# Mathematics / Applied Mathematics MSc

# 5M: Advanced algebraic and geometric topology MATHS5038

**Academic Session:**2019-20**School:**School of Mathematics and Statistics**Credits:**20**Level:**Level 5 (SCQF level 11)**Typically Offered:**Either Semester 1 or Semester 2**Available to Visiting Students:**No**Available to Erasmus Students:**No

#### Short Description

The goals of this course are to study some of the most important tools of algebraic topology and also to study some of the basic objects in the geometric topology of manifolds and submanifolds.

#### Timetable

2 hours of lectures a week, over 11 weeks.

1 hour tutorial a week over 10 weeks (or equivalent)

#### Requirements of Entry

Mandatory Entry Requirements

Recommended Entry Requirements

4H Algebraic and Geometric Topology

4H Differential Geometry

#### Assessment

Assessment

100% Final Exam

Reassessment

In accordance with the University's Code of Assessment reassessments are normally set for all courses which do not contribute to the honours classifications. For non honours courses, students are offered reassessment in all or any of the components of assessment if the satisfactory (threshold) grade for the overall course is not achieved at the first attempt. This is normally grade D3 for undergraduate students, and grade C3 for postgraduate students. Exceptionally it may not be possible to offer reassessment of some coursework items, in which case the mark achieved at the first attempt will be counted towards the final course grade. Any such exceptions are listed below in this box.

**Main Assessment In:** April/May

**Are reassessment opportunities available for all summative assessments?** No

Reassessments are normally available for all courses, except those which contribute to the Honours classification. For non Honours courses, students are offered reassessment in all or any of the components of assessment if the satisfactory (threshold) grade for the overall course is not achieved at the first attempt. This is normally grade D3 for undergraduate students and grade C3 for postgraduate students. Exceptionally it may not be possible to offer reassessment of some coursework items, in which case the mark achieved at the first attempt will be counted towards the final course grade. Any such exceptions for this course are described below.

#### Course Aims

The goals of this course are to study some of the most important tools of algebraic topology and also to study some of the basic objects in the geometric topology of manifolds and submanifolds. In algebraic topology, two important invariants will be introduced: higher homotopy groups and homology groups. Their basic properties will be studied as well as methods of computing them for important examples. In geometric topology, possible topics include classical knot theory and some basic facts about topology in three and four dimensions.

#### Intended Learning Outcomes of Course

At the end of this course students should be able to:

• Demonstrate knowledge of the central definitions and facts of selected topics in algebraic and geometric topology, and use these to solve problems of a numerical or logical nature. Specific topics which may be covered are:

Algebraic topology:

1. Definition and examples of categories and functors

2. Definition and examples of homotopy and homotopy equivalence

3. Definition and examples of CW complexes

4. Definition, properties and examples of homology groups

5. Homotopy invariance of homology

6. State and apply the Mayer-Vietoris sequence and excision

7. Definition and examples of cellular homology

8. State and apply Alexander duality

9. Definition and examples of higher homotopy groups

10. Define/state and apply Hurewicz homomorphism/theorem

11. Definition and examples of fibrations and cofibrations

12. State and apply the long exact homotopy sequence of a fibration

Geometric topology:

1. Definition and examples of Seifert surfaces and associated knot invariants

2. Definition, properties and examples of Alexander module and polynomial of knots and 3-manifolds

3. Know and exhibit basic constructions of 3-manifolds: Heegaard splittings, Dehn surgery, branched covers

4. Construct and derive basic properties of the Poincar´e homology sphere.

5. Understand, state and discuss the statement: Every 3-manifold bounds a 4-manifold.

6. Statement and discussion of Poincar´e and Geometrisation conjectures (now theorems) for 3-manifolds.

7. Reproduce the proofs of the main theorems covered in the course.

#### Minimum Requirement for Award of Credits

Students must submit at least 75% by weight of the components (including examinations) of the course's summative assessment.