NMAA05038U Algebraic Topology (AlgTop)
Homology is one of the central concepts in modern
mathematics with applications in several mathematical disciplines
including algebra, geometry, and topology. Homology now also finds
increasing applications in applied matmematics.
The course is an introduction to singular homology. This functor
assigns to every topological space X a sequence of abelian groups
H(X) reflecting deformation invariant properties of X. In the course
we shall see that singular homology satisfies these 'axioms':
- homotopy invariance
- long exact sequence
- dimension axiom
These axioms actually chararcterize singular homology theory for a
special class of topological spaces called CW-complexes.
We will use homology groups to establish some geometrical aspect
of Euclidean n-space such as topological invariance of dimension,
invariance of domain, Jordan curve theorem in all dimensions, and
vector fields on spheres. We will also introduce Delta- and
CW-complexes and show how to compute homology of these spaces.
The course introduces foundational competences in algebraic topology,
homotopy theory, and homological algebra. Important concepts are
homotopy, homotopy equivalence, chain complex, homology, exactness,
Delta- and CW-complexes.
At the end of the course, the students are expected to be able to:
- define the singular chain complex
- compute homology groups of simple topological spaces
- exploit exact sequences as a computational tool
- compute the Euler characteristic of CW-complexes
The course will strengthen the students competences in
- abstract and precise thinking
- elegance of exposition
- 7,5 ECTS
- Type of assessment
- Oral examination, 20 minutesWith 20 minutes preparation time.
- All aids allowed
- Marking scale
- 7-point grading scale
- Censorship form
- External censorship
Criteria for exam assesment
The student must in a satisfactory way demonstrate that he/she has mastered the learning outcome.
- Theory exercises