Hi,
I have been really stuck for a while on the following:
As you can see below, is part of the proof of the Fundamental group actually being a group. I am aware of the general idea behind the diagrams which 'show' the homotopy. Yet I am confused as to how you give the co-ordinates. As in, below you can see f(2s-t) is used in the left hand side, yet what method was used to know it was 2s-t. I have tried thinking of a general method, something along the lines of continous functions taking the interval [0,1/2] to [1/2,1] and then multiplying this function by t. Yet it didnt seem to work. I am asking as I am trying to find the explicit homotopy required to prove the RHS of the picture, where we need to show f is homotopic to the constant path glued with f.
Thank you for any help I have been stuck for a long time, much appreciated.
I have written a short note on these proofs. Check it out.