Using Taubes’ periodic ends theorem, Auckly gave examples of toroidal and hyperbolic irreducible integer homology spheres which are not surgery on a knot in the three-sphere. We use Heegaard Floer homology to give an obstruction to a homology sphere being surgery on a knot, and then use this obstruction to construct infinitely many small Seifert fibered examples.
"Surgery obstructions and Heegaard Floer homology." Geom. Topol. 20 (4) 2219 - 2251, 2016. https://doi.org/10.2140/gt.2016.20.2219