Fundamental group of a sphere with an attached chord (${\pi }_1(S^2\cup J)\cong \mathbb{Z}$)

Attach a closed interval to a sphere $S^2$ at two distinct endpoints, with its interior disjoint from the sphere. The fundamental group is infinite cyclic group. Remove a point of the sphere away from the endpoints: the remaining plane has a deformation retraction onto an arc between the endpoints, keeping the added closed interval fixed. This gives a circle. An open cap about the removed point is simply connected, and its punctured overlap has trivial map into the first fundamental group because it lies in the plane. The Seifert-van Kampen theorem therefore adds no further relator.