  We use Heegaard Floer homology to define an invariant of homology cobordism.
This invariant is isomorphic to a summand of the reduced Heegaard Floer
homology of a rational homology sphere equipped with a spin structure and is
analogous to Stoffregen's connected Seiberg-Witten Floer homology. We use this
invariant to study the structure of the homology cobordism group and, along the
way, compute the involutive correction terms for certain families of
three-manifolds.
