Almost $\iota$-complexes as immersed curves

TL;DR

The paper introduces a new homomorphism in 3-manifold topology and demonstrates a method to approximate involutive Heegaard Floer complexes using immersed curves, revealing structural properties of the homology cobordism group.

Contribution

It establishes the existence of a new homomorphism $P_{oldsymbol{ extomega}}$ and re-proves the $oldsymbol{bZ}^oldsymbol{ extomega}$ summand in the homology cobordism group using immersed curves.

Findings

Existence of a new homomorphism $P_{oldsymbol{ extomega}}$

Reproof of the $oldsymbol{bZ}^oldsymbol{ extomega}$ summand

Approximation of involutive Heegaard Floer complexes with immersed curves

Abstract

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.