| Unit name | Algebraic Topology |
|---|---|
| Unit code | MATHM1200 |
| Credit points | 20 |
| Level of study | M/7 |
| Teaching block(s) |
Teaching Block 1 (weeks 1 - 12) |
| Unit director | Dr. Mark Hagen |
| Open unit status | Not open |
| Units you must take before you take this one (pre-requisite units) | |
| Units you must take alongside this one (co-requisite units) |
None |
| Units you may not take alongside this one |
None |
| School/department | School of Mathematics |
| Faculty | Faculty of Science |
Unit Aims
The aims of this unit are:
Unit Description
Topology studies those properties of a geometric object that are not affected by continuous deformations.
The unit begins by introducing the basic objects, topological spaces, building on material from Metric Spaces (MATH20006). The main topics covered in this part of the unit include continuity/homeomorphisms, compact and Hausdorff spaces, and connectedness, and important constructions like the quotient and product topologies. The latter constructions are then used to build simplicial complexes (and some small generalisations), a class of topological spaces that serve as concrete examples in the rest of the unit.
The next major theme is homotopy. This will be motivated using various examples to convey the idea and then defined formally. One aim of this part of the unit will be to use simplicial complexes to illustrate how homotopies work and why it is a useful notion, via two important classical results: Dowker’s theorem, which relates the topology on a simplicial complex to a closely related metric space, and Borsuk’s nerve theorem, which, roughly speaking, enables one to replace complicated topological spaces with simpler, more tractable simplicial complexes.
The third theme is homology, and here is where the connection to algebra arises. A basic question is, given two topological spaces, how to tell if they are different. This is generally hard. However, this is often an easier question in algebra: for example, one can certify that two vector spaces are not isomorphic by noticing that they have different dimensions. Homology associates to a topological space a collection of abelian groups, which is often useful for reducing topological questions (like “prove that these two spaces are different”) to algebraic ones.
This part of the unit defines the homology groups associated to a topological space and is then devoted to proving the main results about homology (e.g. homotopy invariance, excision, Mayer-Vietoris sequence) and illustrating these with concrete computations (using simplicial homology, which explains the earlier emphasis on simplicial complexes). Applications of homology will also be discussed: these could include, but are not limited to, Brouwer’s fixed-point theorem, the Jordan curve theorem and generalisations, degree and the fundamental theorem of algebra.
Relation to Other Units
This is one of three Level M units which develop group theory in various directions. The others are Representation Theory and Galois Theory.
Metric Spaces (or the equivalent) is an essential prerequisite. You will be expected to be familiar with basic group theory, although most of the groups encountered will be abelian. Knowledge of first-year linear algebra will be assumed and used
Learning Objectives
Students should absorb the idea of algebraic invariants to distinguish between complex objects, their geometric intuition should be sharpened, they should have a better appreciation of the interconnectivity of different fields of mathematics, and they should have a keener sense of the power and applicability of abstract theories.
Transferable Skills
Lectures, problem sets and discussion of problems, student presentations.
50% Exam, 50% Coursework - assessed problem sheets
If this unit has a Resource List, you will normally find a link to it in the Blackboard area for the unit. Sometimes there will be a separate link for each weekly topic.
If you are unable to access a list through Blackboard, you can also find it via the Resource Lists homepage. Search for the list by the unit name or code (e.g. MATHM1200).
How much time the unit requires
Each credit equates to 10 hours of total student input. For example a 20 credit unit will take you 200 hours
of study to complete. Your total learning time is made up of contact time, directed learning tasks,
independent learning and assessment activity.
See the University Workload statement relating to this unit for more information.
Assessment
The assessment methods listed in this unit specification are designed to enable students to demonstrate the named learning outcomes (LOs). Where a disability prevents a student from undertaking a specific method of assessment, schools will make reasonable adjustments to support a student to demonstrate the LO by an alternative method or with additional resources.
The Board of Examiners will consider all cases where students have failed or not completed the assessments required for credit.
The Board considers each student's outcomes across all the units which contribute to each year's programme of study. For appropriate assessments, if you have self-certificated your absence, you will normally be required to complete it the next time it runs (for assessments at the end of TB1 and TB2 this is usually in the next re-assessment period).
The Board of Examiners will take into account any exceptional circumstances and operates
within the Regulations and Code of Practice for Taught Programmes.
