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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