This is the web-site for the course "Topologie en Meetkunde" (WISB341), 2011. Here you will find all the practical informations about the course, changes that take place during the year, lecture notes, etc.

THE LECTURES: Mondays, 11:00- 12:45, in room Min208.

LECTURERS: Marius Crainic and Ivan Struchiner.

THE WERKCOLLEGES: Wednesdays, 9:00- 10:45, in rooms Min018 and BBL071.

TEACHING ASSISTANTS: R.L.M Wang., K.W. van Urk and Boris Osorno Torres.

EXAMS: two exams (week 16 and 26), at both exams the lecture notes are allowed (but no other books). To pass, one has to pass each exam with at least 5, and have an average of at least 6. There will be no hand in exercises. There will be bonus exercises from time to time (not compulsory but, if solved, they will just be added to the final mark).

PREREQUISITES:

---- Group theory. Note however that we will spend some time reminding some of the notions from group theory that are needed (normal subgroups, quotients, product of groups, free groups, etc).

---- Inleiding Topologie. Due to the "transition fase" for this course, and this year only, we will try as much as possible to make it possible for the students that did not follow "Inleiding Topologie" to follow the course. See also the description of Lecture 1 below and the links therein.

THE AIM OF THE COURSE: one of the main things that will be done in this course will be the classification of surfaces. That includes a "cutting and pasting" part (to show that any surface is homeomorphic to one in the list) and an algebraic topology part, on fundmanetal groups (to show that any two in the list are not homeomorphic). Keywords are: surfaces, triangulations, cutting and pasting, Euler characteristic, CW complexes, fundamental groups, Seifert-van Kampen theorem. Of course, along the way, we will see other things as well (with lots of examples and pictures).

By the way: enjoy the **sphere ** (and not only).

- Lectures 1, 2 and 3 (actually, the last part of these notes was presented in lecture 4).

- Lecture 5.

- Attaching cells (a copy of the lecture notes from last year). This is a ps file.

- The fundamental group (a copy of the lecture notes from last year). This is a ps file. In the lectures, some extra-material on covering spaces has been discussed; it can be found here.

- The Seifert-van Kampen theorem (and a summary of how to compute fundamental groups). Again, these are the notes from the last year.

**WEEK 6/Lecture 1 (February 7):** First of all, I gave a very very rough idea of what this course is going to be about (main objective: classification of surfaces; main tools: cutting and pastig, triangulations, fundamental groups). Strictly speaking, this will start in the next lecture. Since there are quite a few students that did not follow "inleiding topologie", I spent some time with explaining what "topology" is about. The reference is to the lecture notes from last year (included below).
In particular, I pointed out the which parts of these lecture notes are most relevant. Here is a list of them, without the explanations I gave during the lecture:

*** Inl. Top. Chapter 1 (Introduction: Examples): Def 1.1/page 4 (metric spaces), Def. 1.3/23 (metric topology)+ standard examples: circle, sphere, torus, dubble torus, Moebius band, Klein bottle, porojective space etc (but most of them will be discussed again starting from the enxt lecture).

*** Inl. Top. Chapter 2 (The category of topological spaces): top. spaces (Def. 2.1/page 25), convergent sequences (Def. 2.18/30), continuous functions (def. 2.7/27, Th. 2.22/31, Th. 2.35/36 where, in the last theorem, instead of "first countable" read "metric"), homeomorphisms (Def 2.10/28), top. invariants (Def 2.14/29). Closure and interior + Lemma 2.28/32.

*** Inl. Top. Chapter 3 (Constructions of topological spaces)3: subspace topology (pp. 37), embeddings (pp. 37), products (pp. 38), quotients (def. of equivalence relation given in the first chapter, Def. 1.2/22, while the quotient topology in chapter 3 on page 42).

*** Inl. Top. Chapter 4 (Topological properties): connected, path connected (definition in Def. 4.1/51; main properties: cor. 4.5/53, cor. 4.6/53), compactness (abstract definition in Def. 4.12/56+ the many basic properties: prop 4.15/59, theorem 4/16/59, theorem 4.20/60, corollary 4.21/60, theorem 4.23/61, theorem 4.24/61), local compactness (Def 4.30/63).

*** the "removing one point trick" (Ex. 2.15/30; see also the exercises below).

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I advise you to: try to write on 2 sheets of paper (3-4 pages) the main points of "Topology", following the indications above.

To do (e.g. at the werkcollege): from the above lecture notes,

- exercise 2.30.

- exercises about the removing one point trick: Ex. 2.15/30, 4.3/33, 4.38 (iii)/70.

