Homotopy of compact orientable surfaces

This article describes the value (and the process used to compute it) of some homotopy invariant(s) for a topological space or family of topological spaces. The invariant is homotopy group and the topological space/family is compact orientable surface
Get more specific information about compact orientable surface | Get more computations of homotopy group

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.

Fundamental group

Fill this in later

Higher homotopy groups

Fill this in later