I have a possible simple question but I cannot work out the answer by myself.
Suppose we have two cross-caps on a sphere (i.e. two $RP^2$ attached to the sphere by means of a connected sum) and suppose we have two loops each of them linked to each cross-cap. By linked to the cross-cap, I mean that the loop cannot be detached and float free on the sphere because it is not homotopic to the trivial loop (i.e. it is the loop associate to the $Z_2$ component of the $H_1$ homology groups of the $RP^2$).
Are the two loops described above homotopic to each other?
On one hand I know that if I bring the two cross-caps together and I get a Klein Bottle, this has only a "$Z_2-like$" loop in its $H_1$ homology groups and therefore the two loops have to fall in the same path and be homotopic.
On the other hand I see that it is not possible to unlink each loops from its cross-cap and it is not possible to link it to the other cross cap without going twice around any path in it. So I do not see how would be possible to continuously move one loop and bring it to coincide to the other (which is a naïve way of describing homotopy).
Could anyone help me to understand how it works?