Accessibility statement

# Algebraic Topology - MAT00083M

« Back to module search

• Department: Mathematics
• Module co-ordinator: Dr. Eli Hawkins
• Credit value: 10 credits
• Credit level: M
• Academic year of delivery: 2022-23
• See module specification for other years: 2021-22

## Module summary

Algebraic topology relates the shape of topological spaces to the structure of groups. It allows mathematicians to describe the internal structure of spaces, and to tell when two spaces are fundamentally different, using algebra. This module focusses on homology theory, which is one of the crown jewels of 20th century mathematics and provides a paradigm from which substantial parts of modern mathematics have since been developed.

## Related modules

• None

### Prohibited combinations

• None

For MSc students: a first course in topology; a first course in group theory is required.

## Module will run

Occurrence Teaching cycle
A Spring Term 2022-23

## Module aims

Algebraic topology relates the shape of topological spaces to the structure of groups. It allows mathematicians to describe the internal structure of spaces, and to tell when two spaces are fundamentally different, using algebra. This module focusses on homology theory, which is one of the crown jewels of 20th century mathematics and provides a paradigm from which substantial parts of modern mathematics have since been developed.

## Module learning outcomes

Subject content

·Know how both simplicial and singular homology assign an abelian group to a (nice enough) topological space

• Know how maps between spaces relate singular homology groups
• Know how singular homology groups behave when spaces are cut up into pieces

• Further development of analytic abilities and logical reasoning; the ability to synthesise elementary ideas from earlier years into a more complex setting where several tools must be used in a new context.

## Module content

Syllabus

• Delta-complexes
• Simplicial homology
• Singular homology
• Homotopy invariance
• Exact sequences and excision
• Statement of equivalence of simplicial and singular homologies
• Applications

## Assessment

Task Length % of module mark
Online Exam
Algebraic Topology
N/A 100

None

### Reassessment

Task Length % of module mark
Online Exam
Algebraic Topology
N/A 100

## Module feedback

Current Department policy on feedback is available in the student handbook. Coursework and examinations will be marked and returned in accordance with this policy.