620-322 |

Lecturer: Craig Westerland, 168 Richard Berry, phone: 8344 9712, email: C.Westerland@ms.unimelb.edu.au

Lecture: Monday, Thursday, Friday 10-11, 213 Richard Berry. Practice class: Thursday 1-2 Room C, Richard Berry.

Consultation hours: Monday 11-12, Thursday 11-1

This subject introduces the basic concepts and examples of topological spaces, the definition of manifolds and the classification of surfaces, the idea of homotopy of mappings, the concept of covering spaces and their relationship with fundamental groups, and the basic ideas of homology theory. Students should develop the ability to work with the fundamental group and homology groups, to calculate and use the fundamental group, to convert problems involving topological spaces and continuous maps into problems in algebra, to distinguish between different topological spaces, and to construct homeomorphisms and homotopy equivalences between spaces. This subject investigates the basic questions in topology. It demonstrates the power of topological methods in dealing with problems involving shape and position of objects and continuous mappings, and shows how topology can be applied to many areas, including geometry, analysis, group theory and physics.

- Topological spaces and continuous maps
- Quotient spaces
- Homotopy of maps

- The fundamental group
- Seifert-van Kampen theorem
- Covering spaces

- Homology groups

- Categories and functors
- Surfaces and higher dimensional manifolds

- A. Hatcher, Algebraic Topology, Cambridge University Press, 2002 ISBN: 0-521-79540-0, available online at http://www.math.cornell.edu/~hatcher/AT/ATpage.html.
- W. S. Massey, A Basic Course in Algebraic Topology, Springer Verlag, Graduate Texts in Mathematics, 1997, ISBN: 978-0-387-97430-9.

Background:

- J. R. Munkres, Topology, Prentice-Hall, 2000, ISBN 0131784498, 9780131784499.
- T. W. Hungerford, Algebra, Springer Verlag, Graduate Texts in Mathematics, 2003, ISBN: 978-0-387-90518-1.

Up to 60 pages of written assignments (25%), a three-hour written examination (75%, in the examination period).

Homework assignments: (all problems taken from Hatcher)

- First homework (due 27/8): section 0, #1, 9, 10, 12; section 1.1, #12, 15, 17.
- Second homework (due 12/10): section 1.2, #1, 4, 11; section 1.3, #4, 9, 14, 24.

The first set of practice problems, and their solutions.

The final exam will be on Friday, 13 November at 9:30 a.m. in Wilson Hall. The exam will be 3 hours long, with a 15 minute reading period.

The plaigiarism declaration is available here