MMA100, Topology, Spring 18

Latest news

Welcome to the course! The schedule for the course can be found in TimeEdit.

Exam 2018-Marc-15 with solutions



Genkai Zhang,; office, MVH5023, tel. 7725385

Course literature

M. A. Armstrong, Basic Topology, Springer.

Lecture Notes


1.     Fundamental groups and coverings (by Jan-Alve 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 3.1-3.2, very briefly, and it is recommended that you study the Chapt 3.1-3.2 in advance if you’re unsure.






Ch. 1. 2

Euler theorem. Euler characteristic. Classification of surface. General top. spaces.


 Ch. 2

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



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


Ch. 3,4

Quotient spaces, identification spaces/maps.


Ch. 4

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


Ch. 5,
 Chapter 10.4. 

Homotopy. Fundamental groups and covering spaces.


Ch. 5, 6.

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







Recommended exercises




1-7, 10-13, 14-16 (Moebius strip).     17, 23-26


1-8, 10-18, 20-28, 30, 32-34


1-5, 7-18, 20, 21, 26, 28, 30, 32-35, 37-41, 43, 44.


1-7, 9, 10 (11-12, similar problems on  Klein bottle), 13-21, 26-32. 33*

 (This is a bit demanding – read the def. of L(p, q) on p. 82.)


1-7, 10-16, 18-22, 26-28, 30-34, 36, 40, 44-50




1-7, 10-11, 13, 15, 





Demonstration and Discussion Exercises (on Thursdays)

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

Additional  Exercises ( exer-part-1, exer-part-2,              )





Course requirements

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.


Homework 1

Homework 2

Homework 3

Subgroups of the real line and the circle



Written exam on March 15, 2018, 14:00-18:00

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 re-exams 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.







Examination procedures

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 re-examination:
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.

Old exams


2017-August Solution     

2017-March Solution     

2016-March     Solution

2015-June    Solution



2014-Aug      mma100-topo-2014aug21-lsn.pdf 


2014-June      Solution


2012-June      Solution


2012-March   Solution


2011-Aug.       Solution


2010-March   Solut'n




2012-06     Solut'n