Recently, I am giving courses on algebraic and geometric topology. The term “geometric topology”
is understood in a quite restrictive sense. During the courses, the focus is on the study of manifolds, objects which play a central role in geometry and topology. The methods of “algebraic topology” are used to understand manifolds.
Specific themes that I cover within algebraic and geometric topology include the Poincaré duality, the deRham cohomology, bordism theory, group actions on manifolds, characteristic classes, topological K-theory, low dimensional manifolds, and the theory of knots and links.
“The undeniable power of algebraic topology would alone command a leading place
in mathematics; but it has great beauty too, in organizing intuitively different structures
which defied earlier purely analytic or geometric approaches. It has moreover a leading
place in physics, for its success in organizing the most fundamental physical theories.”
— C.T.J. Dodson and Phillip E. Parker,
the authors of the book “A User’s Guide to Algebraic Topology”.