57

数理解析研究所講究録
第2004巻 2016年 57-76

Introduction to Heegaard Floer homology
Motoo Tange

University of Tsukuba

1
InAtiyah‐Floer conjecture and Ozsv \acute{\mathrm{a} \mathrm{t}\mathrm{h}-\mathrm{S}\mathrm{z}\mathrm{a}\mathrm{b}\'{o} \mathrm{s} motivation[25, 26] Ozsváth and Szabó defined topological 3‐manifold invariants by using Floer

homology theory:\hat{HF}(Y,\mathfrak{s}) , HF^{\infty}(Y,\mathfrak{s}) , HF^{+}(Y,\mathfrak{s}) , HF^{-}(Y,\mathfrak{s}).

Those invariants are new invariants in terms of the point that the invariants are categorifi‐
cations of some topological invariants: Casson invariant (with correction term) Alexanderpolynomial and Turaev torsion invariant.

The motivation of defining these invariants isthat it is a symplectic counterpart of Seiberg‐Witten Floer homology via Atiyah‐Floer
conjecture. Original Atiyah‐Floer conjecture for Yang‐Mills equation is the following:

Conjecture 1.1 (Atiyah‐Floer conjecture) Let Y be a closed oriented 3‐manifolds.
The instanton Floer homology on Y and the Lagrangian intersection Floer homology forflat connections are isomorphic each other:

HF^{Inst}(\mathcal{M}_{Y})\cong HF^{Symp}(\mathcal{M}_{ $\Sigma$};\mathcal{L}_{0}, \mathcal{L}_{1}).

The instanton Floer homology HF^{\mathrm{I}\mathrm{n}\mathrm{s}\mathrm{t} (\mathcal{M}_{Y}) is a Morse homology on a moduli space \mathcal{M}_{Y}of Yang‐Mills equation on a 3‐manifold Y up to gauge action. The generators are the flat
connections on Y The differentials are the counting of the moduli space of Yang‐Mills.

solutions on the cylinder Y\times I.

On the other hand for a Heegaard decomposition Y=H_{0}\displaystyle \bigcup_{ $\Sigma$}H_{1} we consider the
symplectic moduli space \mathcal{M}_{ $\Sigma$} on $\Sigma$ of flat connections. Let \mathcal{L}_{0}, \mathcal{L}_{1} in \mathcal{M}_{ $\Sigma$} be Lagrangiansubmanifolds extending to H_{0} and H_{1} The generators of symplectic side are intersection
points of the two manifolds with a suitable general condition. The differentials are thecounting of the holomorphic disks connecting two points. The boundary of a holomorphic
disk lies in union of \mathcal{L}_{0} and \mathcal{L}_{1}.
Heegaard Floer theory corresponds to the opposite side of the Seiberg‐Witten Floer
homology through analogy of Atiyah‐Floer conjecture. Before Heegaard Floer homol‐
ogy appearing, no one would have succeeded a symplectic counterpart of Seiberg‐Witten
theory. Ozsváth and Szabó attacked that natural and essential problem to attain such
Lagrangian intersection Floer homology. Instanton Floer homology had rediscovered topo‐
logical invariants, Casson invariant [45]. Hence they should have expected applications.

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of interesting topological invariants from the Heegaard Floer homology.

pected,More than ex‐

it turns out to be a powerful tool to study low‐dimensional topology and be an

attractable object for low‐dimensional topologists. As a result, a whole lot of results havebeen built by Heegaard Floer homology, some new phenomena are discovered.
This article is a simple overview of Heegaard Floer homology up to now.

But some

interesting topics had to skip because of space limitation. If this article arouses yourinterests in your mind, as present Heegaard Floer topologists were so in the past, thenyou should begin with one in a pile of papers.
One can also read some excellent lecture notes [30] [31] [32] as the first study of Hee‐gaard Floer homology.
2
2.1

Heegaard Floer homology
Definition of Heegaard Floer homologies

We define Heegaard Floer homology in this section. Let
Y=H_{0}\displaystyle \bigcup_{$\Sigma$_{g} H_{1} be a Heegaard

decomposition of a 3‐manifold Y Since H_{0}, H_{1} are two genus g handlebodies, there exist
tuples of g simple closed disjoint curves $\alpha$_{1}, \cdots, $\alpha$_{g}\subset\partial H_{0} and $\beta$_{1},$\beta$_{g}\subset\partial H_{1}each of which is compressing in each handlebody. Hence, on $\Sigma$=\partial H_{0}=\partial H_{1} we can see
the diagram (Heegaard diagram) consisting of 2g curves $\alpha$_{1}, \cdots, $\alpha$_{g}, $\beta$_{1}, \cdots, $\beta$_{g} (these arecalled $\alpha$ ‐curves and $\beta$ ‐curves). Here we assume that all the intersection points between
the $\alpha$ ‐curves and $\beta$ ‐curves are transversal in the general position.
In this situation (our moduli space is the g‐th symmetric product.

two\cdots,Sy \mathrm{m}^{}($\Sigma$_{g}):=$\Sigma$^{g}/S_{g}.
It is a smooth complex manifold by resolving the singular set by the S_{g} ‐action. Let \mathb {T}_{ $\alpha$}and \mathb {T}_{ $\beta$} denote the image of$\alpha$_{1}\times$\alpha$_{2}\times\cdots\times$\alpha$_{g}, $\beta$_{1}\times$\beta$_{2}\times\cdots\times$\beta$_{g}by the quotient map $\Sigma$^{g}\rightarrow$\Sigma$^{g}/S_{g} The images in Sy \mathrm{m}^{}($\Sigma$_{g}) are g ‐dimensional tori.
by $\alpha$=\{$\alpha$_{1}, \cdots, $\alpha$_{g}\} and $\beta$=\{$\beta$_{1}, \cdots, $\beta$_{g}\} Let z be a basepoint in $\Sigma$_{g} in the complement of $\alpha$ and $\beta$ Naturally these are totally real submanifold.

We denote the g curves.

.

(^{\mathrm{d} \Leftrightar ow^{\mathrm{e}\mathrm{f} Y\subset X and i\cdot T_{p}Y\cap T_{p}Y=\{0\} ). Floers theory works for the case of a pair of
totally real submanifolds in a complex (or almost complex) manifold. This tuple of thesedata ($\Sigma$_{g}, $\alpha$, $\beta$, z) is called pointed Heegaard diagram. Since 2 \dim Y=\dim X holds, theintersection points \mathb {T}_{ $\alpha$}\cap \mathb {T}_{ $\beta$} are finite points.
Here we verify the homology of the symmetric product is isomorphic to the homologyof Y.

\displayst le\frac{H_{1}(\mathrm{S}\mathrm{y}\mathrm{ }^{g}($\Sigma$_{g}) {H_{1}(\mathb {T}_{$\alpha$})\oplusH_{1}(\mathb {T}_{$\beta$})\congH_{1}(Y,\mathb {Z})This means that the Sy \mathrm{m}^{}($\Sigma$_{g}) is regarded as an (expounded body of a 3‐manifold. One

direction is the ordinary homology and another sharpener direction is the Heegaard Floer

homology.

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The intersection point\mathrm{x}\in \mathbb{T}_{ $\alpha$}\cap \mathbb{T}_{ $\beta$} and base point z determine a spinC structure.

The point \mathrm{x} implies g Morse trajectories from index 2 to 1 with respect to the Heegaard

splitting. The point z is regarded as the trajectory from index 3 to 0 naturally. Thus
removing the neighborhoods of the g+1 trajectories, we obtain a non‐zero vector field ina holed 3‐manifold. The
homotopy classes of such fields up to homologous coincide withthe spinC structures on Y The homologous means that the two fields are homotopic
after deleting several balls. Hence, we get the map:.

s_{z}:\mathbb{T}_{ $\alpha$}\cap \mathbb{T}_{ $\beta$}\rightar ow \mathrm{S}\mathrm{p}\mathrm{i}\mathrm{n}^{c}(Y).

