Differential Topology M4P54

Spring term 2013


The first part of the course of the course will be concerned with de Rham cohomology, which is a relatively approachable form of cohomology for smooth manifolds. We will prove the Mayer-Vietoris exact sequence, Künneth formula and Poincare duality in this context, and discuss degrees of maps between manifolds.

Time permitting, we will then look at singular homology and cohomology, the relation between singular and de Rham cohomology via the de Rham theorem, and Morse theory.


90% of the mark for the course is based on the final exam, and 5% on a piece of coursework in the middle of term.

The remaining 5% will be based on solutions to the (approximately) biweekly example sheets and class presentations: you should submit a solution to at least one problem from each example sheet and be prepared to present it.

The final example class will take place at 10am on Monday 29 April in Huxley 140.

Other hand-outs


It is expected that you will have taken the courses Algebraic Topology (M3P21) and Manifolds (M4P52), or have learnt the relevant material elsewhere.

While it is by no means necessary, I would recommend taking Dr Reto Müller's course on Riemannian Geometry (M4P51). There will not be very much overlap between the theory in the two courses, but the practice of working with manifolds will be useful (both ways). And Riemannian geometry is a beautiful subject.

Manifolds. You need to be familiar with

Algebraic topology. You should be familiar with homotopies (and the notion of homotopy equivalence). I will also assume that you can work with the fundamental group and covering spaces, although they will not play a large role in this course. While discussing cohomology I will also go over a lot about homology, but it may be useful if you have previously encountered


Materials from spring term 2012