Homology of compact non-orientable surfaces

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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 \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.