# Topologie en Meetkunde, 2013

Schedule The hoorcollege of WISB341 is on mondays from 11:00-to 12:45 uur, in room BBL 169
in room MIN 211, except for week 22 (=27 may) where the class will be held in MIN 012.
Classes are held in English. The werkcollege is wednesday from 9:00 to 10:45 in room BBL 169

Teachers The main teacher is André Henriques. The first week, I will be out of town and Gil Cavalcanti will replace me.
The assistant is Felix Denis (F.L.Denis@students.uuuu.nlnl)

Dictaat We will start by following these notes on the classification of surfaces and then we will follow the first few chapters of Hatcher's book. .

Homework There will be homework, to be handed in during the werkcollege 9 days after the day when it is posted on the website.
The homework counts for 10% ofthe final grade.

Exams There are two exams for this class: one written exam half-way,
and one oral exam at the end. Both count for 45% of the final grade.
Exam questions for the oral exam:
• Define "manifold". Give some examples and counterexamples.
• State the classification of surfaces. Explain the terms "conneced sum", "orientable" and "Euler characteristic".
• Define "CW-complex" with particular emphasis on the topology. Give some examples and counterexamples.
• Define "homotopy equivalence", and prove that it is an equivalence relation. Give some examples.
• Define π1(X) and compute it for as many spaces as you can.
• State van Kampen's theorem and sketch its proof.
• Define "cover". Give some examples and counterexamples. State some classification theorems about covers.
• Define "universal cover" and explain its universal property.
• State and prove the path lifting property for covers.
• Define "homology". Compute it for some familiar examples.
The oral exam will take place in room 613 of the Hans Freudenthalgebouw. Here is the schedule:
 Show up at Monday 24th Tuesday 25th 8:00 Mohamed Hashi 8:40 Wojciech Bizon Oscar van der Heide 9:20 Aleksander Gavus Nick Plantz 10:00 Remie Janssen Jan-Willem Buurlage 10:40 Tami Welker Leonie van Steijn 11:20 Tom Bannik Yiteng Dang 12:30 Tara Drwenski Luuk Hendrikx 13:10 Lauran Toussaint Joep van den Hoven 13:50 Henri van den Pol Tim Coopmans 14:30 Lucet Stefano Maxim van Oldenbeek 15:10 Serop Lazarian Remco de Boer 15:50 Michiel Tel Paul Scharf

Material covered

Monday June 17th: I finished the definition of homology, and introduced the following notions on the way: the standard n-simplex, a singular n-simplex, the group of n-chains, the group of n-cycles, the group of n-boundaries, the n-th homology group. I computed the homology of S1, of S1×S1, and of the Klein Bottle using triangulations of those manifolds (this is treated under the name "simplicial homology" in the book).
Monday June 10th: The classification of connected covers of X via conjugacy classes of subgroups of the fundamental group of X. The definition of higher homotopy groups, and the proof that they are abelian. The (beginning of the) definition of homology.
Homework, hand in Wed June 12th (at the beginning of the class): Consider the CW-complex X with one 0-cell, two 1-cells a and b, and three 2-cells A B C with respective attaching map a2, b2, and aba-1b-1. What is the order of the fundamental group of X? Make a complete list of all connected covers of X (up to isomorphism). For each cover in the list, make a complete description of its natural CW-complex structure (how many cells of each dimension, and what are the attaching maps). Here, the "natural CW-complex structure" is the one such that the covering map sends (interiors of) cells homeomorphically to (interiors of) cells. Identify the universal cover in the list of all covers.
