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
Get more specific information about compact non-orientable surface | Get more computations of homology

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:

$H_{k}(P_{n};\mathbb {Z} )=\lbrace {\begin{array}{rl}\mathbb {Z} ,&k=0\\\mathbb {Z} ^{n-1}\oplus \mathbb {Z} /2\mathbb {Z} ,&k=1\\0,&k\geq 2\\\end{array}}$

Reduced version over the integers

We have:

${\tilde {H}}_{k}(P_{n};\mathbb {Z} )=\lbrace {\begin{array}{rl}0,&k=0\\\mathbb {Z} ^{n-1}\oplus \mathbb {Z} /2\mathbb {Z} ,&k=1\\0,&k\geq 2\\\end{array}}$

Unreduced version over a module

Fill this in later -- basically the behavior is governed by the behavior for the homology of the real projective plane, see homology of real projective space.

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	$\sum _{k=0}^{\infty }(-1)^{k}b_{k}$	$2-n$	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.