Definition of the fundamental group

Question

Why are the elements of the fundamental group of a space equivalence classes? Why isn't the group defined to be the set of all possible loops at a base point with the product operation of paths? What would go wrong if it was defined so? Or is it simply not useful?

Answer 1

We need the homotopy equivalence relation to capture the "essential" parts of the topological space. You can think of it kind like throwing lassos into the space and pulling tightly.

Without the equivalence relation, we fail to get an actual group. We can't even have an identity element, for instance.

To expand: the only obvious candidate for an identity element is the constant map from $[0,1]$ to the basepoint (the "standing around, twiddling one's thumbs" path). If you concatenate this with any other distinct path $\gamma$, then you will not end up with $\gamma$. It will have the same image, to be sure, but it's not just the journey, it's how you get there (to abuse an english phrase).

Answer 2

For a start, it wouldn't be a group because there are no inverses (that is if you use composition of paths as the operation, I am not sure what you mean by product). Also the structure isn't really useful, I suppose. For example your structure would be in general uncountably generated.

Answer 3

You can use paths or loops of variable "length" to get an associative structure with identities, but you do not get strict inverses. One precise definition is that a path of length $r$ in a space $X$ is a map $a: [0,r] \to X$ and this composes with $b:[0,s] \to X$ to give $a+b: [0,r+s] \to X$ if and only if $a(r)=b(0)$. Others, for good reason, prefer to define a path to be a pair $(a,r)$ where $r \geqslant 0$ and $a: [0, \infty) \to X$ is constant on $[r,\infty)$. These are called "Moore paths".

Interestingly, in higher dimensions there is a tendency to talk about the "fundamental $\infty$-groupoid of a space $X$, but this again is not a strict structure, and so, strictly speaking, does not generalise the fundamental groupoid. The higher structures referred to here do generalise the fundamental groupoid, but are defined only for filtered spaces, which turns out not to be a disadvantage.

Later: A question is: why take equivalence classes? As mentioned by Simon, we need more manageable invariants. So there are Seifert-van Kampen Theorems for the fundamental groupoid and group, which allow explicit and useful calculations, given in many books. Analogously, the strict higher groupoids also lead to explicit calculations.