Wednesday June 5th: Exercises from the book (chapter 1, page 79): 10, 14. • Let F2 be the free group on two generators. An F2-set (that is, a set with an action of F2) is the same thing as a set equipped with two permutations. Given a set S equipped with two permutations σ and τ, construct a cover of SvS1 such that p-1(*)=S, and such that the action of the two loops in SvS1 recovers the two given permutations. Apply that construction to the following concrete cases: • S={1,2}, σ=(1,2), τ=e; • S={1,2,3}, σ=(1,2), τ=(1,2,3); • S={1,2,3}, σ=(1,2,3), τ=(1,2,3); • S={1,2,3}, σ=(1,2), τ=(2,3); • An action of ℤ2 on a set S is the same thing as the data of two permutations σ and τ that commute: στ=τσ. Given two such permutations, construct a cover of S×S1 such that the action of the the two loops in S×S1 recovers the two given permutations. Apply the construction to the following concrete cases: • S={1,2}, σ=(1,2), τ=e; • S={1,2,3}, σ=(1,2), τ=(1,2); • S={1,2,3}, σ=(1,2,3), τ=(1,2,3); • S={1,2,3}, σ=(1,2,3), τ=(1,3,2);
Monday June 3st: I recalled the definition of universal cover. I stated and proved the universal property of universal covers. I proved that universal covers are always simply connected. I recalled what it means for a group to act on a set, and I stated the theorem according to which there is an equivalence of categories between the category of G-sets (the category of sets equipped with an action of G -- here, G is the fundamental group of X), and the category of covers of X.
Wednesday May 29th: Exercises from the book (chapter 1, page 79): 4, 8, 9 Extra hints: a. there are no homomorphisms from a finite group into the integers. b. Construct a continuous map from X to the universal cover of S1 (as defined in class). • Describe the universal cover of P2. • Prove that there is an action of π1(X) on the universal cover of X, and that this action is free and transitive on the preimage of the base point. In particular, this shows that the degree of the universl cover is equal to the order of the fundamental group.
Monday May 27th: Proof of the path lifting property. The definition of the universal cover (I defined it as the space whose points are equivalence classes of paths starting from the base point), and some examples: universal cover of torus, of Klein bottle, of genus two surface, of wedge of two circles.
Homework, hand in Wed May 29nd (at the beginning of the class): Given a pointed space X, given a group G, and given a group homomorphism φ:G→ π1(X), show that it is always possible to find a pointed space Y and a basepoint preserving map f:Y → X such that the induced map f*1(Y)→π1(X) is equal to φ.
Wednesday May 15th: Exercises from the book (chapter 1, page 52): 16, 17, 19, 20 (chapter 1, page 79):1, 2, 6. • Make a complete list of all double covers of the wedge of two circles. • Make a complete list of all covers of S1.
Monday May 13th: Fundamental group of an infinite wedge fo circles. Finished the proof of the fact that the presentation of the fundamental group can be read off from from the 1-cells and 2-cells of a CW complex. Covers: definition. Examples of double covers of S1 and of S1×S1.
Wednesday May 8th: Exercises from the book (chapter 1, page 52): 3, 4, 6, 7, 8, 14. • Construct a space whose π1 is a cyclic group of order 3. • Using the isomorphism SL2(ℤ) = ℤ/4*ℤ/2 ℤ/6, construct a space whose π1 is SL2(ℤ).
Monday May 6th: Finished the proof of van Kampen's theorem. Examples of applications of van Kampen's theorem: computation of π1 of the wedge of two circles, and computation of π1 of the torus.
Homework, hand in Wed May 8th (at the beginning of the class): • Prove that the fundamental group of a finite connected graph is a free group on n generators, where n is one minus the Euler characteristic of the graph.
Wednesday May 1st: • finish the exercises from last week that you didn't do. • SL2(ℤ) is the group of 2x2 matrices with integer coefficients and determinant 1. Sow that there are subgroups of SL2(ℤ) isomoprhic to ℤ/4 and ℤ/6. Show that they induce a homomorphism (indeed an isomorphism -- but that's hard) from the amalgamated free product ℤ/4*ℤ/2 ℤ/6 to SL2(ℤ). • Use van Kampen's theorem to compute the fundamental group of P2. • Use van Kampen's theorem to prove that the fundamental group of a wedge of two circles (that is, two circles glued at one point) is a free group on two generators.
Monday April 29th: Presentations of groups. Free products of groups. Amalgamated free products. The statement of van Kampen's theorem (and half of the proof).
Wednesday April 24th: Exercises from the book (chapter 1, page 38): 2, 3, 6, 10, 12, 18 • Use the computation of π1(S1)=ℤ to show that every nonconstant polynomial with coefficients in ℂ has a root in ℂ (this is Theorem 1.8 in Hatcher's book). • Prove that ℝ2 and ℝn (n>2) are not homeomorphic.
