For the post Fundamental group of projective plane is $C_{2}$???, I think there is a mistake in Amos Joshua's answer:

If you rotate the 1 2 loop, it will no longer be connected to the 3 4 loop, and a loop homotopy in the fundamental group requires the base point sit still through the homotopy. What his answer shows is a loop can be homotopy'd to its inverse, but the base point is not held fixed during homotopy.

I'm probably wrong of course but it's quite clear during his homotopy the 2 loops are disconnected. I think if the 1 2 loop is deformed to the left boundary, and 3 4 loop deformed to the right boundary, holding the end points fixed all the time, then the 3 4 loop in the projective plane is just the 1 2 in the opposite direction, i.e. its inverse.

Is this the correct reasoning?

I don't have enough rep to comment and I think no one will read my comment anyway given how old it is. And there seems to be no private messaging option on this site...

• One solution is, get enough points to comment. The world has been waiting a while now for a comment, it can wait a little longer. Oct 13 '19 at 4:59
• Related: Clarify an old answer. Oct 13 '19 at 12:22
• As a side note, if you have sufficient reputation to use chat, you could also ask in one of the chatrooms associated with this site. Oct 13 '19 at 12:45