# MATH3061 Geometry and Topology

## General Information

This page contains information on the Senior mainstream Unit of Study MATH3061 Geometry and Topology.

- Taught in Semester 2.
- Credit point value: 6.
- Classes per week: Three lectures and one tutorial.
- Lecturer(s): Nathan Brownlowe and Laurentiu Paunescu .

Please refer to the Senior Mathematics and Statistics Handbook for all questions relating to Senior Mathematics and Statistics. In particular, see the MATH3061 handbook entry for further information relating to MATH3061.

You may also view the description of MATH3061 in the central units of study database.

Students have the right to appeal any academic decision made by the School or Faculty. For further information, see the Science Faculty web site.

** Enquiries**

All enquiries about this unit of study should be directed to MATH3061@sydney.edu.au.

** Information Sheet **

The MATH3061 Information Sheet contains details of the lecturers and lecture times, consultation times, tutorials, assessment, textbooks, objectives, and learning outcomes for MATH3061.

If you experience problems reading pdf files online, here are some useful tips.

Check your marks Course outline Online resources References Timetable

## Information about the exam

For general information about exams, and the exam timetable, please see the University's examinations web page.

*Please note that the questions for the geometry and topology
sections of the course must be answered in separate booklets*.

The exam is based on all the material covered in lectures, tutorials, assignments, and the quizzes. The best way to prepare for the exam is to read through and think about your lecture notes and then to try to solve as many questions as possible (without looking at the solutions) from the lectures, tutorials, and assignments. You might like to look at the 2013 MATH3061 exam. Here are the solutions to this exam:

Geometry Solutions Topology Solutions.

*We recommend that you do not look at these solutions until you have first attempted to do all of the questions in the exam.*

In addition, here are the questions for the 2014 exam, and here the solutions to the topology section of this exam.

## Course outline

**Geometry**

The course will closely follow the "Geometry" lecture notes by O'Brian.

- Week 1
- Reading for Week 1: pages 1–29, but not the section on Symmetries. [The coordinate formula for a halfturn does not appear in this part of the O'Brian notes. FYI, the image of a point (x,y) under a halfturn about point P(p,q) is (2p−x,2q−y).]
- Week 2
- Reading for Week 2: pages 30–36. Handout on properties of rotations from lecture 6.
- Week 3
- Reading for Week 3: pages 37–40, omit Chapter 3, then pages 51–56.
- Week 4
- Reading for Week 4: pages 57–58 and pages 67–70 of Chapter 4, omit the remainder of Chapter 4 and all of Chapter 5, then pages 97–99.
- Week 5
- Reading for Week 5: pages 100–107.
- Week 6
- Reading for Week 6: page 113 to first half of page 121. Handout end of proof from the last Geometry lecture.

**Topology**

Topology is the study of surfaces under

*continuous deformation*. That is, we allow ourselves to

*stretch*surfaces but not to tear them.

- Week 7
- Graphs, subdivision, sums of degrees = twice the number of edges, connectedness, circuits, trees.
- Week 8
- Disc, annulus, torus, Möbius band, Klein bottle, sphere, projective planes, homeomorphism, stereographic projection.
- Week 9
- Triangulated surfaces, Euler characteristic, invariance under subdivision, cutting, pasting, boundaries, orientation, edge equation.
- Week 10
- Classification of surfaces, genus, oriented closed surfaces in three dimensions, handles, crosscaps.
- Week 11
- Platonic surfaces. Graphs on surfaces: K
_{5}is not planar. Map colouring: the five colour theorem, the Heawood estimate for maps on surfaces. - Week 12
- Knots: Polygonals knots, knots diagrams, the unknot, trefoil knots, figure eight knots, knot colouring.
- Week 13
- Knot determinants, n-colourings, Seifert surfaces, and knot genus.

## Textbook and references

Lecture notes for both parts of MATH3061 are available for purchase as a book from Kopystop at 36 Mountain Street, Broadyway.

See the course information sheet for additional textbook references.

### Further reading and resources (topology)

- Topological equivalence of a torus and a coffee cup
- The Klein bottle (YouTube video)
- Gluing a torus bottle (YouTube video)
- An Introduction to Topology, E. C. Zeeman
- Platonic solids (Wikipedia)
- The Four-Color Problem: Concept and Solution, Steven G. Krantz.
- The Rolfsen Knot Table
- Torus knots
- Seifert surface

## Online resources

**Check your marks**

MATH3061 students: please check that your marks for Geometry Assignment, Geometry Quiz, Topology Assignment and Topology Quiz have been recorded correctly by pressing the "Check marks" button.

Please note that any corrections to Geometry Assignment marks must be made by Friday, Sep 22.

Please note that any corrections to Geometry Quiz marks must be made by Friday, Sep 22.

Please note that any corrections to Topology Assignment marks must be made by Friday, Nov 10.

Please note that any corrections to Topology Quiz marks must be made by Friday, Nov 17.

## Timetable

Show timetable / Hide timetable.