Splicing knot complements and bordered Floer homology
The Heegaard Floer homology group \(\widehat{\text{HF}}(Y)\) of a rational homology sphere \(Y\) is known to have rank at least \(|H_1(Y;\mathbb{Z})|\). When \(\widehat{\text{HF}}(Y)\cong\mathbb{Z}^{|H_1(Y;\mathbb{Z})|}\), \(Y\) is said to be an L-space.NEWLINENEWLINEThe main result of the paper asserts that the splice of two knot complements in integer homology L-spaces is never itself an L-space. The splice of two knot complements is a 3-manifold obtained by gluing the two knot complements (seen as compact manifolds with torus boundary) along their boundaries; the gluing is determined by an orientation-reversing map of the boundaries that takes the meridian (respectively, longitude) of the first knot to the longitude (resp. meridian) of the second.NEWLINENEWLINEThis result is related to a conjecture of \textit{P. Ozsváth} and \textit{Z. Szabó} [Clay Math. Proc. 5, 29--70 (2006; Zbl 1105.57029)], which asserts that the only integer homology sphere L-spaces are connected sums of copies of the Poincaré sphere (with either orientation).NEWLINENEWLINEThe proof of the main result is an application of the bordered Floer homology package developed by R. Lipshitz, P. S. Ozsváth, and D. P. Thurston, and a careful study of the bordered Floer homology groups of knot complements in L-space homology spheres.NEWLINENEWLINEThe paper puts the result in the proper context, makes the comparison with related results of E. Eftekhary, and in the last section it clearly marks the extent to which the techniques can be applied.NEWLINENEWLINETheorem 2.2, based on an idea described to the authors by P. S. Ozsváth, gives an algorithm to compute the bordered Floer group \(\widehat{\text{CFA}}\) from \(\widehat{\text{CFD}}\) and vice-versa, and it is of independent interest.
scientific article; zbMATH DE number 6646161
scientific article