What's the fundamental group of a punctured torus?

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SUMMARY

The fundamental group of a punctured torus is Z x Z, as established in the discussion. Initially, there was confusion regarding whether it could be represented as Z x Z x Z due to a misunderstanding of its topology. The correct interpretation involves recognizing that a punctured torus can be continuously deformed into two cylinders glued to a square patch, confirming that the fundamental group is indeed Z x Z. This clarification resolves the debate on the structure of the fundamental group for this geometric shape.

PREREQUISITES
  • Understanding of fundamental groups in algebraic topology
  • Familiarity with toroidal geometry
  • Knowledge of continuous deformation and homeomorphism concepts
  • Basic grasp of group theory, specifically free groups
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  • Study the properties of fundamental groups in algebraic topology
  • Explore the concept of homotopy and its relation to punctured surfaces
  • Learn about the classification of surfaces and their fundamental groups
  • Investigate the implications of the Seifert-van Kampen theorem in topology
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Mathematicians, particularly those specializing in topology, geometry students, and anyone interested in the algebraic properties of surfaces and their fundamental groups.

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The fundamental group of a torus is Z*Z,then the fundamental group of a punctured torus is Z*Z*Z.

But I've ever done a problem,it said a punctured torus can be continuously deformed into two cylinders glued to a square patch.Really?

If that is right,then the fundamental group of punctured torus is Z*Z.

Which is right?Need help:smile:
 

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Yes, it's true... imagine making the hole bigger and bigger.. make it as big as you can withouth changing the topology. You're left with the two strips glued on a square patch.
 
oh,i see.the fundamental group should be Z*Z. i consider an extra loop,which is the edge circle of the punctured hole. but now I know it's the 2 power of a generator.
 
The fundamental group of the torus is not Z*Z though, it is ZxZ.
 

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