Topology is a branch of mathematics that deals with the fundamental properties of spaces. In the topological world, two spaces are considered identical if they can be continuously transformed into one another or more generally, if they are related by what topologists call homotopy equivalence. For example, a sphere and a pyramid are identical, as are a donut and a coffee cup… A main goal is therefore to classify spaces up to homotopy equivalence.

Algebraic topology uses algebraic methods to solve this and other topological problems. An important method is the search for algebraic homotopy invariants. In particular, we try to assign an algebraic object such as a group to each topological space and to prove that this group is invariant if we replace the space with a homotopy equivalent space. One of the proofs of the fundamental theorem of algebra uses an algebraic invariant.

Algebraic topology is closely related to homological algebra (a toolset for extracting information from a widely used type of sequence of arrows and nodes), which in turn has evolved in parallel with category theory (a language or framework that focuses on the basic components of a given structure).

We will study the basics of category theory and homological algebra in parallel with algebraic topology. The main homotopy invariants we discuss are the ‘singular homology functor’ and the ‘homotopy functor’. An amazing result called the Hurewicz isomorphism, shows that the first singular homology group and the (abelianized) first homotopy group of a space are two variants of the same information. Van Kampen’s theorem allows us to practice our understanding of category theory and to compute homotopy groups of larger spaces from those of smaller spaces from which they are constructed. The final chapter will highlight the relationship between the coverings of a space and subgroups of its first homotopy group.

The objective is to allow the student to familiarize himself with a very active field of mathematics, with broad applications throughout science. Beyond this goal, special emphasis will be put on the Mathematical Method, i.e., the optimal technique to learn and apply mathematics, in particular in order to solve real life problems using mathematical tools. This method is the most important of the objectives of any study program in mathematics.