Recently, I learned the definition of homotopy, and had a question. In all continuous curves(in complex plane) with $a$ as the starting point and $b$ as the end point, if the two curves are homotopic are regarded as the same curve, how many such curves are there?
I would be very happy if someone could answer my question, since this is my first question.
Thank you very much!
There would be only one curve. Suppose $f$ and $g$ are two paths with the same initial and final points. Then define $H:I\times I\to \Bbb C$ by $H(x,t)=(1-t)f(x)+tg(x)$, where $I=[0,1]$. Clearly, $H(x,0)=f(x), H(x,1)=g(x), H(0,t)=f(0)=g(0), H(1,t)=f(1)=g(1)$. Can you check the continuity of $H$?