Let \mathrm{x}, \mathrm{y} be intersection points in \mathb {T}_{ $\alpha$}\cap \mathb {T}_{ $\beta$}
Let $\pi$_{2}(\mathrm{x}, \mathrm{y}) be the homotopy classes
connecting \mathrm{x} and \mathrm{y} The class [u]\in$\pi$_{2}(\mathrm{x}, \mathrm{y}) is represented by a continuous map u:\mathbb{D}\rightarrow
Sym g($\Sigma$_{g}) satisfying.

.

\left{bginary}{lu(\rmRe}gq0\cap rtil\mahb{D})\suetmahb{T}_$\alph} u(i)=\mathr{y}\u( rmRe}\lq0cap\rtilmahb{D})\suetmahb{T}_$\eta u(-i)=\mathr{x},\endary}ight.

where \mathbb{D} is the unit disk in the complex plane.

For any$\phi$\in$\pi$_{2}(\mathrm{x}, \mathrm{y}) let n_{z}( $\phi$) be thealgebraic count \#((z\times \mathrm{S}\mathrm{y}\mathrm{m}^{g-1}($\Sigma$_{g})\cap u(\mathbb{D}) .\hat{\mathcal{M} ( $\phi$)=\mathcal{M}( $\phi$)/\mathbb{R} is the moduli space ofholomorphic disk which is divided by natural \mathbb{R}‐action. Then the differential \hat{\partial} are definedto be\displayst le\hat{\parti l}\mathrm{x}=\sum_{\mathrm{y}\in\mathrm{T}_{$\alpha$}\mathrm{n}^{r\mathrm{F}_{$\beta$}\sum_{$\phi$\n$\pi$_{2}(\mathrm{x},\mathrm{y}),n_{z}($\phi$)=0,$\mu$( \phi$)=1}\hat{\mathcal{M}($\phi$)\cdot\mathrm{y}.
Here $\mu$ is the Maslov index. We put
and Szabó proved the following:

Theorem 2.1\hat{CF}($\Sigma$_{g}, $\alpha$, $\beta$, z)=\mathb {Z}\langle \mathrm{x}|\mathrm{x}\in \mathb {T}_{ $\alpha$}\cap \mathb {T}_{ $\beta$}\rangle Then Ozsváth(\hat{CF}( $\Sigma$, $\alpha$, $\beta$, z),\hat{\partial}) is

.

chain

a

complex and its homology is a topological3‐manifold invariant.

\hat{CF}(Y) and \hat{HF}(Y) respectively. The
proof of this theorem is supported by an analytical argument on the moduli space ofholomorphic disks. The argument is important, however, skip here. Actually, to hold this
topological invariance, we need some general conditions: admissibility in Section 4.2.2 inWe denote the chain complex and homology by[25].
The homology\hat{HF}(Y) is decomposedasfollows:\hat{HF}(Y)=\oplus_{\mathfrak{s}\in \mathrm{S}\mathrm{p}\mathrm{i}\mathrm{n}^{\mathrm{c} (Y)}\hat{HF}(Y,\mathfrak{s})because the differential keep the map s_{z}number of\hat{HF}(Y, \mathfrak{s}),,i.e., s_{z}(\mathrm{x})=s_{z}(\mathrm{y}) when $\pi$_{2}(\mathrm{x}, \mathrm{y})\neq\emptyset The Euler.

is$\chi$(\hat{HF}(Y) =\left\{ begin{ar ay}{l} |H_{1}(Y,\mathb {Z})|&b_{1}(Y)=0\ 0&b_{1}(Y)\neq0. \end{ar ay}\right.

Namely, \hat{HF}(Y) is a categorification of the ordinary homology H_{1}(Y)=H^{2}(Y)

.

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Let

(Y,\mathfrak{s}) be a spinC 3‐manifold with b_{1}> O.

Then the Euler characteristic of HF^{+}agrees with the Turaev torsion.

$\chi$(HF^{+}(Y,\mathfrak{s}))=- $\tau$(Y,\mathfrak{s})This equality makes sense in non‐torsion spinC case, but this equality holds even for any
torsion spinC structure by taking truncated Euler characteristic.

In particular if Y is 0‐

surgery of a knot K , then the i‐th Turaev torsion t_{i}(K) is computed by the coefficientsof the Alexander polynomial of K as in [33].

Furthermore by counting intersection numbers with z\times \mathrm{S}\mathrm{y}\mathrm{m}^{g-1}($\Sigma$_{g})following chain complex,

wedefine theCF^{\infty}($\Sigma$_{g}, $\alpha$, $\beta$, z)=\mathbb{Z}\langle U^{n}\cdot \mathrm{x}|\mathrm{x}\in \mathbb{T}_{ $\alpha$}\cap \mathbb{T}_{ $\beta$}, n\in \mathbb{Z}\}.
The differential \partial^{\infty} is a U ‐equivariant map and\displayst le\partial^{\infty}\mathrm{x}=\sum_{\mathrm{y}\in\mathb {T}_{$\alpha$}\cap\mathb {T}_{$\beta$}\sum_{$\phi$\in$\pi$_{2}(\mathrm{x},\mathrm{y}),$\mu$( \phi$)=1}\# hat{\mathcal{M}($\phi$)U^{n_z}($\phi$)}\mathrm{y}.
There exists a subchain complex CF^{-}( $\Sigma$, $\alpha$, $\beta$, z)=\mathbb{Z}\{U^{n}\cdot \mathrm{x}|\mathrm{x}\in \mathbb{T}_{ $\alpha$}\cap \mathbb{T}_{ $\beta$}, n\in \mathbb{Z}_{<0}\rangleCF^{\infty}/CF^{-} denotes by CF^{+}( $\Sigma$, $\alpha$, $\beta$, z)

quotient complex([25])

Theorem 2.2

.

.

The

Then we have the following:

These chain complexes give topological invariants

HF^{\infty}(Y) HF^{-}(Y) and HF^{+}(Y),.

Furthermore, these homology are \mathbb{Z}[U]\otimes$\Lambda$^{*}H_{1}(Y, \mathbb{Z})/Tors ‐module.
By the definition we have the following short exact sequence:

0\rightarrow CF^{-}(Y)\rightarrow iCF^{\infty}(Y)\rightarrow $\pi$ CF^{+}(Y)\rightarrow 0.
Thus we induce the long exact sequence:.

.

.

\rightarrow HF^{-}(Y)\rightarrow HF^{\infty}(Y)\rightarrow HF^{+}(Y)\rightarrow HF^{-}(Y)\rightarrow\cdots

The cokernel of $\pi$ is denoted by HF_{\mathrm{r}\mathrm{e}\mathrm{d} (Y).

Since the chain complex

in the n=0 level in CF^{\infty}(Y) , the short exact sequence:\hat{CF}(Y) is embedded

0\rightarrow\hat{CF}(Y)\rightarrow CF^{+}(Y)\rightarrow\times UCF^{+}(Y)\rightarrow 0
Here the second is the U‐multiple map. Thus we induce the long exact sequence:.

2.2\rightarrow\hat{HF}(Y)\rightarrow HF^{+}(Y)\rightarrow HF^{+}(Y)\rightarrow\hat{HF}(Y)\rightarrow\cdots

Holomorphic disks

Next step of Heegaard Floer homology is a dimensional reduction of holomorphic disks
in Sy \mathrm{m}^{}($\Sigma$_{g})One of most remarkable points of Heegaard Floer theory is computable byusing words of low‐dimensional topology.
.

