Surgery obstructions for knots in integer homology spheres

For knot surgery in $S^3$, Heegaard Floer homology provides an obstruction due to Hom–Karakurt–Lidman. We extend this obstruction to all integer homology spheres $Y$, for both positive and negative 1/m surgeries. This is used to test infinitely many small Seifert fibered examples and hyperbolic examples. Moreover, we deduce a lower bound on the $b_2(W)$ of smooth cobordism between a pair of integer homology spheres.