Homology of compact 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 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 $S^{2}$when $g=0$). 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.