Module Catalogue 2024/25

Pre Requisite Comment

N/A

Co Requisite Comment

N/A

Aims

To give an introduction to how discrete, algebraic methods can be used to solve continuous, topological problems.

Module Summary
This course is an introduction to algebraic topology. Topology is the study of shape, sometimes referred to as the geometry of rubber sheets. One of the main questions is: given two shapes, can we stretch and shrink one of the shapes so that it becomes the shape of the other? To show that this is possible, we merely need to exhibit a deformation which transforms one into the other. However, to prove that this is not possible, we have to find something about shape which remains unaltered by any deformation. This something can be an algebraic object such as a number, a group, or a vector space.

This module will introduce two such invariants of shapes: the fundamental group, which measures the number and position of loops which can be drawn on the shape, and homology (simplicial and singular), which measures how the shape can be cut up into simplices of different dimensions (lines, triangles etc.). At the same time this provides the first example of a homology theory, which has generalisations to many areas of modern mathematics.

Outline Of Syllabus

Fundamental group: Paths and homotopy. Connectivity and path connectivity. Composition of paths. Fundamental group. The van Kampen Theorem. Covering spaces.

Homology: Delta complexes and simplicial complexes. Chain complexes. Simplicial and singular homology.

Examples and applications

Intended Knowledge Outcomes

Students will have an understanding of the application of discrete, algebraic methods to the study of the topology of surfaces. They will be able to reproduce, explain, and apply concepts and results of algebraic topology as listed in the syllabus.

Intended Skill Outcomes

Students will be able to perform elementary mathematical arguments involving the fundamental group and homology of topological spaces (see knowledge outcomes), calculate the fundamental group and homology of a class of examples, distinguish topological spaces based on their invariants, manipulate simple exact sequences and interpret them.

Students will develop skills across the cognitive domain (Bloom’s taxonomy, 2001 revised edition): remember, understand, apply, analyse, evaluate and create.

Teaching Rationale And Relationship

The teaching methods are appropriate to allow students to develop a wide range of skills, from understanding basic concepts and facts to higher-order thinking.

Lectures are used for the delivery of theory and explanation of methods, illustrated with examples, and for giving general feedback on marked work. Tutorials (within lectures) are used to discuss the course material, identify and resolve specific queries raised by students and to allow students to receive individual feedback on marked work.

Assessment Rationale And Relationship

A substantial formal unseen examination is appropriate for the assessment of the material in this module. The format of the examination will enable students to reliably demonstrate their own knowledge, understanding and application of learning outcomes. The assurance of academic integrity forms a necessary part of the programme accreditation.

Examination problems may require a synthesis of concepts and strategies from different sections, while they may have more than one ways for solution. The examination time allows the students to test different strategies, work out examples and gather evidence for deciding on an effective strategy, while carefully articulating their ideas and explicitly citing the theory they are using.

The coursework assignments allow the students to develop their problem solving techniques, to practise the methods learnt in the module, to assess their progress and to receive feedback; these assessments have a secondary formative purpose as well as their primary summative purpose.

General Notes

N/A