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Next year
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Prerequisites
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Duration
Identical coursesAlgebraic Topology (B-KUL-G0A84A)
This course is taught this academic year, but not next year.Aims
The basic idea of algebraic topology is the following: it is possible to establish a correspondence between certain topological spaces and certain algebraic structures (often groups) in such a way that when there is a topological connection between between two spaces (i.e. a continuous map), then there is also an algebraic connection (i.e. a morphism) between the associated algebraic structures.
In some cases it is possible to translate topological problems into algebraic problems and to solve the latter ones.
The aim of this course is to illustrate this basic idea, by introducing some of these algebraic invariants.
After following this course
- the student unterstands how these algebraic invariants are constructed and understand the main properties (also the proofs) of them,
- the student is able to compute these invariants and apply them to solve some some topological problems
Previous knowledge
The student should have a basic knowledge of some of the most important topological concepts (like topological spaces and continuous maps, open and closed sets and compact spaces). It is useful if the student has followed a course on point set topology. However, it is also possible to follow this course, with only some basic background in topology (e.g. knowledge of metric topology), provided a more general course in topology is followed simultaneously with this course.
Is included in these courses of study
Activities
6 ects. Algebraic Topology (B-KUL-G0A84a)
Content
In this course, we treat two basic algebraic invariants of a topological space and show some applications.
The fundamental group of a topological space.
• Homotopy and the definition of the fundamental group
• Retractions and deformation retracts
• Applications to fixed points (Brouwer fixed point theorem)
• The Seifert – van Kampen Theorem
• Borsuk – Ulam Theorem
• Covering spaces and the connection with the fundamental group:
- lifting of paths and maps
- equivalence of covering spaces
- covering transformations and group actions
The singular homology groups of a topological space
• definition of the singular homology groups Hn(X) of a space
• meaning of H0(X) and H1(X)
• induced morphisms
• The Mayer – Vietoris exact sequence in homology
• Applications towards spheres (degree of a map, vector fields on spheres)
Course material
For part I on the fundamental group, the book Topology of James R. Munkres (Pearson Education International) is used.
Part II on singular homology a text is made available via Toledo.
Evaluation
Evaluation: Algebraic Topology (B-KUL-G2A84a)
Explanation
The exam is an open book exam.