4H: Algebraic and Geometric Topology

Short Description
Topology underlies many branches of geometry and has an enormous impact in mathematics and its applications. The key to its widespread utility lies in the study of topological spaces by focusing on properties which are preserved under continuous deformations such as homotopies. This course will continue the study of topology by considering different ways of distinguishing topological spaces from each other and also attempting to classify continuous mappings. In particular the fundamental group will be introduced: this is the first example of a standard type of construction in algebraic topology, namely a functorial assignment of an algebraic structure to spaces (in this case groups). This will be used to classify surfaces up to homeomorphism and to study covering spaces and their relationships with the fundamental group, illustrating the close links between group theory and geometry and topology via the Galois correspondence between topological objects (covering spaces) and algebraic objects (subgroups of the fundamental group). The course will conclude with a look at the mathematics of knots and braids, seemingly simple objects which have extraordinary importance in mathematics and beyond.

Assessment
100% Examination.

Reassessment
In accordance with the University's Code of Assessment reassessments are normally set for all courses which do not contribute to the honours classifications. For non honours courses, students are offered reassessment in all or any of the components of assessment if the satisfactory (threshold) grade for the overall course is not achieved at the first attempt. This is normally grade D3 for undergraduate students, and grade C3 for postgraduate students. Exceptionally it may not be possible to offer reassessment of some coursework items, in which case the mark achieved at the first attempt will be counted towards the final course grade. Any such exceptions are listed below in this box.
Main Assessment In: April/May

Course Aims
The mains aims are to
h) review basic ideas of topological spaces and continuous mappings, including quotient spaces and topologies;
i) define without using metrics;
j) introduce the notion of homotopy of mappings and homotopy equivalence of spaces;
k) define fundamental groups of based spaces and verify basic properties, including functoriality and algebraic structure;
l) study covering spaces and relations between fundamental groups;
m) apply these ideas to the classification of compact surfaces;
n) study basic properties of knots and braids, introducing some computable invariants of these.
Intended Learning Outcomes of Course
By the end of this course students will be able to:
a) work with topological spaces not necessarily defined using metrics, including quotient spaces;
b) define homotopy of mappings as an equivalence relation, work with homotopy classes
c) and homotopy equivalence of spaces and use these ideas to distinguish spaces and show
d) non-existence of mappings with prescribed properties;
e) give the definition and basic properties of fundamental groups and work with these;
f) work with covering spaces and use relationships between fundamental groups to calculate examples;
g) state and use the classification of compact surfaces up to homeomorphism;
h) work with basic properties of knots and braids, and use suitable invariants to distinguish between pairs of these.