-3132013: mark=7.5

-3042936: mark=7

-3037517: mark=7

-0122432: mark=6

-0243418: Not passed

-0316245: Not passed

-3169944: Not passed

-3117731: Not passed

-0371580: Not passed

**
-002?? 85: Not passed**

This is the web-page for the course Topologie en Meetkunde, 2008. You can also have a look at the web-pages for the 2005-2006 and the 2006-2007 year.

General info:

The course is given every Tuesday, from 9 to 10:45, in MIN 205. Starting date: February 5, 2007. It will end on June 17, 2008.

Lecturer: Marius Crainic.

The exercise classes will take place every Thursday from 13:15 to 15:00, in BBL 105. In charge: Camilo Arias Abad and Maarten Dobbelaar. We are trying to find a third assistant for the werkcollege.

The lecture notes from the last year can be seen here. The lecture notes for this year were made available during the year. Here you can see the table of content, while here you can download the entire file at once. The building pieces are:

- Chapter 1 (for the first 2-3 lectures). Note that some of the pictures use colours.

- Chapter 2 Please note that the notes contain more then will be said in the lectures (and the other way around ...). More specifically, some parts of the notes are meant to explain WHY we do certain things- and those parts are written in small characters (so that they are easy to recognaize).

- Extra-section at the end of Chapter II, including some extra-exercises.

- Connectedness (the first section of Chapter III).

- Compactness and local compactness (sections 3-4 of Chapter III).

- Normal spaces and metrizability (the rest of Chapter III).

- Attaching cells (Chapter IV).

- The fundamental group. (first part of Chapter V).

- Covering spaces (the rest of Chapter V).

- Seifer-van Kampen theorem. .

- How to compute fundamental groups .

- Table of content for the entire lecture notes.

Announcement: the solutions of the **Bonus exercises** will not be accepted after 9AM on Tuesday mornings.

**Enjoy the sphere ** (and not only). (Thanks to Ralph for the link!).

Division into weeks:

**WEEK 6 (February 5):** The first 4 sections of Chapter 1 in the lecture notes, and a bit from section 5 and 6.

**WEEK 7 (February 12).** We continued with Chapter 1 of the notes: Moebius band, torus, Klein bottle, projective plane and started talking about gluings (equivalence relations and quotients)- to which we will return next time.

**WEEK 8 (February 19).** We started with a generous reminder, then continued with chapter 1: quotients, metric spaces, the topology associated to a metric. Then we moved to chapter 2: the general notion of topology and topological space, extreme examples, and then we proved that the topology induced by a metric is, indeed, a topology. The bonus exercise for this week is Exercise 1.28 from Chapter 1, and it is worth 0.25. I remind you that you have to hand it in before next Tuesday at 9AM.

Exercises for the werkcollege's: 1.23, 1.27, 2.2, 2.3, 2.4 (if these are not enough, please look at the others).

**WEEK 9 (February 26).**Continuous functions, neighborhoods, convergence of sequences, basis of neighborhoods, 1-st countability axiom, the Hausdorffness axiom, example: metric spaces, then theorems 2.20 and 2.26.

**WEEK 10 (March 4).** Section 6, 7, 8, 9 in the notes.

Exercises for the werkcolleges (in this order- try to get as far as possible): prove the statements in Example 2.31, then do Ex. 2.19, Ex. 2.21, Ex. 2.25. Then read the second part of section 9. If you are done with these, do Ex. 2.20, Ex. 2.24, Ex. 2.26. If you have more time, look at Example 2.46, and then at Example 2.47.

Bonus Exercise: Ex. 2.23+ Explanation of Example 2.34.

**WEEK 11 (March 11).**Sections 10, 11 in Chapter II of the notes.

**WEEK 12 (March 18):** no lecture (herkans. bolk 2).

**WEEK 13 (March 25).**We finnished Chapter II (see the extra-section on topology basis, with the extra exercises- the link above) and we discussed the first section of Chapter III (connected spaces).

Exercises for the werkcolleges (in this order- try to get as far as possible): 2.35, 2.32, 3.3, 3.4, 3.1, 3.6, 2.33, 3.9, 2.32, 3.7.

**WEEK 14 (April 1).**We continued with Chapter III: we discussed the main definitions and results in section 2 (except for subsection 2.5), we jumped over section 3 (which will not be done in the lectures, and not required at the exam), and we did section 4. More precisely, we looked at the main results: Proposition 3.15, Theorem 3.16, Theorem 3.20, Corollary 3.21, Theorem 3.23 and 3.24 with their corollary, Theorem 3.39, and quite a few examples. We also looked at some proofs, but some of them are left for the next lecture.

