Algebraic Topology - MAT00083M
- Department: Mathematics
- Credit value: 10 credits
- Credit level: M
- Academic year of delivery: 2022-23
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
Additional information
For MSc students: a first course in topology; a first course in group theory is required.
Module will run
| Occurrence | Teaching period |
|---|---|
| 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
Academic and graduate skills
- 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
Indicative assessment
| Task | % of module mark |
|---|---|
| Closed/in-person Exam (Centrally scheduled) | 100 |
Special assessment rules
None
Indicative reassessment
| Task | % of module mark |
|---|---|
| Closed/in-person Exam (Centrally scheduled) | 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.
Indicative reading
A Hatcher, Algebraic Topology, CUP 2001 (S 8.3 HAT)