# "Inleiding Topologie" and "Topologie en Meetkunde", blok 3, 4, 2010

The revised file with the final marks for part A (Inleiding Topologie). The students 3021122 and 3373479: your marks for the retake were too low (4.5).

Here are the marks for part B, after the re-take and the handing in of the last exercises (August 31).

Lecture notes:

- Chapter 1 (Introduction: Some standard spaces). Note that some of the pictures use colours.

- Chapter 2 (Topological spaces).

- Chapter 3 (Constructions of topologies).

- Chapter 4 (Topological properties).

- Chapter 5 (Normal spaces, etc ).

- Chapter 6 (Attaching cells).

- Chapter 7 (The fundamental group).

- Chapter 8-9 (Seifert van Kampen, Examples).

The schedule week by week: WEEK 6/Lecture 1 (February 10): Start of Chapter 1: short presentation of the course, the notion of metric spaces, convergent sequence (in metric spaces), homeomorphisms. The circle, sphere, n-sphere, disks, stereographic projection, first examples of gluings, a first encounter of Moebius band, torus, Klein bottle (by gluings from the square).

Take home exercise: Exercise 1.12. WEEK 7/Lecture 2 (February 17): Reminder + examples of gluings from the unit square: cylinder, Moebius band, sphere, torus (+ double torus), Klein bottle, the projective plane. Reminder from other ocurses: equivalence relations (understood as "gluing data"), the notion of quotient (understood as "the result of gluing"),

Exercises for the werkcolleges: 1.14, 1.13, 1.19, 1.16, 1.12, 1.22, 1.21 (in this order). These are probably more then what most of you can do in 2 hours, but what is important is to do the first three and try to get as far as possible.

Take home exercise: Exercise 1.23. WEEK 8/Lecture 3 (February 24): The notion of topological spaces; opens and closed subset; neighborhoods of points, convergence, continuity, the relationship between convergence and continuity. Examples: metric topologies, trivial, discrete etc topologies, the lower topologyu T_l on the real line. Hausdorffness. The notion of interior, closure and boundary of a subset of a given topological space.

Future exercises for the werkcolleges: topological spaces 2.2, 2.4; continuity: 2.7, 2.8, 2.11; sequences: 2.16, 2.17; interior, closure, etc: 2.22, 2.24, 2.25, 2.27; some more theoretical exercises: 2.1, 2.15, 2.14, 2.29, 2.28. These are a lot. Try to do one from each group; during the next few weeks (when you will receive less exercises) use the remaining time to come back to the exercises from this set that you do not have time for now.

The next take-home exercise: Exercise 2.20. WEEK 9/Lecture 4 (March 3): Reminder; topological properties (e.g. Hausdorffness), basis of neighborohoods (also first countability), closure and interior (the recogniotion lemma); then moved on to chapter 3: subspace topology, embeddings, the definition and explanation of the product topology (to which we will return next time).

Exercises for the werkcollege: 3.1, 3.2, prove the statements from Example 3.5, 3.3, prove the statement from example 3.8, 3.12. 3.15. Also, don't forget the previous (long) list of exercises.

The take home exercise: Exercise 3.6. WEEK 10/Lecture 5 (March 10): Product topology, toplogy bases and induced topologies, quotient topologies: general construction, examples (Moebius, projective spaces), quotients obtained by collapsing a subset to a point; cylinder, cone and suspension of a topological space (quotients modulo group actions is left for next time).

Exercises for the werkcollege: finnish the (long lists of) exercises from the previous two lectures. Then also do Exercise 3.13. Then make sure you understand Example 3.20.

The take-home exercise: Exercise 3.9 (note that this is not the exercise that was announced as "probable" last week). WEEK 11 (March 17): Hertent. blok 2. Note however that, for those interested, there will be a lecture (where I will recall what we have already done and we will look more carefully at some examples and maybe some exercises that you didn't have time for before). If you plan to attend, you may think in advance about questions regarding parts of the course that you did not understand, or exercises that you tried and didn't mannage to do. WEEK 12/Lecture 6 (March 24): Actions of groups on topological spaces, quotients modulo group actions (examples: the projective space). Connectedness, path connectedness, their main properties, examples of their use for proving (in various examples) that two spaces are not homeomorphic.

Exercises for the werkcollege: 4.3, 4.4, 4.5, 4.1, 4.7.

The take-home exercises: Exercise 4.9 (this one is a bit tricky! Hint: concentrate around "the special points"). WEEK 13/Lecture 7 (March 31): Reminder on connectedness; connected components and their main properties. The notion of compact space and first examples (including the proof that [0, 1] is compact); statement of the main properties:

- closed inside compact is compact.

- compact inside Hausdorff is Hausdorff.

- product of two compact spaces is compact.

- continuous functions send compacts to compacts.

from which I mentioned the consequences:

- a subset of R^n is compact if and only if it is closed and bounded.

