# NMAA05038U  Algebraic Topology (AlgTop)

Volume 2015/2016
Education
MSc programme in Mathematics
Content

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
-  excision
-  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.

Learning Outcome

Knowledge:
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.

Skills:
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

Competences:
The course will strengthen the students competences in
- abstract and precise thinking
- elegance of exposition

Knowledge about general topology and abelian groups.
5 hours lectures and 3 hours exercises each week for 7 weeks.
Credit
7,5 ECTS
Type of assessment
Oral examination, 20 minutes
With 20 minutes preparation time.
Aid
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.

• Category
• Hours
• Lectures
• 35
• Theory exercises
• 21
• Preparation
• 120
• Exam
• 30
• Total
• 206