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Proposition 2.3 ([25]) For a continuous map u:\mathbb{D}\rightar ow \mathrm{S}\mathrm{y}\mathrm{m}^{g}($\Sigma$_{g}) there exists a con‐tinuous map from a branched cover û : \hat{\mathb {D} \rightar ow$\Sigma$_{g} satisfying the following commutative
diagram:\hat{\mathbb{D} \rightarrow { (x, v)\in $\Sigma$ \mathrm{x} Sym ($\Sigma$_{g})|x\in v } \rightarrow $\Sigma$,\downar owg\downarow‐fold branched coverSy\mathrm{m}^{}($\Sigma$_{g})\mathbb{D}\rightarrow

This proposition asserts the holomorphic disk in Sy \mathrm{m}^{}($\Sigma$_{g}) is understood as a surface
mapped in $\Sigma$_{g} The condition \partial \mathb {D}\subset \mathb {T}_{ $\alpha$}\cup \mathb {T}_{ $\beta$} corresponds to \^{u}\subset $\alpha$\cup $\beta$ The Maslov
index in the definition of the Heegaard Floer homology is analytical information data ofholomorphic disk.
.

.

Proposition 2.4 ([13]) The Maslov index of a disk u in [u]= $\phi$\in$\pi$_{2}(\mathrm{x}, \mathrm{y}) is determinedby a topological information of the region mapped in $\Sigma$_{g}.
The TQFT viewpoint and absolute grading

2.3

Floer theory is defined in the framework of topological quantum field theory (TQFT).
Any element in Floer homology can be delivered by the cobordism to an element in Floer
homology of another side. Let (W,\mathfrak{s}) be a spinC 4‐dimensional cobordism from Y_{1} to Y_{2}.
Then there exists a U ‐equivariant map:

F_{W,\mathfrak{s} ^{\infty} HF^{\infty}(Y_{1}, \mathrm{t}_{1})\rightarrow HF^{\infty}(Y_{2}, \mathrm{t}_{2}):where \mathfrak{s}|_{Y_{7} =\mathrm{t}_{ $\eta$},

This map is induced to other homologies HF^{+}, HF^{-} andway. For the composition of two cobordisms X=X_{1}\cup X_{2} we have

same

.

\displaystyle\sum_{\ mathfrak{s}\in\mathrm{S}\mathrm{p}\mathrm{i}\mathrm{n}^{\mathrm{c} (X)|\mathfrak{s}|x_{i}=\mathfrak{s}_{i} { _{\} F_{\mathring{X},\mathfrak{s} where 0=\infty, \pm, \wedge.

\hat{HF} in the

F_{\mathring{X}_{1},\mathfrak{s}_{1} oF_{\mathring{X}_{2},\mathfrak{s}_{2} =,Let Y be a rational homology sphere. In general,(Y,\mathfrak{s}) is a torsion spinC 3‐manifold,where c_{1}(\mathrm{s}) is a torsion element. Then the Heegaard Floer homology admits an absolute\mathb {Q}‐grading gr.

This

grading is defined as follows.

First foragenerator \mathrm{x}_{0}\in\hat{HF}(S^{3})

gr(xo) =0 holds. The difference of the gradings by F_{W,\mathfrak{s} ^{\mathrm{o} (0=\infty, \pm, $\Lambda$) is computed by\displaystyle \mathrm{g}\mathrm{r}(F_{W,\mathfrak{s} ^{\mathrm{o} (\mathrm{x}) -\mathrm{g}\mathrm{r}(\mathrm{x})=\frac{ _{1}^{2}(\mathfrak{s})-2 $\chi$(W)-3 $\sigma$(W)}{4}.
The U ‐action lowers the degree by -2
on.

These properties determine the absolute \mathb {Q} ‐grading

the Heegaard Floer homology of (Y, \mathfrak{s}) uniquely.

Heegaard Floer homology relates to smooth 4‐manifold invariant. Ozsváth and Szabó
F_{W,\mathfrak{s}^{\mathrm{ }\mathrm{i}\mathrm{x} : HF^{-}(Y_{1}, \mathrm{t}_{1})\rightarrow HF^{+}(Y_{2}, \mathrm{t}_{2}) by using
3 + 1 dimensional TQFT. First, we cut a 4‐dimensional cobordism into 2 pieces V_{1}, V_{2} alonga suitable 3‐manifold N
Suppose that each of V_{1}, V_{2} has at least one positive eigenvalue
in the intersection form and $\delta$ H_{1}(N) is 0 in H^{2}(W, \partial W) The spinC 4‐manifolds (V_{1},\mathfrak{s}|_{V_{1} )and (V_{2}, \mathfrak{s}|_{V_{2} ) are spinC cobordisms from (Y_{1}, \mathrm{t}_{1}) to (N, \mathfrak{s}|_{N}) and from (N,\mathfrak{s}|_{N}) to (Y_{2}, \mathrm{t}_{2})respectively. Then F_{W,\mathfrak{s}^{\mathrm{ }\mathrm{i}\mathrm{x} is a composition as follows:in [37] defined the mixed invariant as a map.

.

F_{W,\mathrm{t}_1}^{\mathrm{ }\mathrm{i}\mathrm{x}:HF^{-}(Y_{1}, \mathrm{t}_{1})^{F_{V_{1},\mathfrak{s}|_{V_{1} }\rightar ow HF^{+}(N,\mathfrak{s}|_{N})\rightar ow HF_{\mathrm{r}\mathrm{e}\mathrm{d} (N,\mathfrak{s}|_{N})\subset HF^{+}(N,\mathfrak{s}|_{N})^{F_{V_{2},\mathfrak{s}|_{V_{2} ^{+} \rightar ow HF^{+}(Y, \mathrm{t}_{2}).

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The reduced part of HF^{+}(N,\mathfrak{s}|_{N}) is also embedded in HF^{-}(N,\mathfrak{s}|_{N}) naturally.
Let X be a closed smooth 4‐manifold with b_{2}^{+}(X)>1

X we built a cobordism W from S^{3} to S^{3}.

Then deleting two 4‐balls in
Applying U^{n}\cdot$\Theta$^{-} in HF^{-}(S^{3}) to the mixed.

invariant through this cobordism, we obtain an invariant

F_{W,\mathfrak{s} ^{\mathrm{m}\mathrm{i}\mathrm{x} (U^{n}\cdot$\Theta$^{-})=:$\Phi$_{X,\mathfrak{s} \cdot$\Theta$^{+}\in HF^{+}(S^{3})
Here $\Theta$^{-}($\Theta$^{+}) is the top (bottom) generator in HF^{-}(S^{3})(HF^{+}(S^{3})) and n=(c_{1}(\mathfrak{s})^{2}-

2 $\chi$(X)-3 $\sigma$(X))/4 (it agrees with the dimension of the moduli space of Seiberg‐Witten
solutions). The number $\Phi$_{X,\mathfrak{s} \in \mathbb{Z} is a smooth 4‐manifold invariant (smooth OS‐invariant)and conjecturely, the number coincides with the Seiberg‐Witten invariant SW_{X,\mathfrak{s} .
(correction term [33]) Let (Y, \mathfrak{s}) be a spinc 3‐manifold. Then d(Y,\mathfrak{s})defined to be the minimal grading in the image $\pi$ : HF^{\infty}(Y, \mathfrak{s})\rightarrow HF^{+}(Y,\mathfrak{s}) This

Definition 2.5
is.

invariant is called correction term(or d ‐invariant).

In the case of integral homology sphere the invariant is denoted by d(Y)\in 2\mathbb{Z}.

From theproperty of gr the correction term d is a spinC rational homology cobordism invariant.
Suppose that W is a rational homology cobordism between Y_{1}, Y_{2} Let \mathfrak{s} be a spinC.

structure on W with \mathfrak{s}|_{Y_{\mathrm{z} =\mathrm{t}_{ $\iota$}.

Then we haved(Y_{1}, \mathrm{t}_{1})=d(Y_{2}, \mathrm{t}_{2}).

Let (Y, \mathfrak{s}) be a spinC rational homology sphere. Then

HF^{+}(Y,\mathfrak{s})\cong T_{(d(Y,\mathfrak{s}) }^{+}\oplus HF_{\mathrm{r}\mathrm{e}\mathrm{d} (Y, \mathfrak{s})Here.

T_{(d)}^{+} is isomorphic to T^{+} and the minimal grading is d The d‐invariant and Euler.

number of HF_{\mathrm{r}\mathrm{e}\mathrm{d} compute the Casson invariant.

