- #1
- 7,031
- 10,618
Hi, everyone :
I read a problem posted somewhere else on showing that, "for the correct value m--
an integer" , any curve lying on an annulus A={0<a<|z|< b} was homotopic to the
circle re^(i*pi*m*t).
So I thought of this question: what are the homotopy classes of the annulus?.
I know that Pi_1(A)=Z , since the annulus deformation-retracts to S^1 , so that
the two are homotopic.
I wonder if a homotopy H:X-->Y ,in general tells us how to map homotopy
classes in X, to those in Y; specifically:
We have that loops that go around the circle n times form a class, i.e., loops
that go around once are not homotopic to those going around more than once, etc.
So, knowing this , and knowing that S^1 is homotopic to the annulus: can we
figure out what the homotopy classes are in the annulus?
Thanks .
I read a problem posted somewhere else on showing that, "for the correct value m--
an integer" , any curve lying on an annulus A={0<a<|z|< b} was homotopic to the
circle re^(i*pi*m*t).
So I thought of this question: what are the homotopy classes of the annulus?.
I know that Pi_1(A)=Z , since the annulus deformation-retracts to S^1 , so that
the two are homotopic.
I wonder if a homotopy H:X-->Y ,in general tells us how to map homotopy
classes in X, to those in Y; specifically:
We have that loops that go around the circle n times form a class, i.e., loops
that go around once are not homotopic to those going around more than once, etc.
So, knowing this , and knowing that S^1 is homotopic to the annulus: can we
figure out what the homotopy classes are in the annulus?
Thanks .