Compact orientable surface

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This article is about compact orientable surfaces, i.e., two-dimensional compact connected orientable manifolds.

Classification

Further information: classification of compact orientable surfaces

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.

The surface of genus g is sometimes denoted \Sigma_g, S_g or M_g.

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

Particular cases

Value of genus g Surface
0 2-sphere
1 2-torus
2 genus two surface

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

Operations

Connected sum

The connected sum of a surface with genus g_1 and a surface with genus g_2 is a surface with genus g_1 + g_2. If the Euler characteristics of the surfaces are \chi_1 and \chi_2 respectively, the Euler characteristic of the connected sum is \chi_1 + \chi_2 - 2.

Thus, the set of homeomorphism classes of compact orientable surfaces under connected sum is isomorphic to the monoid of nonnegative integers under addition.

Covering spaces

Suppose S_g is a surface of genus g, g > 0. Then, it turns out that for any finite group N of order n, there exists a regular covering map with base S_g and degree d such that the group of deck transformations for the covering map is N. The covering space for this map must also be a compact orientable surface, and have genus h for some h. g,h,n are related as follows:

(2 - 2h) = n(2 - 2g)

or, upon simplification:

h = 1 + n(g - 1)

The justification is as follows: 2 - 2h and 2 - 2g are respectively the Euler characteristics of the compact orientable surfaces, and Euler characteristic of covering space is product of degree of covering and Euler characteristic of base.