Question about the fundamental group of a space

Question

From my understanding the fundamental group is the equivalence class of all homotopic loops. So the group consists of all homotopic closed loops, but if every loop is homotopic in the group, wouldn't that make the group trivial? Or am I missing something here? Please explain in detail if you can what I'm getting wrong. Thank you in advance

Answer 1

If all closed loops are homotopic to one another in your space, then the fundamental group of that space is trivial. The archetypal examples are the Euclidean spaces $\Bbb R^n$, where each closed loop is easily seen to be homotopic to, say, the constant loop at $0$. And since being homotopic is an equivalence relation, all loops are homotopic to one another. Thus we get $\pi_1(\Bbb R^n)\cong0$.

However, there are plenty of spaces where not all loops are homotopic. The circle $S^1$, for one (it may be easier to visualize if you look at an annulus rather than a circle; the result is the same). There any two loops that are homotopic must necessarily go the same number of times in the same direction around the circle. A bit more calculation shows that the converse holds as well, so we get $\pi_1(S^1)\cong \Bbb Z$, with the isomorphism given by the winding number.

Answer 2

The fundamental group can be viewed as the quotient of the set of closed based loops by the based homotopy equivalence relation So, homotopic loops go to the same thing.

Answer 3

Consider the unit circle in the plane, centered at the origin. A loop that goes around the circle once, is not equivalent to a loop that goes around the same circle twice, and none of these two are equivalent to a loop that goes around three times.