Welcome to the course! The schedule for the course can be found in TimeEdit.
Exam 2018Marc15 with
solutions
Genkai Zhang, genkai@chalmers.se;
office, MVH5023, tel. 7725385
M. A. Armstrong, Basic Topology, Springer.
Lecture Notes
1.
Fundamental groups and coverings (by JanAlve Svensson)
2.
Euler theorem and Euler characteristic
3.
Klein’s bottle and its fundamental group
4.
Orthogonal group SO(3) and its double
cover (the spin group)
Some of the students have already taken the courses Real
Analysis/Functional Analysis,
and know the topology of Euclidean space. We shall go over this
part, Chapt 3.13.2, very briefly, and it is
recommended that you study the Chapt 3.13.2 in
advance if you’re unsure.
Lectures
Week 
Chapter 
Contents 

I 
Ch.
1. 2 
Euler
theorem. Euler characteristic. Classification of surface. General top. spaces. 

II 
Ch.
2 
General
topological spaces, continuous functions/mappings, Tietze
extension theorem. 

III 
Ch.3

Compactness/Connectedness.
(Please review and read in advance materials on compact sets in R^n) 

IV 
Ch.
3,4 
Quotient spaces,
identification spaces/maps. 

V 
Ch.
4 
Group
actions and orbit spaces (quotient spaces). Topological groups, subgroups and
coset spaces. 

VI 
Ch.
5, 
Homotopy. Fundamental groups and covering spaces. 

VII 
Ch.
5, 6. 
Applications
of Fundamental groups: Brouwer's fixed point
theorem. (Introduction to triangulation and simplicial complexes). 









Chapters

Excersises 
1 
17,
1013, 1416 (Moebius strip). 17, 2326 
2 
18,
1018, 2028, 30, 3234 
3 
15,
718, 20, 21, 26, 28, 30, 3235, 3741, 43, 44. 
4 
17,
9, 10 (1112, similar problems on Klein bottle), 1321, 2632. 33* (This is a bit demanding – read the def. of L(p, q) on p. 82.) 
5 
17,
1016, 1822, 2628, 3034, 36, 40, 4450 
10 
1828

6 
17,
1011, 13, 15, 





Week 1. Chapt
1. 13, 11(a)(b), 17, 12. Chapt.
7.22. Demo:
Sphere with one/two Mobius cap(s)=P^2/Klein’s b. (along with some symbolic calculus
of surfaces)
Week 2. Chap 2: 3(c), 15, 20, 34, 21.
Week 3/4. Chapt 3: 1, 10, 13, 32, 41, 43
Week 4. Chapt 4: 2, 5, 16, 21, 26, 30
Week 5/6. Chapt 5: 1, 5, 13, 22, 32, 45
Week 7. Chapt 6: 4, 10,
14.
Week 6, 7. Chapt 10: 20, 24 (Notation: H(p) is the orientable surface with n handels/holes)
Chapt. 4, 33, 34
Week 8. Repetition Exercises
The learning goals of the course can be found in
the course
plan.
There will be three assignments (to be uploaded here), and you may
get max 2 bonus points of your solutions.
Subgroups of the real line
and the circle
Written exam on March 15, 2018, 14:0018:00
There will be 8
(or 7) exam problems with a total score of 24 points (pts). The grading limits
are
12 pts for G=Pass (Godkänd) and 18 pts for VG=Very
Good (Väl Godkänd). The
bonus pts
will be added up in the total score for reaching G
however not for VG. The bonus pts
will be valid for the reexams until the next ordinary exam 2019.
Three exam problems will be to test your (1)
understanding of the definitions, theorems and proofs in the course, and (2)
skills and techniques in applying the theory to solve concrete problems.
You are supposed to know
all the definitions in the relevant sections of the textbook and
also the ones that have been mentioned during the lectures. One of the problems
will be
to produce a proof of a theorem in the following list, and the
other two will be to use the methods of the
proofs of theorems in the course and to formulate important definitions and
theorems.
In Chalmers
Student Portal you can read about when exams are given and what rules apply
on exams at Chalmers. In addition to that, there is a schedule
when exams are given for courses at University of Gothenburg.
Before the exam, it is important that you sign up
for the examination. If you study at Chalmers, you will do this by the Chalmers
Student Portal, and if you study at University of Gothenburg, you sign up
via GU's Student Portal,
where you also can read about what rules apply to examination at University of
Gothenburg.
At the exam, you should be able to show valid
identification.
After the exam has been graded, you can see your
results in Ladok by logging on to your Student
portal.
At the annual (regular) examination:
When it is practical, a separate review is arranged. The date of the review
will be announced here on the course homepage. Anyone who can
not participate in the review may thereafter retrieve and review their
exam at the Mathematical
Sciences Student office. Check that you have the right grades and score.
Any complaints about the marking must be submitted in writing at the office,
where there is a form to fill out.
At reexamination:
Exams are reviewed and retrieved at the Mathematical
Sciences Student office. Check that you have the right grades and score.
Any complaints about the marking must be submitted in writing at the office,
where there is a form to fill out.