Homotopy of compact non-orientable surfaces

Statement
This article describes the homotopy groups, including the set of path components $$\pi_0$$, the fundamental group $$\pi_1$$, and the higher homotopy groups $$\pi_k$$ of the compact non-orientable surface $$P_n$$, which is defined as the connected sum of $$n$$ copies of the real projective plane $$\mathbb{P}^2(\R)$$.

Set of path components
Each of the spaces $$P_n$$ is a path-connected space, so its set of path components is a one-point space.