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

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

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

Homology groups with integer coefficients in tabular form

We illustrate how the homology groups work for small values of $n$. Note that $H_{2}$ is zero and all higher $H_{p}$ are zero.

$n$	$P_{n}$	$H_{0}(P_{n})$	$H_{1}(P_{n})$	$H_{2}(P_{n})$
1	real projective plane	$\mathbb {Z}$	$\mathbb {Z} /2\mathbb {Z}$	0
2	Klein bottle	$\mathbb {Z}$	$\mathbb {Z} \oplus \mathbb {Z} /2\mathbb {Z}$	0
3	Dyck's surface	$\mathbb {Z}$	$\mathbb {Z} \oplus \mathbb {Z} \oplus \mathbb {Z} /2\mathbb {Z}$	0

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 torsion-free part 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.