Is Any Closed Curve Homotopic to a Constant Curve in a Simply Connected Region?

[variety]

I have a question about the definition of null homotopic. My textbook (Functions of one Complex Variable I by John B. Conway) defines it as follows: If $$\gamma$$ is a closed rectifiable curve in a region G, then $$\gamma$$ is homotopic to zero if $$\gamma$$ is homotopic to a constant curve. My question is, if G is simply connected, then is $$\gamma$$ homotopic to ANY constant curve in G? This seems obvious to me, but I'm not sure if I should prove it or just state it.

 

[Billy Bob]

Here's my joke: When is a curve that is homotopic to zero not homotopic to zero?

Answer: Let Dn be the open disk of radius 1 centered at n, and let G = union of D0 and D3. Then G is simply connected but not connected.

Let gamma be a curve in D3. Then gamma is homotopic to zero because it is homotopic to the constant curve alpha(t)=3.

However, gamma is not homotopic to the constant curve beta(t)=0, so one might say that gamma is not homotopic to "zero."

 

[Jamma]

Your G would not be simply connected :/ It has to be path connected and therefore connected to be simply connected.

 

[Hurkyl]

I have a question about the definition of null homotopic. My textbook (Functions of one Complex Variable I by John B. Conway) defines it as follows: If $$\gamma$$ is a closed rectifiable curve in a region G, then $$\gamma$$ is homotopic to zero if $$\gamma$$ is homotopic to a constant curve. My question is, if G is simply connected, then is $$\gamma$$ homotopic to ANY constant curve in G? This seems obvious to me, but I'm not sure if I should prove it or just state it.

If G~c and G~c', then you would need to have c~c'...