Heegaard Floer genus bounds for Dehn surgeries on knots (Q2928220)

The article under review gives a new obstruction for a rational homology \(3\)-sphere to arise by Dehn surgery on a knot in terms of Heegaard Floer homology. More precisely, let \(Y\) be a rational homology \(3\)-sphere with \(|H_1(Y)|=p\). If \(Y\) is obtained by \(p/q\)-surgery on a non-trivial knot \(K\) in the \(3\)-sphere, then \(2g(K)-1\geq (p-\ell)/|q|\), where \(\ell\) is the number of \(L\)-structures of \(Y\). An \(L\)-structure of \(Y\) is a spin\(^c\)-structure \(\mathfrak{s}\) for which the associated Heegaard Floer homology \(\widehat{HF}(Y,\mathfrak{s})=\mathbb{Z}\). This inequality is shown to be sharp. It also gives an obstruction for two framed knots to give the same manifold.NEWLINENEWLINEFor a rational homology \(3\)-sphere, its rational and integral Dehn surgery genera are introduced. These are defined to be the minimal genus of a knot which can yield this rational homology 3-sphere by some rational or integral Dehn surgery. There are some calculations of these genera, and it is proved that the difference between two genera can be arbitrarily large.