Lattice homology and Seiberg-Witten-Floer spectra

TL;DR

This paper provides a combinatorial method to explicitly compute the Seiberg-Witten-Floer spectrum for a class of 3-manifolds using lattice homology, enabling calculations of invariants like Manolescu's -invariant.

Contribution

It introduces a new combinatorial approach to determine the Seiberg-Witten-Floer spectrum for almost-rational plumbed homology spheres, including Seifert fibered spaces.

Findings

Explicit combinatorial description of SWF spectrum for the class of manifolds.

Calculation of Manolescu's -invariant for certain connected sums.

Extension of methods to all Seifert fibered rational homology spheres.

Abstract

Using lattice homology, we give an explicit combinatorial description of the Seiberg-Witten-Floer spectrum $\mathit{SWF}(Y)$ for $Y$ an almost-rational plumbed homology sphere. This class of manifolds includes all Seifert fibered rational homology spheres with base orbifold $S^2$. Using our computations, we provide a calculation of Manolescu's $\kappa$-invariant for certain connected sums of these spaces.