MAS435 Algebraic Topology Part A: Semester 1 Exercises

Transcription

1 MAS435 Algebraic Topology Part A: Semester 1 Exercises Dr E. L. G. Cheng Office: J24 e.cheng@sheffield.ac.uk You should hand in your solutions to Exercises #n at the lecture in week n. I will not mark unstapled homework. The exercises are available on the website. Here are some general suggestions for how to study each week. Study the previous week s notes and make sure you have filled in any gaps. Make sure you have learnt any definitions. Do the exercises, which will help you understand the previous week s material and maybe introduce you to the next week s material. If you are stuck on an exercise, you should at least: 1. write down the definitions of everything you are given, and 2. write down the definitions of everything you need to show. Read over the next week s notes in advance of the next lecture, because then you ll get much more out of the lecture. Radical eh? Ask others in the class for help, and help them too. Explaining things to others is one of the best ways to really understand maths. Come to office hours. Use Hatcher to help you. The whole book is available online. 1

2 Exam marking scheme Each question will be given two separate marks out of 4, one for correctness/completeness (C/C) and one for rigour/presentation (R/P). The marks will not be added up, but are to be interpreted as 4 = First 3 = 2:1 2 = 2:2 1 = 3rd 0 = fail/not attempted Note that pass degrees will only be dealt with in the overall classification; there will be no notion of pass standard for individual questions. Under correctness/completeness: 4 = Complete and correct (or almost so) 3 = Either slightly incomplete or slightly incorrect 2 = Either substantially incomplete or incorrect, but with the right ideas 1 = A few correct ideas but little else 0 = Barely any correct ideas, or not attempted Under rigour/presentation: 4 = Logical arguments, well-structured, coherently presented, using an appropriate level of formality and no nonsensical statements. 3 = Here, the right ideas might be presented in a slightly unclear way, possibly with holes in the logic or slightly chaotic organisation, or slightly too informally. There should be no nonsensical statements. 2 = As above but with more drastic problems. At this level a proof is likely to be very sketchy with many unjustified steps; it might go backwards; it might be an informal presentation of ideas rather than a formal proof; it might be so disorganised that it is extremely hard to follow even though it contains some truth; it might contain nonsense alongside true facts. 1 = At this level a proof is likely to contain little more than the correct main idea, informally put. 0 = fail/not attempted An overall judgement in each of these categories will then be made based on the profile of marks for individual questions. The C/C mark will be used as the first indicator of degree class, with the R/P mark giving placement within the class or, in cases where the C/C mark is borderline and the R/P mark extreme, tipping the result over a class boundary in either direction. Some likely combinations are listed below. 2

3 class C/C R/P region First 4444* 4433* low low 2:1 4444* 4332* high high medium low 4433* low 2: high 3222* 3221* medium 3222* 2221* low Homework marking scheme I will mark your homework according to this scheme, where for each set of exercises I will give an overall C/C mark and an overall R/P mark. However the homework won t count to your final grade. 3

4 Exercises #2 Hand in: Week 2 Most of this sheet is an introduction to paths and homotopy. A path in a space X is defined to be a continuous map I X, where I is the unit interval. This definition tries to formalise the idea of literally drawing a path somewhere as a line. So it s not just about the mark you leave with your pen it also tells us how fast we drew the line, whether we stopped and paused anywhere, whether we doubled back on ourselves and so on. 1. The following are some paths in R 2 defined as maps p : I R 2. For each one, draw the image of p in R 2. This is to get you thinking about the difference between a path in a space and its image in the space. i) p(t) = (t, t) ii) p(t) = (t + 1, 1) iii) p(t) = (t 2 + 1, 1) iv) p(t) = (t 3 + 1, 1) v) p(t) = (2t, 1) vi) p(t) = (1, 1) vii) p(t) = (cos 2πt, sin 2πt) viii) p(t) = (cos 4πt, sin 4πt) 2. For the following pairs of points a and b in R 2 construct a path from a to b, that is, a continuous map p : I R 2 such that p(0) = a and p(1) = b. We often have to construct paths from one point to another when we prove things in topology. i) a = (0, 0), b = (1, 0) ii) a = (1, 1), b = (1, 3) iii) a = (1, 1), b = (3, 4) iv) a = (x 1, y 1 ), b = (x 2, y 2 ) 3. Let f, g be continuous maps S 1 R 3. Suppose we have, for every x S 1 a path p x : I R 3 such that p x (0) = f(x) and p x (1) = g(x). Draw a picture that you think describes what is going on here. (Hint: start by drawing the image of f and the image of g.) Can you see how this gives us a map I S 1 R 3? 4. Read pages 1 3 of Hatcher. 5. Look up the definition of topological space on Wikipedia and see how much of the article you understand. Wikipedia is a fantastic resource for mathematics but should of course always be read critically. If you re doing something that matters you should always check the Wikipedia information against something officially published. Furthermore, in rigorous mathematics Wikipedia should never be cited as your source. Some project supervisors may allow Wikipedia citations, but I am not one of them. 4

