数理解析研究所講究録
第 720 巻 1990 年 129-162

129

ON THE STRUCTURE OF THE TORELLI GROUP AND
THE CASSON INVARIANT

東工大 森田 茂之

SHIGEYUKI MORITA

Dedicated to Professor A. Hattori on his sixtieth birthday

1. Introduction

In our previous paper [14], we have investigated the relationship between the

Casson invariant of oriented homology 3-spheres and the algebraic structure of certain

subgroups of the mapping class group of orientable surfaces. More precisely let
be an oriented closed surface of genus $g\geq 2$ and let

We denote$\mathcal{I}_{g}$for the Torelli group of$\Sigma_{g}$on$\Sigma_{g}$$\mathcal{I}_{g}$$\Sigma_{g}$$\mathcal{M}_{g}$consisting oftrivially. Also we denote$\mathcal{K}_{g}$for thegenerated by all the Dehn twists on separating simple closed curves. Now fix a Heegaard embedding

the manifold

be its mapping class group.

, which is the subgroup of

all the elements which act on the homology of

subgroup of$\mathcal{M}_{g}$$\Sigma_{g}$$S_{\varphi}^{3}$$f$

which is obtained from:$\Sigma_{g}arrow S^{3}$$S^{3}$

by first cutting along

reglueing the resulting two pieces by the map

Hence we can define a mapping$\lambda^{*}$:. Then for each element$\varphi$$\mathcal{K}_{g}arrow Z$$f(\Sigma_{g})$$\varphi\in \mathcal{K}_{g}$,and then, is an oriented homology 3-sphere.

by setting$\lambda^{*}(\varphi)=\lambda(S_{\varphi}^{3})$. On theother hand, by making use of the theory of characteristic classes of surface bundles

developed in [11,12,13], we defined a homomorphism
the analogue of the group$\mathcal{K}_{g}$$d$:relative to an embedded disc$\mathcal{K}_{g’ 1}arrow Z$$D^{2}\subset\Sigma_{g}$where$\mathcal{K}_{g1}$is(see \S 2 for the

precise definition). It is the secondary invariant associated with the fact that the firstcharacteristic class$e_{1}$

of surface bundles vanishes on$\mathcal{K}_{g’ 1}$. Now the main theoremof [14] (see also [15]) can be summarized that the two integer valued invariants
and

$d$

$\lambda^{*}$

are essentially equal each other. The precise formulation of this result, whichis rather complicated, as well as its proof was given in the framework of a certain

combination of Johnson’s theory on the structure of the Torelli group [7,8,9,10] withours [op. cit.]. Now the purpose of the present paper is to continue these lines of/130
investigations. As a result we find further close links between the Casson invariant
and the structure of (various subgroups of) the mapping class group.

Now we describe the contents of this paper briefly. In \S 2 we generalize the main

theorem of [14] mentioned above as follows. Namely we consider the general situation
where there is given an embedding $f$ :$\Sigma_{g}arrow M$

of$\Sigma_{g}$into an oriented homology

3-sphere $M$ which need not be Heegaard. We define a mapping$\lambda_{f}(\varphi)=\lambda(M_{\varphi})-\lambda(M)(\varphi\in \mathcal{I}_{g})$cutting $M$ along$f(\Sigma_{9})$where$M_{\varphi}$$\lambda_{f}$:$\mathcal{I}_{g}arrow Z$

byis the homology 3-sphere obtained byand then reglueing the resulting two pieces by the map . We$\varphi$

reformulate the main result of [14] in this more general situation. In \S 3 we consider acertain quotient group of the Torelli group which exactly measures how the elementsof the Torelli group act on the fourth nilpotent quotient of the fundamental groupof$\Sigma_{g}$. It turns out that this quotient group is a central extension of a certain freeabelian group by another one which arise naturally in the work of Johnson and its
extension [7,8,14]. We determne the Euler class of this central extension (Theorem3.1). Using this result, in \S 4 we describe how the mapping$\lambda_{f}$:$\mathcal{I}_{g}arrow Z$a homomorphism. We will give a complete answer in the case where$f$differs fromis a Heegaard

embedding (Theorem 4.3). As a byproduct of this description, we show that the mod2 reduction of$\lambda_{f}$is always a homomorphism, thus generalizing an earlier work by

Birman-Craggs [2] (see Corollary 4.4 and Remark 4.7). In the final section (\S 5), we

determine the set of all homomorphisms$\mathcal{K}_{g’ 1}arrow Z$

which are invariant under taking

conjugates in the whole mapping class group. It turns out that there are essentially

two such homomorphisms (one of which is the homomorphism$d$mentioned above,see Theorem 5.4). We also settle the same problem for the group$\mathcal{K}_{g}$(Theorem5.7). These results will play an important role in our future paper [16] where we willinterpret the invariant$d:\mathcal{K}_{g’ 1}arrow Z$, which is the core of the Casson invariant in thecontext of our approach, as Hirzebruch’s signature defect (see [4]) of certain framed

3-manifolds.

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2. Reformulation of the previous results
In this section we recall our formula proved in [14] which expresses the Casson

invariant of oriented homology 3-spheres in terms of the pasting maps of their Heegaard splittings. In doing so, instead of just summarizing the results of [14] we preferto exhibit the formula somewhat differently in a more general setting.

