Homotopic mappings

Question

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?

Answer 1

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.