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Revision class

Here are some notes on the revision class which took place on Friday 11 January.

Lecture times and places

The classes for this course unit are as follows:

Some of the Tuesday morning two hour sessions will be used as a tutorial session and for discussing the problem sheets.

Queries

Any queries about the course or comments can be emailed to me at pjeccles@manchester.ac.uk. I will try to reply within 48 hours.

Office Hours

I will try and make a point of being in my office (Alan Turing 1.111) available to students during my office hours: Tuesdays from 2 to 3 p.m. and Thursdays from 3.30 to 4.30 p.m. Students are welcome to call at other times but I may be busy and in that case we can arrange another time to meet. I can always be contacted by email.

Books

There are very many books on the basic ideas of topology. The following are some that I have found useful in preparing this course.

• M.A. Armstrong, Basic Topology. The first four chapters of this book covers the material in sections 1 to 7 of the course.


• Stephen Huggett and David Jordan, A Topological Aperitif. This book is a good introduction to topological ideas but does not include most of the course. The first two chapters contains the material on cut-points in section 1 of the course. The library catalogue provides access to an electronic copy of this book.


• C. Kosnoiwski, A First Course in Algebraic Topology. This book contains most of the material in the course and is probably the closest text to the course.
• W.A. Sutherland, Introduction to Metric and Topological Spaces. Chapters 3 to 6 cover most of the material in the first half of the course..

Background material

I have prepared a document summarizing the material on sets and functions (mainly from MATH10101/MATH10111) which will be used in this lecture course. I will not go through these notes in the lectures but will refer to them from time to time. I suggest that students read through these notes as preparation for the course. If you have queries you can raise these when the material is used. The first section of the course uses the last section of these notes. I hope that this summary is useful.

• Background material on sets and functions

Lecture notes

• 1. Topological equivalence and path-connectedness


• 2. Topological spaces


• 3. Constructing new spaces


• 4. Hausdorff spaces


• 5. Compactness


• 6. The fundamental group


• 7. The fundamental group of the circle


• 8. Applications of the fundamental group

Supplementary reading for the level 4 MATH41051 and the MSc MATH61051 course unit

The level 4 and MSc version of this course is rated at 15 credits rather than the 10 credits of the level 3 version. There are three sets of notes to provide supplementary reading for students taking this version of the course. The (three hour) examination for MATH41051/MATH61051 will be the same as the (two hour) examination for MATH31051 with an additional section based on this supplementary reading (with three questions: one question on each set of notes).

• A. Neighbourhoods, interior and closure


• B. Group actions on topological spaces


• C. Covering spaces and the fundamental group

Problem sheets

• 1. Topological equivalence


• 2. Topological spaces


• 3. Subspaces and product spaces


• 4. Quotient spaces


• 5. Hausdorff spaces


• 6. Compactness


• 7. The fundamental group


• 8. The fundamental group of the circle and applications

Solution sheets

• 1. Topological equivalence


• 2. Topological spaces


• 3. Subspaces and product spaces


• 4. Quotient spaces


• 5. Hausdorff spaces


• 6. Compactness


• 7. The fundamental group


• 8. The fundamental group of the circle and applications

• A. Neighbourhoods, interior and closure


• B. Group actions on topological spaces


• C. Covering spaces and the fundamental group

Assessment

Coursework

The coursework assignment is available here. Solutions and feedback are available here.

Examination

For MATH31051 the examination rubric is as follows: Answer ALL four questions in Section A (40 marks in all) and THREE of the four questions in Section B (15 marks each). The total number of marks on the paper is 85. A further 15 marks come from work during the semester making a total of 100. It is a two hour examination.

For MATH41051/MATH61051 the examination rubric is as follows: Answer ALL four questions in Section A (40 marks in all), THREE of the four questions in Section B (15 marks each) and ALL three questions in Section C (50 marks in all). The total number of marks on the paper is 135. A further 15 marks come from work during the semester making a total of 150. It is a three hour examination.

• MATH31051 January 2010


• MATH31051 January 2011 and some solutions


• MATH41051 January 2010


• MATH41051 January 2011 and some solutions

It should be noted that the syllabus of the second half of the course has changed this year. The material on the fundamental group hsa replaced material on the classification of surfaces. It should be obvious which questions for previous years are relevant to this year's course.

Diary of amendments to and additions to this webpage