Suppose then that we are given an embedding$f$:$\Sigma_{g}arrow M$

where

is an

$\Sigma_{g}$oriented closed surface of genus $g\geq 2$ and $M$ is an oriented homology 3-sphere(although the most important example is the case where $M$ is the 3-sphere$S^{3}$

and $f$is a Heegaard embedding, we do not assume these conditions). Since $M$ is a homology
sphere, the embedded surface$f(\Sigma_{g})$splits $M$ into two parts $V+andV_{-}$ , where $V+$is the one such that the induced orientation on the boundarywith the given one on$\Sigma_{g}$

via

$f$

. We may call

negative) piece of $M$ with respect to the embeddingwhere$\iota$:$\partial V_{+}arrow\partial V_{-}$(resp.

$V+$

is the “identity”. Now let$f$elements which act on the homology of$\mathcal{I}_{g}$$\Sigma_{g}$$V_{-}$

coincides) the positive (resp.

. We can write$\mathcal{I}_{g}$Namely it is the subgroup of the mapping class group$\partial V+=f(\Sigma_{g})$$M=V+ \bigcup_{\iota}V_{-}$

be the Torelli group of$\mathcal{M}_{g}$oftrivially. Also let$\Sigma_{g}$$\mathcal{K}_{g}$$\varphi\in \mathcal{I}_{9}$, we consider the manifold

be the subgroup of$M_{\varphi}=V+ \bigcup_{\iota\varphi}V_{-}$

obtained by first cutting $M$ along the embedded surface$f(\Sigma_{g})$the two pieces $V+andV_{-}$ along their boundaries by the map$M_{\varphi}$$\iota\varphi$.

consisting of all thegenerated by all the Dehn twists on separating simple closed curves oneach element$\Sigma_{g}$$\Sigma_{g}$. Forwhich is the one

and then reglueing. It is easy to see thatis an oriented homology 3-sphere so that we have the Casson invariant$\lambda(M_{\varphi})$.

Now define a mapping$\lambda_{f}$by$\lambda_{f}(\varphi)=\lambda(M_{\varphi})-\lambda(M)(\varphi\in \mathcal{I}_{g})$:$\mathcal{I}_{g}arrow Z$. We also consider the restricted map

Z. (Recall here that any homology 3-sphere can be expressed aswith $f$ :$\Sigma_{9}arrow S^{3}$

closed curve$\omega$on$\Sigma_{g}$definition the group$S_{\varphi}^{3}$$\lambda_{f}$for some:$\mathcal{K}_{g}arrow$$\varphi\in \mathcal{K}_{g}$a Heegaard embedding, see [14]). For each separating simple, we denote$\mathcal{K}_{g}$$D_{tt}\in \mathcal{K}_{g}$for the right handed Dehn twist on . By$\omega$is generated by such elements. Now almost the same proof

as that of Proposition 3.5 of [14] yields
3

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Proposition 2.1. The mapping
each generator$D_{\omega}\in \mathcal{K}_{g}$invarian of the knot$t$

we have$f(\omega)$:$\lambda_{f}$is a homomorphism. Moreover for$\mathcal{K}_{g}arrow Z$$\lambda_{f}(D_{\omega})=-\lambda’(f(\omega))$, where$\lambda’(f(\omega))$is the Casson

in $M$ .

Motivated by the fact that Casson’s invariant of a knot in an oriented homology
3-sphere can be expressed as a polynomial of degree two on the linking numbers

among various homology classes of its Seifert surface (see Proposition 3.2 of [14]), we
define a commutative algebra $A$ overintegral homology group of$\Sigma_{g}$as follows. Let $H=H_{1}$ ($Z$. Then$A$$\Sigma_{g}$; Z) be the firstis defined to be the one generated by thesymbol $l(u, v)$ for any two elements $u,v\in H$ with the relations(i) $\ell(v, u)=P(u, v)+u\cdot v$(ii) $l(n_{1}u_{1}+n_{2}u_{2}, v)=n_{1}l(u_{1},v)+n_{2}l(u_{2}, v)(n_{1}, n_{2}\in Z)$

where

is the inersection number of$u\cdot v$$u$and . In some sense $A$ is the universal$v$model for the linking pairing on $H$ . More precisely, given an embedding$f$:$\Sigma_{g}arrow M$

as before, we have the “evaluation map”$\epsilon_{f}$:$\mathcal{A}arrow Z$

defined by $\epsilon_{f}(\ell(u, v))=\ell k(f_{*}(u),f_{*}(v)^{+})$ where$M\backslash f(\Sigma_{g})$(so thatis the homology class in

obtained by pushing the homology class $f_{*}(v)$ to the positive direction$f_{*}(v)^{+}$

bedded disc$D^{2}$$f_{*}(v)^{+}$

is supported in$D^{2}\subset\Sigma_{g}$and let$V_{-}$). Now for a technical reason we choose an em-$\mathcal{M}_{g’ 1}$

be the mapping class group of. We have a natural surjective homomorphism

canonically identified with$\pi_{1}(T_{1}\Sigma_{g})$We consider the Torelli group$\mathcal{I}_{g’ 1}$

where$T_{1}\Sigma_{g}$$\mathcal{M}_{g’ 1}arrow \mathcal{M}_{g}$$\Sigma_{g}$and its kernel can beis the unit tangent bundle ofwhich is the subgroup of

the elements which act on $H$ trivially and also let$\mathcal{K}_{g1}$$\mathcal{M}_{g’ 1}$$\omega$on$\Sigma_{g}\backslash D^{2}$

