Homotopy of compact orientable surfaces: Difference between revisions

Revision as of 01:30, 28 July 2011

This article describes the value (and the process used to compute it) of some homotopy invariant(s) for a topological space or family of topological spaces. The invariant is homotopy group and the topological space/family is compact orientable surface
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Statement

This article describes the homotopy groups, including the set of path components $π_{0}$ , the fundamental group $π_{1}$ , and the higher homotopy groups $π_{k}$ of the compact orientable surface $Σ_{g}$ , which can be defined as the connected sum of $g$ many copies of the 2-torus. For $g = 0$ , we obtain the 2-sphere, and for $g = 1$ , we get the 2-torus.

Fundamental group

The fundamental group is a free product of $g$ copies of the free abelian group of rank two. In other words, it is given as:

$(Z \times Z) * (Z \times Z) * \dots (Z \times Z)$

where the number of copies of $Z \times Z$ is $g$ .

Higher homotopy groups

Fill this in later

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 invariant = homotopy group|
 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}}