## MATH3024 Homotopy and Surfaces

### 15 creditsClass Size: 60

Module manager: Professor John Wood
Email: j.c.wood@leeds.ac.uk

Taught: Semester 2 View Timetable

Year running 2011/12

### Pre-requisite qualifications

MATH1022 and (MATH1035 or MATH1050)

This module is approved as an Elective

### Module summary

This module develops some intuitive geometrical ideas about the way surfaces 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. Results can be applied to obtain non-trivial, and even surprising, results in various branches of mathematics.

### 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

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

 Delivery type Number Length hours Student hours Lecture 33 1.00 33.00 Private study hours 117.00 Total Contact hours 33.00 Total hours (100hr per 10 credits) 150.00

### Opportunities for Formative Feedback

Regular problem solving assignments

### Methods of assessment

Exams
 Exam type Exam duration % of formal assessment Standard exam (closed essays, MCQs etc) 2 hr 30 mins 100.00 Total percentage (Assessment Exams) 100.00

Normally resits will be assessed by the same methodology as the first attempt, unless otherwise stated