Undergraduate Programme and Module Handbook 2024-2025

# Module MATH3491: Geometric Topology III

## Department: Mathematical Sciences

### MATH3491: Geometric Topology III

Type | Open | Level | 3 | Credits | 20 | Availability | Available in 2024/2025 | Module Cap | Location | Durham |
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#### Prerequisites

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

#### Corequisites

- None

#### Excluded Combination 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

Activity | Number | Frequency | Duration | Total/Hours | |
---|---|---|---|---|---|

Lectures | 42 | 2 per week for 20 weeks and 2 in term 3 | 1 hour | 42 | |

Problems Classes | 8 | 4 in each of terms 1 and 2 | 1 hour | 8 | |

Preparation and Reading | 150 | ||||

Total | 200 |

#### Summative Assessment

Component: Examination | Component Weighting: 100% | ||
---|---|---|---|

Element | Length / duration | Element Weighting | Resit Opportunity |

End of year written examination | 3 hours | 100% |

#### Formative Assessment:

Eight assignments to be submitted.

■ 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