How to compute the fundamental group of this space?

Question

enter image description here

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?

The disk is contractible, right? What will you get if you collapse it to a point? — Connor, Oct 23 '14 at 17:50

score 1 · Answer 1 · answered Oct 24 '14 at 00:25

1

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 $S^2\cup L$.

answered Oct 24 '14 at 00:25

Dan Rust

30,108

I tried this one, but the intersection of U and V doesn't seem to be path connected. – Keith Oct 24 '14 at 00:32
The intersection is just $D^2$. – Dan Rust Oct 24 '14 at 00:37

score 0 · Answer 2 · answered Oct 23 '14 at 17:53

0

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...

answered Oct 23 '14 at 17:53

Timbuc

34,191

So, that means the fundamental group of X is the free group of two generators? – Keith Oct 24 '14 at 00:19

How to compute the fundamental group of this space?

2 Answers2