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andlook

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So I've been using Seifert-Van Kampen (SVK) to calculate the fundamental group of the torus. I haven't done any formal group theory, hence my problem ...

I have T^2=(S^1)x(S^1)

If A= S^1, B=S^1, A intersection B is 0. And T^2 = union of A and B.

Then fundamental group of (A intersection B) = 0

And I already have fundamental group of (S^1) = Z

Then using SVK the fundamental group of the torus is the free product of S^1 with S^1 over 0.

Which I think is isomorphic to the ZxZ.

In the literature this is written as the direct sum of Z+Z. Why is this the direct sum and not the cross product?

Since I don't know group theory better I don't know if it is possible to just ask the simpler: Why is the free product of Z*Z is isomorphic to direct sum Z+Z not the product of ZxZ.