Almost $\iota$-complexes as immersed curves

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