Topology is a subject of fundamental importance in many branches of modern mathematics.  Basically, it concerns properties of objects which remain unchanged under continuous deformation, which means by squeezing, stretching and twisting, but not cutting.  Apples and oranges are topologically the same, but you can’t deform an orange into a doughnut! More precisely, we can never deform in a continuous way a sphere (the surface of an orange) into a torus (the surface of a doughnut). Knots are also examples of topological objects, where a trefoil knot can never be deformed into an unknotted piece of string. It's the business of topology to describe more precisely such phenomena.

The aim of this module is to explore properties of topological spaces. To distinguish topological spaces we will consider topological invariants such as the fundamental group, which is a powerful way of using an algebraic invariant to detect topological features of spaces.

Prerequisite Modules: MA0213 Groups

• Understand the fundamental abstract notions of general topology including topological spaces, continuous maps, subspaces, connectedness, compactness, homeomorphisms, and separation properties.

• Understand homotopies of maps, homotopy equivalence and the construction of the fundamental group of a space. Understand basic properties of the fundamental group. Be able to compute the fundamental group of simple spaces.

• Understand applications of topological invariants to proving results in algebra and analysis

27 fifty-minute lectures

Some handouts will be provided in hard copy or via Learning Central, but students will be expected to take notes of lectures.

Students are also expected to undertake at least 50 hours private study including preparation of worked solutions for tutorial classes.

Skills:
 

• Logical analysis and the ability to deal with new ideas and challenges with confidence

• Use of mathematical language in a logically and rhetorically correct way

• Understanding of the interactions between different mathematical disciplines (in this case Algebra and Topology)

• Application of abstract topological concepts in other areas of Mathematics

• Translation of graphical arguments into mathematical proofs

Transferable Skills:
 

• Assimilation and use of abstract ideas.

Formative assessment is by means of regular problem sheets.  Feedback to students on their solutions and their progress towards learning outcomes is provided during lectures and tutorial classes.

The summative assessment is the written examination at the end of the module.  This gives students the opportunity to demonstrate their overall achievement of learning outcomes.  It also allows them to give evidence of the higher levels of knowledge and understanding required for above average marks.

The examination paper has a choice of three from four equally weighted questions.

• Definition of a topological space, and examples.

• Continuous maps and homeomorphisms.

• Topological constructions: subspaces, products and quotients.

• Connectedness, Hausdorff spaces and compact spaces.

• Homotopy, and homotopy equivalence

• The fundamental group and some basic properties: moving the base point, products, functoriality, homotopy-invariance.

• Applications to: Brouwer fixed point theorem (for a disc), Borsuk-Ulam theorem, Ham sandwich theorem, fundamental theorem of algebra.

• Homology (if time permits)

• Basic Topology, M. A. Armstrong, Springer UTM, 1983.

• Topology and Geometry, Bredon, G.E., Springer GTM, 1997.

• Algebraic Topology, Allen Hatcher, CUP, 2002. Electronic version available at: https://www.math.cornell.edu/~hatcher/AT/AT.pdf