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Unit of study_

MATH4311: Algebraic Topology

One of the most important aims of algebraic topology is to distinguish or classify topological spaces and maps between them up to homeomorphism. Invariants and obstructions are key to achieve this aim. A familiar invariant is the Euler characteristic of a topological space, which was initially discovered via combinatorial methods and has been rediscovered in many different guises. Modern algebraic topology allows the solution of complicated geometric problems with algebraic methods. Imagine a closed loop of string that looks knotted in space. How would you tell if you can wiggle it about to form an unknotted loop without cutting the string? The space of all deformations of the loop is an intractable set. The key idea is to associate algebraic structures, such as groups or vector spaces, with topological objects such as knots, in such a way that complicated topological questions can be phrased as simpler questions about the algebraic structures. In particular, this turns questions about an intractable set into a conceptual or finite, computational framework that allows us to answer these questions with certainty. In this unit you will learn about fundamental group and covering spaces, homology and cohomology theory. These form the basis for applications in other domains within mathematics and other disciplines, such as physics or biology. At the end of this unit you will have a broad and coherent knowledge of Algebraic Topology, and you will have developed the skills to determine whether seemingly intractable problems can be solved with topological methods.

Code MATH4311
Academic unit Mathematics and Statistics Academic Operations
Credit points 6
Prerequisites:
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None
Corequisites:
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None
Prohibitions:
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None
Assumed knowledge:
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Familiarity with abstract algebra and basic topology, e.g., (MATH2922 or MATH2961 or equivalent), (MATH3961 or equivalent) and (MATH2923 or equivalent).

At the completion of this unit, you should be able to:

  • LO1. Demonstrate a coherent and advanced understanding of the key concepts of fundamental group, covering spaces, homology and cohomology.
  • LO2. Apply the fundamental principles and results of algebraic topology to solve given problems.
  • LO3. Distinguish and compare the properties of different types of topological spaces and maps between them.
  • LO4. Formulate topological problems in terms of algebraic invariants and determine the appropriate framework to solve them.
  • LO5. Communicate coherent mathematical arguments appropriately to student and expert audiences, both orally and through written work.
  • LO6. Devise computational solutions to complex problems in algebraic topology.
  • LO7. Compose correct proofs of unfamiliar general results in algebraic topology.