SF2724: Topics in mathematics IV: Applied topology,
- Notes for lectures 1 to 8 are available on the course schedule.
- Comments to lectures 9 to 16 are available here.
In topology one uses homology and cohomology
translate geometrical properties of spaces into algebraic
statements. They give computable ways of finding algebraic
characteristics of spaces, such as the number of components or
the number of "holes" of different types.
In this course we will study two specific cohomology theories: K-theory
and De Rham cohomology. They have both the advantage of being
elementary, and can be studied with no particular previous knowledge
(except the prerequisites stated below). We will see how these theories
are used to compute invariants of spaces, and we will try to give
The aim of the first part on K-theory is to present a proof of the
periodicity theorem for complex vector bundles. Vector bundles have
been an important source of invariants for topological spaces. During
the last 50 years a big part of algebraic topology has been motivated
and inspired by understanding vector bundles and related invariants.
The periodicity theorem has been the key reason why these
invariants have been so useful. Our aim is not only to learn
about vector bundles and necessary background but also to learn how to
read a research article. This part of the course is going to be based
on an article of M. Atiyah and R. Bott "On the periodicity theorem for
complex vector bundles", Acta Math. 112 1964, 229–247.
In the second part of the course we study De Rham cohomology. This
theory takes it starting point from classical methods of vector
analysis. In vector analysis we learn that a conservative vector field
has no rotation, but the converse holds only if the vector field is
defined on a simply connected domain. This observation can be
generalized to a full theory where the vector fields are replaced by
differential forms and the div, grad, and rot operators find their
natural common interpretation as the d-operator. The resulting De Rham
cohomology theory gives a way of computing the numbers of holes of
different dimensions for open subsets of R^n, or more generally for
smooth manifolds. Using De Rham cohomology we will study the degree of
maps, linking numbers of curves, and the index of vector fields. The
subject of characteristic classes connects the vector bundles of the
first part with De Rham cohomology. In this part of the course we use
material from the book "From Calculus to Cohomology" by I. Madsen and
Prerequisites: SF2729 Groups and Rings, and SF2713 Foundations of Analysis
(or equivalent courses).