Theorem 2.6 Let Y be an integral homology sphere. Then the Casson invariant $\lambda$(Y) is

computed by$\lambda$(Y)= $\chi$(HF_{red}(Y) -\displaystyle \frac{1}{2}d(Y).

The Heegaard Floer homology of (branched)covering space $\Sigma$_{n}(K) is less known so far.
Casson invariant formula of double branched cover is known by Mullins [18] as follows:$\lambda$($\Sigma$_{2}(K) =\displaystyle \frac{1}{8}\mathrm{s}\mathrm{i}\mathrm{g}\mathrm{n}(K)-\frac{1}{12}V_{K}'(-1)/V_{K}(-1),

where $\Sigma$_{2}(K) is a homology sphere and V_{K} is the Jones polynomial. On the other handOzsváth and Szabó show the following in [35]:Theorem 2.7 Let Lalink in S^{3}.

There existsaspectral sequence with E_{2} ‐term the

Khovanov homology of L such that the sequence converges to the \mathb {F}_{2} ‐coefficient homology\hat{HF}($\Sigma$_{2}\{L), \mathbb{F}) Here \mathrm{F} is.

afield isomorphic to \mathbb{Z}/2\mathbb{Z}.

The Khovanov homology is a categorification of Jones polynomial, and \hat{HF} and Kh have
surgery exact sequence. Do this spectral sequence interpret Mullins formula?

Definitely,

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covering space is a weak point for Casson invariant and Heegaard Floer homology. Dehn

surgery formula for these invariants have been studied ever, however there are not so muchresearch for covering space.

For a rational homolog sphere Y the renormalized Euler characteristic is defined as,\displaystyle \hat{ $\chi$}(Y,\mathfrak{s})= $\chi$(HF_{\mathrm{r}\mathrm{e}\mathrm{d} (Y,\mathfrak{s}) -\frac{1}{2}d(Y,\mathfrak{s}).

Then due to [43] we have\displaystyle\sum_{\mathfrak{s}\in\mathrm{S}\mathrm{p}\mathrm{i}\mathrm{n}^{c}(Y)}\hat{$\chi$}(Y,\mathfrak{s})=|H_{1}(Y)|\cdot$\lambda$_{CW}(Y),where $\lambda$_{CW} is the Casson‐Walker invariant.
The correction term has the 4‐dimensional information as follows:
Theorem 2.8 Let (Y, \mathrm{t}) be a spinc rational homology sphere.

definite bounding (W,\mathfrak{s}) with \mathfrak{s}|_{Y}=\mathrm{t},If (Y, \mathrm{t}) admits a negative

then we have the following inequality:c_{1}(\mathfrak{s})^{2}+b_{2}(W)\leq 4d(Y, \mathrm{t}).

The extended inequality of this theorem to a 3‐manifold with b_{1}>0 can prove the Thom
conjecture again [33].
2.4\mathrm{L} ‐spaces

Let Y be a rational homology sphere. If

called an L ‐space.

\hat{HF}(Y,\mathfrak{s})=\mathbb{Z} for any spinC structures, then Y is

Examples of \mathrm{L} ‐space are lens spaces. The genus one Heegaard diagramof a lens space L(p, q) consists of single $\alpha$ ‐curve and $\beta$ ‐curve. Then the pointed Heegaard

digram is (T^{2}, $\alpha$, $\beta$, z).

Thus the symmetric product is T^{2} itself and we have \mathbb{T}_{ $\alpha$}= $\alpha$ and\mathbb{T}_{ $\beta$}= $\beta$ Hence, the generators $\alpha$\cap $\beta$ are p points. The differentials are all zero, because.

the p points belong to distinct spinC structures. Thus we have the following:

Proposition 2.9 Let Y be a lens space and \mathfrak{s} any spinc structure on Y.

\hat{HF}(Y, \mathfrak{s})\cong \mathbb{Z}, HF^{\infty}(Y,\mathfrak{s})=\mathbb{Z}[U, U^{-1}], HF^{+}(Y,\mathfrak{s})\cong \mathbb{Z}[U, U^{-1}]/U\mathbb{Z}[U], HF^{-}(Y,\mathfrak{s})\cong \mathbb{Z}[U]We put T^{\infty}=\mathbb{Z}[U, U^{-1}], T^{+}=\mathbb{Z}[U, U^{-1}]/U\cdot \mathbb{Z}[U], T^{-}=\mathbb{Z}[U].

Another component as a\mathbb{Z}[U] ‐module is \mathbb{Z}[U]/U^{n} , which is a finite rank \mathb {Z}‐module and we denote it by T^{+}(n).

Any elliptic manifolds (i.e., it is a finite fundamental group) are an \mathrm{L} ‐space. In otherwords \mathrm{L} ‐space is a generalization of such a manifold. In addition, \mathrm{L} ‐spaces contain notelliptic 3‐manifolds. Suppose that a hyperbolic knot K is an \mathrm{L} ‐space knot (i.e., an integral
Dehn surgery is an \mathrm{L} ‐space), e.g., the (-2 3, 7 ) ‐pretzel knot. Then a sufficiently large,

Dehn surgery of the knot is also an \mathrm{L} ‐space. This is an easy result due to the surgery exact

sequence of Heegaard Floer homology. In other examples, double cover of any alternatingknot is also \mathrm{L} ‐space [35]. [35] shows that the double cover of quasi‐alternating link is\mathrm{L} ‐space. The quasi‐alternating knots \mathcal{Q} is the smallest set defined inductively as follows:(1) \mathrm{u}\mathrm{n}\mathrm{k}\mathrm{n}\mathrm{o}\mathrm{t}\in \mathcal{Q};(2) if L_{0}, L_{1}\in \mathcal{Q}, \det(L_{0})\det(L_{1})\neq 0 and L_{0}, L_{1} are two types of theresolution of a crossing of L and \det(L)=\det(L_{0})+\det (L1), then L\in \mathcal{Q}.
\mathrm{L} ‐space includes interesting geometric properties besides:

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Theorem 2.10([29]) Any L ‐space does not admit co‐oriented taut foliation.

This result is useful, because the nonexistence of taut foliation is hard to prove with‐out \mathrm{L} ‐space property.

We do not find a method in Heegaard Floer theory to prove thenonexistence of tight contact structure, so far.

Recently developing noticeable property is left‐orderbility of the fundamental group.
(Left‐orderable (LO) means that the existence of a total order on a group that any left
action of the group keeps the order relation between any two elements. A group is LO
if and only if it is a subgroup in \mathrm{H}\mathrm{o}\mathrm{m}\mathrm{e}\mathrm{o}^{+}(\mathb {R}) We believe that the following mysterious
relationship:.

Conjecture 2.11 Let Y be a rational homology sphere.
if Y is an L ‐space.

Then Y is not LO ,if and only

It is well‐known that LO group is a non‐torsion group. Then any elliptic manifold is not

LO. Furthermore, Boyer, Gordon, and Watson [2] proved that for any Seifert manifold or

Sol manifold this conjecture is true. Any other hyperbolic manifold is not so known. For
example for not all double covers of (quasi‐)alternating link this relationship is verified.
2.5

Surgery exact triangle

In general Floer theory an exact triangle among three homologies is inherent. In Heegaard

Floer theory, the three homologies correspond to replacements of $\alpha$ or $\beta$ ‐curves. In thecontext of a low‐dimensional topology, the replacements are 0 ‐surgery, and p‐surgery. Let

Y_{n} be a n‐surgery of a 3‐manifold Y along a null‐homologous knot K Then the following.

exact triangle holds:.

.

.

