The fundamental group and higher homotopy groups

In this video, we introduce one of the most important tools in algebraic topology, the fundamental group, and more generally, the higher homotopy groups. This can be thought of as a tool for studying topology by probing topological spaces with spheres. We go through the basic definition and show how it gives groups which can be considered algebraic invariants for studying topological spaces. In particular, functoriality means that it can be used to distinguish non-homeomorphic spaces. We go through an example of such by looking at higher dimensional spheres and the real projective space.