Homotopy of compact non-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 and the topological space/family is compact non-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 non-orientable surface $P_n$, which is defined as the connected sum of $n$ copies of the real projective plane ${\mathbb{P}}^2(\mathbb{R})$.

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.

Fundamental group

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