4

, let

$D_{\omega}$$\Sigma_{g}$.

consisting of all

be the subgroup ofgenerated by all the Dehn twists on bounding simple closed curves oneach bounding simple closed curve

relative to$\mathcal{M}_{g’ 1}$$\Sigma_{g}\backslash D^{2}$. For

denote the Dehn twist on$\omega$

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and letof$u_{1},$ $\cdots$$\Sigma_{g}\backslash D^{2}$$u_{h},$ $v_{1},$ $\cdots$which$\omega$,

be a symplectic basis of the homology of the subsurface$v_{h}$

bounds (hereafter, following Johnson, we call such element a

BSCC map of genus ). Then we can reformulate one of the main results of [14] as$h$Theorem 2.2. The assignment$\mathcal{K}_{g1}\ni D_{\omega}-\sum^{h}\{l(u_{i}, u_{i})l(v_{i},v;)-p(u_{i}, v_{i})\ell(v_{i}, u;)\}$$i=1$$-2 \sum_{i<j\leq h}\{l(u_{i}, u_{j})l(v_{i},v_{j})-f(u_{i}, v_{j})l(u_{j}, v_{i})\}$

for each generator$D_{\omega}\in \mathcal{K}_{g1}$extends to a well deffied homomorphism$p:\mathcal{K}_{g1}arrow \mathcal{A}$and the following diagram is commutative$\mathcal{K}_{g’ 1}arrow^{\rho}\mathcal{A}$$\downarrow\epsilon_{f}$$\downarrow$$arrow^{\lambda_{f}}Z$$\mathcal{K}_{g}$where$\mathcal{K}_{g’ 1}arrow \mathcal{K}_{g}$is th $enat$ ural urjection.
$s$

Roughly speaking, for each generator$D_{\omega}\in \mathcal{K}_{g’ 1}$

the element$\rho(D_{\omega})\in A$

can beconsidered as the “universal model” for the Casson invariant of the bounding simple
closed curve$f$:$\Sigma_{g}arrow M$$-\epsilon_{f}(\rho(D_{\omega}))$$\omega$on$\Sigma_{g}\backslash D^{2}$(up to signs). Namely if there is given an embedding, then we can consider

as a knot in $M$ and we have$f(\omega)$$\lambda’(f(\omega))=$. The following is an immediate corollary to the above theorem which

we present here for later use.
Corollary 2.3. (i)(ii)$\lambda_{f}([\varphi, \psi])=0$$\lambda_{f}(\varphi\psi\varphi^{-1})=\lambda_{f}(\psi)$

for any$\varphi\in \mathcal{I}_{g}$andfor any$\psi\in \mathcal{K}_{g}$$\varphi\in \mathcal{I}_{g}$and$\psi\in \mathcal{K}_{9}$.

Now if there were an algorithm to express any given elementof Dehn twists on separating simple closed curves on
5.

$\Sigma_{g}$$\varphi\in \mathcal{K}_{g}$as a product, then we can explicitly

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calculate the element$\rho(\varphi)\in \mathcal{A}$and hence the desired value$\lambda_{f}(\varphi)$by using Theorem

2.2. However unfortunately there is no such one and our next main result of [14]is that the homomorphism
computable ones:$\overline{\rho}$:$\mathcal{K}_{g’ 1}arrow A\otimes Q$:$\rho$$\rho=\frac{1}{24}d+\overline{\rho}$$\mathcal{K}_{g’ 1}arrow \mathcal{A}$, wherecan be decomposed as a sum of two more

$d:\mathcal{K}_{g’ 1}arrow Z$

is a modification of . The point here is that the homomorphism$\rho$

can be read off from Johmson’s homomorphism
below) and the homomorphism

characteristic class $e_{1}\in H^{2}$ (homomorphism$d$

is a certain homomorphism and$d$$\tau_{3}$:$\overline{\rho}$(which will be recalled$\mathcal{K}_{g’ 1}arrow\overline{T}$is the secondary invariant associated with the first$\mathcal{M}_{g’ 1}$; Z) of surface bundles introduced in [14]. Thisshould be considered as the core of the Casson invariant from ourpoint of view.
Now, in order to define the homomorphim

the definition of Johnson’s homomorpisms$\tau_{k}$$\overline{\rho}$:$\mathcal{K}_{g’ 1}arrow \mathcal{A}\otimes Q$, we briefly recall, which will be also needed in later

sections (see [7,8,14] for details).
We denote$\Gamma_{1}$forand let$\pi_{1}(\Sigma_{g}\backslash IntD^{2})$be the lower central series of$\{\Gamma_{k}\}_{k\geq 1}$; $\Gamma_{k}=[\Gamma_{k-1}, \Gamma_{1}](k\geq 2)$ . The mapping class groupquotient groupon$\Gamma_{1}$$N_{k}$$N_{k}=\Gamma_{1}/\Gamma_{k}$trivially}. Let

naturally and we define$\mathcal{L}=\oplus_{k\geq 1}\mathcal{L}_{k}$$\mathcal{M}_{g’ 1}$

acts on the nilpotent$\mathcal{M}(k)=\{\varphi\in \mathcal{M}_{g’ 1}$;$\varphi$acts

be the free graded Lie algebra on $H$ (over Z).

