I have this exercise:

Compute the fundamental group $pi_1(X)$ of the space $X=S^2 cup { (x,0,0) : xin (-1,1)

} cup {(0,y,0):yin (-1,1) } cup {(0,0,z):zin (1,1) }$.

I tried that via Seifert-Van Kampen:

Let $I_x = { (x,0,0) : xin (-1,1)

}$, $I_y = {(0,y,0):yin (-1,1) }$, $I_z = {(0,0,z):zin (1,1) }$.

Then $pi_1(X) = pi_1(S^2 cup I_x cup I_y cup I_z) = pi_1((S^2 cup I_x) cup (I_y cup I_z)) cong pi_1(S^2 cup I_x) ast_{pi_1((S^2 cup I_x) cap (I_y cup I_z))} pi_1(I_y cup I_z) cong (pi_1(S^2) ast_{pi_1(S^2cap I_x)} pi_1(I_x)) ast_{pi_1((S^2 cup I_x) cap (I_y cup I_z))} (pi_1(I_y) ast_{I_ycap I_z} pi_1(I_z)) cong ({0 } ast_{pi_1(S^0)} { 0 }) ast_{pi_1((0,0,0))} ({0 } ast_{pi_1((0,0,0))} { 0 })=({0} ast_{pi_1(S^0)} {0}) ast ({0} ast {0})$.

I know it looks ridiculous.

I don’t know how to compute this product with amalgamation.