Unit information: Algebraic Topology in 2012/13

Unit name	Algebraic Topology
Unit code	MATHM1200
Credit points	20
Level of study	M/7
Teaching block(s)	Teaching Block 2 (weeks 13 - 24)
Unit director	Professor. Rickard
Open unit status	Not open
Pre-requisites	Level 2 Analysis. Level 3 Group Theory.
Co-requisites	None
School/department	School of Mathematics
Faculty	Faculty of Science

Description including Unit Aims

Many problems about continuous mappings between geometrical sets can be very difficult, because there are so many possible maps - consider how many maps there are from the reals to the reals. One such problem is answered by Brouwers fixed point theorem: every continuous transformation from a disc (including the boundary) to itself has some point of the disc as a fixed point. The aim of algebraic topology is to tackle such problems by turning them into more manageable problems in algebra. For example, we shall prove Brouwers theorem by transforming it into a very trivial question about group theory. The methods of the course will be mainly algebraic, involving some of the elementary theory of groups, (mostly abelian), but we shall apply this algebra to several specific problems in geometry.

Aims

The aim of the unit is to give an introduction to algebraic topology with an emphasis on cell complexes, fundamental groups and homology.

Syllabus

• Topological spaces – open sets, closed sets, product, quotient and subspace topologies, connectedness, path-connectedness, continuous maps, homeomorphisms, Hausdorff spaces
• Homotopy
• The definition of the fundamental group and its calculation for the circle
• The Fundamental Theorem of Algebra
• Brouwer's Fixed Point Theorem
• Covering Spaces
• van Kampen's Theorem
• Graphs and free groups
• The Classification of Surfaces

Relation to Other Units

This is one of three Level M units which develop group theory in various directions. The others are Representation Theory and Galois Theory.

Intended Learning Outcomes

Students should absorb the idea of algebraic invariants to distinguish between complex objects, their geometric intuition should be sharpened, they should have a better appreciation of the interconnectivity of different fields of mathematics, and they should have a keener sense of the power and applicability of abstract theories.

Transferable Skills:

The assimilation of abstract and novel ideas.

Geometric intuition.

How to place intuitive ideas on a rigorous footing.

Presentation skills.

Teaching Information

Lectures, problem sets and discussion of problems, student presentations.

Assessment Information

There will be no final examination.

The final assessment mark for Algebraic Topology is calculated from:

• 80% for coursework (problem sets).

• 20% based on seminar presentations given by students during the semester. These will be graded on the understanding and insight they demonstrate, and on the clarity, quality, lucidity and style of the delivery. Additionally, some credit will be awarded for active participation in discussions, in lectures and in seminars throughout the course.

Reading and References

W. A. Sutherland, Introduction to metric and topological spaces, Clarendon Press, Oxford.

Munkres, Topology (2nd Edition), Pearson Education

Hatcher, Algebraic Topology, Chapters 0,1,2.

O.Ya.Viro, O.A.Ivanov, V.M.Kharlamov, N.Y.Netsvetaev, Elementary Topology