Almost $\iota$-complexes as immersed curves

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.