- a continuous bijection from a compact to a Hausdorff space is a homeomorphism.

I will come back to these next time.

The take-home exercises: Exercise 4.40, parts (i)- (iii). WEEK 14/Lecture 8 (April 7):

Exercises for the werkcollege: 4.22, 4.23, 4.24, 4.25, 4.27, 4.28, 4.29, 4.30, 4.33, 4.41, 4.21, 4.19.

The take-home exercises: Exercise 4.39 (in part (i) skip the question: "But normal?"). This is the last take-home exercise before exam A. The deadline for handing it in is April 21 (when we meet at the exam). WEEK 15/Lecture 9 (April 14): Due to an emergency, M. Crainic cannot be present. The lecture is given by Ivan Struchiner.

Note: there is no take-home exercise for this lecture.

For the werkcollege: look back at the various exercises that you could not solve. If you've done it all, look at the exercises in the last paragraph of chapter 4 (except for Exercise 4.39 which is a take-home from last week). WEEK 16/(April 21): Exam A. It will take place from 9AM to 12 in the room Educatorium Alfa. Recall: you are allowed to use the lecture notes during the exam. WEEK 17/Lecture 10 (April 28): Attaching an n-cell, examples, attaching more cells, cell complexes, Euler characteristic (with examples). The notes can be found above.

The take-home exercises: Exercise 6.18. WEEK 18/(May 5): No lecture (Liberation Day). WEEK 19/Lecture 11 (May 12): We finnished the chapter on cell complexes; this time we discussed the more theoretical parts and proofs. We paid special attention to the universal property of spaces obtained by attaching one cell and to "universal properties in general" (in particular we mentioned the universal property of products of sets, the one of quotients and I gave a hint about the universal property of quotients of groups modulo normal subgroups- note that these universal properties are not mentioned in the lecture notes so, if you did not attend, I advise you to look at the hand-written notes of some colleague that did attend).

The take-home exercises: for a normal subgroup N of a group G, state the universal property of the quotient group G/N, and prove that it determines the quotient uniquely up to group isomorphisms (so that one can talk about "the abstract quotient"). Then convince yourself that the explicit construction of G/N that you have seen before (e.g.in group theory) does satisfy the universal property.

Hint: look at the hand-written notes for this course (before writing the exercise on the blackboard, some rather helpful hint was given).

Deadline: since this exercise was announced on the web-page very late (and it seems that quite a few have missed the class), he deadline for handing in this exercise is prolonged by one week (so it will be May 26). WEEK 20/Lecture 12 (May 19): We start with Chapter 7 (available on the upper part of the page): homotopies, homotopy equivalence, path homotopy, the fundamental group.

Exercises for the werkcollege: 7.1, 7.3, 7.6, 7.9, then read Example 7.10 and solve 7.12, 7.13.

The take-home exercise: 7.14. WEEK 21/Lecture 13 (May 26): Reminder on the fundamental group; main properties (invariance of the base point, topological invariance, homotopy invariance). Proof of the fact that the fundamental group of the circle is isomorphic to the group of integers; consequences (that there is no retraction of the 2-disk on its boundary circle amd the Brower fixed point theorem).

Exercises for the werkcollege: 7.2, 7.5, 7.8, 7.19, 7.22, 7.23.

The take-home exercise: here. Errata: in the file, the number n (in z to the power n) is equal to 2. WEEK 22/Lecture 14 (June 2): The Seifert-van Kampen theorem: the algebraic notion of amalgamated product, with particular cases A, B, C (quotients) in discussed in detail and case D (free product) only briefly touched (will get back to it next time). Then the statement of the Seifert- van Kampen theorem, with consequences \$A\$, \$B\$ and \$C\$ (hence also: the vanishing of the fundamental group of the n-sphere with \$n\$ greater or equal to 2 + the fact that attaching an n-cell with n greater or equal to 3 does not affect the fundamental group).

The take-home exercise: here WEEK 23/Lecture 15 (June 9): Reminder on Seifert-van Kampen (and corollaries associated to cases A, B and C). Discussion of case D: construction of free products of groups (via reduced words), free groups on an arbitrary number of generators, groups given by generators and relations (and some examples that are not in the notes). Back to Seifert van Kampen and Corollary D; ex: bouquet of two circles. Final example: a torus and a spehere touching in one point (computation based on applying Seifert van Kampen theorem three times, namely: once Corollary B, then the corollary to Corollary C- the one on attaching a 2-cell, then Corollary D).

The take-home exercise: here WEEK 24/Lecture 16 (June 16): WEEK 25/ (June 23): No lecture. WEEK 26/ (June 25) Exam B (9:00-12:00 in EDU Beta). Here are the similar exams from 2006, 2007 and 2009. WEEK 34 (27 August): hertent (9:00- 12:00).

Enjoy the sphere (and not only).