PROJECT III 2017-18
COVERING SPACES
As
is well known, apart from at 0, the square root function is 2-valued: the
square root of 4 is 2 and -2; that of
-1 is i and -i.
Thought of as a function on the complex numbers, it is possible to define
Òsquare rootÓ continuously for a while, but not globally: as you move round the
unit circle from 1 taking square roots as you go, when you return to 1 you are
forced to have chosen the ÒotherÓ root from the one you chose at the start. The
way out of this, and for discussing many other functions which naturally have
multiple values, is through the concept of a covering space. In the example just outlined, the square root
function on the non-zero complex numbers is modeled as taking values in an
extended complex plane that wraps twice round the usual one.
More
generally, given any space X (eg, circle, sphere, torus, Klein bottle, punctured complex
plane, etc) one can ask what objects cover it n to 1: that is, find spaces Y
with a continuous function from Y to X which is n to 1, but Y locally
looks the same as X? Here n is any positive integer, or even
infinity.
Such
objects naturally occur in many areas of mathematics: topology, complex
geometry, differential equations, dynamics, and so on. So while a project on this
topic needs to start with the basic theory, it can subsequently be explored in
any number of ways, depending on your interest. The project
will be jointly supervised, by Andrew Lobb in Michaelmas and John Hunton in Epiphany.
PREREQUISITES
Though
there is no need to follow this project into functions of a complex variable,
the module Complex Analysis II
(MATH2011) would still be useful, not least for the general notions of space
discussed there. The module Geometric
Topology II (MATH2627) would represent the general nature of the topic
(though the projectÕs content is not reliant on that module). Topology III (MATH3281) would be a
sensible choice to study alongside the project and there would be useful
material to be found there.
RESOURCES
A
good place to start is the Wikipedia page on covering spaces
https://en.wikipedia.org/wiki/Covering_space
which
gives a general overview of some of the basic ideas, problems and applications,
and a lot of links for further study. For a more detailed treatment of the core
material, see the book
Allen
Hatcher Algebraic Topology, CUP,
especially Section 1.3. This book is available free of charge from www.math.cornell.edu/~hatcher.
The
applications to complex functions, ultimately leading to notions of Riemann Surfaces is covered in many
texts, see, for example
A F Beardon, A primer on
Riemann Surfaces, LMS lecture note series 78, CUP.
EMAIL
John
Hunton (mailto:john.hunton@durham.ac.uk)
Andrew
Lobb (mailto:andrew.lobb@durham.ac.uk)