Results for the retake exam.

Results for Exam B as well as the final marks.

Please fill in the cursusevaluatie.

Herkansing: August 21, 9:00- 12:00, room to be announced (it is when you can do the retake- one exam covering the entire course).

This is the web-page for the course Topologie en Meetkunde, 2009. You can also have a look at the web-pages for the 2005-2006, 2006-2007 and the 2007-2008 year.

Place: Min 208 (for the weeks 6-11, 13-15) and AardKlein (for the weeks 17-20, 22-end).

Language: English.

Lecturer: Marius Crainic.

Place: BBL 503 for Group 1, Min 202 for Group 2.

In charge: Ionut Marcut, Gijs Heuts, Sebastiaan Klein.

- they have to be handed in no later then 9AM of the Wednesday that follows the lecture during which the exercise was given.

- they will not be discussed in the Werkcollege during the week the exercise was given.

- for each such exercise you may receive a certain number of points (usually 0.25 or 0.50) which will be added to the mark you receive at the exam. (hence, in principle, some of the students may add up with marks bigger then the maximum 10! For those, the official mark will still be 10.)

- Chapter 1. Note that some of the pictures use colours.

- Chapter 2.

- Chapter 3. Note that normal spaces (and what follows after them in this chapter) will not be discussed (in detail) in the lectures and will not be required for Exam A.

- Chapter 4: Attaching cells.

- Chapter 5: The fundamental group.

- Chapter 6: The Seifert van Kampen theorem.

- Chapter 7: A summary on how to compute the fundamental group and examples.

- Table of contents5 of the entire lecture notes.

Division into weeks:

**WEEK 6/Lecture 1 (February 4):** Introduction and first examples of topological spaces (see Chapter 1 of the lecture notes).

**WEEK 7/Lecture 2 (February 11):** More examples: stereographic projection, various constructions of the sphere, the Moebius band (by gluing, then models in R^3), torus,double torus, the Klein bottle.

Exercises for this and next week: 1.8 1.9, Look at the computations in the notes (the explicit formulas) for Moebius band and the torus, 1.14, 1.15, 1.19, 1.17, 1.18, 1.20, 1.21, 1.25, 1.26. This is quite a lot, but try to do as many as possible.

Bonus exercise for this week (see above): 1.10 (to be delivered by Feb 18, 9AM).

**WEEK 8/Lecture 3 (February 18):** finnishing Chapter 1 of the lecture notes.

**WEEK 9/Lecture 4 (February 25):** metric spaces and metric topologies, then we moved to Chapter 2: the general notion of topological space, continuity, convergent sequences, Hausdorffness.

Exercises: 2.1, 2.2, 2.3, 2.8, 2.5, 2.6, 2.9.

**WEEK 10/Lecture 5 (March 4):** we started withe the general constructions of topologies: subspace topology, product topology, topology bases. At the end, bases of neighborhoods of a point and the first countability axiom was briefly discussed.

Exercises: prove the claims made in Example 2.31, then do 2.19, 2.22, Example 2.34, 2.30, then look at the Exercise 2.4 and also point out the relationship with the notion of "topology basis", Exercise 2.35.

**WEEK 11/Lecture 6 (March 11):** Quotient topologies, special classes of quotients- including quotients modulo group actions; examples, the projective space. Definition of interior, closure, boundary.

Bonus exercise: prove Theorem 2.43.

**WEEK 12 (March 18):** no offical lecture (herkans. blok 2). Nevertheless, for those interested, there was a lecture in which we will look back to some of the things we have already done ("recap") and some exercises from the Exam A of previous years.

**WEEK 13/Lecture 7 (March 25):**

**WEEK 14/Lecture 8 (April 1):**

**WEEK 15/Lecture 9 (April 8):**

**WEEK 16 (April 15):** exam A.
Here you can see the exams of the previous three years: 2008, 2007,
2006.

**WEEK 17/Lecture 10 (April 22):** We went through Chapter 4. More precisely: the definitons of cells, attaching one n-cell, attaching several n-cells, cellular decomposition, Euler number, examples.

Bonus exercise: Exercise 4.11.

**WEEK 18/Lecture 11 (April 29):** Reminder on attaching cells and cell decompositions. Then we discussed in more detail the notion and importance of attaching maps.

**WEEK 19/Lecture 12 (May 6):** Homotopies, homotopy equivalences, path homotopies, the fundamental group, main properties.

**WEEK 20/Lecture 13 (May 13):** Reminder on homotopies, fundamental group, the main properties. Then covering spaces and computation of the fundamental group of the circle.

**WEEK 21 (May 20):** Seifert-van Kampen theorem: explanation of the statement and first consequences.

**WEEK 22/Lecture 14 (May 27):** Seifert-van Kampen again.

**WEEK 23/Lecture 15 (June 3):** Seifert-van Kampen again.

Note: for the last lectures, the list of exercises for the werkcollege was pretty much the same as the entire list of exercises from the notes (most importantly: all the explicit exercises regarding cell decompositions, homotopy equivalences, computations of fundamental groups).

**Enjoy the sphere ** (and not only).