- #1
Zetta
Gold Member
- 11
- 0
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.
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.