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.

34 x 1 hr lectures and 12 x 1 hr tutorials in a semester

4H: Algebraic and Geometric Topology (MATHS4112)

Assessment

90% Examination, 10% Coursework.

Reassessment

There will opportunities to resit for MSc students.

Reassessments are normally available for all courses, except those which contribute to the Honours classification. 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 for this course are described below. 

The mains aims are to

a) review basic ideas of topological spaces and continuous mappings, including quotient spaces and topologies;

b) define without using metrics;

c) introduce the notion of homotopy of mappings and homotopy equivalence of spaces;

d) define fundamental groups of based spaces and verify basic properties, including functoriality and algebraic structure (with additional reading material);

e) study covering spaces and relations between fundamental groups (with additional reading material);

f) apply these ideas to the classification of compact surfaces (with additional reading material);

g) study basic properties of knots and braids, introducing some computable invariants of these

h) Study additional material on homology through directed reading.

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.

i)   to compute homology of 1 and 2-dimensional complexes using various approaches.

Students must submit at least 75% by weight of the components (including examinations) of the course's summative assessment.