# Gwybodaeth Modiwlau

Module Identifier
MAM2620
Module Title
Algebraic Topology
2022/2023
Co-ordinator
Semester
Semester 1
Pre-Requisite
Pre-Requisite

#### Assessment

Assessment Type Assessment length / details Proportion
Semester Exam 3 Hours   100%
Supplementary Exam 3 Hours   100%

### Learning Outcomes

On successful completion of this module students should be able to:

Demonstrate knowledge of examples of topologies.

Determine basic topological characteristics of sets.

Determine whether a function is continuous.

Determine whether two topological spaces are homeomorphic.

Determine whether a topological space is compact.

Determine whether a topological space is connected.

Demonstrate an ability to construct new topological spaces from old.

Demonstrate an understanding of advanced topics from algebraic topology.

### Brief description

Commonly known as 'rubber-sheet geometry?, Topology is a subject, which has arisen essentially from abstracting certain characteristics of space, provides an extremely general framework that is used in many areas of science, including mathematics, physics and computer science. This module develops ideas found in earlier mathematical modules on analysis to provide a rigorous and, in a sense, elegantly simple set of concepts that will be extremely useful to those pursuing a career in science. Algebraic Topology is concerned with measuring in some way characteristics of abstract spaces using algebraic techniques.

### Aims

To develop certain notions of space that students will have already been exposed to in a much more general and abstract framework that can be applied to many areas of science.

### Content

Topological properties of subsets in n-dimensional Euclidean space: open sets; topological equivalence of sets; convergence; continuity; development of topological axioms (generalisation to metric spaces).
Topological spaces: definition of a topological space; bases; relative topology; product topology.
Elementary topological properties: closed sets; interior points; closure; isolated points; boundary; Hausdorff separation axiom.
Continuity: definition of continuity; open maps; homeomorphisms; simple topological invariants
Compactness: definition of compactness; compact sets in n-dimensional Euclidean space.
Connectivity: connected and disconnected spaces; path connected spaces; Jordan curve theorem.
Identification spaces: examples include circle; torus; Mobius band; Klein bottle.

Specialised mini courses from a selection of:
Compactness: Heine-Borel Theorem; Tychonoff'r Theorem; Compactifcation Theorems.
CW-complexes: construction and examples of applications.
Homotopy Theory: deformation retractions; fundamental group
Homology: simplicial and singular homology; covering spaces.
Cohomology: the Universal Coefficient Theorem.

### Module Skills

Skills Type Skills details
Application of Number Throughout the module.
Communication Students will be expected to submit clearly written solutions to set exercises.
Improving own Learning and Performance Feedback via marked exercises and examples classes. Students will be expected to develop their own approach to time-management.
Information Technology Use of the internet and mathematical packages will be encouraged to enhance their understanding of the module content and examples of application
Personal Development and Career planning Students will be exposed to an area of application that they have not previously encountered.
Problem solving The assignments will give the students opportunities to show creativity in finding solutions and develop their problem solving skills.
Research skills Students will be encouraged to consult various books, journals and recommended material on the internet to enhance their understanding of the module content and examples of application.
Subject Specific Skills The application of topological techniques to solve problems.
Team work Students will be encouraged to work in groups to solve problems.

### Notes

This module is at CQFW Level 7