# Topology and Groups

## Module overview

### What is *Topology and Groups* about?

*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.

We will:

- 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 be unlinked.
- 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.

### Flipped lectures

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).

#### Pre-class

**Before** each in-class session, you will need to prepare by:

- watch a certain number of videos or read those sections of the notes
- do the corresponding pre-class questions.

You can see which videos are required for each in-class session by referring to the week-by-week plan below.

#### In-class

In class, we will:

- discuss the pre-class questions and consolidate our understanding of the videos/notes,
- work through more complicated and interesting examples,
- work out the details of some of the proofs omitted from videos.

### Prerequisites

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*) including:- homomorphisms, kernels, normal subgroups, quotient groups, the first isomorphism theorem;
- if you have seen group actions, so much the better.

### Assessment

#### 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 to improve.

#### Coursework

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 marks.

#### Examination

There will be an examination, worth 90% of the final grade.

### Workload

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.

## Week-by-week plan

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)

### Week 2 *(1h 16m 5s)*

### Week 3 *(1h 37m 1s)*

### Week 4 *(1h 25m 31s)*

### Week 5 *(1h 42m 4s)*

### 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)*

### Week 8 *(1h 31m 9s)*

### Week 9 *(1h 6m 46s)*

## Index of all video lectures

- 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.

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
below similarly.

**1. Fundamental group***(2h 32m 50s)*- 1.01 Winding numbers and the fundamental theorem of algebra
*(12.33)*. - 1.02 Paths, loops, and homotopies
*(14.51)*. - 1.03 Concatenation and the fundamental group
*(17.08)*. - 1.04 Examples and simply-connectedness
*(21.21)*. - 1.05 Basepoint dependence
*(12.59)*. - 1.06 Fundamental theorem of algebra, reprise
*(14.42)*. - 1.07 Induced maps
*(7.32)*. - 1.08 Brouwer's fixed point theorem
*(19.31)*. - 1.09 Homotopy equivalence
*(20.53)*. - 1.10 Homotopy invariance
*(11.20)*.

- 1.01 Winding numbers and the fundamental theorem of algebra
**2. Topological spaces***(2h 1m 13s)*- 2.01 Topological spaces, continuous maps
*(16.42)*. - 2.02 Bases, metric and product topologies
*(19.46)*. - 2.03 Subspace topology
*(19.05)*. - 2.04 Connectedness, path-connectedness
*(18.56)*. - 2.05 Compactness
*(16.43)*. - 2.06 Hausdorffness
*(14.50)*. - 2.07 Homeomorphisms
*(15.11)*.

- 2.01 Topological spaces, continuous maps
**3. Quotients***(54m 10s)*- 3.01 Quotient topology
*(21.16)*. - 3.02 Quotient topology: continuous maps
*(10.43)*. - 3.03 Quotient topology: group actions
*(22.11)*.

- 3.01 Quotient topology
**4. CW complexes and the homotopy extension property***(53m 21s)*- 4.01 CW complexes
*(20.12)*. - 4.02 Homotopy extension property (HEP)
*(14.28)*. - 4.03 CW complexes and the HEP
*(18.41)*.

- 4.01 CW complexes
**5. Van Kampen's Theorem***(1h 33m 41s)*- 5.01 Theorem, examples
*(19.13)*. - 5.02 Fundamental group of a CW complex
*(15.09)*. - 5.03 Fundamental group of a mapping torus
*(17.43)*. - 5.04 Proof of Van Kampen's theorem
*(41.36)*.

- 5.01 Theorem, examples
**6. Braids***(43m 32s)*- 6.01 Braid group
*(17.43)*. - 6.02 Artin action
*(12.27)*. - 6.03 Wirtinger presentation
*(13.24)*.

- 6.01 Braid group
**7. Covering spaces***(2h 31m 22s)*- 7.01 Covering maps
*(27.41)*. - 7.02 Path-lifting, monodromy
*(19.29)*. - 7.03 Path-lifting: uniqueness
*(15.44)*. - 7.04 Homotopy-lifting, monodromy
*(22.15)*. - 7.05 Fundamental group of the circle
*(20.37)*. - 7.06 Group actions and covering spaces, 1
*(23.36)*. - 7.07 Group actions and covering spaces, 2
*(22.00)*.

- 7.01 Covering maps
**8. Galois theory of covering spaces***(2h 8m 1s)*- 8.01 Lifting criterion
*(24.56)*. - 8.02 Covering transformations
*(26.05)*. - 8.03 Normal covers
*(25.25)*. - 8.04 The deck group
*(15.16)*. - 8.05 The Galois correspondence, 1
*(16.03)*. - 8.06 The Galois correspondence, 2
*(20.16)*.

- 8.01 Lifting criterion

## Solutions

Solutions to all worksheets.