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
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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 ${\mathrm{\Sigma }}_g$, 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

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Higher homotopy groups

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