Homotopy in Teichmüller space definition: to be or not to be? That is the question

Question

In the book Introduction to Teichmüller Spaces, by Imayoshi & Taniguchi, we finde the following definition of the Teichmüller space of a Riemann surface $R$, denoted $T(R)$:

I want to draw attention for the term homotopic in this definition. I have tried to show that such relation as defined above is an equivalence relation (as it should be). I could show it is reflexive and transitive, but I could not to show it is symetric. What I have done:

Suppose $(S_1,f_1)\sim (S_2,f_2)$. Then, exists a conformal mapping $f:S_1\to S_2$ and a continuous fuction (this is the homotopy) $H:[0,1]\times S_1\to S_2$ such that $H_0=f_2\circ f_1^{-1}$ and $H_1=f$, where $H_t(p)=H(t,p)$, $p\in S_1$. Then, my first (and, I think, the most natural) thought was trying to show that $f_1\circ f_2^{-1}:S_2\to S_1$ is homotopic to $f^{-1}:S_2\to S_1$ (which is also a conformal mapping). If I do so, then I would have $(S_2,f_2)\sim (S_1,f_1)$.

I wish I could write $K:[0,1]\times S_2\to S_1$ as $K_t=H_t^{-1}$ because I would have $K_0=H_0^{-1}=f_1\circ f_2^{-1}$ and $K_1=H_1^{-1}=f^{-1}$. But I can't do that since I don't know if the inverse $H_t^{-1}$ exists, for all $t\in [0,1]$. I mean, I don't know what happens "along the process" of the homotopy $H$... Even I could write it, I still would have to show that $K$ is a homotopy.

Then I have started to think:

1) Is the

"If a map $g$ is homotopic to a map $h$ then $g^{-1}$ is homotopic to $h^{-1}$"

result TRUE and maybe a basic fact in homotopy theory?; and

2) Instead of HOMOTOPY at that definition, could not be ISOTOPY, maybe? Or asking for $H_t$ be quasiconformal, for all $t\in [0,1]$, anything like this?

Anyway, I need to show that such relation is an equivalence relation: HOW DO I SHOW THAT THE RELATION IS SYMMETRIC?

Thank you!

Answer 1

I got an answer here. It's pretty simple and indeed 1) holds:

Consider topological spaces $X$ and $Y$, two homotopic homeomorphisms $f,g:X\to Y$ and $I=[0,1]\subset \mathbb{R}$. Let $H:I\times X\to Y$ be the homotopy with $H(0,x)=f(x)$ and $H(1,x)=g(x)$. Then define $$\begin{array}{llll}K:&I\times Y&\to&X\\ &(t,y)&\mapsto &f^{-1}\big(H(1-t,g^{-1}(y))\big)\end{array}.$$

This is clearly continuous, $K(0,y)=f^{-1}(y)$ and $K(1,y)=g^{-1}(y)$.

All the credit to John Rognes, author of the answer.