Heegaard-Floer homology

In mathematics , Heegaard-Floer homology is an invariant of a closed spin ^c -3 manifold . It is constructed using the Heegaard decomposition of Lagrange-Floer homology. Several homology groups are obtained, which are related to one another through exact sequences. ${\displaystyle Y}$ ${\displaystyle Y}$

The Heegaard-Floer homology was developed in a long series of works by Peter Ozsváth and Zoltán Szabó .

By constructing suitable filters, invariants can be constructed. An example of this is the knot homology associated with a knot in a 3-manifold . Another example is the so-called contact homology , an invariant of contact structures. ${\displaystyle K}$ ${\displaystyle Y}$

construction

Preparations

Let be a closed , orientable 3-manifold and ${\displaystyle Y}$

${\displaystyle Y=H_{1}\cup _{\Sigma }H_{2}}$

a Heegaard decomposition of with a Heegaard surface and a Heegaard diagram . ${\displaystyle Y}$ ${\displaystyle \Sigma _{g}}$ ${\displaystyle (\alpha _{1},\ldots ,\alpha _{g},\beta _{1},\ldots ,\beta _{g})}$

Consider the symmetrical product

${\displaystyle Sym^{g}(\Sigma _{g})=(\Sigma _{g}\times \ldots \times \Sigma _{g})/S_{g}}$,

where the symmetric group acting on the product of identical factors is on elements. It is a smooth manifold and a complex structure on induces a complex structure on the symmetric product. ${\displaystyle S_{g}}$ ${\displaystyle g}$ ${\displaystyle g}$ ${\displaystyle \Sigma _{g}}$

From the Heegaard diagram one obtains two totally real -dimensional tori in the complex manifold . ${\displaystyle (\alpha _{1},\ldots ,\alpha _{g},\beta _{1},\ldots ,\beta _{g})}$ ${\displaystyle g}$ ${\displaystyle T_{\alpha }=\alpha _{1}\times \ldots \times \alpha _{g},T_{\beta }=\beta _{1}\times \ldots \times \beta _{g}}$ ${\displaystyle Sym^{g}(\Sigma _{g})}$

For two intersections, choose two connecting paths . The difference is a loop in and thus represents an element ${\displaystyle x,y\in T_{\alpha }\cap T_{\beta }}$ ${\displaystyle a\subset T_{\alpha },b\subset T_{\beta }}$ ${\displaystyle a-b}$ ${\displaystyle Sym^{g}(\Sigma _{g})}$

${\displaystyle \epsilon (x,y)\in H_{1}(Sym^{g}(\Sigma _{g}))/(H_{1}(T_{\alpha })\oplus H_{1}(T_{\beta }))\cong H_{1}(Y)}$.

Means Morse theory can be (to a selected base point ) each intersection of a [[spin ^c structure]], and thus a spin ^c structure uniquely corresponding element assign, so that for all pairs of intersections of each Poincaré dual to is. ${\displaystyle z\in \Sigma _{g}\setminus (\alpha _{1}\cup \ldots \cup \beta _{g})}$ ${\displaystyle x\in T_{\alpha }\cap T_{\beta }}$ ${\displaystyle s_{z}(x)\in H^{2}(Y)}$ ${\displaystyle s_{z}(x)-s_{z}(y)}$ ${\displaystyle \epsilon (x,y)}$

Denote the set of homotopy classes of pictures which and on and as well as the circular arcs according to and according to depict so-called Whitney slices . For the modular space of the holomorphic mappings in this homotopy class, which can be realized as a smooth manifold using small perturbations . It comes with an effect through the effect by means of and preserving complex automorphisms of . Denote . One can calculate with the Atiyah-Singer index rate. Let the number of intersections of with (for the selected base point ) . Finally, we define as the (signed) number of points in if , and if . ${\displaystyle \pi _{2}(x,y)}$ ${\displaystyle u\colon D^{2}\to Sym^{g}(\Sigma _{g})}$ ${\displaystyle i}$ ${\displaystyle -i}$ ${\displaystyle x}$ ${\displaystyle y}$ ${\displaystyle S^{1}\cap \left\{z\colon Re(z)\geq 0\right\}}$ ${\displaystyle T_{\alpha }}$ ${\displaystyle S^{1}\cap \left\{z\colon Re(z)\leq 0\right\}}$ ${\displaystyle T_{\beta }}$ ${\displaystyle \phi \in \pi _{2}(x,y)}$ ${\displaystyle \mathbb {R} }$ ${\displaystyle \mathbb {R} }$ ${\displaystyle i}$ ${\displaystyle -i}$ ${\displaystyle D^{2}}$ ${\displaystyle {\widehat {M}}(\phi )=M(\phi )/\mathbb {R} }$ ${\displaystyle \mu (\phi )=dim(M(\phi ))}$ ${\displaystyle n_{z}(\phi )}$ ${\displaystyle z}$ ${\displaystyle \phi }$ ${\displaystyle \left\{z\right\}\times Sym^{g-1}(\Sigma _{g})}$ ${\displaystyle c(\phi )}$ ${\displaystyle {\widehat {M}}(\phi )}$ ${\displaystyle \mu (\phi )=1}$ ${\displaystyle c(\phi )=0}$ ${\displaystyle \mu (\phi )\not =0}$

