Homotopy of compact non-orientable surfaces

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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(\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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