Module MATH4021: TOPOLOGY IV

Department: MATHEMATICAL SCIENCES

MATH4021: TOPOLOGY IV

Prerequisites

• Complex Analysis II (MATH2011), Linear Algebra II (MATH2021) and Analysis in Many variables II (MATH2031).

Corequisites

• None.

Excluded Combination of Modules

• Topology III (MATH3281)

Aims

• To provide a balanced introduction to Point Set, Geometric and Algebraic Topology, with particular emphasis on surfaces and knots.

Content

• Topological Spaces and Continues Functions: Topology on a set, open sets, closed sets, limit points and closure, examples of topologies.
• Compactness and Connectedness.
• Topological groups and group actions.
• The Orthogonal groups The Fundamental Group: calculation for circle, homotopy type, homotopy equivalence.
• Generators and relations of groups, Tietze theorem, Van Kampen's theorem.
• Compact surfaces, classical knots, basic knot invariants.
• Reading material on higher homotopy groups.

Learning Outcomes

• By the end of the module students will: be able to solve novel and/or complex problems in Topology.
• have a systematic and coherent understanding of theoretical mathematics in the field of Topology.
• have acquired a coherent body of knowledge of these subjects demonstrated through one or more of the following topic areas: Topological spaces.
• Topological Groups and group actions.
• Fundamental group, homotopy type.
• Group presentations and Van Kampen's Theorem.
• Surfaces and Knots.
• Knowledge and understanding in one topic of higher homotopy groups.

• In addition students will have specialised mathematical skills in the following areas which can be used with minimal guidance: Spatial awareness.
• Ability to read independently to acquire knowledge and understanding in the area of higher homotopy groups.

Modes of Teaching, Learning and Assessment and how these contribute to the learning outcomes of the module

• Lectures demonstrate what is required to be learned and the application of the theory to practical examples.
• Subject material assigned for independent study develops the ability to acquire knowledge and understanding without dependence on lectures.
• Assignments for self-study develop problem-solving skills and enable students to test and develop their knowledge and understanding.
• Formatively assessed assignments provide practice in the application of logic and high level of rigour as well as feedback for the students and the lecturer on students' progress.
• Summative examination assesses acquired knowledge. The Subject material assigned for independent study will form part of the examined material.

Teaching Methods and Learning Hours

Summative Assessment

Formative Assessment:

At least two written assignments in each of the first two terms. No collections.

■ Attendance at all activities marked with this symbol will be monitored. Students who fail to attend these activities, or to complete the summative or formative assessment specified above, will be subject to the procedures defined in the University's General Regulation V, and may be required to leave the University