Monday April 22nd: Computation of the fundamental group of Sn for n>2, and for n=1. Statement without proof: how to get a presentation of π1(X) for X a CW complex by taking one generator per 1-cell and one relation per 2-cell: the attaching maps ofthe 2-cells tell you what relations to impose between the generators.

Wednesday April 10th: Preparation for the exam. Here's a practice exam.
Monday April 8th: Definition of the fundamental group of a pointed topological space. It's a group. The homomorphism π1(X,x) → π1(Y,f(x)) induced by a continuous map f:X → Y. π1 of a product is the product of the π1's. The isomorphism π1(X,x) → π1(X,y) induced by a path between x and y.
Wednesday March 27th: • Show that the "infinite dimenisonal sphere" S=⋃Sn (equipped with the CW-complex topology) is contractible. Hint: show that it is homeomorphic to the "boundary" of an infinite dimensional cube, and use a straight line homotopy. • Recall the "official" definition of CPn as equivalence classes of points of Cn+1-{0}. Prove that CPn is a manifold. • Consider the map from the unit sphere in Cn to CPn-1 given by f(z1,...,zn) := [z1:...:zn:0] and let X:=CPn-1f D2n. Prove that the space X is homeomorphic to CPn (write down a formula for a map X→CPn; prove that the formula defines a continuous map; show that it's a homeomorphism). • The quaternions (H=SpanR{1,i,j,k}) form a non-commutative algebra in which division by non-zero elements makes sense. Define HPn as a quotient of Hn+1-{0} and show that it is a manifold. Show (using the same argument as for the case of CPn) that HPn is a cell complex with one cell in each dimension 0,4,8,...,4n. • The octonions (O=SpanR{1,e1,...,e7}) form a non-commutative and non-associative algebra in which division by non-zero elements makes sense. Show that the naive definition of OPn as a quotient of On+1 does not make sense (for n>2, OPn indeed does not exist). Show that the definition of OP2 by gluing a 16-cell onto S8 does make sense.
Monday March 25th: The definition of CW complexes. Many examples.
Homework, hand in Wed March 27th (at the beginning of the class): Let X be a topological space, let f:Sn-1 → X be a continuous map, and let X∪f Dn be the result of attaching an n-dimensional cell along the map f. Prove that the inclusion of X into X∪f Dn is an NDR (definition given in class).
Wednesday March 20th: Consider the standard embedding of S1 into S3: show that the open complement S3-S1 deformation retracts onto S1 (not the same S1). Prove that a torus minus a point is homotopy equivalent to a Klein bottle minus a point. Prove that the notion of homotopy equivalence is compatible with the operation of taking the products of two spaces. Let X:=D2, and let f:S1 → X be the map z↦ z2 that goes "twice around the boundary": show that the resulting topological space X∪f D2 is homotopy equivalent (but not homeomorphic) to S2. Study Bing's house with two rooms (definition in the book) and show that it's contractible. Show that CP2 can be obtained by gluing a 4-dimensional cell onto S2.
Monday March 18th: Proof that homotopy equivalence is an equivalence relation. Definition of contractible. Definition of "defomation retract" and of "neighborhood defomation retract" (NDR). Proof that if A⊂X is an NDR pair and A is contractible, then the projection map from X to X/A is a homotopy equivalence. Definition of the operation of gluing cells.
Wednesday March 6th: Prove that if M and N are surfaces, then χ(M#N)=χ(M)+χ(N)-2. Compute for every surface M the value of χ(M). Write down a random word (say of length 20) as in exercise (*), and determine to which surface it corresponds. Show that MB×Dn-2 is homeomorphic to the manifold Dn-1×[0,1]/(x,0)~(r(x),1) that I defined in class. Given an example of a closed non-orientable 3-manifold. Let r:S1→S1 be the reflection along the x-axis; show that there exists a map f:S1→KB such that f is homotopic to rof. Show that {*} and Dn are homotopy equivalent. Show that S1 and MB are homotopy equivalent. Show that Sn-1 and ℝn\{0} are homotopy equivalent. Show that the theta-graph (two vertices a and b, and three edges from a to b) is homotopy equivalent to the barbell graph (two vertices a and b, one loop from a to a, one loop from b to b, and one edge from a to b).