\rightarrow HF^{+}(Y)\rightarrow HF^{+}(Y_{0}, [i])\rightarrow HF^{+}(Y_{p}, i)\rightarrow HF^{+}(Y)\rightarrow\cdots

Read [26] for surgery exact sequence. The decategorification of this exact triangle is the
Dehn surgery formula of Casson invariant as follows:$\lambda$_{CW}(Y_{p})= $\lambda$(Y)-s(1,p)+\displaystyle \frac{1}{2p}\triangle_{K}' (1),where s(1,p) is the Dedekind sum, which is equivalent to the Casson‐Walker invariant ofL(p, 1) Unlike Casson invariant, generally, for determining Floer homology of Y_{p} we needhomologies of Y and Y_{0} exact triangle and a more extra information. The other triads
of surgery exact sequence are \{Y, Y_{n}, Y_{n+1}\} or \{Y, Y_{0}, Y_{1/q}\} These correspond to surgery
formulas for Casson invariant. The surgery triangle is applied on the variable situation.
.

,

.

2.6

Let

Contact structure(Y, $\xi$) be a contact 3‐manifold, which the 2‐plane filed $\xi$ is nowhere integrable. Any

contact 3‐manifold is decomposed into two types of non‐isotopic contact structures:tightand overtwisted. The classification of tight contact structures is more diffcult than the oneof overtwisted structures. The latter structures agrees with the classification of homotopy

types of 2‐plane fields due to Eliashberg [4]. The situations which a tight contact structure

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naturally appear is the cases where it is the boundary of a symplectic 4‐manifold withboundary condition. This contact structure is called symplectic fillable. The
boundary of Stein manifold is also a tight contact structure, which is called Stein fillable.
Ozsváth and Szabó define in [36] an isotopy invariant of contact structure. The classsome suitablec( $\xi$) is an element in \hat{HF}(-Y,\mathfrak{s}( $\xi$))/\pm 1 and has the following property:

Theorem 2.12 If $\xi$ is an overtwisted contact structure,
fillable contact structure, then c( $\xi$)\neq 0.
The converse of this theorem does not holds.

thenc( $\xi$)= O. If $\xi$ is Stein

This theorem implies that the invariantc( $\xi$) detects the tight‐ness and Stein fillability. Since the detection of tight‐ness by using
contact topology is difficult in general, after this theorem, the computation of c( $\xi$) has
been a reasonable method. The most simple example in [15] is the contact +1 ‐surgery
of a Legendrian right‐handed trefoil which is topologically - $\Sigma$(2,3,4)
It admits no.

symplectic fillable but a tight contact structure. The former is due to an application of
the Seiberg‐Witten invariant and the latter part is due to the computation of c( $\xi$)In general, invariants of some tight contact structures are vanishing (structures withsome symplectic fillable or positive Giroux torsionetc.) However if one take a suitable
twisted system, one can sometimes give a non‐vanishing class [29]. For other examples
related to contact structures, symplectic topology, Lefschetz fibration, and Stein manifold.

readers should take a look at [22].
2.7Graph manifoldsIn[34], Ozsváth and Szabó gave a method to compute HF^{+} for plumbed 3‐manifolds.

Inshort, this method is a machinery put surgery exact sequence together.

Némethi[19] gives a systematical algorithm (the tau function and the graded root) to compute

Heegaard Floer homology for any plumbed 3‐manifold with at most one bad vertex. This
computation gives the module structure of HF^{+} completely, and it is used as a first usefultrial to explore Heegaard Floer behavior for some topological phenomenon.
The more general 3‐manifolds are hyperbolic 3‐manifolds. Does Heegaard Floer ho‐mology capture hyperbolic structure? This is a natural and challenging question.
3
3.1

Knot Floer homology
Definition and an example

The most attractable point of Heegaard Floer theory is what it is able to build
variations of Floer

One of variation is knot Floer

somehomology.
homology.
homology was defined by Ozsváth and Szabó [27] and independently by Rasmussen [42].
Let K be a knot in S^{3} We take \mathrm{a}(g, 1) ‐decompositionS^{3}=H_{0}\displaystyle \bigcup_{$\Sigma$_{g} H_{1} of (S^{3}, K)In other words K is transversal about $\Sigma$_{g} and the union of the two arcs A_{i}=K\cap H_{i}(i=0,1) Furthermore, we assume that A_{i} does not intersect compressing disks of $\alpha$curves and $\beta$ ‐curves. Such a Heegaard splitting
always exists for any knot K From theassumption K intersects two points \partial A_{0}=\partial A_{1} in $\Sigma$_{g} with $\alpha$ ‐curves and $\beta$ ‐curves disjoint.
We denote the points by z, w We call such a diagram double pointed Heegaard diagram

Knot Floer

.

.

.

.

.

66

complex of knot Floer homology CFK^{\infty}(S^{3}, K, i) is isomorphic toa\mathbb{Z}[U] ‐module. We denote by Mas(x) the absolute grading on
CF^{\infty}($\Sigma$_{g}, $\alpha$, $\beta$, w)
CF^{\infty}(Y) which is defined in the previous section. The grading satisfies:
Then the chainas

Mas(\mathrm{x})-Mas(\mathrm{y})= $\mu$( $\phi$)-2n_{w}( $\phi$)
where $\phi$\in$\pi$_{2}(\mathrm{x}, \mathrm{y}),.

We will introduce a filtration (Alexander filtration Alex (\mathrm{x}) ) on the module by using aknot in the S^{3}.

The filtration is defined as follows. For \mathrm{x}, \mathrm{y}\in \mathb {T}_{ $\alpha$}\cap \mathb {T}_{ $\beta$}

Alex (\mathrm{x})-Alex(\mathrm{y})=n_{z}( $\phi$)-n_{w}( $\phi$)
where$\phi$ is a disk with $\phi$\in$\pi$_{2}(\mathrm{x}, \mathrm{y})This definition is.

amake an absolute filtration we impose a symmetry.

,

relative \mathb {Z} ‐filtration only.

To\#\{\mathrm{x}|Alex(\mathrm{x})=i\}=\#\{\mathrm{x}|Alex(\mathrm{x})=-i\}\mathrm{m}\mathrm{o}\mathrm{d} 2.
We denote such a filtered chain complex of CF^{\infty}($\Sigma$_{g}, $\alpha$, $\beta$, z) by CFK^{\infty}( $\Sigma$, $\alpha$, $\beta$, z, w).

Here we give an example of double pointed Heegaard diagram and the filtered chain

complex. Consider the trefoil K=3_{1} In Figure 1 we describe two digging arcs and two.

\rightarrow

Figure 1: The trefoil knot and the Heegaard decomposition of S^{3}.

2‐handles on a 3‐ball.
meridian disk of K.

The circles $\beta$_{1}, $\beta$_{2}

arethe attaching circles.

The circle $\beta$_{2} is theMoving the circles and holes on the 3‐ball we obtain the first picturein Figure 2. The dashed arc presents a longitude of a knot K In this case the double
points z, w are two points close to the meridian circle $\beta$_{2} The position of curves $\alpha$_{2}, $\beta$_{2}
can be made cancel the two criticalpoints in terms of Morse theory. Hence, the diagram
moves to the next picture and it is isotopic to the last
diagram in Figure 2.
From the last diagram we compute the Alexander filtration of the trefoil. The inter‐
section points are 3 points: \mathrm{x}_{1}, \mathrm{x}_{2} X3. The non‐trivial disks on the diagram are D_{1}, D_{2}.
D_{1} is a disk connecting from \mathrm{x}_{2} to \mathrm{x}_{1} and D_{2} is a disk connecting from \mathrm{x}_{2} to X3. Thus,
the filtration definition implies.

.

,

Alex (\mathrm{x}_{2})-Alex(\mathrm{x}_{1})=-1,

Alex (\mathrm{x}_{2})-Alex(\mathrm{x}_{3})=1.

From the symmetric condition we haveAlex (\mathrm{x}_{1})=1,

Alex (\mathrm{x}_{2})=0,

Alex (\mathrm{x}_{3})=-1.

67

Figure 2: A double pointed Heegaard diagram of the trefoil knot and a move.

