Topology is concerned with the way in which geometric objects can be continuously deformed to one another. It can be thought of as a variation of geometry where there is a notion of points being "close together" but without there being a precise measure of their distance apart.
Examples of topological objects are surfaces which we might imagine to be made of some infinitely malleable material. However much we try, we can never deform in a continuous way a torus (the surface of a bagel) into the surface of the sphere. Other kinds of topological objects are knots, i.e. closed loops in 3-dimensional space. Thus, a trefoil or "half hitch" knot can never be deformed into an unknotted piece of string. It's the business of topology to describe more precisely such phenomena.
In topology, especially in algebraic topology, we tend to translate a geometrical, or better said a topological problem to an algebraic problem (more precisely, for example, to a group theoretical problem). Then we solve that algebraic problem and try to see what that solution tells us of our initial topological problem. So to do topology you need to work equally well with both geometric and algebraic objects.
Pre-requisites: (MATH2003 AND MATH2039 AND MATH2049) OR (MATH2003 AND MATH2039 AND MATH2046)
Aims and Objectives
Having successfully completed this module you will be able to:
- Understand and apply the notion of homology and compute homology groups for some examples..
- Understand and apply the notion of homotopy and compute homotopy groups for some examples
- Homotopy and homotopy type
- Paths and homotopy, definition fundamental group, simply connected spaces.
- Fundamental group in general: induced homomorphisms, calculations and applications (R^2 is not homeomorphic to R^n for n not equal to 2.)
- Covering maps, covering spaces, universal cover, deck transformations.
- Simplical complexes.
- Simplicial and singular homology.
- Homotopy invariance.
- Exact sequences and excision, applications: Brouwer fixed point theorem, distinguishing spheres.
- Seifert-van Kampen Theorem.
If time permits one or more of the following topics:
- Euler characteristic
- Lefschetz fixed point theorem
Learning and Teaching
Teaching and learning methods
Lectures, problem sheets, private study
|Total study time||90|
Resources & Reading list
A. HATCHER. Algebraic topology. Cambridge University Press.
M.A. ARMSTRONG. Basic Topology. Springer.
This is how we’ll formally assess what you have learned in this module.
This is how we’ll assess you if you don’t meet the criteria to pass this module.
Repeat type: Internal & External