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

Prerequisite areas

- Algebraic Topology: for Algebraic Topology
- Differential Geometry: for Differential Geometry

## Differential Geometry

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 and Topology, formerly 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 Analysis and Topology, formerly 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. See e.g. IB Geometry. You can check if you are at the required level by doing the following exercises: Geometry 2022-23, Example Sheet 2 (Question 10) and Example Sheet 3 (Questions 1, 8).

**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**

- Standard undergraduate group theory, including the structure theorem for finitely generated abelian groups. See e.g. IA Groups and IB Groups, Rings and Modules.
- Fluency with definitions in basic point-set topology, such as compactness and connectedness. See e.g. IB Analysis and Topology.

**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

Part II notes for Algebraic Topology on Henry Wilton's teaching page