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Algebraic Topology (MATH M1200)

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Administrative Information

  1. Unit number and title: MATH M1200 Algebraic Topology
  2. Level: M/7
  3. Credit point value: 20 credit points
  4. Year: 12/13
  5. First Given in this form: 1997-98
  6. Unit Organiser: Jeremy Rickard
  7. Lecturer: Prof J Rickard
  8. Teaching block: 2
  9. Prerequisites: MATH 20200 Metric Spaces 2 and MATH33300 Group Theory

Unit aims

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

General Description of the Unit

Algebraic Topology concerns constructing and understanding topological spaces through algebraic, combinatorial and geometric techniques. In particular, groups are associated to spaces to reveal their essential structural features and to distinguish them.  In cruder terms, it is about adjectives that capture and distinguish essential features of spaces.  

The theory is powerful. We will give applications including proofs of The Fundamental Theory of Algebra and Brouwer's Fixed Point Theorem (which is important in economics).

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.

Teaching Methods

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

Learning Objectives

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.

Assessment Methods

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.

Award of Credit Points

Credit points for the unit are gained by:

  • either passing the unit (i.e. getting a final assessment mark of 50 or over),
  • or getting a final assesment mark of 30 or over AND handing in satisfactory attempts at the set homework questions.

Transferable Skills

The assimilation of abstract and novel ideas.

Geometric intuition.

How to place intuitive ideas on a rigorous footing.

Presentation skills.

Texts

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

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

Unit Webpage

http://www.maths.bris.ac.uk/~majcr/algtop.html