Topology is the study of properties of spaces that are invariant under continuous deformations. An often cited example is that a cup is topologically equivalent to a torus, but not to a sphere. In general, topology is the rigorous development of ideas related to concepts such nearness, neighbourhood, and convergence.
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The roots of topology go back to the work of Leibniz and Euler in the 17th and 18th century. It was only towards the end of the 19th century, through the work of Poincaré, that topology began taking shape as a subject in its own right. His seminal paper ``Analysis Situs’’ from 1895 introduced, among other things, the idea of a homeomorphism and the fundamental group. Nowadays, topological ideas are an indispensable part of many fields of mathematics, ranging from number theory to partial differential equations.
In this course we will introduce topological spaces and study their properties. We will embark on a study of homotopy and introduce the fundamental group as an important tool to classify topological spaces. This page will feature weekly updates and pointers to additional material.
- Allen Hatcher. Algebraic Topology. Cambridge University Press.
An electronic version of the book is freely available on the author’s web page, and a printed version should be available in the library or the campus bookshop.
Intended Learning Outcomes
Upon completion of this course, you will know how to distinguish spaces by means of topological invariants. You will be able to construct spaces by gluing and to prove that in certain cases the result is homeomorphic to a standard space. In addition, you will be able to construct examples of spaces with given properties (e.g., compact but not connected or connected but not path connected).
Brief lecture notes will be published regularly, usually in the days after each lecture. They will be available on the dedicated Lectures page.
Weekly problem sheets can be found on the Exercises page. Assessed work will be 15% of your mark. Of this, 2% (at most) may be earned every week (starting the second week) by turning in one of the three indicated exercises. These will be marked by a TA with a score of 0, 1, or 2. Please let me know if any of the problems are unclear or have typos.
Homework solutions must be placed in the dropoff box (near the front office), by 12:00 on Thursdays. No late work will be accepted. Please write your name, the date, and the module code (MA3F1) at the top of the page. If you collaborate with other students, please include their names.
Solutions typeset using LaTeX are preferred. Each problem should require at most one side of one page. If you find you need more space then write out a complete solution and then rewrite with conciseness in mind.
The following references are not needed for the course, but can provide additional information and perspective for those interested.