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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

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George Revingston
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  • Yes exactly. That's why ${\pi_1(\mathbb{R})}$ is trivial, for example – Riemann'sPointyNose Mar 22 '21 at 00:05
  • So the fundamental group does not consist of all loops that are homotopic to each other? What exactly does it consist of then? – George Revingston Mar 22 '21 at 01:12
  • The fundamental group consists of all the classes of homotopic loops. Like, you can think of taking all the loops in a topological space around some point ${x_0}$ and trying to put them into different classes. We put them in the same class if they are homotopic. And the fundamental group is all of these classes of loops based around a given point. ${\pi_1(\mathbb{R})}$ is trivial since there is only one class of loops, since every loop is homotopic to each other. Note that the fundamental group does depend on the point you choose to look at loops around; however – Riemann'sPointyNose Mar 22 '21 at 15:02
  • continued comment the reason we can, for example, not worry about writing the point we are taking loops around when we say ${\pi_1(\mathbb{R})}$ is because ${\mathbb{R}}$ is path-connected. This tells us that it doesn't matter what point we consider loops around, the fundamental group is always the same. So we don't bother writing any specific point. With a lot of spaces though, it does matter. For example, if you have a disconnected space that consists of two parts that have completely different fundamental groups, you must say which point you are taking the fundamental group around – Riemann'sPointyNose Mar 22 '21 at 15:04
  • Oh yes, that makes sense, thank you so much for the detailed explanation this helped a lot – George Revingston Mar 23 '21 at 05:13

3 Answers3

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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.

Igor Rivin
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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.

Arthur
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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.

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