Homology 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 homology and the topological space/family is compact non-orientable surface
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Statement

Suppose $k$ is a positive integer. We denote by $P_n$ (not standard notation, should try to find something) the connected sum of the real projective plane with itself $n$ times, i.e., the connected sum of $n$ copies of the real projective plane.

Unreduced version over the integers

We have:

$H_k(P_n;\mathbb{Z})=\{\begin{array}{cc}\mathbb{Z},& k=0\\ {\mathbb{Z}}^{n-1}\oplus \mathbb{Z}/2\mathbb{Z},& k=1\\ 0,& k\ge 2\\ \end{array}$

===Reduced version over the inte