Almost $\iota$-complexes as immersed curves

Abstract

This paper uses immersed curves to study homology 3-spheres and their corresponding almost $\iota$-complexes. A new proof of the classification of almost $\iota$-complexes is given using a geometric argument. This is then used to reprove the existence of an infinite rank summand in the homology cobordism group, ${\Theta }_{\mathbb{Z}}^3$. This paper further demonstrates a new integer homomorphism from the homology cobordism group, $P_{\omega }:{\Theta }_{\mathbb{Z}}^3\to \mathbb{Z}$, conjectured by Dai, Hom, Stoffregen, and Truong. The application of immersed curves to the study of almost $\iota$-complexes admits several streamlined proofs and could inspire applications to similar problems.