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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 $$\bar{d}$$ and $$\text{}\underline{d}$$ for certain families of three-manifolds.more » « less
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We construct infinitely many compact, smooth 4-manifolds which are homotopy equivalent to $$S^{2}$$ but do not admit a spine (that is, a piecewise linear embedding of $$S^{2}$$ that realizes the homotopy equivalence). This is the remaining case in the existence problem for codimension-2 spines in simply connected manifolds. The obstruction comes from the Heegaard Floer $$d$$ invariants.more » « less
