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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?

How do I go about asking such a question?

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...

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In very specific situations, you could ask a new question on which you ask for clarification/explanation. But this is often not useful, since the issue is probably an oversight or minor misunderstanding. The most productive thing to do is wait until you can comment and then do so.


In this specific case, it is not clear what your doubt is. The loop in question is the entire 1-2-3-4 loop. The base point (the center of the image) is indeed fixed throughout the entire rotation. The movement of the center throughout the second part of the illustration is not the movement of the base point, it is the movement of the point which corresponds to half the timespan of the loop. The base point does not move at all during the entire homotopy.

You say that "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 do you mean by "the 1 2 loop (...) will no longer be connected to the 3 4 loop"? I can only surmise that you mean that their images do not completely overlap, but this is not related whatsoever to the base point being fixed or not. Whatever you mean, it indeed is true that the base point sits still through the entire homotopy.

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    $\begingroup$ Its worth pointing out that the user who posted the answer has only 286 rep, and this answer, from over a year ago, was the last interaction they have had with the site. So its fair to assume that a comment will not be answered. [Although, according to their profile, they were "last seen yesterday"...] $\endgroup$ – user1729 Oct 14 at 11:45
  • $\begingroup$ In a general case, what if the author has last seen years ago? Then the probability the followup question gets answered is too low. Will that be reasonable to ask another question by linking the existing one? It will be helpful to link the new question to the existing one so that some other users facing the same issue can get them clarified. $\endgroup$ – Intellex Oct 19 at 8:43

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