##
**Knots, sutures, and excision.**
*(English)*
Zbl 1208.57008

The authors defined monopole Floer homology \(HM(Y)\) for any closed 3-manifod \(Y\) in [P. Kronheimer and T. Mrowka, Monopoles and three-manifolds. New Mathematical Monographs 10. Cambridge: Cambridge University Press (2007; Zbl 1158.57002)]. The homology is a topological invariant of closed oriented 3-manifolds coming from Seiberg-Witten gauge theory. Here they define two varieties of the homology: (balanced) sutured monopole Floer homology (SHM) and knot monopole Floer homology (KHM). In addition, in the latter part of the article they give the definition of (balanced) sutured instanton Floer homology (SHI) and knot instanton Floer homology (KHI).

The definition of sutured Floer homology is based on Juhász’s construction in [A. Juhász, Geom. Topol. 12, No. 1, 299–350 (2008; Zbl 1167.57005)]. Along his philosophy the authors also give the definition of SHM. It is that the surface decomposition by Gabai along a surface \(S\subset (M,\gamma)\) is interpreted as taking a subset of \(\text{spin}^c\) structures. Thus for a surface \(R\subset Y\) let \(HM(Y|R)\) be the subgroup of \(\text{spin}^c\) structures satisfying \(\langle c_1(s),[R]\rangle=2g(R)-2\). Then SHM of a sutured manifold \((M,\gamma)\) is defined to be such a subgroup \(HM(Y|R)\) in \(HM(Y)\), where \(Y\) is a closed manifold obtained by attaching an auxiliary surface and gluing both surfaces \(R_+\) and \(R_-\). The authors use auxiliary surfaces to obtain a closed 3-manifold. \(R\) is the embedded surfaced glued in the end. The SHM does not depend on the auxiliary surface and the gluing. The well-definedness is based on the excision theorem of monopole Floer homology, which is proved in the article.

Let \((Y,\Sigma=\Sigma_1\cup\Sigma_2)\) be a pair of a 3-manifold and two embedded surfaces. \(Y\) has one or two components. In the two components case, each surface is embedded in each component. Cutting \(Y\) open along \(\Sigma_i\), gluing \(\Sigma_2\) to \(-\Sigma_1\) and \(\Sigma_1\) to \(-\Sigma_2\), we obtain a 3-manifold \((\widetilde{Y},\widetilde{\Sigma})\). In this setting the excision theorem is that \(HM(Y|\Sigma)\) is isomorphic to \(HM(\widetilde{Y}|\widetilde{\Sigma})\). This kind of excision theorem was first proved by A. Floer in the context of instanton homology as presented in [P. J. Braam and S. K. Donaldson, The Floer memorial volume. Hofer, Helmut (ed.) et al., The Floer memorial volume. Basel: Birkhäuser. Prog. Math. 133, 195–256 (1995; Zbl 0996.57516)].

KHM is defined to be the SMH for the sutured manifold obtained from the complement of a knot in a 3-maniold. The authors prove that KHM admits some fundamental properties: Euler characteristic, genus detection and fiberedness detection. The Euler characteristic of KHM is the symmetrized Alexander polynomial. The maximal degree \(\max\{i|KHM(K,i)\neq 0\}\) coincides with the Seifert genus \(g(K)\) of \(K\). A knot \(K\) is fibered, if and only if \(KHM(K,g(K))\cong {\mathbb Z}\) holds. These two detections are supported by horizontal and annulus decomposition theorems and the rank of SHM of a sutured manifold with two taut foliations with non-torsion difference element. In the Heegaard Floer setting the former one is due to Y. Ni [Invent. Math. 170, No. 3, 577–608 (2007); erratum ibid. 177, No. 1, 235–238 (2009; Zbl 1138.57031)] and A. Juhász [Geom. Topol. 12, No. 1, 299–350 (2008; Zbl 1167.57005)] and the latter one is due to P. Ghiggini [Am. J. Math. 130, No. 5, 1151–1169 (2008; Zbl 1149.57019)]. As mentioned above, SHM of the sutured manifold \((M',\gamma')\) decomposed by a surface of \((M,\gamma)\) is a direct summand of \(SHM(M,\gamma)\) with respect to a set of \(spin^c\) structures for the surface.

