Instanton Floer homology of almost-rational plumbings

TL;DR

This paper proves an isomorphism between framed instanton Floer homology and Heegaard Floer homology for a class of 3-manifolds called almost-rational plumbings, using lattice homology and recent cobordism map decompositions.

Contribution

It establishes the isomorphism for almost-rational plumbings, including all Seifert fibered rational homology spheres, expanding the understanding of Floer homologies for these manifolds.

Findings

Isomorphism between instanton and Heegaard Floer homology for almost-rational plumbings.

Includes all Seifert fibered rational homology spheres with base orbifold S^2.

Extends to Seifert fibered rational homology spheres with base RP^2.

Abstract

We show that if $Y$ is the boundary of an almost-rational plumbing, then the framed instanton Floer homology $\smash{I^\#(Y)}$ is isomorphic to the Heegaard Floer homology $\smash{\widehat{\mathit{HF}}(Y; \mathbb{C})}$. 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$\unicode{x2014}$with base $\mathbb{RP}^2$$\unicode{x2014}$directly). Our proof utilizes lattice homology, and relies on a decomposition theorem for instanton Floer cobordism maps recently established by Baldwin and Sivek.