Loops in $\mathbb RP^2$

Question

We know that $\pi_1(\mathbb RP^2)=Z_2.$ How do non-trivial loops in $\mathbb RP^2$ look like?

(If $\mathbb RP^2$ is the upper hemisphere $D$ with antipodal points of $\partial D$ identified.)

Answer 1

Roughly speaking, a nontrivial loop in $\mathbb R P^2$ is a loop that crosses through the boundary $\partial D$ an odd number of times, while a trivial loop is one that crosses through $\partial D$ an even number of times.

This assumes that the loop only crosses the boundary finitely many times (which is true up to $\epsilon$-homotopy), and it only counts "crossings" of the boundary, i.e. sudden jumps from one end of $D$ to the other.

Answer 2

Take a curve -an embedding of an interval- that wanders in the interior of the dome, but with initial and end points being antipodal in the ecuator.

Is it easy to see that a regural neighbourhood in the identification ${\Bbb{R}}P^2$ has boundary connected?

Use the picture as an aid

The red curve is the core of a Möbius band!!