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Please take this page in conjunction with the Part III Guide to Courses Geometry and Topology section.

Prerequisite areas

Differential Topology

The course generally starts from scratch, and since it is taken by people with a variety of interests (including topology, analysis and physics) it is usually fairly accessible. It is an important stepping stone for many other geometry courses.

You will find this helpful for the following Part III courses:

  • Complex Manifolds
  • (Algebraic Topology)
  • Other geometry and geometric analysis courses which change from year to year (eg Riemannian Geometry)
  • Theoretical Physics courses (eg General Relativity, Symmetries, Fields and Particles, Applications of Differential Geometry to Physics)

Relevant undergraduate courses are:

First level prerequisites

Linear algebra: abstract vector spaces and linear maps, bilinear forms. See e.g. Ib Linear Algebra

Multi-variable calculus: derivatives of functions as linear maps, the chain rule, partial derivatives, Taylor's theorem in several variables. See e.g.Ib Analysis II. You can check if you are at the required level by doing the following exercises: Analysis II 2015-16 Sheet 4 (Questions 4, 5, 11).

Solutions to first-order differential equations (Picard's theorem)

Elementary point-set topology: topological spaces, continuity, compactness etc. See e.g. Ib Metric and Topological Spaces. You can check if you are at the required level by doing the following exercises: Met & Top 2015-16 Example Sheet 1

Second level prerequisites

Some exposure to ideas of classical differential geometry, e.g. Riemannian metrics on surfaces, curvature, geodesics.

Useful books and resources

Notes from the Part II Course.

Milnor's classic book "Topology from the Differentiable Viewpoint" is a terrific introduction to differential topology as covered in Chapter 1 of the Part II course. It is quite different in feel from the Part III course but would be great to look at in preparation.

Nakahara "Geometry, Topology and Physics". This is not a pure maths book, so comes with a warning that it is not always completely precise and rigorous. It also covers lots of material outside the Part III course. However, it is excellent for giving an intuitive picture of the concepts, and may be particularly helpful to physicists taking the course.

Algebraic Topology

Relevant undergraduate courses are:

First level prerequisites

Second level prerequisites

Some experience of some version of homology in algebraic topology. For example you should know about:

  • Homotopic maps and homotopy equivalence of spaces.
  • Chain complexes and exact sequences.
  • Simplicial homology. (Or another type of homology.)

Useful books and resources

Chapter 1, Algebraic Topology, Allen Hatcher, CUP, 2009

Part II notes for Algebraic Topology on Oscar Randal-Williams’ teaching page