Rank inequalities for the Heegaard Floer homology of Seifert homology spheres

We establish three rank inequalities for the reduced flavor of Heegaard Floer homology of Seifert fibered integer homology spheres. Combining these inequalities with the known classifications of non-zero degree maps between Seifert fibered spaces, we prove that a map f: Y ′ → Y f: Y’ Y between Seifert homology spheres yields the inequality | deg ⁡ f | r a n k H F r e d (Y) ≤ r a n k H F r e d (Y ′) | f| rank HF ₑ₄₃ (Y) rank HF ₑ₄₃ (Y’). These inequalities are also applied in conjunction with an algorithm of Némethi to give a method to solve the botany problem for the Heegaard Floer homology of these manifolds.