Exercises for the next two werkcolleges (in this order): 3.19, 3.25, 3.24, 3.29, 3.30, 3.31, 3.34, 3.35, 3.22, 3.23, 3.21,3.33, 3.32, 3.20, 3.36 (just try to do as much as you can- but it is more important that you do a few properly then to do all of them in a rush!).

**WEEK 15 (April 8).**

**WEEK 16 (April 15):** no lecture; instead, one can give the Exam A.

**WEEK 17 (April 22):** The rest of chapter III (normal spaces, metrizability). Exercises: 3.38, 3.39, 3.40, 3.41, 3.43.

**WEEK 18 (April 29).**We started with Chapter IV: cells, attaching one n-cell, the intuitive idea behind "reconstruction" and various examples.

Exercises: look again at Example 4.12, 4.13 the do yourself Exercises: 4.2, 4.3, 4.1.

**WEEK 19 (May 6):**attaching more cells, cell complexes.

Exercises (please note that other exercises may be handed in during the lecture- in particular: a bonus exercise): 4.12, 4.13, 4.14, 4.15, 4.19, 4.21, 4.22, 4.23, 4.24, 4.25, 4.16, etc (try to do as much as you can).

**WEEK 20 (May 13):**Start with Chapter V (please note that other exercises may be handed in during the lecture- in particular: a bonus exercise): The fundamental group.

Exercises (please note that other exercises may be handed in during the lecture- in particular: a bonus exercise): 5.2, 5.5, 5.6, 5.7, 5.9, 5.10, 5.11, 5.17, 5.18, etc.

**WEEK 21 (May 20):**Covering spaces and the fundamental group of the circle.

Exercises: look at the previous exercises, then do 5.19, 5.20, 5.21, 5.23, 5.25.

**WEEK 22 (May 27):** no lecture (herkans. blok 3).

**WEEK 23 (June 3).**

**WEEK 24 (June 10).**

**WEEK 25 (June 17).**

Rules for the examniation: there are two exams, A and B. Both of them have to be passed, with a minimum mark of 5 for each one of them, and with an average of at least 6. One part of the exam is not valid for more then a year. There will be one chance to repeat the exam.

(Tentamenregeling: er zijn twee deeltentamens, A en B. Deze hebben geen zelfstandige betekenis: om het vak te halen moet over beide tentamens een gemiddelde van minstens 6 worden gehaald; met dien verstande dat de uitslag van beide tentamens minstens 5 moet zijn. Een deeltentamen blijft niet geldig tot volgend jaar. Er is een herkansing over de gehele stof.)

General presentation:

Given a set of points X, a topology on X is some "extra-data" on X which allows us to make sense of statement such as:

- two points of X are "close to each other"

- a sequence of points (in X) converges to another point.

- a function f: X --> |R is continuous

or other "topological statements" which you might have seen in analysis courses (such as compactness, connectedness).

Familiar sets of points such as the circle, the sphere, the torus, although they are all "isomorphic" as sets (i.e. there are bijections between them), our intuition tells us that they look quite different. That is because our intuition sees not only the bare sets of points, but also the natural topologies on them (the meaning of "a point gets closer to another one" beeing clear in each of these examples). And our intuition is right: the circle, the sphere and the torus are not isomorphic as topological spaces.

Apart from "geometric examples" of topological spaces (circle, tory, cube, etc). you have probably seen (or you will see) other interesting examples coming from analysis, functional analysis, distribution theory, etc. Such as: the space of bounded continuous functions, the space of compactly supported smooth functions, the space of distributions on an open set in the euclidean space.

Question A: A central question in topology is: given two topological spaces X and Y, how do we decide if they are isomorphic as topological spaces? Example: how do you PROVE that the circle and the sphere are not? Or that [0, 1] and [0, 1) are not?

Question B: Another important question is: given two spaces X and Y, when can one obtain one from the other by "pushing" and "pulling" them, without "breaking them". Particular case: given a space X and a subspace A, when can one "push X inside A", without "breaking it", and without leaving X. Question B is often very useful for answering Question A, and it is at the origin of "Algebraic Topology".

** The aim** of the course is to make the students know, understand, and feel what topological spaces are, give them the tools to answer questions as the two posed above and, in particular, discuss in detail the case of surfaces.

** The content** of the course is as follows. The first few courses are dedicated to the notion of topology and immediate topological notions, to examples and to constructions which allow us to produce new examples out of old ones (direct product, direct sum, quotient topology, etc). Then we concentrate on deeper topological properties such as compactness, local compactness, connectedness, normality, which allow us to attack Question A in many examples. In the second part we discuss more refined tools for answering the two questions mentioned above and for understanding the classification of surfaces: the fundamental group, computations using covering spaces, computations using the Seifert-van Kampen theorem and cell-attachements.