Homotopy of compact orientable surfaces - Revision history

Vipul: /* Fundamental group */

2011-07-29T01:59:25Z

Fundamental group

← Older revision		Revision as of 01:59, 29 July 2011
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	===Fundamental group===		===Fundamental group===

	~~The [[fundamental group]] is a [[free product]] of <math>g</math> copies of the free abelian group of rank two. In other words, it is given as:~~		{{fillin}}

	~~<math>(\mathbb~~{~~Z} \times \mathbb{Z}) * (\mathbb{Z} \times \mathbb{Z}) * \dots (\mathbb{Z} \times \mathbb{Z})</math>~~

	~~where the number of copies of <math>\mathbb~~{Z} ~~\times \mathbb{Z~~}~~</matH> is <math>g</math>.~~

	===Higher homotopy groups===		===Higher homotopy groups===

	{{fillin}}		{{fillin}}

2011-07-28T01:30:56Z

← Older revision		Revision as of 01:30, 28 July 2011
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	invariant = homotopy group\|		invariant = homotopy group\|
	space = compact orientable surface}}		space = compact orientable surface}}

			==Statement==

			This article describes the [[homotopy group]]s, including the [[set of path components]] <math>\pi_0</math>, the [[fundamental group]] <math>\pi_1</math>, and the higher [[homotopy group]]s <math>\pi_k</math> of the [[compact orientable surface]] <math>\Sigma_g</math>, which can be defined as the [[connected sum]] of <math>g</math> many copies of the [[2-torus]]. For <math>g = 0</math>, we obtain the [[2-sphere]], and for <math>g = 1</math>, we get the [[2-torus]].

			===Fundamental group===

			The [[fundamental group]] is a [[free product]] of <math>g</math> copies of the free abelian group of rank two. In other words, it is given as:

			<math>(\mathbb{Z} \times \mathbb{Z}) * (\mathbb{Z} \times \mathbb{Z}) * \dots (\mathbb{Z} \times \mathbb{Z})</math>

			where the number of copies of <math>\mathbb{Z} \times \mathbb{Z}</matH> is <math>g</math>.

			===Higher homotopy groups===

			{{fillin}}

2011-04-02T21:03:08Z

Created page with "{{homotopy invariant computation| invariant = homotopy group| space = compact orientable surface}}"

New page

{{homotopy invariant computation|
invariant = homotopy group|
space = compact orientable surface}}