The Maslov grading is

Mas(\mathrm{x}_{2})-Mas(\mathrm{x}_{1})=-1, Mas(\mathrm{x}_{2})-Mas(\mathrm{x}_{3})=1
The differential \partial^{\infty} works as follows:\partial^{\infty}\mathrm{x}_{1}=0, \partial^{\infty}\mathrm{x}_{2}=U\cdot \mathrm{x}_{1}+\mathrm{x}_{3}, \partial^{\infty}\mathrm{x}_{3}=0Here we compute hat Heegaard Floer homology.

Since the version is embedded in the

0 ‐level in CF^{\infty} with U=0 , then we have\hat{\partial}\mathrm{x}_{1}=0, \hat{\partial}\mathrm{x}_{2}=\mathrm{x}_{3}, \hat{\partial}\mathrm{x}_{3}=0,\hat{HF}(S^{3})\cong \mathbb{Z}\cdot \mathrm{x}_{1}.
By the definition of the absolute grading, gr(xl) =0 Since U‐action decreases \mathrm{b}\mathrm{Y}2
Maslov gradings are computed as Mas(\mathrm{x}_{1})=0, Mas(\mathrm{x}_{2})=-1, Mas(\mathrm{x}_{3})=-2.
.

Here we define differential \hat{\partial}^{K} of knot Floer homology\overline{CFK}(S^{3}, K)

as

,

the

follows:\displayst le\hat{\partial}^{K}\mathrm{x}=\sum_{\mathrm{y}\in\mathrm{T}_{$\alpha$}\cap\mathrm{T}_{$\beta$}\sum_{$\phi$\n$\pi$_{2}(\mathrm{x},\mathrm{y}),n_{\mathrm{z}($\phi$)=n_{w}($\phi$)=0}\# hat{\mathcal{M}($\phi$)\mathrm{y}.
The chain complex

ferentials are \hat{\partial}^{K}.

\overline{CKF}($\Sigma$_{g}, a, $\beta$, z, w) is isomorphic to \hat{CF}($\Sigma$_{g}, $\alpha$, $\beta$, w) and the dif‐

68

Theorem 3.1(Knot Floer homology)

The homology of(\overline{CKF}($\Sigma$_{g}, $\alpha$, $\beta$, z),\hat{\partial}^{K}) isanisotopy invariant of a knot K. This homology decomposes as follows:\overline{HFK}(S^{3}, K)=\oplus_{i}\overline{HFK}(S^{3}, K, i),\overline{HFK}(S^{3}, K, i) is the homology with the Alexander grading i.
Furthermore, the Euler characteristic of \overline{HKF}(S^{3}, K) is the i‐th Alexander polynomialwhere$\chi$(\overline{HFK}(S^{3}, K, j))=a_{i}.
In the case of our trefoil, the differentials \hat{\partial}^{K} are computed as follows:\hat{\partial}^{K}\mathrm{x}_{1}=0, \hat{\partial}^{K}\mathrm{x}_{2}=0, \hat{\partial}^{K}\mathrm{x}_{3}=0.
Hence, we obtain\overline{HFK}(S^{3},Kj)=\left\{ begin{ar y}{l \mathb{Z}_(0)}&j=1\ \mathb{Z}_(-1)}&j=0\ \mathb{Z}_(-2)}&j=-2. \end{ar y}\right.

Taking the Euler characteristic of these complexes, we have\displaystyle \sum_{j=-g(K)}^{g(K)} $\chi$(\overline{HFK}(S^{3}, K, g) \cdot ^{j}=t-1+t^{-1}=\triangle_{K}(t).

We denote for any \mathrm{x}\in \mathbb{T}_{ $\alpha$}\cap \mathbb{T}_{ $\beta$} the map CFK^{\infty}(S^{3}, K)\ni U^{n}\cdot \mathrm{x}\mapsto-n\in \mathbb{Z} by Alg.

Then one can visualize the double complex Alex‐filtration and Alg‐filtration on the plane.
In the case of K=3_{1} , the double complex is described as Figure 3.

\overline{HFK}(K, j)\cong \mathb {Z}_{(s+\frac{ $\sigma$}{2}) ^{|a_{ $\gamma$}|In this case theGenerally, due to [40] if K is alternating, thenhomology is determined by the Alexander polynomial and the knot signature. However,for non‐alternating knot, it is different in general, see Theorem 3.7.
Other important variations of definitions of knot Floer homology are the ones by grid
diagram [17] and Kauffman state [40]. By the former, knot Floer homology is computedfrom a purely combinatorial information of a knot. By the latter, on the knot Floer
homology, a geometric meaning of a state sum is given.
3.2.

Genus and fiberness.

In this section we give the characterizations of Seifert genus and fiberness on knot Floer

homology. The knot Floer homology detects the Seifert genus g(K)Theorem 3.2([29]) Let K be a knot in S^{3}.

Then we have.

g(K)=\displaystyle \max\{s| HFK (S^{3}, K, s)\neq\{0\}\}.
Furthermore, Heegaard Floer homology detects the fibered‐ness of knot.
Theorem 3.3([20]) Let K be a knot in S^{3}\overline{HFK}(S^{3}, K, g(K))\cong \mathbb{Z}.

.

Then K isafibered knot if and only if

69

CFK^{\infty}(S^{3}, K)
Figure 3: Differentials of CFK^{\infty}(S^{3}, K) and \overline{CFK}(S^{3}, K) in it.

3.3

Legendrian and transverse knot.

Legendrian knot (or transverse knot) in a contact 3‐manifold is a curve tangent (or trans‐verse) on any plane of the standard contact S^{3} at any point on the knot. Isotopy invariants
for those knots are classically the Thurston‐Bennequin invariant (tb), and the rotation
number (rot) for Legendrian knots, and the self‐linking number (s1) for transverse knots.

These and smooth isotopy type are called classical invariants. Classical invariants dis‐tinguish these knots but not complete. For example Chekanov [3] and Eliashberg foundnon‐isotopic Legendrian knots (CE‐pairs) with the same classical invariants.
In the Heegaard Floer theory, (Legendrian or transverse) (unclassical) isotopy invari‐ants in (S^{3}, $\xi$_{std}) are included.
Let L be a Legendrian knot in the standard contactS^{3} Ozsváth, Szabó and Thurston defined in [41] defined Legendrian isotopy invariant$\lambda$(L)\in HFK^{-}(S^{3}, L) using the grid diagram. After that, in [14] Lisca, Ozsváth, Stipsicz,
and Szabó defined another Legendrian invariant \mathfrak{L}(L)\in HFK^{-}(S^{3}, L) using open bookdecomposition compatible with the contact structure, which it is proved these invariants
are equivalent to each other in[1]. For the transverse knot invariant \mathfrak{T} which is definedby Legendrian approximation, the same history has been traced. This equivalence is so
fruitful in terms of what some deeper information may be appeared. The image of thehat version homology \overline{HFK}(L) is referred to as \hat{\mathfrak{L} (L) The invariants reprove CE‐pair
is not isotopic and a link 6_{3}^{2} is not simple (i.e., the knot is determined uniquely fromclassical invariants), and transverse CE‐pair, which is trivial DGA invariant are also not(transverse) isotopic [23]. The DGA invariant is sometimes not useful for Legendrian
approximation of transverse knot because it vanishes for a stabilized knot. However theinvariant \mathfrak{L} is sensitive for stabilized knots (it is multiplied by U or identity depending
.

,

.

70

stabilization). By using \mathfrak{T} it is proven that the CEs twist knots are transverse noton,

simple. The Alexander and Maslov gradings of \mathfrak{L} and \mathfrak{T} compute as follows:Alex(\displaystyle \mathfrak{L}(L) =\frac{1}{2}(\mathrm{t}\mathrm{b}(L)-\mathrm{r}\mathrm{o}\mathrm{t}(L)+1)

2Alex(\mathfrak{L}(L))-Mas(\mathfrak{L}(L))=d_{3}( $\xi$).

