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

Statement

Suppose g is a nonnegative integer. We denote by \Sigma_g the compact orientable surface of genus g, i.e., \Sigma_g is the connected sum of g copies of the 2-torus T = S^1 \times S^1(and it is the 2-sphere S^2when g = 0). In other words, \Sigma_0 = S^2, \Sigma_1 = T, \Sigma_2 = T \# T, and so on.

Unreduced version over the integers

We have:

H_k(\Sigma_g;\mathbb{Z}) := \lbrace\begin{array}{rl} \mathbb{Z}, & k = 0,2 \\ \mathbb{Z}^{2g}, & k = 1 \\ 0, & k > 2\end{array}

In other words, the zeroth and second homology groups are both free of rank one, and the first homology group is \mathbb{Z}^{2g}, i.e., the free abelian group of rank 2g.

Reduced version over the integers

We have:

\tilde{H}_k(\Sigma_g;\mathbb{Z}) := \lbrace\begin{array}{rl}  0, & k = 0 \\\mathbb{Z}^{2g}, & k = 1 \\ \mathbb{Z}, & k = 2 \\ 0, & k > 2 \end{array}

Unreduced version with coefficients in M

With coefficients in a module M over a ring R, we have:

H_k(\Sigma_g;M) := \lbrace\begin{array}{rl} M, & k = 0,2 \\ M^{2g}, & k = 1 \\ 0, & k > 2\end{array}

Reduced version with coefficients in M

\tilde{H}_k(\Sigma_g;M) := \lbrace\begin{array}{rl} 0, & k = 0 \\ M^{2g}, & k = 1 \\ M, & k = 2 \\   0, & k > 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 genus g surface 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 = b_2 = 1, b_1 = 2g, all higher b_k are zero
Poincare polynomial Generating polynomial for Betti numbers 1 + 2gx + x^2
Euler characteristic \sum_{k=0}^\infty (-1)^k b_k 2 - 2g In particular, this means that the Euler characteristic is negative for g > 1, and also that the Euler characteristic and genus can be computed in terms of each other -- given the genus, we can find the Euler characteristic, and vice versa.

Facts used in homology computation

  1. Homology of spheres: This tackles the g = 0 case.
  2. Homology of torus: This tackles the g = 1 case.
  3. Homology of connected sum: This tackles the inductive procedure of taking connected sums.