On the Heegaard Floer homology of branched double-covers.(English)Zbl 1076.57013
For a closed, oriented three-manifold \(Y\) and a \(\text{Spin}^{c}\) structure \(\mathfrak{s}\in\text{Spin}^{c}(Y)\), the authors defined the Heegaard Floer homology \(\widehat{H\!F}(Y,\mathfrak{s})\) [Ann. Math. (2) 159, No. 3, 1027–1158 (2004;Zbl 1073.57009)]. They showed in [Ann. Math. (2) 159, No. 3, 1159–1245 (2004;Zbl 1081.57013)] that its Euler characteristic \(\chi(\widehat{H\!F}(Y,\mathfrak{s}))\) is one if the first Betti number of \(Y\) is zero. Therefore one has \(\chi(\widehat{H\!F}(Y))=|H^{2}(Y;\mathbb{Z})|\), where \(\widehat{H\!F}(Y):=\bigoplus_{\mathfrak{s}\in\text{Spin}^{c}(Y)}\widehat{H\!F}(Y,\mathfrak{s})\). Note that since \(\widehat{H\!F}\) has relative \(\mathbb{Z}/2\mathbb{Z}\)-grading, its Euler characteristic is well defined. For a knot in \(Y\), denote by \(Y_{0}\) (\(Y_{1}\), respectively) the three-manifold obtained from \(Y\) by Dehn surgery along the knot with framing corresponding to the meridian (\(\text{meridian}+\text{longitude}\), respectively). The authors also showed that there exists a long exact sequence \(\cdots\rightarrow\widehat{H\!F}(Y)\rightarrow\widehat{H\!F}(Y_{0})\rightarrow \widehat{H\!F}(Y_{1})\rightarrow\cdots\).
Now let us consider a link \(L\) in the three-sphere \(S^{3}\). M. Khovanov introduced an invariant \(K\!h^{i,j}(L)\) whose ‘Euler characteristic’ \(\sum_{i,j}(-1)^{i}\dim\left(K\!h^{i,j}(L)\right)q^{j}\) is equal to (a version of) the Jones polynomial. Note that if we replace \(q\) with \(-1\) we have \[|H^{2}(\Sigma(L))|=\bigl| \sum_{i,j}(-1)^{i+j}\dim\left(K\!h^{i,j}(L)\right)\bigr|,\] where \(\Sigma(L)\) is the double branched cover of \(S^{3}\) branched along \(L\). Let \(L_{0}\) (\(L_{1}\), respectively) denote the ‘\(A\)-resolution’ (‘\(A^{-1}\)-resolution’, respectively) of \(L\) at a crossing appearing in the definition of the Kauffman bracket. Then there exists a long exact sequence \(\dots\rightarrow K\!h(L)\rightarrow K\!h(L_{0})\rightarrow K\!h(L_{1})\rightarrow\cdots\) [O. Viro, Remarks on definition of Khovanov homology, arXiv: math.GT/0202199].
The main result of the paper under review is to show where the similarity between \(\widehat{H\!F}\) and \(K\!h\) described above comes from. In fact, the authors show that there exists a spectral sequence with \(E^{2}\) term \(K\!h(L;\mathbb{Z}/2\mathbb{Z}):=\bigoplus_{i,j}K\!h^{i,j}(L;\mathbb{Z}/2\mathbb{Z})\) that converges to \(\widehat{H\!F}(\Sigma(L);\mathbb{Z}/2\mathbb{Z})\).
Now let us consider a link \(L\) in the three-sphere \(S^{3}\). M. Khovanov introduced an invariant \(K\!h^{i,j}(L)\) whose ‘Euler characteristic’ \(\sum_{i,j}(-1)^{i}\dim\left(K\!h^{i,j}(L)\right)q^{j}\) is equal to (a version of) the Jones polynomial. Note that if we replace \(q\) with \(-1\) we have \[|H^{2}(\Sigma(L))|=\bigl| \sum_{i,j}(-1)^{i+j}\dim\left(K\!h^{i,j}(L)\right)\bigr|,\] where \(\Sigma(L)\) is the double branched cover of \(S^{3}\) branched along \(L\). Let \(L_{0}\) (\(L_{1}\), respectively) denote the ‘\(A\)-resolution’ (‘\(A^{-1}\)-resolution’, respectively) of \(L\) at a crossing appearing in the definition of the Kauffman bracket. Then there exists a long exact sequence \(\dots\rightarrow K\!h(L)\rightarrow K\!h(L_{0})\rightarrow K\!h(L_{1})\rightarrow\cdots\) [O. Viro, Remarks on definition of Khovanov homology, arXiv: math.GT/0202199].
The main result of the paper under review is to show where the similarity between \(\widehat{H\!F}\) and \(K\!h\) described above comes from. In fact, the authors show that there exists a spectral sequence with \(E^{2}\) term \(K\!h(L;\mathbb{Z}/2\mathbb{Z}):=\bigoplus_{i,j}K\!h^{i,j}(L;\mathbb{Z}/2\mathbb{Z})\) that converges to \(\widehat{H\!F}(\Sigma(L);\mathbb{Z}/2\mathbb{Z})\).
Reviewer: Hitoshi Murakami (Tokyo)
MSC:
| 57M27 | Invariants of knots and \(3\)-manifolds (MSC2010) |
| 57R58 | Floer homology |
| 57M12 | Low-dimensional topology of special (e.g., branched) coverings |
| 57M25 | Knots and links in the \(3\)-sphere (MSC2010) |
| 57N10 | Topology of general \(3\)-manifolds (MSC2010) |
Software:
MathematicaCite
References:
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