M(\mathfrak{T}(T))=2A(\mathfrak{T}(T))=\mathrm{s}1(T)+1.
Here d_{3}( $\xi$) is the invariant of 2‐plane field on 3‐dimensional obstruction defined in [7].
3.4\mathrm{L} ‐space surgery

The lens space knot K is a knot yielding a lens space by an integral Dehn surgery. This
research field is the one which was remarkably developed by Heegaard Floer homology.

Ozsváth and Szabó obtained the Alexander polynomial restriction for lens space knot (orfor \mathrm{L} ‐space knot more strongly).

Theorem 3.4 Let K be an L ‐space knot.

Then the Alexander polynomial is of form:\displaystyle \triangle_{K}(t)=(-1)^{m}+\sum_{j=1}^{m}(-1)^{m-j}(t^{n_{j} +t^{-n_{j} )where the exponents n_{j},give an increasing sequence:0<n_{1}<n_{2}<\cdots<n_{m}=d.

As a corollary, this theorem and Theorem 3.3 imply that any \mathrm{L} ‐space knot must be afibered knot.

By this theorem many knots are ruled out from \mathrm{L}‐space knot, in particular
Besides, if K yields a lens space L(p, q) then due to [12], we havelens space knot.

,

2g(K)-1\leq p(1).

This inequality follows from HF^{+}(S_{0}^{3}(K), i)\cong T^{+}(t_{i})(i\neq 0) and g(K)=d (degree of\triangle_{K}(t))

follows:.

Greene in [8] improved the inequality (1) in the case of lens space knot in S^{3} as

2g(K)-1\leq p-2\sqrt{(4p+1)}/5
by using Theorem 3.4 essentially.
in[8] classifies all lens spaces obtained by Dehn surgeriesS^{3} by extracting some information of a lattice embedding coming from embeddingof the resolution of a lens space in a definite 4‐manifold.

This implies this property in

Theorem 3.4 is so strong among the other \mathrm{L} ‐space knot restrictions.
3.5

Dehn surgery formula

We introduce the Heegaard Floer homology of a Dehn surgery

S_{p/q}^{3}(K) in [27], [38], [39].

Briefly speaking, the computation of the homology is the mapping cone technique of thechain complex. Let K be a knot in S^{3} We denote Alex and Alg by j‐ and i ‐coordinate.
.

71

Let C be CFK^{\infty}(K).

Let A_{s}^{+}=C\displaystyle \{\min\{i, j-s\}\geq 0\} and B^{+}=C\{i\geq 0\}

We define.

\mathb {A}_{i}^{+}, \mathbb{B}^{+}, v^{+}, h^{+} as follows:\mathbb{A}_{i}^{+}=\oplus_{s\in \mathbb{Z} (s, A_{\lfloor(i+ps)/q\rfloor}^{+}) , \mathbb{B}_{i}^{+}=\oplus_{s\in \mathbb{Z} (s, B^{+})v^{+}:(s, A_{\lfloor(i+ps)/q\rfloor}^{+})\rightarrow(s, B^{+}) , h^{+}:(s, A_{\lfloor(i+ps)/q\rfloor}^{+})\rightarrow(s+1, B^{+})The second components v_{s}^{+} and h_{s}^{+} of v^{+} and h^{+}on.

the level s are defined as follows:

The map v_{s}^{+} is defined to be the quotient to i\geq O. The map h_{s}^{+} is defined to be theand the identification with i\geq O. Here wecomposition of the quotient todefine \mathbb{A}_{i}^{+}\rightar ow \mathbb{B}^{+} to be

j\displaystyle \geq \mathrm{L}\frac{i+ps}{q}\rflo r

D_{i,p/q}^{+}(s, a_{s})=\{(s, v_{\mathrm{L}\frac{i+ps}{q}\rfloor}^{+}(a_{s})+h_{\mathrm{L}_{q}^{i+p(s-1)}\rfloor}^{+}(a_{s-1})\}.
HF^{+}(S_{p/q}^{3}(K), i) is isomorphic to the homology of the mapping\mathrm{X}_{i,p/q}^{+}with
D_{i,p/q}^{+}.
The mappingwith respect toD_{i,p/q}^{+} is the chain complex which the generators
Theorem 3.5

conerespect to

cone

are

\mathb {A}_{i}^{+}\oplus \mathb {B}_{i}^{+} and the differential is(_{D_{i )}p/q}^{+ i} \partial_{\mathrm{A}^{+} \partial_{\mathrm{B}+_{i}0)

In the case where q=1 and p\geq 2g(K)-1or

S_{-p}^{3}(K)is isomorphic to

,

as

.

in

[27], the module of HF^{+} of S_{p}^{3}(K)HF_{*}^{+}(S_{p}^{3}(K), [s])\cong H_{*+\frac{(2s-p)^{2}-p}{4p}(A_{s}^{+},[S])}, HF_{*}^{+}(S_{-p}^{3}(K), [s])\cong H_{*+\frac{p-(2s+p)^{2} {4p}}(^{b}A_{s}^{+}, [s])wherebA^{+}=C\displaystyle \{\min\{i, j-s\}\geq 0\} Further the homology of \^{A}_{s} :=C\displaystyle \{\max\{i, j-s\}=0\}:=C\displaystyle \{\min\{i, j-s\}=0\} compute the hat version of S_{p}^{3}(K) and S_{-p}^{3}(K) in the.

and b\hat{A}_{s}
same

way.

Ni and Wu[21] gave the correction term formula of rational surgery of a knot.

The

spinC structures of Dehn surgery of a knot are identified with \mathbb{Z}/p\mathbb{Z} naturally.

Theorem 3.6 Let K be a knot in S^{3} and p, q positive integers. For 0\leq i<p we haved(S_{p/q}^{3}(K), i)=d(L(p, q), i)-2\displaystyle \max\{V_{\mathrm{L}\frac{i}{q}\rfloor}, V_{\mathrm{L}\frac{p+q-1-i}{q}\rfloor}\}.
Here V_{i} is a concordance invariant will be defined as below.
3.6

Satellite knots

Hedden computed HFK of the ( + )‐Whitehead double D_{+}(K, n) of a knot K

ficient is \mathbb{F}=\mathbb{Z}/2\mathbb{Z}Theorem 3.7.

Let g be the genus of K.

([9]) Let K be a knot in S^{3}.

The case of n\geq 2 $\tau$(K) :\overlin{HFK}(D_{+}(K,n)i=\left{\begin{ar y}{l \mathb{F}_(1)}^{n-2g }\oplus_{j=-g}^{ [H_{*-1}(\mathcl{F}(K,j)]^{2}&i=1\ mathb{F}^2n-4g3}(\oplus_{j}^g[H_{*}(\mathcl{F}(K,j)]^{4}&i=0\ mathb{F}_(-1)}^{n-2g }\oplus_{j=-g}^{ [H_{*+1}(\mathcl{F}(K,j)]^{2}&i=-1. \end{ar y}\right.

.

The coef‐

72

The case of n<2 $\tau$(K) :\overlin{HFK}(D_{+}(K,n)i=\left{\begin{ar y}{l \mathb{F}_(1)^{2$\tau(K)-2g }\oplus\mathb{F}_(0)^{2$\tau(K)-n}\oplus_{j=-g}^ [H_{*-1}(\mathcl{F}(K,j)]^{2}&i=1\ mathb{F}_(0)^{4$\tau(K)-4g }\oplus\mathb{F}_(-1)^{4$\tau(K)-2n1}\oplus_{j=-g}^ [H_{*}(\mathcl{F}(K,j)]^{4}&i=0\ mathb{F}_(-1)^{2$\tau(K)-2g }\oplus\mathb{F}_(-2)^{ $\tau(K)-n}\oplus_{j=-g}^ [H_{*+1}(\mathcl{F}(K,j)]^{2}&i=-1. \end{ar y}\ight.

Here $\tau$(K) (concordance invariant defined in the knot Floer theory) will be defined in thenext section. Since the Alexander polynomial of

D_{+}(K, n) is -nt+(2n+1)-nt^{-1} this,theorem says that knot Floer homology is strictly stronger invariant than Alexander poly‐
nomial. For example, it follows immediately that if D_{+}(K, n) is fibered non‐trivial knot,

then the double is figure‐8 knot only ( K= the unknot and n=1 ) by using Theorem 3.3
and the reduced knot filtration formula of $\tau$(K) in [44].

