  Here the existence of a new homomorphism $P_{\omega} : \Theta_{\mathbb{Z}}^3
\to \mathbb{Z}$ is proven and the existence of a $\mathbb{Z}^{\infty}$ summand
in $\Theta_{\mathbb{Z}}^3$ is reproven. This is done by approximating the
involutive Heegaard Floer complexes of homology 3-spheres with immersed curves
on the twice punctured disk.