5 Exercises #3 Hand in: Week 3 1. i) Construct a continuous map R 2 \ {0} S 1 which is the identity on S 1, where here S 1 is the standard circle embedded in R 2. ii) Is there a continuous map R 2 S 1? 2. Express the torus S 1 S 1 as a quotient of R 2. Hint: the map R R S 1 S 1 that we did in the lecture is the quotient map. 3. Restate the definition of topological space in terms of closed sets instead of open sets. 4. Let X be any set. Put a metric on X so that the associated metric topology is the same as the discrete topology on X, in which every subset of X is open. 5. i) Given paths a γ1 b γ2 c in a space X, we can define a new path γ 2 γ 1 : a c by going along γ 1 twice as fast and then along γ 2 twice as fast. Make this definition precise. ii) Is the operation on paths in a space associative? That is, given paths in a space X, do we have a γ1 b γ2 c γ3 d (γ 3 γ 2 ) γ 1 = γ 3 (γ 2 γ 1 )? iii) Let X be a space and a X. Try to make the set of loops on a into a group. 5

6 Exercises #4 Hand in: Week 4 1. i) Show that if X has the discrete topology and Y is any topological space, all maps X Y are continuous. ii) Show that if Y has the indiscrete topology and X is any topological space, all maps X Y are continuous. 2. i) Express the torus S 1 S 1 as a quotient of I 2. ii) Express the sphere S 2 as a quotient of I 2 by collapsing the boundary to a point. iii) Find two circles on a torus such that, if we collapse them to a single point, we get the sphere S Show that the map [0, 1) S 1 given by is not a homeomorphism. t (cos2πt, sin2πt) 4. Show that the disc D 2 is homotopy equivalent to a point. Hint: it may help to take D 2 to be the standard disc embedded in R 2 but expressed in polar coordinates, and take the single point to be the origin. 5. Let γ : a b be a path in X, and γ its reverse path. Show that γ γ is homotopic to the constant path at a. 6. How many different loops can you find in the space that looks like a figure 8? (That is, two copies of S 1, joined at a single point.) 6

7 Exercises #5 Hand in: Week 5 1. Let X = [0, 1) as a subspace of R with the metric topology. i) Show that [0, 1 2 ) is open in X although it is not open in R. ii) Find a subset U X such that U is closed in X although it is not closed in R. 2. Show that the group operation of the fundamental group is well-defined. That is, show that if [f 1 ] = [f 2 ] and [g 1 ] = [g 2 ] then [f 1 ].[g 1 ] = [f 2 ].[g 2 ]. 3. i) Prove that if X is contractible then it is path-connected. Hint: if X is contractible then there is a homotopy from id X to cst a for some point a X. Show that this gives a path from any x to a, and use these paths to construct a path from any x to any y. ii) Is the converse true? Prove it or find a counter-example. 4. Let n N. Construct a continuous map S 1 S 1 such that the pre-image of any point is a set of n points, justifying your answer carefully. Hint: wind the circle n times round itself. 5. This question is revision on groups, which we will soon be needing. i) Find all the subgroups of Z 2 Z 2, justifying your answer carefully. We will later be seeing that the subgroups of the fundamental group of a space X correspond in a precise way to covering spaces of X, that is, spaces that wrap around X nicely. So it s important to be able to find all the subgroups of a group. ii) Find all the subgroups of Z, justifying your answer carefully. iii) Write down the definitions of: group homomorphism, kernel, image, normal subgroup, quotient group, Cartesian product of groups. Also write down the statement of the First Isomorphism Theorem. These are all concepts that we will be using when we use fundamental groups to study spaces, so it s important that you remind yourself what they are. iv) Find a concept from group theory that is an anagram of a British supermarket name. 7

