Continuous deformation

[ZaidAlyafey]

Prove that If two loops \(\displaystyle \Gamma_1 , \Gamma_2\) in a domain \(\displaystyle D\) can be continuously deformed to a point then they can be continuously deformed to each other .

I have no experience in topology , I found this note in a complex analysis book .

 

[Ackbach]

Re: continuous deformation

I think you must mean that the two loops smoothly deform to the same point. Is that correct? If so, can you think of a smooth deformation that takes one loop to the other?

 

[ZaidAlyafey]

Re: continuous deformation

Assume that we have two closed contours \(\displaystyle \Gamma_1 , \Gamma_2 \) then we can continuously deform \(\displaystyle \Gamma_1 \) to \(\displaystyle \Gamma_2\) if we can find a deformation function \(\displaystyle z(s,t)=(1-s)z_1(t)+s z_2(t) \) where \(\displaystyle s \in [0,1] , t \in [0,1]\) and \(\displaystyle z_1(t),z_2(t) \) are parametrization for \(\displaystyle \Gamma_1,\Gamma_2 \) respectively .

 

[Ackbach]

Re: continuous deformation

Assume that we have two closed contours \(\displaystyle \Gamma_1 , \Gamma_2 \) then we can continuously deform \(\displaystyle \Gamma_1 \) to \(\displaystyle \Gamma_2\) if we can find a deformation function \(\displaystyle z(s,t)=(1-s)z_1(t)+s z_2(t) \) where \(\displaystyle s \in [0,1] , t \in [0,1]\) and \(\displaystyle z_1(t),z_2(t) \) are parametrization for \(\displaystyle \Gamma_1,\Gamma_2 \) respectively .

True, but you need more than that, right? Is the wording of the original problem such that both contours can be smoothly deformed to the same point? If so, how can you leverage that information to get the deformation you need?

 

[ZaidAlyafey]

Re: continuous deformation

True, but you need more than that, right? Is the wording of the original problem such that both contours can be smoothly deformed to the same point? If so, how can you leverage that information to get the deformation you need?

Clearly we don't need the two contours to deform to the same point . I never heard of smooth deformation , it is never mentioned in the context . But, by definition these contours are finite sequence of smooth curves.

 

[Ackbach]

You're in a domain, so going from one point to the other is no issue. This proof is one of those "what do you know" kind of proofs. By the time you finish writing down all the stuff you've been given, the answer kind of pops out at you. Can you write down all the deformation functions you know you have?

 

[ZaidAlyafey]

Since we are given that both loops deform to a point in $D$ , let \(\displaystyle \Gamma_1 \) deforms to the point $z_1 \in D$ and \(\displaystyle \Gamma_2 \) deforms to the point $z_2 \in D$ so we can describe these deformations as follows

\(\displaystyle z_1(s,t) = (1-s)z_1(t) +s z_1 \) for \(\displaystyle \Gamma_1\)

\(\displaystyle z_2(s,t) = (1-s)z_2(t) +s z_2 \) for \(\displaystyle \Gamma_2\)

But I cannot see how to deduce the deformation from \(\displaystyle \Gamma_1 \) to \(\displaystyle \Gamma_2\)?

 

[Ackbach]

Since we are given that both loops deform to a point in $D$ , let \(\displaystyle \Gamma_1 \) deforms to the point $z_1 \in D$ and \(\displaystyle \Gamma_2 \) deforms to the point $z_2 \in D$ so we can describe these deformations as follows

\(\displaystyle z_1(s,t) = (1-s)z_1(t) +s z_1 \) for \(\displaystyle \Gamma_1\)

\(\displaystyle z_2(s,t) = (1-s)z_2(t) +s z_2 \) for \(\displaystyle \Gamma_2\)

But I cannot see how to deduce the deformation from \(\displaystyle \Gamma_1 \) to \(\displaystyle \Gamma_2\)?

Right. And there's one more deformation you can write down. How about $z_{1}$ to $z_{2}$? Why can you do that one?

 

[ZaidAlyafey]

Since any domain is connected then any two points in $D$ can be transformed to each other because they can be connected through a finite set of straight lines , is that right ?

 

[Ackbach]

Since any domain is connected then any two points in $D$ can be transformed to each other because they can be connected through a finite set of straight lines , is that right ?

Right. So you have a deformation from $\Gamma_{1}$ to $z_{1}$, from $z_{1}$ to $z_{2}$, and from $\Gamma_{2}$ to $z_{2}$. Any ideas from here?

 

[ZaidAlyafey]

Right. So you have a deformation from $\Gamma_{1}$ to $z_{1}$, from $z_{1}$ to $z_{2}$, and from $\Gamma_{2}$ to $z_{2}$. Any ideas from here?

Done, I see what you mean . Thanks for your time .

 

[Ackbach]

Done, I see what you mean . Thanks for your time .

You're very welcome, as always!