MX4546: ALGEBRAIC TOPOLOGY (2017-2018)

• Elementary concepts of homotopy theory.
• The fundamental group and its naturality properties.
• Fundamental groups and covering spaces.
• Free groups and subgroups of free groups.
• The Seifert-VanKampen theorem.
• Presentations of groups.
• The concept of a surface.
• Triangulations.
• The classification of compact surfaces without boundary.
• If time allows, an introduction to homology theory.

Syllabus

• Revision of topological spaces
• Topological equivalence, homotopy and homotopy equivalence, deformation retraction.
• The fundamental group, homomorphisms induced by continuous maps, and homotopy invariance.
• The fundamental group of a circle and introduction to covering spaces
• Applications: The fundamental theorem of algebra, Brauer fixed point, and Borsuk-Ulam.
• Covering spaces: Concept, existence and classification.
• Desk transformation and group actions.
• The Seifert Van-Kampen theorem.
• Computation of the fundamental group and applications to classification of closed surfaces.

Course Aims

The course is an introduction to algebraic topology. The overall aim of the course is to give an understanding of algebraic topology as the study of the properties of topological spaces which are invariant under continuous deformations. The main aim is the study of the fundamental group and related tools, such as covering spaces and the Seifert-van Kampen theorem. The course also aims to make the student familiar with compact surfaces.

Learning Objectives

By the end of the course the student should:

- be able to describe some simple spaces such as: circle, 2-dimensional sphere, torus, projective plane, Klein bottle,

- understand what is the fundamental group of a topological space,

- know about covering spaces and their relation to the fundamental group,

- know about the Seifert-van Kampen theorem and how to apply it,

- tell apart simple cases of non-homeomorphic spaces,

- recognize simple cases of deformation retracts,

- be able to compute the fundamental group of some basic topological spaces.

Information on contact teaching time is available from the course guide.

More Information about Week Numbers

1st attempt: 1 two-hour written examination (80%) and in-course assessment (20%).

Resit: If required and permitted by Regulations, there will be 1 two-hour written examination. The CAS mark will be based on the maximum of examination (100%) and examination (80%) together with in-course-assessment (20%).

Only the marks obtained at the first attempt can count towards Honours classification.

In-course assignments will normally be marked within one week and feedback provided to students in tutorials. Students will be invited to contact the Course Coordinator for feedback on the final examination. Students do practice questions in tutorials allowing formative assessment and feedback from tutors.