What's the fundamental group of a punctured torus?

[kakarotyjn]

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

 

[quasar987]

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.

 

[kakarotyjn]

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.

 

[quasar987]

The fundamental group of the torus is not Z*Z though, it is ZxZ.