Manifolds with small Heegaard Floer ranks (Q982167)

The main result claims that the \(0\)-surgery on the trefoil is the only closed orientable irreducible \(3\)-manifold with positive first Betti number and Heegaard Floer homology of rank two. The argument goes roughly this way. The rank assumption implies that the \(3\)-manifold contains a nonseparating torus \(F\). Then the manifold is shown to fiber over the circle with fiber \(F\) by using the main theorem of \textit{Y. Ai} and \textit{Y. Ni} [Int. Math. Res. Not. 2009, No.~19, 3726--3746 (2009; Zbl 1192.57009)], which is the detection of torus fibration by Heegaard Floer homology. Finally, the rank assumption also implies that the Betti number equals one and the first homology has no torsion. There are only three possible monodromies for such a torus bundle, and then the manifold is uniquely determined by the rank of Heegaard Floer homology again. Furthermore, the authors determine the links whose branched double cover gives rise to this manifold, the \(0\)-surgery on the trefoil. There are only two links, which are certain \(2\)-cables of the Hopf link. Based on this result, the last part of the paper proves that Khovanov homology detects the unknot if and only if it detects the two component unlink, by using a spectral sequence from Khovanov homology of a link to the Heegaard Floer homology of its double branched cover by \textit{P. Ozsváth} and \textit{Z. Szabó} [Adv. Math. 194, No.~1, 1--33 (2005; Zbl 1076.57013)]. In fact, \textit{P. Kronheimer} and \textit{T. Mrowka} [Khovanov homology is an unknot-detector, \url{arXiv: 1005.4346}] recently proved that Khovanov homology detects the unknot. However, the equivalence above does not follow from it.