Then as is well known there is a natural isomorphism$\Gamma_{k}/\Gamma_{k+1}\cong \mathcal{L}_{k}$. Now Johnson’s

homomorphism$\tau_{k}$is defined as$\mathcal{M}_{g’ 1}arrow Hom(H, \mathcal{L}_{k})$$\tau_{k}(\varphi)([\gamma])=[\varphi_{*}(\gamma)\gamma^{-1}](\varphi\in \mathcal{M}(k),\gamma\in\Gamma_{1})$homology class of$\gamma$andwhich is contained in$H$:$\Gamma_{k}$$[\varphi_{*}(\gamma)\gamma^{-1}]$

denotes the image inby the assumption$\varphi\in \mathcal{M}(k)$$\mathcal{L}_{k}$, where $[\gamma]\in H$ is theof the element. The intersection pairing on

defines a natural isomorphism $H\cong H^{*}$ so that we can write

If we choose a symplectic basis$x_{1},$$\cdots$,$x_{g},$ $y_{1},$$\cdots y_{g}$$\varphi_{*}(\gamma)\gamma^{-1}$$\tau_{k}$:$\mathcal{M}(k)arrow \mathcal{L}_{k}\otimes H$.

of $H$ , then explicitly we have$\tau_{k}(\varphi)=\sum^{g}\{\tau_{k}(\varphi)(x_{i})\otimes y_{i}-\tau_{k}(\varphi)(y;)\otimes x_{i}\}\in \mathcal{L}_{k}\otimes H$

$i=1$

where

$\tau_{k}’ s$in the right hand side are the old ones. Now for $k=2,$6$\mathcal{M}(2)$is nothing

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but the Torelli group$\tau_{2}$:$\mathcal{I}_{g1}$and$\mathcal{L}_{2}$is naturally isomorphic to. Johnson proved that$\mathcal{I}_{g’ 1}arrow\Lambda^{2}H\otimes H$$a\wedge b\wedge c\in\Lambda^{3}H(a, b, c\in H)$

is identified with

For $k=3$ , according to Johnson [9],$\mathcal{M}(3)$define a module$T$

to be the submodule ofthe elements of the forms$H)$$(a\wedge b)\otimes(a\wedge b)$$\tau_{3}$:is equal to$T$

and it can be shown that the kernel
Hence if we write$\overline{T}$for $T/T_{0}$ , thenas follows. There is a$\Lambda^{2}H\otimes\Lambda^{2}H\subset\Lambda^{2}H\otimes H^{2}$

and$(a\wedge b)^{\otimes 2}$$\overline{T}$. We

generated by all

and $(a\wedge b)\otimes(c\wedge d)+(c\wedge d)\otimes(a\wedge b)(a,$$b,$ $c,$$d\in$$a\wedge brightarrow c\wedge d$$Tarrow \mathcal{L}_{3}\otimes H$of this homomorphism is the submodule ofgenerated by all the elements of the form.

and in [14] we have

, we have a natural homomorphism$T_{0}$, where$\mathcal{L}_{3}\otimes H\cong\Lambda^{2}H\otimes H^{2}/\Lambda^{3}H\otimes H$(henceforth these elements will be denoted byrespectively). By the definition of$\mathcal{K}_{g’ 1}$$\mathcal{K}_{g’ 1}arrow \mathcal{L}_{3}\otimes H$

so that$\mathcal{L}_{3}\cong\Lambda^{2}H\otimes H/\Lambda^{3}H$$\Lambda^{3}H\subset\Lambda^{2}H\otimes H$$(a\wedge b)\otimes c+(b\wedge c)\otimes a+(c\wedge a)\otimes b\in\Lambda^{2}H\otimes H$

determined the image of the homomorphism
natural isomorphism

is equal to${\rm Im}\tau_{2}$so that we can write$\Lambda^{2}H$$T$.

$a\wedge brightarrow c\wedge d-a\wedge crightarrow b\wedge d+a\wedge drightarrow b\wedge c$can be considered as a submodule ofIn these terminologies we have proved in [14] that${\rm Im}\tau_{3}$is a subgroup of$\overline{T}$$\mathcal{L}_{3}\otimes H$.

