The homotopy group of the projected space DP1

[wdlang]

take a circle in a plane

identify two opposite points

we get the projected space DP1

about the homotopy group of DP1, i have two answers

first, we take the upper semicircle from 0 to pi, and identify 0 with pi, by this way, we get a circle again. So the homotopy of DP1 should be identical to that of the circle, which is Z.

second, it is often shown that the only loop in DP1 that cannot be shrank to a point is the open route from 0 to pi

i am really puzzled

 

[lavinia]

take a circle in a plane

identify two opposite points

we get the projected space DP1

about the homotopy group of DP1, i have two answers

first, we take the upper semicircle from 0 to pi, and identify 0 with pi, by this way, we get a circle again. So the homotopy of DP1 should be identical to that of the circle, which is Z.

second, it is often shown that the only loop in DP1 that cannot be shrank to a point is the open route from 0 to pi

i am really puzzled

you are right - the circle with opposite points identified is another circle.