D_{+}(K, n) is following:ofThus, due to [9] the $\tau$ ‐invariant$\tau$(D_{+}(K,n) =\left\{ begin{ar ay}{l 0n\geq2$\tau$(K)\ 1n<2$\tau$(K). \end{ar ay}\right.

Hence, untwisted (n=0) Whitehead double with positive $\tau$(K) is not smoothly slice. On[5] says that any knot with trivial Alexander polyno‐
Then, for example for any positive torus knot K, D_{+}(K, 0) is
topologically slice but not smoothly slice knot. Such a knot gives an exotic \mathbb{R}^{4} by Freed‐mans result [5] and Gompfs result [6]. The result of 0 ‐framed attachment of the knot
on the 4‐ball is embeddable in a 4‐manifold
homeomorphic to \mathbb{R}^{4} but not in \mathbb{R}^{4}.
the other hand, Freedmans result

mial is topologically slice.

Concordance invariants

4

Two knots K_{0}, K_{1} are defined to be (knot) concordant if there exists a smoothly embedded
annulus f :S^{1}\times[0, 1]\mapsto S^{3}\times[0 1 ] such that f|_{S^{1}\times\{i\}}=K_{i},.

The set C of the equivalentclasses of knots by knot concordance is an abelian group so that the connected‐sum is the
addition. The zero element in C corresponds to the equivalent class of slice knot. We say aknot K to be slice, if there exists a proper smoothly embedded disk in 4‐ball such that theboundary is isotopic to K

In the Heegaard Floer package there exist some concordance.

invariants. We give the definitions of those. In this section we denote CFK^{\infty}(K) by C.

The $\tau$ ‐invariant and variations.

4. 1

Let \mathcal{F}(K, j) be C\{Alex\leq j, Alg=0\}.

Then we have\subset \mathcal{F}(K, i-1)\subset \mathcal{F}(K, i)\subset \mathcal{F}(K, i+1)\subset\cdots\displaystyle \bigcup_{i\in \mathbb{Z} \mathcal{F}(K, i)=\hat{CF}(S^{3}) Knot Floer homology \overline{HFK}(S^{3}, K, j)\underline{\mathrm{i}\mathrm{s}\mathrm{t} \mathrm{h}\mathrm{e} graded homology
H_{*}(\mathcal{F}(K_{j)/\mathcal{F}(K_{j}},-1\sim The inclusion map $\iota$^{m} \mathcal{F}(K, m)\subset CF(S^{3}) induces map

and

.

:

$\iota$_{*}^{m}:

H_{*}(\mathcal{F}(K, m))\rightarrow HF(S^{3})\cong \mathbb{Z}onthe homology.

a

If m issufficiently small, then\mathcal{F}(K, m) is the 0 ‐map. If m is sufficiently large, then \mathcal{F}(K, m) is isomorphic to
The $\tau$ ‐invariant is defined as the following minimal:$\tau$(K) :=\displaystyle \min { m|$\iota$_{*}^{m} is non‐trivial}.

\hat{CF}(S^{3}).

73

Theorem 4.1([28])

The $\tau$ ‐invariant gives a group homomorphism $\tau$ : C\rightarrow \mathbb{Z}.

This invariant $\tau$ gives a bound of the 4‐ball genus

Theorem 4.2([28]) Let K be a knot.

g_{4}(K).

Then the following inequality holds:| $\tau$(K)|\leq g_{4}(K).

The knot signature $\sigma$(K) is also similar inequality | $\sigma$(K)|\leq 2g_{4}(K) For any alternatingknot, $\sigma$(K)=-2 $\tau$(K) Generally, this inequality on $\tau$ is sharper than the one on $\sigma$.
For $\tau$ various sharpener invariants exist. To compute $\tau$ of a cable knot, \mathrm{H}\mathrm{o}\mathrm{m} in [10].

.

,introduced $\epsilon$ ‐invariant:$\epsilon$=

Theorem 4.3\{\hat{CF}(S^{3})\rightar ow b\hat{A}_{ $\tau$(K)} trivial\hat{A}_{ $\tau$(K)}\rightar ow\hat{CF}(S^{3}) trivial

1

on

:

-1

:

on

the homology,
the homology,

otherwise.

0([10]) Let K_{p,q} be the (p, q) ‐cable knot of K with p>0 Then, we have.

$\tau$(K_{p,q})=\left\{ begin{ar ay}{l} p$\tau$(K)+(p-1)(q-$\epsilon$(K) /2& $\epsilon$(K)\neq0\ (p-1)(q-sgn(q) /2=$\tau$(T_{p,q})& $\epsilon$(K)=0 \end{ar ay}\right.
Here T_{p,q} is the (p, q) ‐torus knot.
Ni and Wu defined knot concordance invariant V_{k} to compute the correction term of Dehnv_{k}^{+}

A_{k}^{+}\rightarrow B^{+}be a map
which is defined in the similar way
surgery of a knot in [21]. Let
to the previous section (note that the degree of the map is slightly different from theprevious section). The restriction to the T^{\infty} ‐part gives a U‐power map. We define theexponent to be V_{k} The same invariant H_{k} replacing v_{k}^{+} with h_{k}^{+} : A_{k}^{+}\rightarrow B^{+} is given. V_{k}is a decreasing \mathb {Z}_{\geq 0} ‐valued function on \mathb {Z} and V_{k+1}=V_{k} or V_{k}-1 The following minimal.

.

value $\nu$(K) :=\displaystyle \min\{k|V_{k}=0\} is a concordance invariant and $\tau$(K)\leq $\nu$(K)\leq g_{4}(K) holds
in [11]. This
4.2$\nu$(K) is a sharpener invariant than $\tau$(K).

$\Upsilon$ ‐invariant.

Recently, Ozsváth, Stipsicz and Szabó in [24] (also Livingston [16]) defined the concor‐

dance invariant$\Upsilon$_{K}(t),whose value is a continuous function on the interval$\Upsilon$ is a group homomorphism[0 2 ] And
C\rightarrow Cont([0,2 $\Upsilon$_{K}(t) is a piecewise linear function with,.

finite non‐smooth points, which is the number of the smooth points is also concordance
invariant. We have $\Upsilon$_{K}'(0)=- $\tau$(K).

Here we define $\Upsilon$ according to [16]. Let \mathcal{F}_{t} be an s‐filtered chain complex with \mathcal{F}_{t,s}=C\displaystyle \{\frac{t}{2}Alex+(1-\frac{t}{2})Alg\leq s\} Then $\Upsilon$ is defined to be $\Upsilon$_{K}(t)=-2 $\nu$(K, \mathcal{F}_{t}) where.

,$\nu$(K, \displaystyle \mathcal{F}_{t})=\min { s|\mathrm{I}\mathrm{m}\mathrm{a}\mathrm{g}\mathrm{e}(H_{*}(\mathcal{F}_{t,s})\rightar ow H_{*}(C) contains the non‐trivial element of grading 0 }.
$\Upsilon$_{K}(t) give a 4‐ball genus bound:
Theorem 4.4([24]) For any 0<t<1,|$\Upsilon$_{K}(t)|/t\leq g_{4}(K).

74This invariant $\Upsilon$ is weaker than \{V_{k}\} invariant but is useful for the linear independence
onconcordance group of a family of knots. For example, consider torus knots \{T_{p,p+1}\}.

Let G_{0} be generators in C with grading O.
convexarehull of G_{0} in \mathbb{R}^{2}.

Any element in G_{0} lies on the boundary of theSince $\Upsilon$_{T_{P,P+1}} has a convex function with p singular points, theselinearly independent in Cont ([0,2]) and the knots are also linearly independent.

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Institute of Mathematics

University of Tsukuba
Ibaraki 305‐8571

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