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:

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

Reduced version over the integers

We have:

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

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:

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

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.