8 Exercises #6 Hand in: Week 6 This sheet is not as long as it looks, but it does go over the page. 1. Which of the following maps f : X Y has the property: There is a natural number n such that for all y Y, f 1 (y) is a set of n points. i) f : I 2 S 1 S 1 by identifying the edges in the usual way ii) f : I S 1 by identifying the endpoints. iii) f : [0, 1) S 1 by t (cos2πt, sin2πt) iv) f : S 1 Möbius Band by inclusion into the boundary v) The quotient map S 2 RP 2 which identifies antipodal points. This property is going to be one of the important properties of covering maps that we ll see. You construct a map S 1 S 1 with this property on the last sheet. 2. In this question you do not need to define the maps precisely; just describe them informally. But you should make sure they re continuous, at least informally. i) Can you think of a map from the 11-holed torus to the 3-holed torus that satisfies the above property? What value of n do you get? Hint: draw the 11-holed torus as below: ii) Can you think of a map between the spaces as below, satisfying the above property? What value of n do you get? 3. i) Define a map f : S 1 S 1 that wraps the first circle three times around the second. ii) We know that there is one loop on S 1 for every integer. We know that f maps loops to loops. Which loops on the second circle get hit by this map? (That is, which loops on the second circle are images of loops on the first circle under this map?) This will give us an important correspondence between fundamental groups of spaces and fundamental groups of their covering spaces. 4. i) Show that if F : C D is a functor then F preserves isomorphisms, that is, if f is an isomorphism in C then F f is an isomorphism in D. ii) Provide an example to show that the converse is not true. Hint: try the fundamental group functor π 1. More overleaf. 8

9 5. An initial object in a category C is an object I such that given any object X C there is a unique morphism I X. A terminal object in C is an object T such that given any object X C there is a unique morphism X T. i) Show that the empty set is an initial object in Set, and any one-element set is a terminal object in Set. ii) What are the initial and terminal objects in Top and Grp? iii) Does the fundamental group functor π 1 : Top Grp preserve initial and terminal objects? Hint: F preserves initial objects if whenever I is initial, F I is also initial. Similarly for terminal objects. iv) Does the forgetful functor U : Grp Set preserve initial and terminal objects? 6. Try and invent the definition of fundamental groupoid. It should be like fundamental group but should use all paths in a space, not just loops. (You can look it up, if you think that s easier than inventing it.) 9

10 Exercises #8 Hand in: Week 8 (after reading week) 1. Let X be a Möbius Band. Now X is homotopy equivalent to its central circle so π 1 (X) is Z. Let f : S 1 X be the inclusion of S 1 into the boundary of X. What is the group homomorphism π 1 f : π 1 (S 1 ) π 1 (X)? 2. What space is formed by glueing two Möbius Bands together along their boundary? 3. (Informal.) The picture below depicts a space X = A B and a loop γ in X based at x A B. (The black blob is supposed to be a hole. So A is a disk with a small hole off to right, B is a disk with a small hole off to the left, and the union matches up the holes so that the result is also homotopy equivalent to an annulus.) Show that γ is homotopic to a product γ 2 γ 1 of loops γ 1 in A and γ 2 in B. 4. i) Prove directly that if X is contractible then it is simply-connected. Hint: you need to prove that every loop based at a is homotopic to the constant loop at a. ii) Is the converse true? Prove it or give a counter-example. 5. Let X be the based space formed by glueing S 1 and S 2 at the basepoint, as depicted below. Find all the connected covering spaces of X, indicating which one is simply-connected. Hint: there should be one for each subgroup of π 1 (X), which is Z. Please turn over. 10

11 6. Let X be the based space formed by glueing two (based) copies of RP 2 at the basepoint. We will depict this as below: Find covering maps from each of the following spaces to X, and say how many sheets each has. Recall the number of sheets how many times the covering spaces covers X, or formally, how many point are in the pre-image of any given point of X. i) ii) iii) Can you think of a covering space of X that is simply-connected? 7. How many covering spaces can you dream up for the Hawaiian earring (depicted below)? This is open-ended. If you work out how to make a whole lot of quite similar ones, see if you can make some very different ones. 11

