The Fundamental Group — A Primer

Our eventual goal is to get comfortable with the notion of the “homology group” of a topological space. That is, to each topological space we will associate a group (really, a family of groups) in such a way that whenever two topological spaces are homeomorphic their associated groups will be isomorphic. In other words, we will be able to distinguish between two spaces by computing their associated groups (if even one of the groups is different). In general, there may be many many ways to associate a group with an object (for instance, it could be a kind of symmetry group or a group action). But what we want to do, and what will motivate both this post and the post on homology, is figure out a reasonable way to count holes in a space. Of course, the difficult part of this is determining what it means mathematically to have a “hole” in a space.