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 integers

We have:

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

Unreduced version over a module

If we consider the homology with coefficients in a module $M$ over a ring $R$ where 2 is invertible, then we have:

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

Related invariants

These are all invariants that can be computed in terms of the homology groups.

Invariant	General description	Description of value for connected sum of $n$ copies of real projective plane	Comment
Betti numbers	The $k^{th}$ Betti number $b_k$ is the rank of the $k^{th}$ homology group.	$b_0=1$, $b_1=n-1$, all higher $b_k$ are zero
Poincare polynomial	Generating polynomial for Betti numbers	$1+(n-1)x$
Euler characteristic	$\sum _{k=0}^{\mathrm{\infty }}(-1{)}^k{b}_k$	$2-n$	In particular, this means that the Euler characteristic is negative for $n>2$. Note that if the Euler characteristic of a compact surface is odd and at most $1$, then the surface must be non-orientable and its homeomorphism type can be computed (using $2-n=\chi$. If the Euler characteristic is even and at most $0$, then there is a unique possibility for a compact orientable surface and a unique possibility for a compact non-orientable surface. For an Euler characteristic of $2$, there is a unique compact orientable surface and no compact non-orientable surface. For an Euler characteristic bigger than $2$, there is no (connected) compact surface possible.