  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 $\mathbb{Z}^\infty$-subgroup and a $\mathbb{Z}$-summand. Our
proof proceeds by introducing an algebraic variant of the involutive Heegaard
Floer package of Hendricks-Manolescu and Hendricks-Manolescu-Zemke. This is
inspired by an analogous argument in the setting of knot concordance due to the
second author.
