Homotopy of compact 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 orientable surface $$\Sigma_g$$, which can be defined as the connected sum of $$g$$ many copies of the 2-torus. For $$g = 0$$, we obtain the 2-sphere, and for $$g = 1$$, we get the 2-torus.