How to compute the fundamental group of this space?

Question

I know that without the closed disk, a sphere with the a diameter deformation retracts onto the wedge sum of a circle and a sphere. But I can't figure out how to deform the disk to a suitable space....

Could anyone help me?

Answer 1

An idea: "squeeze" the disk to the point $\;(0,0,0)\;$ , so that you get two tangent spheres joined at this point and with a "stick" (the line $\;\{(0,0,z)\;:\;-1\le z\le 1\}\;$) as common diameter through their tangency point...

Answer 2

Let $U$ be $X$ with the top half of $L$ and the north hemisphere of $S^2$ removed. Let $V$ be similar but bottom and south. Use Van Kampen's theorem and the result that you already mentioned for $S^2\cup L$.