MATH3061 Geometry and Topology
General Information
This page contains information on the senior mainstream unit of study MATH3061.
 Taught in Semester 2.
 Credit point value: 6.
 Classes per week: Three lectures and one tutorial.
 Lecturer(s): Anne Thomas and Andrew Mathas .
Please refer to the Senior Mathematics and Statistics Handbook for all questions relating to Senior Mathematics and Statistics. In particular, see the handbook entry for MATH3061 for further information relating to MATH3061.
You may also view the description of MATH3061 and the description of in the University's course search database.
For enrolled students or other authorized people only, here is a link to the Canvas page for MATH3061.
Students have the right to appeal any academic decision made by the School or Faculty: see sydney.edu.au/students/academicappeals.html.
Consultation
Lecturer  Availability  Time  Location 

Anne Thomas  Weeks 1 – 6  Thursday 9am  zoom 
Andrew Mathas  Weeks 7 – 12  Thursday 9am  zoom 
Assessment
Date^{*}  Description  Better mark  Weighting 

23:59 September 10  Geometry Assignment  10%  
9:00 September 16  Geometry Quiz  10%  
23:59 October 22  Topology Assignment  10%  
9:00 November 4  Topology Quiz  10%  
Exam  60% 
References
 Nigel O'Brian and Jonathan Hillman, Lecture notes for MATH3061: Geometry and Topology
 (G) Judith N. Cederberg,, A course in modern geometries. New York: SpringerVerlag, c1989, 513 73.
 (G) George E. Martin, Transformation geometry: an introduction to symmetry. New York: SpringerVerlag, c1982, 516.55 8.
 (G) William P. Thurston, The geometry and topology of threemanifolds. Princeton, NJ: Princeton University, 1979
 (T) Colin Adams, The knot book : an elementary introduction to the mathematical theory of knots. Providence, R.I., American Mathematical Society, 2004, 514.2242 3
 (T) Donald W.Blackett, Elementary topology: a combinatorial and algebraic approach. London; New York: Academic Press, 1982, 513.83 102
 (T) P.A. Firby and C.F. Gardiner, Surface topology. New York: Ellis Horwood, 1991, 513.83 110 A.
Check your marks EdStem Lecture recordings Online resources References
Enquiries
All enquiries about this unit of study should be directed to MATH3061@sydney.edu.au. Any mathematical questions sent to this email address will be redirected to the EdStem forum (NOTE: to register for this course on EdStem you first need to do it through Canvas). Please give your name and SID when emailing us. We reserve the right not to reply to anonymous emails. We will use Canvas for assignment submission.
If you experience problems reading pdf files online, here are some useful tips.
Unit outline
Canvas webpage
The MATH3061 Canvas webpage contains details about consultations, assessment, textbooks, objectives, and learning outcomes for MATH3061.
This part of the course will explore geometric transformations and properties that are invariant under these transformations.
 Week 1
 Linear algebra review. The Euclidean plane. Transformations. Isometries.
 Week 2
 Transformation groups. Reflections. Fixed points. Rotations.
 Week 3
 Involutions. Glidereflections. Classification of isometries. Parity.
 Week 4
 Symmetry groups. Affine transformations. Derivative of an isometry.
 Week 5
 The projective plane. Projective lines. Collineations.
 Week 6
 Conics. Classification of Conics.
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. Knot determinants, ncolourings, Seifert surfaces, and knot genus.
Textbook and references
Lecture notes will be published in the resource table below.
Supplementary notes for both parts of math3061 are available online: Nigel O’Brian's Geometry lecture notes.
 Jonathan Hillman's Topology lecture notes.
See the course Canvas webpage 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 FourColor Problem: Concept and Solution, Steven G. Krantz.
 The Rolfsen Knot Table
 Torus knots
 Seifert surface
Online resources
Lectures  Tutorials  Assessment  

Week 1 9/813/8 
Geometry 1A Geometry 1B Geometry 1C Geometry 1D Geometry 1E 
No tutorials in week 1  
Week 2 16/820/8 
Geometry 2A Geometry 2B Geometry 2C Geometry 2D Geometry 2E 
Tutorial 1 questions Tutorial 1 solutions 

Week 3 23/827/8 
Geometry 3A Geometry 3B Geometry 3C Geometry 3D Geometry 3E 
Tutorial 2 questions Tutorial 2 solutions 

Week 4 30/83/9 
Geometry 4A Geometry 4B Geometry 4C Geometry 4D Geometry 4E 
Tutorial 3 questions Tutorial 3 solutions 
Geometry Assignment  questions 
Week 5 6/910/9 
Geometry 5A Geometry 5B Geometry 5C Geometry 5D Geometry 5E 
Tutorial 4 questions Tutorial 4 solutions 
Geometry Assignment (10%) Due 23:59 September 10 
Week 6 13/917/9 
Geometry 6A Geometry 6B Geometry 6C Geometry 6D Geometry 6E 
Tutorial 5 questions Tutorial 5 solutions 
Geometry Quiz (10%) 9:00 September 16 
Week 7 20/924/9 
Topology 7A Topology 7A unfilled Topology 7B Topology 7B unfilled Topology 7C Topology 7C unfilled Topology 7D Topology 7D unfilled Topology 7E Topology 7E unfilled 
Tutorial 6 questions Tutorial 6 solutions 
Geometry Assignment  solutions 
Midsemester break  
Week 8 4/108/10 
Labour Day Topology 8A unfilled Topology 8B Topology 8B unfilled Topology 8C Topology 8C unfilled Topology 8D Topology 8D unfilled Topology 8E Topology 8E unfilled 
Tutorial 7 questions Tutorial 7 solutions 
Labour Day 
Week 9 11/1015/10 
Topology 9A Topology 9A unfilled Topology 9B Topology 9B unfilled Topology 9C Topology 9C unfilled Topology 9D Topology 9D unfilled Topology 9E Topology 9E unfilled 
Tutorial 8 questions Tutorial 8 solutions 
Topology Assignment  questions 
Week 10 18/1022/10 
Topology 10A Topology 10A unfilled Topology 10B Topology 10B unfilled Topology 10C Topology 10C unfilled Topology 10D Topology 10D unfilled Topology 10E Topology 10E unfilled 
Tutorial 9 questions Tutorial 9 solutions 
Topology Assignment (10%) Due 23:59 October 22 
Week 11 25/1029/10 
Topology 11A Topology 11A unfilled Topology 11B Topology 11B unfilled Topology 11C Topology 11C unfilled Topology 11D Topology 11D unfilled Topology 11E Topology 11E unfilled 
Tutorial 10 questions Tutorial 10 solutions 

Week 12 1/115/11 
Topology 12A Topology 12A unfilled Topology 12B Topology 12B unfilled Topology 12C Topology 12C unfilled Topology 12D Topology 12D unfilled Topology 12E Topology 12E unfilled 
Tutorial 11 questions Tutorial 11 solutions 
Topology Assignment  solutions Topology Quiz (10%) 9:00 November 4 
Week 13 8/1112/11 
Tutorial 12 questions Tutorial 12 solutions 
Topology Quiz  solutions 
Timetable
Show timetable / Hide timetable.