of index

a power of two.
Now there is a uniquely defined homomorphism

symplectic subbasis$u_{1},$$\theta$: $Tarrow A$ such that for anyof $H$ , the equality$\cdots u_{h},$ $v_{1},$ $\cdots v_{h}$$\theta((u_{1}\wedge v_{1}+\cdots u_{h}\wedge v_{h})^{\otimes 2})=\sum^{h}\{p(u_{i}, u_{i})p(v_{i}, v_{i})-p(u_{i}, v_{i})p(v;, u_{i})\}$$i=1$$+2 \sum_{i<j\leq h}\{l(u_{i}, u_{j})p(v_{i}, v_{j})-\ell(u_{i}, v_{j})P(u_{j}, v_{i})\}$

holds. Let$D_{\omega}\in \mathcal{K}_{g’ 1}$

closed curve on

be a BSCC map of genus$\Sigma_{g}\backslash D^{2}$

$h$

where

. Choose a symplectic basis

homology of the subsurface which$-(u_{1} Av_{1}+\cdots+u_{h}\wedge v_{h})^{\otimes 2}$$\omega$$\tau_{3}$:, then we have$\mathcal{K}_{g’ 1}arrow\overline{T}$is a bounding simple$\cdots$$u_{h},$ $v_{1},$ $\cdots$,$v_{h}$of thebounds. If we define an element $t\in T$ by $t=$$\rho(D_{\omega})=\theta(t)$. Hence the element contains

all the information of the “universal Casson invariant”
homomorphism$u_{1},$$\omega$$t$$\rho(D_{\omega})$computes only the image ofNow the definition of the homomorphism

7$\overline{\rho}$:$t$. However Johnson’sin$\mathcal{K}_{g1}arrow \mathcal{A}\otimes Q$.

$\overline{T}:\tau_{3}(D_{\omega})=\overline{t}\in\overline{T}$is defined as follows.

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There exists a certain homomorphismof the homomorphism$d$$\overline{d}$: $Tarrow Z$ , which is a formal counterpart, and for each generator

element $t’\in T\otimes Q$ such that the image of$\overline{\rho}(D_{\omega})=\theta(t’)+\frac{1}{3}\overline{d}(t’)\in \mathcal{A}\otimes Q$

$t’$

$t’$

$D_{\omega}\in \mathcal{K}_{g’ 1}$

as above, choose any

in Z7 is equal to$\tau_{3}(D_{\omega})$. Now set

, which turns out to be independent of the choice of. This is the definition of the homomorphism :$\overline{\rho}$$\mathcal{K}_{g’ 1}arrow \mathcal{A}\otimes Q$. Now we summarize

the above as
Theorem 2.4. We have the equality$\rho(\varphi)=\frac{1}{24}d(\varphi)+\overline{\rho}(\varphi)$

for any$\varphi\in \mathcal{K}_{g’ 1}$.

3. A quotient group of the Torelli group$|In$

this section we determine the structure of a certain quotient group of theTorelli group$\mathcal{I}_{g1}$. Roughly speaking it is the group which contains exactly the

informations carried by the homomorphisms$\tau_{2}$and$\tau_{3}$. More precisely recall that wehave a short exact sequence

1 $arrow \mathcal{K}_{g’ 1}arrow \mathcal{I}_{g’ 1}arrow^{\tau_{2}}\Lambda^{3}Harrow 1$

and the homomorphism

group$\mathcal{I}_{g’ 1}/Ker\tau_{3}$$\tau_{3}$:$\mathcal{K}_{g’ 1}arrow\overline{T}$. Hence if we define, then we have an extension

to be the quotient$1arrow{\rm Im}\tau_{3}arrow\overline{\mathcal{I}}_{g,1}arrow\Lambda^{3}Harrow 1$

Theorem 3.1. The extension$1arrow{\rm Im}\tau_{3}arrow\overline{\mathcal{I}}_{g,1}arrow\Lambda^{3}Harrow 1$

8$\overline{\mathcal{I}}_{g,1}$.

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is a central extension and its Euler class$\chi\in H^{2}(\Lambda^{3}H;{\rm Im}\tau_{3})$is given by$\chi(\xi, \eta)=$$-(a\cdot d)b\wedge crightarrow e\wedge f-(a\cdot e)b\wedge crightarrow f\wedge d-(a\cdot f)b\wedge crightarrow d\wedge e$$-(b\cdot d)c\wedge arightarrow e\wedge f-(b\cdot e)c\wedge arightarrow f\wedge d-(b\cdot f)c\wedge arightarrow d\wedge e\in{\rm Im}\tau_{3}\subset\overline{T}$.

$-(c\cdot d)a\wedge brightarrow e\wedge f-(c\cdot e)a\wedge brightarrow f\wedge d-(c\cdot f)a\wedge brightarrow d\wedge e$

where$\xi=a\wedge b\wedge c,$ $\eta=d\wedge e\wedge f\in\Lambda^{3}H(a, b, c, d, e,f\in H)$

and$\xi\wedge\eta\in H_{2}(\Lambda^{3}H;Z)$.

Before proving the above theorem, we prepare several technical results. Recall
that we write$\Gamma_{1}$for$\pi_{1}(\Sigma_{g}\backslash IntD^{2}),$$\{\Gamma_{k}\}_{k\geq 1}$

the subgroup of the mapping class groupact on the nilpotent quotient(ii) For any elements$\mathcal{M}_{g1}$$\varphi\in \mathcal{I}_{g’ 1},$$\varphi\in \mathcal{M}(k)$$\psi\in \mathcal{M}(k)$$\mathcal{M}(k)$forconsisting of all the elements which

trivially.

$N_{k}=\Gamma_{1}/\Gamma_{k}$Lemma 3.2. (i) For any element

for its lower central series and

and

and

$\gamma\in\Gamma_{f}$$\gamma\in\Gamma_{f}$, we have$\varphi(\gamma)\gamma^{-1}\in\Gamma_{k+\ell-1}$.

