	Department of Mathematics & Computer Science

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MA3151 Topology

MA3151(=MC356) Topology

Credits: 20

Convenor: Dr. F. Neumann

Semester: 1

Prerequisites:	essential: MA1102(=MC145), MA2151(=MC240), MA2161(=MC255)	desirable:
Assessment:	Coursework: 10%	Three hour examination in January: 90%

Lectures:	36	Problem Classes:	10
Tutorials:	none	Private Study:	104
Labs:	none	Seminars:	none
Project:	none	Other:	none
Surgeries:	none	Total:	150

Explanation of Pre-requisites

This module will draw on some basic ideas from both algebra and analysis. In algebra knowledge will be assumed of basic number systems and of the idea of a group and associated ideas. In analysis the central topics drawn on are the ideas of topological spaces, of closed or compact subsets of ${\bf R}^n$ and continuous functions.

Course Description

Many geometric ideas turn out not really to depend on size or distance - thus for example, a Möbius band has only one side however large or small it is; a knotted loop of string remains knotted however much the string is pulled and twisted about, and the surface of a donut or bicycle inner-tube cannot be deformed into the surface of a ball without cutting and pasting parts of it together. All these observations are about underlying properties of the objects and the study of such ideas is called topology.

This module introduces the main ideas, in particular the central concepts of homotopy and the fundamental group.

This module represents a branch of mathematics which brings together many of the basic ideas of algebra and analysis met before in the course as well as linking many of the later topics. Ideas in the module will particularly complement the modules MA3121 Complex Analysis, MA4111 Differential Geometry as well as the level 4 module MA4101 Algebraic Topology run in the alternate years.

Aims

To explore geometric objects such as knots or surfaces and how elementary algebra and analysis can provide a rigorous language for discussing their properties.

Objectives

To know the definitions of and understand the key concepts introduced in this module.

To be able to understand, reproduce and apply the main results and proofs in this module.

To be able to solve some routine topological problems.

Transferable Skills

The ability to present arguments and solutions in a coherent and logical form.

The ability to use the techniques taught within the module to solve problems.

The ability to apply taught principles and concepts to new situations.

Syllabus

Topological spaces, continuous maps, examples, group actions, homotopy, the fundamental group and its properties, calculations of $\pi_1(-)$ , covering spaces, applications such as knot theory or classification of surfaces, as time allows.

Reading list

Background:

K. Jaehnich, Topology, Springer.

H. Sato, Algebraic Topology: an intuitive approach, Springer.

Details of Assessment

There will be a series of pieces of work set during the semester which together will count for 10% of the final mark.

There will be eight questions on the examination paper; any number of questions may be attempted but full marks may be gained from answers to five questions. All questions will carry equal weight.