- read carefully the story about the Moebius band: Subsection 4.1/page 10 (Chapter 1), Example 3.17/page 43, Example 4.28/page 62. Imagine the same story about the torrus. How does the story look for the Klein bottle?

**WEEK 7/Lecture 2 (February 14):** I began by stating the classification problem for compact surfaces. Then we saw the definition of a topological manifold with several examples. We discussed the connected sum of two surfaces and then stated Thm. 1.9 which gives a complete list of compact surfaces. We introduced polygonal regions, and we saw that a labeling of the polygon along with an orientation on each edge gives rise to a quotient space. We then saw how to put together this information together into a labeling scheme.

Werkcolleges: all the exercises from tne notes.

**WEEK 8/Lecture 3 (February 21):** In this lecture we described the operations on polygonal regions (and the associated operation on labeling schemes) which leave the quotient space unchanged. We saw how to construct connected sums of tori and connected sums of projective spaces out of a polygonal region with identifications on its edges. We suggest that you prove that the operations leave the quotient space unchanged, and that you play around with them to get a feeling of what is going on. Try to see what the connected sum of a torus and a projective space corresponds to (we will come back to this in the next lecture).

Werkcolleges: all the exercises from tne notes.

**WEEK 9/Lecture 4 (February 28):** In this lecture proper labeling schemes were introduced. We showed that every proper labeling scheme could be put into a normal form. In this way, we concluded that any surface which is obtained from a polygonal region by identifying its edges in pairs is homeomorphic to one of the surfaces in the list of Theorem 1.9.

Werkcolleges: all the exercises from tne notes.

**WEEK 10/Lecture 5 (March 7):**
We discussed triangulated surfaces and showed that any triangulated compact connected surface is homeomorphic to one of the surfaces in the list of Thm 1.9.

**WEEK 11 (March 14):** No course (Hertent.).

**WEEK 12/Lecture 6 (March 21):** n-cells, attaching n-cells, started the paragraph on the universal property (stated it and its consequences).

For the werkcollege: 6.2, 6.3, 6.4, 6.5 from the notes on "Attaching Cells" (see above).

**WEEK 13/Lecture 7 (March 28):** Reminder on n-cells and the universal property. Proof that the characteristic map of an n-cell determines the adjunction space uniquely. Attaching more cells; cell decompositions (CW complexes), the Euler characteristic, examples. Then started the chapter on the fundamental group: homotopies between maps, homotopy equivalences.

For the werkcollege: 6.14, 6.15, 6.16, 6.17, 6.18, etc.

**WEEK 14/Lecture 8 (April 4):** Homotopies, homotopy equivalences, path homotopies, concatenation of paths, the main properties, the definition of the fundamental groups, independence of the base point.

For the werkcollege: 7.2, 7.4, 7.5,7.8,7.14. Prove theorem 7.22 yourself in the way we started doing it in the class (do not look at the proof in the notes).

**WEEK 15/Lecture 9 (April 11):** Finnished the basic properties of the fundamental group, then started the section on the fundamental group of the circle. We started with the end: deriving some of the consequences of the fact that the last mentioned group is non-trivial (no retractions of the disk into the circle, Brower's fixed point theorem + other stuff that is not in the lecture notes: boundary points). Then I briefly describied how the degree map is defined.

For the werckollege: 7.6, 7.7, 7.9, 7.11, 7.15, 7.18, 7.19, 7.25, 7.22.

**WEEK 16 (April 18):** No course (Tentamen Week 3).

**WEEK 17 (April 25):** No course (Pasen).

**WEEK 18/Lecture 10 (Mei 2):** We discussed the lifting properties of coverings and deduced that there is an action of the fundamental group of the base on the fiber of the covering. We then saw that if the covering space is simply connected, then there is a bijection between the fiber the covering map and the homotopy group of the base. We discussed properly discontinuous actions and showed that they induce covering maps. For
Then we started with the Seifert - van Kampen theorem. We stated it in terms of a universal property and used it to deduce that the spheres of dimension greater or equal to 2 are simply connected.
BONUS EXERCISE: Exercise 12 of the "Addendum to Chapter 7". To be handed in next monday (09.05) during the lecture.

**WEEK 19/Lecture 11 (Mei 9):**

**WEEK 20/Lecture 12 (Mei 16):**

**WEEK 21/Lecture 13 (Mei 23):**

**WEEK 22 (Mei 30):** No course (Hertentamen blok 3).

**WEEK 23/Lecture 14 (June 6):**

**WEEK 24 (June 13):** No course (Pinksteren).

**WEEK 25 (June 20):** No course (Studieweek).

**WEEK 26 (June 27):** Exam (tentamen week).