Instanton Floer homology of almost-rational plumbings

Abstract

We show that if $Y$ is the boundary of an almost-rational plumbing, then the framed instanton Floer homology $I^{\mathrm{\#}}(Y)$ is isomorphic to the Heegaard Floer homology $\widehat{HF}(Y;ℂ)$. This class of $3$–manifolds includes all Seifert fibered rational homology spheres with base orbifold $S^2$ (we establish the isomorphism for the remaining Seifert fibered rational homology spheres — with base ${ℝ\mathbb{P}}^2$ — directly). Our proof utilizes lattice homology, and relies on a decomposition theorem for instanton Floer cobordism maps recently established by Baldwin and Sivek.