the fundamental group of the immersed image of Klein bottle in $\mathbb{R}^3$

Question

how to get the fundamental group of the immersed image of Klein bottle in $\mathbb{R}^3$?

I just try to use the Van-Kampen theorem to prove it is $\mathbb{Z}$. i am not sure.

if u know something about it, thanks for helping me .

Answer 1

by "collapsing subspace"(see Hatcher chapter 0) the immersed image is homotopy equivalence to the wedge sum of $S^1,S^1,S^2$. so the fundamental group is $Z*Z$