- #1
PsychonautQQ
- 784
- 10
Well, for starters, ##\pi(T)##, the fundamental group of the torus, is ##\pi(S^1)x\pi(S^1)=## which is in turn isomorphic to the direct product of two infinite cyclic groups. Before I tackle the case of n connect tori with one point removed, I'm trying to just understand a torus with a point removed.
I'm imagining the torus as a donut. With no points removed, given any base point, the two loop equivalence classes are obvious (three counting the class containing trivial loops). Removing a point, we will now have a new class of loops that go around that missing point. We will also still have the class that goes around through the 'hole' of the donut. I'm trying to think about what happens to the path that goes around the outside of the donut when we take away a point... Does this get split into two different equivalence classes or just stay one? The more I think about it and try to visualize I'm pretty sure it stays one.
Cool, so perhaps the fundamental group of a torus with a point removed is isomorphic to ##ZxZxZ##?
Now, as for a connected sum of n tori... I think all that would happen is we would pick up another ##Z## for each additional 'donut hole'.
How far off am I on this?
I'm imagining the torus as a donut. With no points removed, given any base point, the two loop equivalence classes are obvious (three counting the class containing trivial loops). Removing a point, we will now have a new class of loops that go around that missing point. We will also still have the class that goes around through the 'hole' of the donut. I'm trying to think about what happens to the path that goes around the outside of the donut when we take away a point... Does this get split into two different equivalence classes or just stay one? The more I think about it and try to visualize I'm pretty sure it stays one.
Cool, so perhaps the fundamental group of a torus with a point removed is isomorphic to ##ZxZxZ##?
Now, as for a connected sum of n tori... I think all that would happen is we would pick up another ##Z## for each additional 'donut hole'.
How far off am I on this?