12 Exercises #9 Hand in: Week 9 1. i) Let X be the space formed from a unit square by glueing its edges together with orientations shown below. Find a loop γ in X such that [γ] id in π 1 X but [γ] 4 = 1. ii) Now let n be any natural number. Generalise the space X above to make a space with a loop γ such that [γ] id in π 1 X but [γ] n = 1 2. i) Find all the subgroups of Z 4. Justify your answer. ii) Let X be the space as above, which has fundamental group Z 4. Find all its connected covering spaces. Justify your answer. 3. Find all the connected covering spaces of the Möbius Band. 4. i) Find all the subgroups of Z Z. Justify your answer. ii) Hence classify the connected covering spaces of a torus. Justify your answer. (Do I have to say justify your answer every time??) 5. In this question we will prove that π 1 (X Y ) = π 1 (X) π 1 (Y ). We need to show that loops in the product space correspond to pairs of loops, one in X and one in Y. If we consider a loop γ in the product space we see that we have the following maps: We define a map I γ X Y p X q Y π 1 (X Y ) π 1 (X) π 1 (Y ) [γ] ([pγ], [qγ]) Show that this map is well-defined, injective and surjective. (Note that we take the basepoint of X Y to be (x, y) where x and y are the basepoints of X and Y respectively.) 6. Let (X, x) be a based space and ( X, p x) (X, x) be a covering space. Recall that we have a map π 1 (X, x) φ p 1 (x) [γ] γ(1) where γ is the unique lift of the loop γ to a path in X starting at x. i) Show that if X is path connected then φ is surjective. Hint: first carefully write down what each of these things means. ii) Show that if X is simply-connected then φ is injective. Hint: first carefully write down what each of these things means. Then use the fact that X is simply-connected, so any two paths with the same endpoints must be homotopic. Then use the fact that you can compose this upstairs homotopy with p to get a downstairs homotopy. 12

13 Exercises #10 Hand in: Week 10 This week s homework is shorter I hope, as last week s was hard, and many of you should be busy with project drafts this week. 1. What does Van Kampen s Theorem tell us about: i) the construction of S 2 as in Example 7.5, and ii) the construction of S 1 as in Example 7.6? 2. What is the fundamental group of the space formed from S 2 by identifying the north and south pole? Hint: try and deform this space into something homotopy equivalent, whose fundamental group is more obvious. 3. Let X be the space given by identifying the edges of a square as below: a a b Show that this is in fact the Klein Bottle. b Hint: cut it up and stick it back together to make the usual construction of a Klein Bottle from a square. 4. Make sure you understand the example of Van Kampen s theorem where we glued a disc onto a Möbius Band. 13

14 Exercises #11 Hand in: Week Let X be the space formed from S 2 by identifying the north and south pole. Express this space as a cell complex, and hence calculate its fundamental group. Hint: In Ex#9, q.2, we found that the fundamental group of this space is Z, so you should get the same answer by this different method! 2. Here are three spaces we have seen that are homotopy equivalent but not homeomorphic: S 2 S 1, the space X above, and the space Y given by the union of a sphere S 2 with one of its diameters, as depicted below. Compare the connected covering spaces of these three spaces. Hint: the covering spaces will not be the same, as the spaces themselves are not homemorphic to one another. But they will be similar in certain ways. 3. For each of the following groups G, construct a topological space with G as its fundamental group, justifying your answer. (i) G = Z Z (ii) G = Z 2 (iii) G = Z 3 (iv) G = Z Z Z. (v) G = a, b, c, d abacad 1 a 1 4. Let X be the space formed by glueing two annuli along their outer boundary circle. Calculate the fundamental group of X twice once using Van Kampen s Theorem, and once by showing that X is homotopy equivalent to a familiar space. What important result do you need to invoke in the second case? Moral: if you re trying to calculate the fundamental group of something, it s always a good idea to do it by two different methods as a way of double checking your answer. 5. Prove the induction part of Theorem That is, if we write mt for T # #T m times, and nrp 2 similarly, then mt #nrp 2 (2m + n)rp 2 given that the result is true for m = n = 1. More overleaf. 14

15 6. In this question we are going to calculate the fundamental group of the two-holed torus as T #T, starting with a torus with a disc cut out, and then glueing two of those together using Van Kampen s theorem. i) Let G be the group Draw a picture of X G as a cell complex. a, b, c aba 1 b 1 c. ii) Let X be the space formed by cutting a disc out of a torus. Show that X G = X, and thus π 1 X = a, b, c aba 1 b 1 c. Hint: start with the usual expression of a torus as a cell complex (with a and b on its edges), and then cut out a disc with boundary c. Which disc must this be? iii) Express the two-holed torus M 2 as a union of two copies of X glued along the boundary circle. Use Van Kampen s theorem to calculate the fundamental group of M 2. 15