Definition of rational spheres of homology

Be a rational sphere of homology , i.e. h., is finite . As above, a Heegaard decomposition, a base point and a spin ^c structure ( corresponding to a unique element of ) are given . Let be the free Abelian group generated by the points with . Define the boundary operator through ${\displaystyle Y}$ ${\displaystyle H_{1}(Y)}$ ${\displaystyle z}$ ${\displaystyle H^{2}(Y)}$ ${\displaystyle {\mathfrak {t}}}$ ${\displaystyle {\widehat {CF}}}$ ${\displaystyle x\in T_{\alpha }\cap T_{\beta }}$ ${\displaystyle s_{z}(x)={\mathfrak {t}}}$ ${\displaystyle \partial \colon {\widehat {CF}}\to {\widehat {CF}}}$

${\displaystyle \partial x=\Sigma _{y\in T_{\alpha }\cap T_{\beta },\phi \in \pi _{2}(x,y)\mid s_{z}(y)={\mathfrak {t}},n_{z}(y)=0}c(\phi )y}$.

The Heegaard-Floer homology is defined as the homology of . Ozsváth-Szabó prove that it does not depend on the choice of the Heegaard decomposition, the base point, the complex structure and the perturbations and thus actually defines an invariant . One defines . The homology groups have a relative grading through for any one . ${\displaystyle {\widehat {HF}}}$ ${\displaystyle ({\widehat {CF}},\partial )}$ ${\displaystyle {\widehat {HF}}}$ ${\displaystyle {\widehat {HF}}(Y,{\mathfrak {t}})}$ ${\displaystyle {\widehat {HF}}(Y)=\bigoplus _{{\mathfrak {t}}\in H^{2}(Y)}{\widehat {HF}}(Y,{\mathfrak {t}})}$ ${\displaystyle gr(x,y)=\mu (\phi )-2n_{z}(\phi )}$ ${\displaystyle z\in \pi (x,y)}$

Next is the free Abelian group generated by pairs of with . Let be the sub-complex created by pairs with and . A relative gradation is defined by and an edge operator by ${\displaystyle CF^{\infty }}$ ${\displaystyle \left[x,i\right]}$ ${\displaystyle x\in T_{\alpha }\cap T_{\beta },i\in \mathbb {Z} }$ ${\displaystyle s_{z}(x)={\mathfrak {t}}}$ ${\displaystyle CF^{-}}$ ${\displaystyle \left[x,i\right]}$ ${\displaystyle i<0}$ ${\displaystyle CF^{+}=CF^{\infty }/CF^{-}}$ ${\displaystyle gr(\left[x,i\right],\left[y,j\right])=gr(x,y)+2(i-j)}$

${\displaystyle \partial \left[x,i\right]=\Sigma _{y\in T_{\alpha }\cap T_{\beta }}\Sigma _{\phi \in \pi _{2}(x,y)}c(\phi )\left[y,i-n_{z}(\phi )\right]}$.

The groups are defined as the homology groups of the complexes with the boundary operator . Ozsváth-Szabó prove that for rational spheres of homology is always isomorphic to for the given morphism of , and that the homology groups do not depend on the choice of the Heegaard decomposition, the base point, the complex structure and the perturbations, i.e. actually invariants of the rational homology sphere and define a spin ^c structure . After all, you define . ${\displaystyle HF^{\infty },HF^{+},HF^{-}}$ ${\displaystyle CF^{\infty },CF^{+},CF^{-}}$ ${\displaystyle \partial }$ ${\displaystyle HF^{\infty }}$ ${\displaystyle \mathbb {Z} \left[U,U^{-1}\right]}$ ${\displaystyle U(\left[x,i\right])=\left[x,i-1\right]}$ ${\displaystyle CF^{\infty }}$ ${\displaystyle HF^{\pm }}$ ${\displaystyle HF^{\pm }(Y,{\mathfrak {t}})}$ ${\displaystyle Y}$ ${\displaystyle {\mathfrak {t}}}$ ${\displaystyle HF^{+}(Y)=\bigoplus _{{\mathfrak {t}}\in H^{2}(Y)}HF^{+}(Y,{\mathfrak {t}})}$