, we have the $equ$ ali $ty$$[\varphi\psi\varphi^{-1}(\gamma)\gamma^{-1}]=[\psi(\gamma)\gamma^{-1}]$in$\mathcal{L}_{k+t-1}$.

Proof. To prove (i), we use the induction on . If $p=1$ , then the claim follows$p$

immediately from the definition of the group $M(k)$ . Now we assume that the claim

holds up to e-l and prove it for . Suppose that an element$p$$\gamma=[\gamma_{1},\gamma_{2}]$with$\varphi(\gamma_{1})=\gamma_{1}\alpha$and$\gamma_{1}\in\Gamma_{l-1}$

and$\varphi(\gamma_{2})=\gamma_{2}\beta$$\gamma_{2}\in\Gamma_{1}$for some$\gamma\in\Gamma_{f}$is expressed as. Then by the induction assumption, we have$\alpha\in\Gamma_{k+\ell-2}$

and$\beta\in\Gamma_{k}$$\varphi(\gamma)\gamma^{-1}=[\varphi(\gamma_{1}), \varphi(\gamma_{2})][\gamma_{2}, \gamma_{1}]$$=\gamma_{1}\alpha\gamma_{2}\beta\alpha^{-1}\gamma_{1}^{-1}\beta^{-1}\gamma_{1}\gamma_{2}^{-1}\gamma_{1}^{-1}$$=\gamma_{1}[\alpha, \gamma_{2}\beta]\gamma_{2}[\beta, \gamma_{1}^{-1}]\gamma_{2}^{-1}\gamma_{1}^{-1}$9.

. Then we compute

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Now both of the elements$\varphi(\gamma)\gamma^{-1}\in\Gamma_{k+l-1}$$[\alpha,\gamma_{2}\beta]$

and$[\beta)\gamma_{1}^{-1}]$

are contained inas required. In general any element

product of elements of the form

andelements$\varphi\in \mathcal{I}_{g’ 1},$for some$\gamma_{1}\in\Gamma_{t+1},\gamma_{2}\in\Gamma_{k+t-1}$$\psi\in \mathcal{M}(k)$$\gamma$. Hencecan be expressed as agiven above and an easy inductive argument$[\gamma_{1},\gamma_{2}]$shows that the claim also holds for$\gamma\in\Gamma_{f}$$\Gamma_{k+1-1}$. Next we prove (ii). If we apply (i) to the$\gamma\in\Gamma_{\ell}$, we can write. It follows that$\varphi(\gamma)=\gamma_{1}\gamma,$ $\psi(\gamma)=\gamma_{2}\gamma$$\varphi^{-1}(\gamma)=\varphi^{-1}(\gamma_{1}^{-1})\gamma$. Then wecompute$\varphi\psi\varphi^{-1}(\gamma)\gamma^{-1}=\varphi\psi(\varphi^{-1}(\gamma_{1}^{-1})\gamma)\gamma^{-1}$$=\varphi(\psi\varphi^{-1}(\gamma_{1}^{-1})\gamma_{2}\gamma)\gamma^{-1}$$=\varphi\psi\varphi^{-1}(\gamma_{1^{-1}})\varphi(\gamma_{2})\gamma_{1}$Now again by $(!)$ , we can conclude that$\gamma_{2}(mod \Gamma_{k+t})$$\psi(\gamma)\gamma^{-1}$

.

Hence

.

$\varphi\psi\varphi^{-1}(\gamma_{1}^{-1})\equiv\gamma_{1}^{-1}(mod \Gamma_{k+l})$.

$\varphi\psi\varphi^{-1}(\gamma)\gamma^{-1}\equiv\gamma_{1}^{-1}\gamma_{2}\gamma_{1}\equiv\gamma_{2}(mod \Gamma_{k+\ell})$and$\varphi(\gamma_{2})\equiv$

Since$\gamma_{2}=$, This finishes the proof.

As a corollary to Lemma 3.2,(i), we have

Corollary 3.3. For any two elemen ts$[\varphi, \psi]$$\varphi\in \mathcal{M}(k)$and$\psi\in \mathcal{M}(\ell)$, the commutator

is contained in $\mathcal{M}(k+P-1)$ .

Proof. Clearly it is enough to prove the as sertion for the caseassume this condition. Now let

can write$\gamma\in\Gamma_{1}$$k\geq P$. Henceforth webe any element.Then by the assumption wefor some$\psi(\gamma)=\gamma_{1}\gamma$

and$\varphi(\gamma)=\gamma_{2}\gamma$$\psi^{-1}(\gamma)=\psi^{-1}(\gamma_{1^{-1}})\gamma$

and$\varphi^{-1}(\gamma)=\varphi^{-1}(\gamma_{2}^{-1})\gamma$$\gamma_{1}\in\Gamma_{f}$and$\gamma_{2}\in\Gamma_{k}$. Now we compute$[\varphi, \psi](\gamma)=\varphi\psi\varphi^{-1}(\psi^{-1}(\gamma_{1}^{-1})\gamma)$$=\varphi\psi(\varphi^{-1}\psi^{-1}(\gamma_{1}^{-1})\varphi^{-1}(\gamma_{2}^{-1})\gamma)$$=[\varphi, \psi](\gamma_{1}^{-1})\varphi\psi\varphi^{-1}(\gamma_{2}^{-1})\varphi(\gamma_{1})\gamma_{2}\gamma$

10.

