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
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
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
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.
You need to be familiar with
- the definition of smooth manifolds, tangent bundles, vector fields
- the exterior algebra, forms on manifolds, exterior differentiation
- orientation, integration, Stokes' theorem
- derivatives of smooth maps, pull-backs on forms
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
- chain complexes
the definition of some form of homology (simplicial, cellular or singular)
- functoriality and homotopy invariance of homology
- Madsen and Tornehave, From Calculus to Cohomology
Achieves a lot using only de Rham cohomology (making no mention of homology or
singular cohomology), and provides
a nice introduction to the first part of the course.
The starting point (calculus on R3)
may be more elementary than necessary for this course; the later parts of the
book have a nice approach to characteristic classes, which are sadly outside
the scope of the course.
- Bott and Tu, Differential Forms in Algebraic Topology
Also begins with the de Rham approach to cohomology, but goes on to discuss
singular homology/cohomology (with integral and other coefficients).
Reference for when you want to track down details about singular
cohomology that I gloss over.
- Massey, A Basic Course in Algebraic Topology
Nice presentation of the material on the fundamental group and covering
spaces required for the course. Unlike the shorter version
Algebraic Topology: An Introduction, it also covers singular and
cellular homology (but still not cohomology).
- Milnor, Topology from the Differentiable Viewpoint
Elegantly and briefly proves many interesting results (e.g. about degree
and applications) without employing any algebraic topology.
- Milnor, Morse Theory
Concise, yet covers far more than I will have time to say about Morse theory.
Materials from spring term 2012