Compact orientable surface - Revision history

Vipul: /* Classification */

2011-07-27T16:38:45Z

Classification

← Older revision		Revision as of 16:38, 27 July 2011
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	This article is about compact orientable surfaces, i.e., two-dimensional [[compact connected orientable manifold]]s.		This article is about compact orientable surfaces, i.e., two-dimensional [[compact connected orientable manifold]]s.
	==Classification==		==Classification==

			{{further\|[[classification of compact orientable surfaces]]}}

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

Vipul at 19:00, 2 April 2011

2011-04-02T19:00:45Z

← Older revision		Revision as of 19:00, 2 April 2011
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	~~We first discuss~~ compact orientable surfaces, i.e., two-dimensional [[compact connected orientable manifold]]s.		This article is about compact orientable surfaces, i.e., two-dimensional [[compact connected orientable manifold]]s.
	==Classification==		==Classification==

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

			The surface of genus <math>g</math> is sometimes denoted <math>\Sigma_g</math>, <math>S_g</math> or <math>M_g</math>.

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

			==Particular cases==

			{\| class="sortable" border="1"
			! Value of genus <math>g</math> !! Surface
			\|-
			\| 0 \|\| [[2-sphere]]
			\|-
			\| 1 \|\| [[2-torus]]
			\|-
			\| 2 \|\| [[genus two surface]]
			\|}
	==Algebraic topology==		==Algebraic topology==

Vipul: moved Compact orientable surfaces to Compact orientable surface

2011-04-02T18:32:38Z

moved Compact orientable surfaces to Compact orientable surface

← Older revision	Revision as of 18:32, 2 April 2011
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Vipul at 16:46, 2 April 2011

2011-04-02T16:46:12Z

← Older revision		Revision as of 16:46, 2 April 2011
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	{{further\|[[homotopy of compact orientable surfaces]]}}		{{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]].

Vipul: moved Classification of compact orientable surfaces to Compact orientable surfaces

2011-04-02T16:32:19Z

moved Classification of compact orientable surfaces to Compact orientable surfaces

← Older revision	Revision as of 16:32, 2 April 2011
(No difference)

Vipul at 16:32, 2 April 2011

2011-04-02T16:32:07Z

← Older revision		Revision as of 16:32, 2 April 2011
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	We first discuss compact orientable surfaces, i.e., two-dimensional [[compact connected orientable manifold]]s.		We first discuss compact orientable surfaces, i.e., two-dimensional [[compact connected orientable manifold]]s.
	==~~Statement~~==		==Classification==

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

Vipul: Created page with "We first discuss compact orientable surfaces, i.e., two-dimensional compact connected orientable manifolds. ==Statement== These are classified by the nonnegative integers. I..."

2011-04-02T16:31:54Z

Created page with "We first discuss compact orientable surfaces, i.e., two-dimensional compact connected orientable manifolds. ==Statement== These are classified by the nonnegative integers. I..."

New page

We first discuss compact orientable surfaces, i.e., two-dimensional [[compact connected orientable manifold]]s.
==Statement==

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

Nonnegative integers <math>\leftrightarrow</math> Homeomorphism classes of compact orientable surfaces

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

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

==Algebraic topology==

===Homology===

{{further|[[homology of compact orientable surfaces]]}}

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

In particular, the [[Betti number]]s are <math>b_0 = 1, b_1 = 2g, b_2 = 1</math>, the [[Poincare polynomial]] is <math>1 + 2gx + x^2</math>, and the [[Euler characteristic]] is <math>2 - 2g</math>.

We see from this that the surfaces of genus <math>g</math> 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 <math>g = 1 - (\chi/2)</math>.

===Cohomology===

{{further|[[cohomology of compact orientable surfaces]]}}

===Homotopy===

{{further|[[homotopy of compact orientable surfaces]]}}