An infinite-rank summand of the homology cobordism group

Abstract

We show that the three-dimensional homology cobordism group admits an infinite-rank summand. It was previously known that the homology cobordism group contains a ${\mathbb{Z}}^{\mathrm{\infty }}$-subgroup and a $\mathbb{Z}$-summand. Our proof proceeds by introducing an algebraic variant of the involutive Heegaard Floer package of Hendricks, Manolescu, and Zemke. This is inspired by an analogous argument in the setting of knot concordance due to the second author.