## MATH3024 Homotopy and Surfaces

### 15 creditsClass Size: 100

Taught: Semester 2 (Jan to Jun) View Timetable

Year running 2005/06

### Pre-requisites

(MATH1031 or MATH1050) and MATH1021 or equivalent.

This module is approved as an Elective

### Objectives

To provide a first introduction to elementary ideas from algebra and topology that are linked in modern mathematical developments. The basic concept studied is that of the fundamental group of a polyhedral surface. On completion of this module, students should be able to: (a) classify a given surface; (b) calculate the fundamental group of some simple surfaces; (c) use a knowledge of the fundamental group to obtain results in algebra, analysis and topology.

### Syllabus

This module introduces a completely different aspect to topology from that offered in the module Topology. It has little to do with formal analysis but develops some intuitive geometrical ideas about the way surface may be described and shows how to develop algebraic methods for classifying them. This leads to the idea of associating a group, the fundamental group, with any topological space. However, in this module we only compute the fundamental group of a circle and show how to use this to calculate the fundamental group of a polyhedral surface and other simple spaces, such as those obtained by gluing together polygons along their edges. Nevertheless, it is shown that these results can be applied to obtain non-trivial, and even surprising, results in various branches of mathematics. The topics covered are: 1. Subsets of , products and quotients of such subsets, continuous maps and homeomorphisms between these. 2. Polyhedral surfaces. Representation by sentences. Equivalent sentences. Classification of sentences by canonical words. 3. Geometrical description of surfaces represented by canonical words, Euler characteristic. 4. Fundamental group. Path space. Homotopy of paths. Composition of paths. 5. Homotopies of maps. Deformation retracts. 6. Calculation of ?(S1). Path lifting theorem. Homotopy lifting theorem. 7. Applications from among: Brouwer's fixed point theorem. Borsuk-Ulam theorem. Fundamental theorem of algebra. Jordan curve theorem, Pancake and Ham Sandwich theorems. . 8. Computations of fundamental groups. Finitely presented groups. Van Kampen's theorem. Classification of surfaces.

### Teaching methods

Due to COVID-19, teaching and assessment activities are being kept under review - see module enrolment pages for information

Lectures: 26 hours. One video class plus 6 examples classes.

### Methods of assessment

Due to COVID-19, teaching and assessment activities are being kept under review - see module enrolment pages for information

One 3 hour examination at end of semester (100%).