. Then we have

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Hence
.

$[\varphi, \psi](\gamma)\gamma^{-1}=[\varphi,\psi](\gamma_{1}^{-1})\gamma_{1}\cdot\gamma_{1}^{-1}\varphi\psi\varphi^{-1}(\gamma_{2}^{-1})\gamma_{2}\gamma_{1}\cdot\gamma_{1}^{-1}\gamma_{2}^{-1}\varphi(\gamma_{1})\gamma_{1}^{-1}\gamma_{2}\gamma_{1}\cdot[\gamma_{1}^{-1}, \gamma_{2}^{-1}]$

Now by Lemma 3.2,(i), we havebecause certainly we have$[\varphi, \psi](\gamma_{1}^{-1})\gamma_{1}\in\Gamma_{k+l-1}$$[\varphi, \psi]\in \mathcal{M}(k)$. Similarly we haveand of course$[\gamma_{1}^{-1}, \gamma_{2}^{-1}]\in\Gamma_{k+\ell}$$\varphi\psi\varphi^{-1}(\gamma_{2}^{-1})\gamma_{2}\in\Gamma_{k+\ell-1},$. Hence$\varphi(\gamma_{1})\gamma_{1}^{-1}\in\Gamma_{k+1-1}$

as required. This$[\varphi, \psi](\gamma)\gamma^{-1}\in\Gamma_{k+\ell-1}$

completes the proof.

Recall that for any element$\mathcal{L}_{k}$$\varphi\in \mathcal{M}(k)$we have a homomorphism. Now by virtue of Lemma 3.2,(i), for each element

integer$p$

we can define a similar homomorphism$\varphi\{[\gamma]\}=[\varphi(\gamma)\gamma^{-1}]\in \mathcal{L}_{k+t-1}$$\gamma\in\Gamma_{k}$, where$[\gamma]\in \mathcal{L}_{k}$$\varphi\{\}$$\mathcal{L}_{f}arrow \mathcal{L}_{k+l-1}$

is the image in: $Harrow$and a positive$\varphi\in \mathcal{M}(k)$:$\tau_{k}(\varphi)$by settingof any element$\mathcal{L}_{k}$. It is easy to check that this correspondence in fact defines a well definedhomomorphism.

Proposition 3.4. Let(see Corollary 3.3). If$\varphi\in \mathcal{M}(k)$

$k$

and

$p$

and

$\psi\in \mathcal{M}(l)$so that we have $[\varphi, \psi]\in \mathcal{M}(k+P-1)$

are greater than one, then we have$\tau_{k+t-1}([\varphi)\psi])(u)=\varphi\{\tau_{\ell}(\psi)(u)\}-\psi\{\tau_{k}(\varphi)(u)\}(u\in H)$

Proof. Letsome$\gamma\in\Gamma_{1}$$\gamma_{1}\in\Gamma_{f}$andbe any element. Then we can write$\gamma_{2}\in\Gamma_{k}$Corollary 3.3, the term

. Now in the computation of$[\varphi, \psi](\gamma_{1}^{-1})\gamma_{1}$is contained in$\psi(\gamma)=\gamma_{1}\gamma,\varphi(\gamma)=\gamma_{2}\gamma$$\Gamma_{k+2l-2}\subset\Gamma_{k+t}$$[\varphi, \psi](\gamma)\gamma^{-1}\equiv\varphi\psi\varphi^{-1}(\gamma_{2}^{-1})\gamma_{2}\cdot\varphi(\gamma_{1})\gamma_{1}^{-1}(mod \Gamma_{k+\ell})$forin the proof of$[\varphi, \psi](\gamma)\gamma^{-1}$

3.3 and the assumption $P\geq 2$ ). Hence we have

11

.

.

(by Corollary

140
On the other hand by Lemma 3.2,(ii), we have$[\varphi\psi\varphi^{-1}(\gamma_{2^{-1}})\gamma_{2}]=[\psi(\gamma_{2}^{-1})\gamma_{2}]$$=-\psi\{\tau_{k}(\varphi)([\gamma])\}$

as elements of$\mathcal{L}_{k+l-1}$

can conclude that

. Also we have

. Hence we$[\varphi(\gamma_{1})\gamma_{1}^{-1}]=\varphi\{\tau_{\ell}(\psi)[\gamma]\}\in \mathcal{L}_{k+t-1}$$\tau_{k+t-1}([\varphi, \psi])([\gamma)])=\varphi\{\tau_{f}(\psi)([\gamma])\}-\psi\{\tau_{k}(\varphi)([\gamma])\}$

as required.

This completes the proof.

Proposition 3.5. For each element$\mathcal{L}_{k+\ell-1}$$\varphi\in \mathcal{M}(k),$the homomorphism$\varphi\{\}$:$\mathcal{L}_{1}arrow$is given explicitly as$\varphi\{\xi\}=\sum_{i=1}^{f}[\cdots[u_{1}, u_{2}],$ $\cdots$

where $\xi=[\cdots[u_{1}, u_{2}],$$\cdots$]$u_{f}$$,$]$\in \mathcal{L}_{f}(u;\in H)$]$\tau_{k}(\varphi)(u_{i})$$,$]$\cdots$
$,$

]

$,$

$u_{f}$].