Let \(w\) be a line bundle over \(Y\) such that \(c_1(w)\) has odd pairing with an integer homology class and let \(E\) be a \(U(2)\) bundle with an isomorphism \(\wedge^2E\to w\). Let \({\mathcal B}={\mathcal C}/{\mathcal G}\) be the set of \(SO(3)\) connections in \(ad(E)\) up to the transformation of a gauge group \({\mathcal G}\). The Chern-Simons functional on \({\mathcal C}\) can give the definition of instanton Floer homology \(I_\ast(Y)_w\) as in [S. K. Donaldson, Floer homology groups in Yang-Mills theory. Cambridge Tracts in Mathematics. 147. Cambridge: Cambridge University Press (2002; Zbl 0998.53057)]. For a surface \(R\subset Y\) let \(\xi_R\) be the real line bundle with \(w_1(\xi_R)\) dual to \(R\). The map \(E\mapsto E\otimes \xi_R\) induces an involution \(\iota_R\) on \({\mathcal B}\). Instanton homology using \({\mathcal B}/\iota_R\) in place of \({\mathcal B}\) is written as \(I_\ast(Y)_{w,R}\). Actions of two images of the \(\mu\)-map, \(\mu(R)\) and \(\mu(y)\) for \(y\in Y\), give the eigenspaces on \(I_\ast(Y)_w\) and \(I_\ast(Y)_{w,R}\). The degrees are \(-2\) and \(-4\) respectively. \(I_\ast(Y|R)_w\) is defined to be the simultaneous eigenspace of the operators \(\mu(R)\) and \(\mu(y)\) with the eigenvalues \((2g(R)-2,2)\), since \(\mu(R)\) and \(\mu(y)\) commute. The subspace plays the same role in \(HM(Y|R)\). Sutured Floer homology SHI and knot Floer homology KHI are defined in the same way as SHM and KHM. The properties of the following items are similarly proven here: the excision theorem for \(I_\ast(Y|R)_w\), SHI for surface decompositions, SHI for taut sutured manifolds, and the detection of the genus and fiberedness for KHI. However whether the Euler characteristics of \(KHI(K)\) is the symmetrized Alexander polynomial of \(K\) is still open. Also, instanton Floer homology can prove the Property P conjecture for any knot from non-vanishing of \(I_\ast(Y|\bar{R})_w\) and Floer’s exact triangle.

Monopole, instanton and Heegaard Floer homology theories possess a sutured and a knot version, which have common properties. The question whether these homologies are mutually isomorphic remains as a conjecture. However recently the equivalence between monopole Floer homology and Heegaard Floer homology for any closed 3-manifold has been proven by C. Kutluhan, Y.-J. Lee and C. H. Taubes [HF=HM I : Heegaard Floer homology and Seiberg–Witten Floer homology; arXiv:1007.1979], [HF=HM II: Reeb orbits and holomorphic curves for the ech/Heegaard-Floer correspondence; arXiv:1008.1595], and [HF=HM III: Holomorphic curves and the differential for the ech/Heegaard Floer correspondence; arXiv:1010.3456].

At the end of the article the authors also raise interesting future applications: non-vanishing of the Donaldson invariant for any symplectic 4-manifold in this context, monopole Floer homology of Dehn surgery and KHM, recovering problem of Floer homology of closed 3-manifolds from sutured Floer homology.

The definition of sutured Floer homology is based on Juhász’s construction in [A. Juhász, Geom. Topol. 12, No. 1, 299–350 (2008; Zbl 1167.57005)]. Along his philosophy the authors also give the definition of SHM. It is that the surface decomposition by Gabai along a surface \(S\subset (M,\gamma)\) is interpreted as taking a subset of \(\text{spin}^c\) structures. Thus for a surface \(R\subset Y\) let \(HM(Y|R)\) be the subgroup of \(\text{spin}^c\) structures satisfying \(\langle c_1(s),[R]\rangle=2g(R)-2\). Then SHM of a sutured manifold \((M,\gamma)\) is defined to be such a subgroup \(HM(Y|R)\) in \(HM(Y)\), where \(Y\) is a closed manifold obtained by attaching an auxiliary surface and gluing both surfaces \(R_+\) and \(R_-\). The authors use auxiliary surfaces to obtain a closed 3-manifold. \(R\) is the embedded surfaced glued in the end. The SHM does not depend on the auxiliary surface and the gluing. The well-definedness is based on the excision theorem of monopole Floer homology, which is proved in the article.

Let \((Y,\Sigma=\Sigma_1\cup\Sigma_2)\) be a pair of a 3-manifold and two embedded surfaces. \(Y\) has one or two components. In the two components case, each surface is embedded in each component. Cutting \(Y\) open along \(\Sigma_i\), gluing \(\Sigma_2\) to \(-\Sigma_1\) and \(\Sigma_1\) to \(-\Sigma_2\), we obtain a 3-manifold \((\widetilde{Y},\widetilde{\Sigma})\). In this setting the excision theorem is that \(HM(Y|\Sigma)\) is isomorphic to \(HM(\widetilde{Y}|\widetilde{\Sigma})\). This kind of excision theorem was first proved by A. Floer in the context of instanton homology as presented in [P. J. Braam and S. K. Donaldson, The Floer memorial volume. Hofer, Helmut (ed.) et al., The Floer memorial volume. Basel: Birkhäuser. Prog. Math. 133, 195–256 (1995; Zbl 0996.57516)].

