Topology and groups
Topology and Groups is about the interaction between
topology and algebra, via an object called the
fundamental group. This allows you to translate certain
topological problems into algebra (and solve them) and vice versa.
- introduce formal definitions and theorems for studying
topological spaces, which are like metric spaces but without
a notion of distance (just a notion of open sets).
- meet the fundamental group, a group associated to any
topological space which encodes information about deformation
classes of loops.
- use the fundamental group to prove some great theorems:
- the fundamental theorem of algebra (any nonconstant polynomial has
a root over the complex numbers),
- Brouwer's fixed point theorem (any continuous self-map of the disc
has a fixed point),
- the trefoil knot cannot be unknotted; the Borromean rings cannot
- a subgroup of a free group is free.
- introduce some powerful calculational tools for computing the
fundamental group, including:
- Van Kampen's theorem
- the theory of covering spaces.
- study the beautiful Galois correspondence between covering
spaces and subgroups of the fundamental group.
This module will be different from most modules you will have taken at
UCL. Instead of me standing up and lecturing for 3 hours a week, I
have pre-recorded your lectures and put them online (together with
lecture notes for each lecture). Your weekly homework will be to watch
the relevant videos for the next lecture and to do any of the
(hopefully short) associated pre-class questions (as well as finishing
off any of the work you started but didn't finish in-class).
Before each in-class session, you will need to prepare by:
You can see which videos are required for each in-class session by
referring to the week-by-week plan below.
- watch a certain number of videos or read those sections of the notes
- do the corresponding pre-class questions.
In class, we will:
- discuss the pre-class questions and consolidate our understanding of
- work through more complicated and interesting examples,
- work out the details of some of the proofs omitted from videos.
I will assume you are familiar with:
- equivalence relations,
- metric spaces (on the level of Analysis 4) including:
- compactness and continuity characterised in terms of open sets;
- group theory (on the level of Algebra 4 or Geometry and Groups)
- homomorphisms, kernels, normal subgroups, quotient groups, the
first isomorphism theorem;
- if you have seen group actions, so much the better.
- Classwork portfolio: In class, you will produce a portfolio
of classwork (solutions to problems, proofs of lemmas and theorems,
worked examples, etc.) which will be partly your own work, partly
group-work. You may of course continue to work on classwork at home
if you don't have time to finish it in class.
I want to see your classwork portfolios at least every fortnight so I
can give you feedback on how to improve your proof-writing skills, any
misconceptions you might have, that kind of thing. This is formative
assessment: it does not contribute to your grade, but it will help you
- Short projects: 10% of the final grade will be based on
coursework. This is not your portfolio of classwork, instead
this will be two short projects which I will set during term
(one in week 4, one in week 7). For each project, I will assign a
mark out of five and your coursework mark will be the sum of these
- Examination: There will be an examination, worth 90% of the
Some weeks will involve more videos than others, averaging about an
1.5 hours per week. Of course, it will take you longer than the stated
time to watch and absorb each video, as you may need to pause and
think, make coffee, rewind, etc. How you fit this in is up to you, as
long as you have watched the videos/read the notes before the
corresponding session. If you find it easier/quicker to just read the
notes (referring only to the videos when you get stuck) that's fine:
we all learn in different ways.
Each week there is a 2-hour session on Monday 12--2 and a 1-hour
session on Thursday 9-10. Below I indicate what videos we will be
focusing on in which sessions, along with the total watching time (to
help you plan your lives). Each link is to a page with an embedded
video (usually 15-20 minutes long) along with notes and pre-class
questions for the video.
In preparation for each session, I expect you to have either
watched the video or read the notes (or both) and to have thought
about the pre-class questions enough that you would happily discuss
your thoughts on each question with the class (even if you haven't
completely figured it out).
Week 1 (44m 34s)
- Monday: (12.33) (worksheet)
- 1.01 (motivation: fundamental theorem of algebra)
- Thursday (32.01) (worksheet)
- 1.02, 1.03 (Paths, loops, homotopy, concatenation,
Week 2 (1h 16m 5s)
- Monday (49.02) (worksheet)
- 1.04, 1.05, 1.06 (Examples, basepoint dependence, fundamental
theorem of algebra revisited)
- Thursday: (27.03) (worksheet)
- 1.07, 1.08 (Induced maps and Brouwer's fixed point theorem)
Week 3 (1h 37m 1s)
Week 4 (1h 25m 31s)
- Monday (53.18) (worksheet)
- 2.04, 5.01, 5.02 (connectedness/path-connectedness; Van Kampen's
theorem: statement and applications)
- I will give out Assessed Project 1.
- Thursday (32.13) (worksheet)
- 1.09, 1.10 (Homotopy equivalence and invariance)
Week 5 (1h 42m 4s)
- Monday (55.20) (worksheet)
- 3.02, 4.02, 4.03 (Homotopy extension property)
- The second hour on Monday will be a catch-up session, for you to
ask about anything from the past four weeks which has confused
- Thursday (46.44) (worksheet)
- 2.05, 2.06, 2.07 (Compact, Hausdorff spaces; homeomorphisms)
Reading week (41m 36s)
No sessions, you have the chance to watch the (nonexaminable) proof of
Van Kampen's theorem 5.04 (41.36) at your leisure.
Week 6 (1h 1m 15s)
Week 7 (1h 25m 9s)
- Monday (47.10) (worksheet)
- 7.01, 7.02 (Covering spaces, path-lifting, monodromy)
- I will give out Assessed Project 2.
- Thursday (37.59) (worksheet)
- 7.03, 7.04 (Uniqueness of lifts, homotopy-lifting)
Week 8 (1h 31m 9s)
- Monday (70.13) (worksheet)
- 7.05, 7.06, 7.07, 8.01 (Fundamental group of the circle, group
actions and covering spaces, lifting criterion -- first four
- Thursday (20.56) (worksheet)
- 8.01 (Lifting criterion: proof -- from 4 minutes on)
Week 9 (1h 6m 46s)
Week 10 (36m 19s)
- Monday (36.19) (worksheet)
- Thursday (worksheet)
- No videos, we will be studying groups acting on trees.
Each page below contains:
The notes are annotated with times (these annotations should look like
(8.30)) which indicate whereabouts in the video you can see that
section of the notes. I have also indicated the length of the videos
- an embedded video (usually lasting 15-20 minutes),
- a set of notes for the video,
- some pre-class questions you should do in preparation for class
after watching the video/reading the notes.
- 1. Fundamental group (2h 32m 50s)
- 2. Topological spaces (2h 1m 13s)
- 3. Quotients (54m 10s)
- 4. CW complexes and the homotopy extension property (53m 21s)
- 5. Van Kampen's Theorem (1h 33m 41s)
- 6. Braids (43m 32s)
- 7. Covering spaces (2h 31m 22s)
- 8. Galois theory of covering spaces (2h 8m 1s)
Solutions to all worksheets.