Those sitting the examination in MA3427 are advised to consult the Module MA3427, Michaelmas Term 2014: Preparing for Examination.
The lecture course in Michaelmas Term 2012 included a section on Quaternions and Rotations that established basic properties of the algebra of quaternions, and showed how each quaternion of unit modulus determines a rotation of 3-dimensional Euclidean space. The theory was developed to establish the homomorphism from the group Sp(1) of unit quaternions to the group SO(3) of rotations of 3-dimensional Euclidean space that is a covering map, where each rotation in SO(3) is the image of a pair of unit quaternions. The group Sp(1) is isomorphic to SU(2), and is homeomorphic to a 3-dimensional sphere. These results show that SO(3) is homeomorphic to 3-dimensional real projective space, and therefore the fundamental group of SO(3) is the cyclic group of order two.
The lecture course in Michaelmas Term 2012 included a section on the topological classification of closed surfaces (i.e., compact two-dimensional manifolds without boundary).
Prior to 2009, Algebraic Topology was taught in Course 421. The course website contains worked solutions to examination papers in recent years. Lists of material that is or is not examinable in a particular year is applicable to that year only: it should not be taken as a guide to what will be examinable in subsequent years.
In Michaelmas Term 2008, the lecture course included a section on Covering Maps and Discontinuous Group Actions that extends material taught in subsequent years, covering the theory of deck transformations of covering spaces, and establishing necessary and sufficient conditions for covering maps over a given topological space to be isomorphic.
Also, lecture notes for part of course 421 (Algebraic topology), taught at Trinity College, Dublin, in Michaelmas Term 1988 are available:
In 1988 the course included material on the construction of covering maps over locally simply-connected topological spaces. In particular, it was shown that, given any of subgroup of the fundamental group of a locally simply-connected topological space, one can construct a corresponding covering space whose fundamental group is isomorphic to that subgroup.
Back to D.R. Wilkins: Lecture Notes
Dr. David R. Wilkins, School of Mathematics, Trinity College Dublin.