# Differential Topology M4P54

Spring term 2013

### Overview

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.

### Assessment

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.

### Prerequisites

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

• 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

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

• chain complexes
• the definition of some form of homology (simplicial, cellular or singular)
• functoriality and homotopy invariance of homology

### References

• 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).
• Hatcher, Algebraic Topology
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.