving the following: Theorem 1. The map of simplicial
More informationHomework 3 MTH 869 Algebraic Topology
Homework 3 MTH 869 Algebraic Topology Joshua Ruiter February 12, 2018 Proposition 0.1 (Exercise 1.1.10). Let (X, x 0 ) and (Y, y 0 ) be pointed, path-connected spaces. Let f : I X y 0 } and g : I x 0 }
More informationMath 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 informationHOMOTOPY 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 informationA GLIMPSE OF ALGEBRAIC K-THEORY: Eric M. Friedlander
A GLIMPSE OF ALGEBRAIC K-THEORY: Eric M. Friedlander During the first three days of September, 1997, I had the privilege of giving a series of five lectures at the beginning of the School on Algebraic
More informationNotation. For any Lie group G, we set G 0 to be the connected component of the identity.
Notation. For any Lie group G, we set G 0 to be the connected component of the identity. Problem 1 Prove that GL(n, R) is homotopic to O(n, R). (Hint: Gram-Schmidt Orthogonalization.) Here is a sequence
More informationMath 396. Bijectivity vs. isomorphism
Math 396. Bijectivity vs. isomorphism 1. Motivation Let f : X Y be a C p map between two C p -premanifolds with corners, with 1 p. Assuming f is bijective, we would like a criterion to tell us that f 1
More informationEILENBERG-ZILBER VIA ACYCLIC MODELS, AND PRODUCTS IN HOMOLOGY AND COHOMOLOGY
EILENBERG-ZILBER VIA ACYCLIC MODELS, AND PRODUCTS IN HOMOLOGY AND COHOMOLOGY CHRIS KOTTKE 1. The Eilenberg-Zilber Theorem 1.1. Tensor products of chain complexes. Let C and D be chain complexes. We define
More informationThe Hurewicz Theorem
The Hurewicz Theorem April 5, 011 1 Introduction The fundamental group and homology groups both give extremely useful information, particularly about path-connected spaces. Both can be considered as functors,
More informationMetric spaces and metrizability
1 Motivation Metric spaces and metrizability By this point in the course, this section should not need much in the way of motivation. From the very beginning, we have talked about R n usual and how relatively
More informationHomotopy 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 information7.2 Conformal mappings
7.2 Conformal mappings Let f be an analytic function. At points where f (z) 0 such a map has the remarkable property that it is conformal. This means that angle is preserved (in the sense that any 2 smooth
More information5.3 The Upper Half Plane
Remark. Combining Schwarz Lemma with the map g α, we can obtain some inequalities of analytic maps f : D D. For example, if z D and w = f(z) D, then the composition h := g w f g z satisfies the condition
More informationGEOMETRY 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 informationAlgebraic Topology. Len Evens Rob Thompson
Algebraic Topology Len Evens Rob Thompson Northwestern University City University of New York Contents Chapter 1. Introduction 5 1. Introduction 5 2. Point Set Topology, Brief Review 7 Chapter 2. Homotopy
More informationLemma 1.3. The element [X, X] is nonzero.
Math 210C. The remarkable SU(2) Let G be a non-commutative connected compact Lie group, and assume that its rank (i.e., dimension of maximal tori) is 1; equivalently, G is a compact connected Lie group
More information(FALL 2011) 225A - DIFFERENTIAL TOPOLOGY FINAL < HOPF DEGREE THEOREM >
(FALL 2011) 225A - DIFFERENTIAL TOPOLOGY FINAL < HOPF DEGREE THEOREM > GEUNHO GIM 1. Without loss of generality, we can assume that x = z = 0. Let A = df 0. By regularity at 0, A is a linear isomorphism.
More informationMath 676. A compactness theorem for the idele group. and by the product formula it lies in the kernel (A K )1 of the continuous idelic norm
Math 676. A compactness theorem for the idele group 1. Introduction Let K be a global field, so K is naturally a discrete subgroup of the idele group A K and by the product formula it lies in the kernel
More informationMath 541 Fall 2008 Connectivity Transition from Math 453/503 to Math 541 Ross E. Staffeldt-August 2008
Math 541 Fall 2008 Connectivity Transition from Math 453/503 to Math 541 Ross E. Staffeldt-August 2008 Closed sets We have been operating at a fundamental level at which a topological space is a set together
More informationSOLUTIONS 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 information1. 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 informationCELLULAR HOMOLOGY AND THE CELLULAR BOUNDARY FORMULA. Contents 1. Introduction 1
CELLULAR HOMOLOGY AND THE CELLULAR BOUNDARY FORMULA PAOLO DEGIORGI Abstract. This paper will first go through some core concepts and results in homology, then introduce the concepts of CW complex, subcomplex
More informationTHE 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 informationINVERSE LIMITS AND PROFINITE GROUPS
INVERSE LIMITS AND PROFINITE GROUPS BRIAN OSSERMAN We discuss the inverse limit construction, and consider the special case of inverse limits of finite groups, which should best be considered as topological
More information10. Smooth Varieties. 82 Andreas Gathmann
82 Andreas Gathmann 10. Smooth Varieties Let a be a point on a variety X. In the last chapter we have introduced the tangent cone C a X as a way to study X locally around a (see Construction 9.20). It
More informationExercise: Consider the poset of subsets of {0, 1, 2} ordered under inclusion: Date: July 15, 2015.
