Hi So I've been using Seifert-Van Kampen (SVK) to calculate the fundemental 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 fundemental group of (A intersection B) = 0 And I already have fundemental group of (S^1) = Z Then using SVK the fundemental 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.