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MATH3491: Geometric Topology III

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Type Open
Level 3
Credits 20
Availability Available in 2024/2025
Module Cap
Location Durham
Department Mathematical Sciences

Prerequisites

  • Topology II (MATH2727) AND Complex Analysis II (MATH2011) AND Algebra II (MATH2581)

Corequisites

  • None

Excluded Combinations of Modules

  • None

Aims

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

Content

  • 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.

Learning Outcomes

Subject-specific Knowledge:

  • By the end of the module, students will:
  • 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:

  • In addition students will have specialised mathematical skills in the following areas which can be used with minimal guidance: Spatial awareness.

Key Skills:

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

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.
  • 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.

Teaching Methods and Learning Hours

ActivityNumberFrequencyDurationTotalMonitored
Lectures422 per week for 20 weeks and 2 in term 31 hour42 
Problems Classes84 in each of terms 1 and 21 hour8 
Preparation and Reading150 
Total200 

Summative Assessment

Component: ExaminationComponent Weighting: 100%
ElementLength / DurationElement WeightingResit Opportunity
End of year written examination3 hours100 

Formative Assessment

Eight assignments to be submitted.

More information

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