Monday March 4th: Euler characteristic χ(M)=V-E+F pseudo-proof that χ(M) is an invariant of M. Orientability: definition an n-manifold M is not orientable if it is possble to embed MB×Dn-2 in it. Number of boundary components, Euler characteristic, and orientability form a full set of invariants for connected 2-manifolds. From Hatcher chapter 0: Homotopic maps. Homotopy equivalent spaces.
Homework, hand in Wed March 6th (at the beginning of the class): Prove that ℂP2 := (ℂ3-{(0,0,0)})/(a,b,c)~(λa,λb,λc), λ∈ℂ,λ≠ 0 is a manifold. Show that there is a way of embedding S2 into ℂP2 (hint: fix c=0) so that the complement is homeomorphic to an open 4-ball {x∈ℝ4:||x||<1}.
Wednesday Feb 27th: Let Tn=S1×...×S1 be the n-dimensional torus. Show that the two possible versions of the connected sum Tn#Tn (in one of them, we connect the two tori using a handle -- in the other one, we use a cross-handle) are homeomorphic to each other. Show that attaching a handle to an n-sphere produces a manifold that is homeomoprhic to Sn-1×S1. Identify which surface this is in the classification theorem. How about these: a b c d e f g? (note to Felix: use the beamer to show the images of those surfaces during the wercollege -- try to open them in multiple windows so that they are all simultanously visible).
Monday Feb 25th: The real projective plane P2=ℝP2, and the complex projective plane ℂP2. The operation of connected sum is not well defined: the difference between connecting two n-manifolds using a handle or using a cross-handle. Example where the above problem shows up: the connected sum of two complex projective planes. Lemma: cross-handle = two cross caps. Lemma: in the presence of a cross-cap, we can trade a handle for a corss-handle. The classification theorem of compact surfaces with boundary.
Homework, hand in Wed Feb 27th (at the beginning of the class): Let M be a closed n-dimensional manifold ("closed" means "without boundary"), and let N be an n-dimensional manifold whose boundary is Sn-1. Prove that N/(∂N~*) [by that, I mean the topological space obtained by crushing the whole boundary of N to a point] is homeomorphic to M if and only if and only if N is homeomorphic to the open subset of M obtained by removing an open disc (embedded in a way that is locally like the standard embedding of Dn in ℝn).
Wednesday Feb 20th: Recall the classification of surfaces. Which surface does one get by gluing a hexagon according the the following pattern: (*) • aabcb-1c-1 • abcabc • abccb-1a • aabb-1c-1c-1. Prove that the connected sum of a torus and a projective plane is the homeomorphic to the connected sum of three projective planes. Show that by gluing two Möbius bands along their boundary, one gets a Klein bottle. Show that the space obtained by collapsing the whole boundary of S1x[0,1] to a point is not a manifold.
Monday Feb 18th: Basic properties of compact spaces and of maps between them: the image of something compact is compact; inside a compact space closed <=> compact. Various descriptions of P2: as a quotient of S2; as a quotient of D2. Various descriptions of the Klein bottle: as a quotient of [0,1]x[0,1]; as P2#P2. Handles, cross-handles, and cross-caps. Lemma: adding a cross-handle is the same as adding two cross-caps. The standards way of building P2#...#P2 by gluing the sides of a 2n-gon. The standards way of building T2#...#T2 by gluing the sides of a 4n-gon. Classification theorem of surfaces: the statement.
Homework, hand in Wed Feb 20th (at the beginning of the class): Prove that the following two constructions of the connected sum operation yield the same manifold up to homeomorphism. Let M and N be n-dimensional manifolds. Pick continuous embeddings f:Dn→M and g:Dn→N that are locally like the standard embedding of Dn into ℝn. Let Mo be M minus the interior of f(Dn), and let No be N minus the interior of g(Dn). We define X to be (Mo⊔No)/f(x)~g(x) for x ∈ Sn-1, and define Y to be (Mo⊔(Sn-1×[0,1])⊔No)/f(x)~(x,0),g(x)~(x,1) for x ∈ Sn-1. Prove that X and Y are homeomorphic to each other by exhibiting a map X→Y and proving that it is a homeomorphism. [For those who don't know them, I recall some useful facts about compact spaces and continuous maps between them: • If K is compact and f:K→X is continuous, then f(K) is compact. • If K and L are compact spaces and f:K→L is a continuous map that is also a bijection, then its inverse f-1:L→K is automatically continuous.]