07-13-2015 Contents 1. Dimension 1 2. The Mayer-Vietoris Sequence 3 2.1. Suspension and Spheres 4 2.2. Direct Sums 4 2.3. Constuction of the Mayer-Vietoris Sequence 6 2.4. A Sample Calculation 7 As we
More informationHungry, Hungry Homology
September 27, 2017 Motiving Problem: Algebra Problem (Preliminary Version) Given two groups A, C, does there exist a group E so that A E and E /A = C? If such an group exists, we call E an extension of
More informationExercise Sheet 3 - Solutions
Algebraic Geometry D-MATH, FS 2016 Prof. Pandharipande Exercise Sheet 3 - Solutions 1. Prove the following basic facts about algebraic maps. a) For f : X Y and g : Y Z algebraic morphisms of quasi-projective
More informationMaster Algèbre géométrie et théorie des nombres Final exam of differential geometry Lecture notes allowed
Université de Bordeaux U.F. Mathématiques et Interactions Master Algèbre géométrie et théorie des nombres Final exam of differential geometry 2018-2019 Lecture notes allowed Exercise 1 We call H (like
More informationSimplicial Homology. Simplicial Homology. Sara Kališnik. January 10, Sara Kališnik January 10, / 34
Sara Kališnik January 10, 2018 Sara Kališnik January 10, 2018 1 / 34 Homology Homology groups provide a mathematical language for the holes in a topological space. Perhaps surprisingly, they capture holes
More informationBare-bones outline of eigenvalue theory and the Jordan canonical form
Bare-bones outline of eigenvalue theory and the Jordan canonical form April 3, 2007 N.B.: You should also consult the text/class notes for worked examples. Let F be a field, let V be a finite-dimensional
More informationThus, X is connected by Problem 4. Case 3: X = (a, b]. This case is analogous to Case 2. Case 4: X = (a, b). Choose ε < b a
Solutions to Homework #6 1. Complete the proof of the backwards direction of Theorem 12.2 from class (which asserts the any interval in R is connected). Solution: Let X R be a closed interval. Case 1:
More informationDIFFERENTIAL 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 information1 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 informationTangent spaces, normals and extrema
Chapter 3 Tangent spaces, normals and extrema If S is a surface in 3-space, with a point a S where S looks smooth, i.e., without any fold or cusp or self-crossing, we can intuitively define the tangent
More informationHomology 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 information1 Differentiable manifolds and smooth maps
1 Differentiable manifolds and smooth maps Last updated: April 14, 2011. 1.1 Examples and definitions Roughly, manifolds are sets where one can introduce coordinates. An n-dimensional manifold is a set
More information1 )(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 informationStratification 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 informationarxiv: 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 informationS n 1 i D n l S n 1 is the identity map. Associated to this sequence of maps is the sequence of group homomorphisms
ALGEBRAIC TOPOLOGY Contents 1. Informal introduction 1 1.1. What is algebraic topology? 1 1.2. Brower fixed point theorem 2 2. Review of background material 3 2.1. Algebra 3 2.2. Topological spaces 5 2.3.
More informationarxiv: v2 [math.ag] 24 Jun 2015
TRIANGULATIONS OF MONOTONE FAMILIES I: TWO-DIMENSIONAL FAMILIES arxiv:1402.0460v2 [math.ag] 24 Jun 2015 SAUGATA BASU, ANDREI GABRIELOV, AND NICOLAI VOROBJOV Abstract. Let K R n be a compact definable set
More informationEQUIVARIANT COHOMOLOGY. p : E B such that there exist a countable open covering {U i } i I of B and homeomorphisms
EQUIVARIANT COHOMOLOGY MARTINA LANINI AND TINA KANSTRUP 1. Quick intro Let G be a topological group (i.e. a group which is also a topological space and whose operations are continuous maps) and let X be
More informationMath 757 Homology theory
Math 757 Homology theory March 3, 2011 (for spaces). Given spaces X and Y we wish to show that we have a natural exact sequence 0 i H i (X ) H n i (Y ) H n (X Y ) i Tor(H i (X ), H n i 1 (Y )) 0 By Eilenberg-Zilber
More informationTOPOLOGY 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````` The Belt Trick
````` The Belt Trick Suppose that we keep top end of a belt fixed and rotate the bottom end in a circle around it. Suppose that we keep top end of a belt fixed and rotate the bottom end in a circle around
More information