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Let $$ f_t(z)= \left\{ \begin{array}{ll} z & \mbox{if}\ \ 0<arg(z)< \pi/2-m_0 \\ z\ e^{-t/2m_0} & \mbox{if }\ \ \pi/2-m_0\leq arg(z) \leq \pi/2+m_0\\ z\ \exp(-t) & \mbox{if }\ \ 0<arg(z)< \pi/2-m_0 \end{array} \right. $$

be a map from the upper-half plane to itself. Here $t$ is a real number and $0<m_0<\pi/2$

I just found this map in a paper of S. Wolpert where he constructs a quasi-conformal mapping to represent the Fenchel-Nielsen twist parameter.

My question: is this mapping homotopic to the identity or not?

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Hint: the upper half plane is contractible. –  Baby Dragon Feb 24 at 4:58
    
Oh God! I forgot that. Thanks! –  yaa09d Feb 24 at 5:02
    
You could answer your own question now! –  Baby Dragon Feb 24 at 5:03

1 Answer 1

up vote 2 down vote accepted

First of all, thank you @BabyDragon.

I forgot that the upper half plane is contractible. Hence, $f_t$ and the identity are homotopic.

In fact, Wikipedia says the following:

For a topological space $X$ the following are all equivalent (here $Y$ is an arbitrary topological space):

  • $X$ is contractible (i.e. the identity map is null-homotopic).
  • $X$ is homotopy equivalent to a one-point space.
  • $X$ deformation retracts onto a point. (However, there exist contractible spaces which do not strongly deformation retract to a point.)

  • Any two maps $f,g: Y → X$ are homotopic.

  • Any map $f: Y → X$ is null-homotopic.
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