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 ${\mathrm{\Sigma }}_g$ the compact orientable surface of genus $g$, i.e., ${\mathrm{\Sigma }}_g$ 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, ${\mathrm{\Sigma }}_0=S^2$, ${\mathrm{\Sigma }}_1=T$, ${\mathrm{\Sigma }}_2=T\#T$, and so on.

Unreduced version over the integers

We have:

$H_k({\mathrm{\Sigma }}_g;\mathbb{Z}):=\{\begin{array}{cc}\mathbb{Z},& k=0,2\\ {\mathbb{Z}}^{2g},& k=1\\ 0,& k>2\end{array}$

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:

${\tilde{H}}_k({\mathrm{\Sigma }}_g;\mathbb{Z}):=\{\begin{array}{cc}0,& k=0\\ {\mathbb{Z}}^{2g},& k=1\\ \mathbb{Z},& k=2\\ 0,& k>2\end{array}$

Unreduced version with coefficients in $M$

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

$H_k({\mathrm{\Sigma }}_g;M):=\{\begin{array}{cc}M,& k=0,2\\ M^{2g},& k=1\\ 0,& k>2\end{array}$

Reduced version with coefficients in $M$

${\tilde{H}}_k({\mathrm{\Sigma }}_g;M):=\{\begin{array}{cc}0,& k=0\\ M^{2g},& k=1\\ M,& k=2\\ 0,& k>2\end{array}$

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	$\sum _{k=0}^{\mathrm{\infty }}(-1{)}^k{b}_k$	$2-2g$	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.