Homotopy of compact orientable surfaces: Difference between revisions

Latest revision as of 01:59, 29 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
Get more specific information about compact orientable surface | Get more computations of homotopy group

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

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Higher homotopy groups

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 ===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===
 {{fillin}}