Last modified: 02 Oct 2025 16:16
Algebraic topology is a tool for solving topological or geometric problems with the use of algebra. Typically, a difficult geometric or topological problem is translated into a problem in commutative algebra or group theory. Solutions to the algebraic problem then provide us with a partial solution to the original topological one.
| Study Type | Undergraduate | Level | 4 |
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| Term | Second Term | Credit Points | 15 credits (7.5 ECTS credits) |
| Campus | Aberdeen | Sustained Study | No |
| Co-ordinators |
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Elementary concepts of homotopy theory.
The fundamental group and its naturality properties.
Fundamental groups and covering spaces.
Free groups and amalgamated products
The Seifert-Van Kampen theorem
Presentations of groups.
Syllabus:
Revision of topological spaces
Topological equivalence, homotopy and homotopy equivalence, deformation retraction.
The fundamental group, homomorphisms induced by continuous maps, and homotopy invariance.
The fundamental group of a circle and introduction to covering spaces
Applications: The fundamental theorem of algebra, Brauer fixed point, and Borsuk-Ulam.
Covering spaces: Concept, existence and classification.
Desk transformation and group actions.
The Seifert Van-Kampen theorem.
Computation of the fundamental group
Information on contact teaching time is available from the course guide.
| Assessment Type | Summative | Weighting | 50 | |
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| Assessment Type | Summative | Weighting | 50 | |
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| Knowledge Level | Thinking Skill | Outcome |
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There are no assessments for this course.
| Assessment Type | Summative | Weighting | ||
|---|---|---|---|---|
| Assessment Weeks | Feedback Weeks | |||
| Feedback | ||||
| Knowledge Level | Thinking Skill | Outcome |
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| Knowledge Level | Thinking Skill | Outcome |
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| Procedural | Remember | Be able to describe some simple space such as: circle, 2-dimensional sphere, torus, projective plane, Klein bottle. |
| Conceptual | Understand | Understand what is the fundamental group of a topological space. |
| Conceptual | Understand | Know about covering spaces and their relation to the fundamental group. |
| Procedural | Apply | Know about the Seifert-van Kampen theorem and how to apply it. |
| Procedural | Analyse | Tell apart simple cases of non-homeomorphic spaces. |
| Procedural | Analyse | Recognise simple cases of deformation retracts. |
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