Compact orientable surface: Difference between revisions

Revision as of 16:46, 2 April 2011

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 $R^{3}$ 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 $Z$ , and $H_{1}$ is isomorphic to $Z^{2 g}$ .

In particular, the Betti numbers are $b_{0} = 1, b_{1} = 2 g, b_{2} = 1$ , the Poincare polynomial is $1 + 2 g x + x^{2}$ , and the Euler characteristic is $2 - 2 g$ .

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 $g = 1 - (χ / 2)$ .

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 $χ_{1}$ and $χ_{2}$ respectively, the Euler characteristic of the connected sum is $χ_{1} + χ_{2} - 2$ .

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:

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

or, upon simplification:

$h = 1 + n (g - 1)$

The justification is as follows: $2 - 2 h$ and $2 - 2 g$ 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.

@@ Line 29: / Line 29: @@
 {{further|[[homotopy of compact orientable surfaces]]}}
+==Operations==
+===Connected sum===
+The connected sum of a surface with genus <math>g_1</math> and a surface with genus <math>g_2</math> is a surface with genus <math>g_1 + g_2</math>. If the Euler characteristics of the surfaces are <math>\chi_1</math> and <math>\chi_2</math> respectively, the Euler characteristic of the connected sum is <math>\chi_1 + \chi_2 - 2</math>.
+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 <math>S_g</math> is a surface of genus <math>g</math>, <math>g > 0</math>. Then, it turns out that for any [[finite group]] <math>N</math> of order <math>n</math>, there exists a regular covering map with base <math>S_g</math> and degree <math>d</math> such that the group of deck transformations for the covering map is <math>N</math>. The covering space for this map must also be a compact orientable surface, and have genus <math>h</math> for some <math>h</math>. <math>g,h,n</math> are related as follows:
+<math>(2 - 2h) = n(2 - 2g)</math>
+or, upon simplification:
+<math>h = 1 + n(g - 1)</math>
+The justification is as follows: <math>2 - 2h</math> and <math>2 - 2g</math> are respectively the [[Euler characteristic]]s of the compact orientable surfaces, and [[Euler characteristic of covering space is product of degree of covering and Euler characteristic of base]].