Understanding the Fundamental Group: Exploring Pi1(X.x) and Its Definition

[Zetta]

What I understand from the definition of the fundamental group is:
Pi1(X.x) is "the set of rel {0,1} homotopy classes [a] of closed paths"

Ok, when I think about one [a] it consists of all:
1.Closed paths like a and b with a(0)=a(1)=x & b(0)=b(1)=x --->since
they are closed.
2.And since they are rel {0,1} homotopic, a(0)=b(0)= x =a(1)=b(1).
So it seems to me that all paths with the two above conditions belong to
ONE class say [a], so what I conclude is that for ONE particular "x"
Pi1(X,x) consists of only one class!
How can any other path like c say starts from "x" and end to "x" and not
to be in [a]?
I would be thankful if anyone can help me.

 

[dx]

I don't think you've understood what 'homotopic' paths are. Imagine a plane with a hole in it, and pick a point x. Let a path 'a' start from x and wind around the hole and return to x. Let a path 'b' start from x and return to x without winding around the hole. These two paths are not homotopic, i.e. they cannot be continuously deformed into each other, and therefore the equivalence classes [a] and are not the same.

 

[Zetta]

Thank you for the great help, brief and efficient!
Just to check that I am on the right track now,
1.Can I say this is related to (because of) "discontinuaty of the choice of paths" between A and B?
2. Can I say in an "8" shape for example the top part and the bottom parts belong to two different classes (hard to imagine for me but I feel this might be the case)?

 

[dx]

1. Yes, but that wouldn't be acceptable as a rigorous proof. To prove that they don't belong to the same equivalence class, you must first specify the topology on the space X, and then prove that there does not exist a homotopy from a to b, i.e. a continuous function ψ : [0,1] x [0,1] → X such that ψ(a, 0) = ψ(a, 1) = x and ψ(0,y) = a(y), ψ(1,y) = b(y).

2. No, the relation of 'homotopic to' is defined between different paths, not different parts of the same path, whether it be '8' shaped or not.

 

[Zetta]

Sorry, You did answer to the question number 1.

 

[madness]

I think he means the space is 8 shaped, in which case the two circles are different path components and the fundamental group is the direct sum of the fundamental group of two circles.

 

[Zetta]

I meant this, please see the attachement.

 

[dx]

In the space you've drawn, [α₁] = [α₂] = [β₁] = [β₂], since there are no holes. Each pair of these paths are homotopic, since nothing prevents us from continuously deforming any of them into any of the others.

 

[Zetta]

Yes, I got that with your very nice and clear explanation. "Holes"!
And what I learned from your answer is that when thinking about definitions don't let your brain to be lazy and bring the simplest possible space like R^2 (for example).
Thank you very much

 

[Zetta]

I have a question to clear up definitions for myself:
When we calculate (or talk about) Pi1(X,x) do we actually calculate |Pi1(X,x)|?
I mean "the number of equivalent classes like [a] w.r.t x in X"?
For example if that is the case I can imagine what Pi1(S^2,(1,0)) means, but if that is not the case then what do we do precisely when we calculate Pi1(X,x)?
Thank You

 

[dx]

When we calculate (or talk about) π₁(X,x) do we actually calculate |π₁(X,x)|?

What is |π₁(X,x)|?

π₁(X,x) is a group. When you are asked to find π₁(X,x), what you are looking for is the group π₁(X,x). First you start out with the space of loops at x, denoted Ω(X,x), and called the "loop space at x". Then you find the set of equivalence classes in Ω(X,x) modulo homotopy. Composition of loops is defined in Ω(X,x), and this induces a product structure on the set of equivalence classes, making them a group. This group is called π₁(X,x).

 

[Zetta]

I mean for example in Pi1(S^2,1)={1}, this "{1}" exactly refers to what?

 

[dx]

{1} is the point at which the loops start out and end. For path connected spaces, the fundamental group doesn't depend on the starting point. For example, the the punctured plane is path connected.

Let's find the fundamental group of the puncured plane, X = R \ {0}. Let [a_n] be the equivalence class of loops that wind around the hole n times in the clockwise direction, and [b_n] the equivalence class of loops that wind around the hole n times in the anti-clockwise direction. If a_n is a representative loop of [a_n] and b_n is a representative of [b_n], then a_nb_n (composition of loops), will be a representative of the zero loop. By such arguments, we can see that the fundamental group of the punctured plane is isomorphic with the additive group of integers Z = { ..., -1, 0, 1, ...}

So, π₁(R \ {0}) = Z.

 

[Zetta]

Sorry for these easy quastions, but if you see my lec notes(pdf) you might agree that there are some ambguities there for some one like me.

 

[madness]

{1} is the identity of the group, i.e. the constant loop. Every based loop on the space is homotopic to the constant loop which stays at the base point for the whole path. I'm a little confused by your response dx where you say {1} is the point where the loop starts and ends.

 

[dx]

Sorry I misread, I thought he was talking about the basepoint 1 in π₁(S², 1). The 1 in the fundamental group {1} of S² refers to the identity element. Thanks madness.

 

[dx]

Some explanation of why it is the identity element:

In a simply connected space, all loops are homotopic to the identity loop. Thus we havea single equivalence class: [1]. This element is the identity element in the fundamental group, since composition of a loop with the identity loop gives you the same loop back.

 

[Zetta]

So, can I say that e.g. "Pi1(S^1)=Z" means that:
"The set of homotopy rel {0,1} classes of closed paths starting and ending at any point say x " in S^1 (which is a group) is homomorphic (actually isomorphic) to Z i.e. there is a homomorphism of groups between Pi1(S^1) and Z where "+ in Z" correspondes to "."(concatination of paths in Pi1(S^1))?

if I can, then the exact equivalent wording like above for "Pi1(S^n)={1} for n>1" would be what? this is what I am trying to understand.

 

[dx]

π₁(S²) = {1} simply means that the fundamental group of the sphere S² is isomorphic to the group with a single element 1 (the identity).

 

[Zetta]

Thank you dx like before, you are giving me the most precise answer that I possibly can get, it seems even before I post my new question you knew what is in my mind!