Definition of general 3-manifolds

For 3-manifolds with is larger and one has infinitely many homotopy classes with in the definition of the boundary operator . One can prove that there are holomorphic disks only in a finite number of these homotopy classes, which is why one again receives a finite sum. For this you have to restrict yourself to special Heegaard diagrams. With this restriction, the definitions work exactly as in the case of rational spheres of homology. ${\displaystyle b_{1}(Y)\not =0}$ ${\displaystyle \pi _{2}(x,y)}$ ${\displaystyle \mu (\phi )=1}$

The different homology groups are related to one another via natural long exact sequences :

${\displaystyle \ldots \rightarrow HF^{-}(Y,{\mathfrak {t}})\rightarrow HF^{\infty }(Y,{\mathfrak {t}})\rightarrow HF^{+}(Y,{\mathfrak {t}})\rightarrow \ldots }$

and with the morphism defined above ${\displaystyle U\colon HF^{\infty }(Y,{\mathfrak {t}})\to HF^{\infty }(Y,{\mathfrak {t}})}$

${\displaystyle \ldots \rightarrow {\widehat {HF}}(Y,{\mathfrak {t}})\rightarrow HF^{+}(Y,{\mathfrak {t}})\rightarrow HF^{+}(Y,{\mathfrak {t}})\rightarrow \ldots }$

Calculations

Examples

• For is and . ${\displaystyle Y=S^{3}}$ ${\displaystyle HF^{+}(Y)=\mathbb {Z} \left[U,U^{-1}\right]/\mathbb {Z} \left[U\right]}$ ${\displaystyle {\widehat {HF}}(Y)=\mathbb {Z} }$

• An L-space is a rational sphere of homology for which a free Abelian group is of rank . This is the case for and all lens chambers . ${\displaystyle Y}$ ${\displaystyle {\widehat {HF}}(Y)}$ ${\displaystyle \sharp H^{2}(Y)}$ ${\displaystyle S^{3}}$

• For the Brieskorn sphere is for straight , and otherwise. ${\displaystyle Y=\Sigma (2,3,5)}$ ${\displaystyle HF_{k}^{+}(Y)=\mathbb {Z} }$ ${\displaystyle k\geq 2}$ ${\displaystyle HF_{k}^{+}(Y)=0}$

• For the Brieskorn sphere is for and straight , and otherwise. ${\displaystyle Y=\Sigma (2,3,7)}$ ${\displaystyle HF_{k}^{+}(Y)=\mathbb {Z} }$ ${\displaystyle k=-1}$ ${\displaystyle k\geq 0}$ ${\displaystyle HF_{k}^{+}(Y)=0}$

Surgery exact triangle

Let be a knot in a closed, orientable 3-manifold , with a meridian and a longitude . Let the 3-manifold obtained by surgery on and the 3 manifold obtained by surgery on . Then you have exact sequences ${\displaystyle K}$ ${\displaystyle Y}$ ${\displaystyle m}$ ${\displaystyle l}$ ${\displaystyle Y_{0}}$ ${\displaystyle l}$ ${\displaystyle K}$ ${\displaystyle Y_{1}}$ ${\displaystyle (m+l)}$ ${\displaystyle K}$

${\displaystyle \ldots {\widehat {HF}}(Y)\rightarrow {\widehat {HF}}(Y_{0})\rightarrow {\widehat {HF}}(Y_{1})\rightarrow \ldots }$

and

${\displaystyle \ldots HF^{+}(Y)\rightarrow HF^{+}(Y_{0})\rightarrow HF^{+}(Y_{1})\rightarrow \ldots }$.

literature

• Ozsváth-Szabó: Holomorphic disks and invariants for closed 3-manifolds , Ann. of Math. (2) 159 (2004) no. 3, 1027-1158.
• Ozsváth-Szabó: Holomorphic disks and three-manifold invariants: properties and applications , Ann. of Math. (2) 159 (2004) no. 3, 1159-1245.

Web links

• Ozsváth-Szabó: An introduction to Heegaard Floer homology
• Ozsváth-Szabó: Lectures on Heegaard Floer homology

Individual evidence

1. ↑ Ozsváth-Szabó: Holomorphic disks and invariants for closed three-manifolds