Almost ι-complexes as immersed curves

This paper uses immersed curves to study homology 3-spheres and their corresponding almost ι -complexes. A new proof of the classification of almost ι -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, ΘZ³ . This paper further demonstrates a new integer homomorphism from the homology cobordism group, Pω: ΘZ³ → Z , conjectured by Dai, Hom, Stoffregen, and Truong. The application of immersed curves to the study of almost ι -complexes admits several streamlined proofs and could inspire applications to similar problems.