KHM is defined to be the SMH for the sutured manifold obtained from the complement of a knot in a 3-maniold. The authors prove that KHM admits some fundamental properties: Euler characteristic, genus detection and fiberedness detection. The Euler characteristic of KHM is the symmetrized Alexander polynomial. The maximal degree \(\max\{i|KHM(K,i)\neq 0\}\) coincides with the Seifert genus \(g(K)\) of \(K\). A knot \(K\) is fibered, if and only if \(KHM(K,g(K))\cong {\mathbb Z}\) holds. These two detections are supported by horizontal and annulus decomposition theorems and the rank of SHM of a sutured manifold with two taut foliations with non-torsion difference element. In the Heegaard Floer setting the former one is due to Y. Ni [Invent. Math. 170, No. 3, 577–608 (2007); erratum ibid. 177, No. 1, 235–238 (2009; Zbl 1138.57031)] and A. Juhász [Geom. Topol. 12, No. 1, 299–350 (2008; Zbl 1167.57005)] and the latter one is due to P. Ghiggini [Am. J. Math. 130, No. 5, 1151–1169 (2008; Zbl 1149.57019)]. As mentioned above, SHM of the sutured manifold \((M',\gamma')\) decomposed by a surface of \((M,\gamma)\) is a direct summand of \(SHM(M,\gamma)\) with respect to a set of \(spin^c\) structures for the surface.

Let \(w\) be a line bundle over \(Y\) such that \(c_1(w)\) has odd pairing with an integer homology class and let \(E\) be a \(U(2)\) bundle with an isomorphism \(\wedge^2E\to w\). Let \({\mathcal B}={\mathcal C}/{\mathcal G}\) be the set of \(SO(3)\) connections in \(ad(E)\) up to the transformation of a gauge group \({\mathcal G}\). The Chern-Simons functional on \({\mathcal C}\) can give the definition of instanton Floer homology \(I_\ast(Y)_w\) as in [S. K. Donaldson, Floer homology groups in Yang-Mills theory. Cambridge Tracts in Mathematics. 147. Cambridge: Cambridge University Press (2002; Zbl 0998.53057)]. For a surface \(R\subset Y\) let \(\xi_R\) be the real line bundle with \(w_1(\xi_R)\) dual to \(R\). The map \(E\mapsto E\otimes \xi_R\) induces an involution \(\iota_R\) on \({\mathcal B}\). Instanton homology using \({\mathcal B}/\iota_R\) in place of \({\mathcal B}\) is written as \(I_\ast(Y)_{w,R}\). Actions of two images of the \(\mu\)-map, \(\mu(R)\) and \(\mu(y)\) for \(y\in Y\), give the eigenspaces on \(I_\ast(Y)_w\) and \(I_\ast(Y)_{w,R}\). The degrees are \(-2\) and \(-4\) respectively. \(I_\ast(Y|R)_w\) is defined to be the simultaneous eigenspace of the operators \(\mu(R)\) and \(\mu(y)\) with the eigenvalues \((2g(R)-2,2)\), since \(\mu(R)\) and \(\mu(y)\) commute. The subspace plays the same role in \(HM(Y|R)\). Sutured Floer homology SHI and knot Floer homology KHI are defined in the same way as SHM and KHM. The properties of the following items are similarly proven here: the excision theorem for \(I_\ast(Y|R)_w\), SHI for surface decompositions, SHI for taut sutured manifolds, and the detection of the genus and fiberedness for KHI. However whether the Euler characteristics of \(KHI(K)\) is the symmetrized Alexander polynomial of \(K\) is still open. Also, instanton Floer homology can prove the Property P conjecture for any knot from non-vanishing of \(I_\ast(Y|\bar{R})_w\) and Floer’s exact triangle.

Monopole, instanton and Heegaard Floer homology theories possess a sutured and a knot version, which have common properties. The question whether these homologies are mutually isomorphic remains as a conjecture. However recently the equivalence between monopole Floer homology and Heegaard Floer homology for any closed 3-manifold has been proven by C. Kutluhan, Y.-J. Lee and C. H. Taubes [HF=HM I : Heegaard Floer homology and Seiberg–Witten Floer homology; arXiv:1007.1979], [HF=HM II: Reeb orbits and holomorphic curves for the ech/Heegaard-Floer correspondence; arXiv:1008.1595], and [HF=HM III: Holomorphic curves and the differential for the ech/Heegaard Floer correspondence; arXiv:1010.3456].

At the end of the article the authors also raise interesting future applications: non-vanishing of the Donaldson invariant for any symplectic 4-manifold in this context, monopole Floer homology of Dehn surgery and KHM, recovering problem of Floer homology of closed 3-manifolds from sutured Floer homology.

Reviewer: Motoo Tange (Kyoto)

### MSC:

57M27 | Invariants of knots and \(3\)-manifolds (MSC2010) |

57R58 | Floer homology |

53D40 | Symplectic aspects of Floer homology and cohomology |