Wednesday Feb 13th: • Show that for any manifold M, the connected sum of M with Sn is again M. • Prove that the quotient of T2=S1× S1 by the equivalence relation (x,y)~(±x,±y) is again homeomorphic to T2. • Prove that the quotient of D2 by the equivalence relation x~-x for x∈∂D2 is the projective plane P2. • Show that S1, T1, and P1 are all the same manifold. • Let M be a manifold with boundary. Prove that M⊔M/(m,0)~(m,1), m∈∂M is again a manifold, this time without boundary. This is called the "double" of M. • Apply the doubling procedure to the solid torus (=the "interior" of usual embedding of the torus in ℝ3), which manifold does one get as a result of this doubling procedure? (it can be expressed as the product of two known manifolds). • Apply the doubling procedure to the Möbius band. Which well-known surface does one get?
Monday Feb 11th: Definition of compact. Examples of non-compact manifolds. Explanation of the term "paracompact". Definition of manifolds with boundary. The operations of disjoint union, of cartesian product, and of connected sum. Examples of manifolds: the sphere Sn, the torus Tn=S1×...×S1, projective space Pn = Sn/x~-x, the disc Dn, the Möbius band, the Klein bottle.
Wednesday Feb 6th:...
Monday Feb 4th: the definition of topological spaces, continuous maps and of homeomorphisms. of the product topology, of the subspace topology, of the quotient topology (here I had to first recall equivalence relations and quotients of sets), Definition of Hausdorff spaces, base of a topology and locally Euclidean spaces Definition of (topological) manifold Examples: ℝn, open set of ℝn, Sn ⊂ ℝn+1 with subspace topology: I parametrized Sn using stereographic projection and showed that these projections furnished the local Euclidean structure. Hausdorffness and second countability follow from the same property for ℝn+1. Since second countability is a funny and difficult to grasp property I recalled that those who took manifolds last term saw that it was used to produce partitions of unity and that partitions of unity in turn implied that manifolds are metrizable, allowed us to define integrals and were used in the proof of the theorem stating that de Rham cohomology and Cech cohomology with real coefficients are isomorphic. Since half of the class did not take manifolds I hope that they got the message "second countability => nice things".
Results: midterm exam / oral exam / homework average // final grade
3470814: 4.5 / --
3688143: 9.75 / 9.5 / 9 // 9.5
F120138: 7.5 / 8 / 9.5 // 8.0
3688364: 6 / 6.5 / 7 // 6.5
3668983: 6.25 / 7.5 / 9.5 // 7.0
3695921: 6.5 / 5 / 8 // 6.0
3688356: 8.75 / 8.5 / 7.5 // 8.5
3892565: 6.5 / 8 / 8.5 // 7.5
F120139: 7 / 9 / 10 // 8.0
3345041: 7.75 / 10 / 9.5 // 9.0
3149420: 8.75 / 9.5 / 8 // 9.0
3684385: 7.25 / 8.5 / 9.5 // 8.0
3691373: 9.75 / 8 / 4 // 8.5
3587037: 8.5 / 9.5 / 7.5 // 9.0
3894347: 4.5 / --
3655792: 3.75 / 4 / -- // 4.0
4032608: 5.5 / 7.5 / 9 // 7.0
3588904: 4.75 / --
3647978: 9.75 / 8.5 / 6.5 // 9.0
3379416: 4.75 / --
3989534: 7.5 / 8 / 9.5 // 8.0
3689239: 5.25 / 5 / 8 // 5.0
3683745: 7 / 4 / -- // 5.0
3993466: 9 / 7 / 8 // 8.0
3704823: 6.75 / 6.5 / 5.5 // 6.5
3706915: 9.5 / 10 / 9.75 // 10.0
3701174: 5.25 / --
3590542: 5.25 / 8 / 6.5 // 6.5