MATH33020: Geometric Topology

• Topology; Complex Analysis; Algebra.

• None

• None

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

• Homotopies and homotopy equivalence.
• Simplicial complexes and simplicial homology.
• The fundamental group: calculation for circle, homotopy invariance.
• Generators and relations of groups, Tietze theorem, Van Kampen's theorem.
• Covering spaces and their classification.
• Compact surfaces and their classification.
• Classical knots, basic knot invariants.
• Linking numbers, Seifert surfaces.

Subject-specific Knowledge:

• By the end of the module students will:
• Be able to solve novel and/or complex problems in Geometric Topology, have a systematic and coherent understanding of theoretical mathematics in the field of Geometric Topology, and have acquired a coherent body of knowledge of this subject demonstrated through one or more of the following topic areas:
• Simplicial complexes and simplicial homology.
• Fundamental group, homotopy type.
• Group presentations and Van Kampen's Theorem.
• Covering spaces.
• Surfaces and Knots.

Subject-specific Skills:

• Students will have basic mathematical skills in the following areas: problem solving, abstract reasoning.

Key Skills:

• Students will have basic mathematical skills in the following areas: problem solving, abstract reasoning.

• This module will be delivered by the Department of Mathematical Sciences.
• Lectures demonstrate what is required to be learned and the application of the theory to practical examples.
• 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.
• The end-of-year examination assesses the knowledge acquired and the ability to solve predictable and unpredictable problems.

Eight written assignments to be assessed and returned. Other assignments are set for self-study and complete solutions are made available to students.

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