I find it counter-intuitive to visualize finite fundamental groups.

For example, the fundamental group of the real projective plane is a group with two elements. That means there are two kinds of loops, one which you cannot shrink to a point, and the other, you can. But then, if you traverse this non-trivial loop twice, you end up with something that can be homotoped to the trivial loop.

Is there some way I can visualize this? Is there a way to visualize the projective plane? (I used to think the projective plane is like the pacman universe, but the pacman universe is more like a torus). Is there a visualizable surface where I can see this effect easily?

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    $\begingroup$ Regarding the projective plane. There is a model $S^1/_\pm$ where you identify antipodal points on a sphere. This model can also be viewed as a half-sphere where the antipodal points of the border are identified, thus when you travel along a big circle and you traverse the border you jump on the opposite side of the sphere. Both models helped me a lot. $\endgroup$ May 3, 2018 at 14:41
  • $\begingroup$ @PatrickAbraham What is the difference between this and the pacman universe, i.e, old videogames where the screen wraps around? $\endgroup$ May 3, 2018 at 14:42
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    $\begingroup$ mathoverflow.net/a/38220 this might be helpful $\endgroup$
    – tschih
    May 3, 2018 at 14:43
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    $\begingroup$ If you use an arrow with a left- and a right-hand mark, you will see when you cross the border the left- and right-hand mark will swap places. This is also called non-orientable. $\endgroup$ May 3, 2018 at 14:45
  • $\begingroup$ See jstor.org/stable/2318771?seq=1#page_scan_tab_contents $\endgroup$ May 3, 2018 at 17:37

1 Answer 1



The real projective plane can be viewed as a a square [0,1]x[0,1] with the edges identified as in the diagram above. Let b be our base point. The green loop going across the square from the 'b' in the lower left to the 'b' in the top right (which are identified) cannot be homotoped relative to its endpoints to the trivial loop. However if we compose it with the blue loop then the end of the green line no longer has to stick to 'b' during a homotopy relative to endpoints so we can 'detach' it as in the right figure and then this clearly homotopes to the trivial loop.

To clarify, the blue loop (although it doesn't look like it in my picture) is the same loop as the green one, if you can imagine drawing the green one, then rotating yourself to the opposite side of the square, then drawing the same loop.


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