Compact orientable surface

We first discuss compact orientable surfaces, i.e., two-dimensional compact connected orientable manifolds.

Classification

These are classified by the nonnegative integers. In other words, there is a correspondence:

Nonnegative integers $\leftrightarrow$ Homeomorphism classes of compact orientable surfaces

The correspondence, in the forward direction, is as follows: given a nonnegative integer $g$, the corresponding compact orientable surface, called the surface of genus $g$, is defined as a connected sum of $g$ copies of the 2-torus. Two special cases are of note: for $g=0$, we take the corresponding surface to be the 2-sphere, and for $g=1$, we take the corresponding surface to be the 2-torus. After that, each time we increment $g$ by $1$, we take the connected sum with a new 2-torus.

Pictorially, the surface of genus $g$ can be embedded in ${\mathbb{R}}^3$ with as many holes as the genus.

Algebraic topology

Homology

Further information: homology of compact orientable surfaces

The homology groups of the surface with genus $g$ are given as follows: $H_0$ and $H_2$ are both $\mathbb{Z}$, and $H_1$ is isomorphic to ${\mathbb{Z}}^{2g}$.

In particular, the Betti numbers are $b_0=1,b_1=2g,b_2=1$, the Poincare polynomial is $1+2gx+x^2$, and the Euler characteristic is $2-2g$.

We see from this that the surfaces of genus $g$ are all in different homotopy classes and are in fact not even homology-equivalent. We can in fact recover the genus of a compact orientable surface simply from its Euler characteristic, by $g=1-(\chi /2)$.

Cohomology

Further information: cohomology of compact orientable surfaces

Homotopy

Further information: homotopy of compact orientable surfaces