Proof. We assume that $k\geq 2$ because for $k=1$ the as sertion is empty. By thedefinition of the homomorphism$(^{*})$For any element, it suffices to prove the following statement$\varphi\{\}$$\gamma=[\cdots[\gamma_{1}, \gamma_{2}],$ $\cdots$$\varphi(\gamma)\gamma^{-1}\equiv\prod_{i1}^{\ell_{=}}[\cdots[\gamma_{1},\gamma_{2}],$ $\cdots$]]$\gamma_{l}$$,$$\varphi(\gamma_{i})\gamma_{i}^{-1}$
$,$

]

]

$\in\Gamma_{f}(\gamma_{i}\in\Gamma_{1})$$\cdots$$,$]$,\gamma_{\ell}$, we have] $(mod \Gamma_{k+l})$ .

We prove the above assertion by the induction on . If $p=1$ , then the assertion is$p$clear. We assume that
where$\alpha=[\cdots[\gamma_{1}, \gamma_{2}],$$(^{*})$$\cdots$]

holds up to $P-1$ and prove it for . Write$\ell$$\gamma_{\ell-1}$$,$]$\in\Gamma_{l-1}$$\gamma=[\alpha, \gamma_{f}]$. By the induction assumption we can write$\varphi(\alpha)\equiv\overline{\alpha}\alpha$$\equiv\alpha\overline{\alpha}(mod \Gamma_{k+l-1})$

where$\overline{\alpha}=\prod_{i=1}^{f-1}[\cdot\cdot , [\gamma_{1},\gamma_{2}], \cdots],$$\varphi(\gamma_{i})\gamma_{i^{-}}$]$\cdot$

$,$

12

. .],

$\gamma_{1-1}$]$\in\Gamma_{k+1-2}$. Also we can write

141
for some$\varphi(\gamma\ell)=\gamma_{f}\beta$$\beta\in\Gamma_{k}$. Then we compute$\varphi(\gamma)\gamma^{-1}=[\varphi(\alpha), \varphi(\gamma\ell)][\gamma_{l}, \alpha]$$=\alpha\overline{\alpha}\gamma_{f}\beta\overline{\alpha}^{-1}\alpha^{-1}\beta^{-1}\alpha\gamma_{f}^{-1}\alpha^{-1}$$=\alpha[\alpha\sim,\gamma_{f}]\alpha^{-1}\cdot\alpha\gamma_{f}[\overline{\alpha},\beta]\gamma_{l^{-1}}\alpha^{-1}\cdot\alpha\gamma_{f}\alpha^{-1}[\alpha, \beta]\alpha\gamma_{1^{-1}}\alpha^{-1}$$\equiv[\tilde{\alpha},\gamma_{f}][\alpha,\beta](mod \Gamma_{k+1})$. Now using the fact that

because$[\overline{\alpha},\beta]\in\Gamma_{k+t}$

see that$[\alpha, \beta]\equiv[\alpha, \varphi(\gamma_{f})\gamma_{1^{-1}}](mod \Gamma_{k+1})$that$[ \overline{\alpha},\gamma_{f}]\equiv\prod_{i=1}^{t-1}[\cdots[\gamma_{1},\gamma_{2}],$ $\cdots$]$\varphi(\gamma_{i})\gamma_{i^{-1}}$$\cdots$]

it is easy to. Also it can be easily shown by induction$,$$\varphi(\gamma)\gamma^{-1}\equiv\prod^{f}[\cdots[\gamma_{1},\gamma_{2}],$$\beta\equiv\varphi(\gamma_{f})\gamma_{\ell^{-1}}(mod \Gamma_{k+1})$]$\cdots$$,$]$,\gamma_{1-1}$$\varphi(\gamma_{i})\gamma_{i^{-1}}$
$,$

]

]

$\cdots$$,$$,\gamma_{f}$]] $(mod \Gamma_{k+f})$ . Hence$\gamma_{f}$$,$] $(mod \Gamma_{k+l})$$i=1$

as required. This completes the proof.
Now we are ready to prove the main result of this section.

Proof of Theorem 3.1. In general we have$\mathcal{K}_{g’ 1}$). Hence if$\varphi$

is contained in

ately that the extension$\mathcal{I}_{g’ 1}$, then$\tau_{3}(\varphi\psi\varphi^{-1})=\varphi_{*}\tau_{3}(\psi)(\varphi\in \mathcal{M}_{g’ 1},$$\tau_{3}(\varphi\psi\varphi^{-1})=\tau_{3}(\psi)$$\chi\in H^{2}(\Lambda^{3}H;{\rm Im}\tau_{3})$. It follows immedi-

is a central extension. Next

$1arrow{\rm Im}\tau_{3}arrow\overline{\mathcal{I}}_{g,1}arrow\Lambda^{3}Harrow 1$

we determine the Euler class$\psi\in$of this central extension. To dothis we briefly recall the definition of the “Euler class” of a general central extension$0arrow Aarrow Garrow Qarrow 1$(see [3] for details). Choose a set map$s$: $Qarrow G$ such that$s(1)=1$ and set $c(f, g)=s(f)s(g)s(fg)^{-1}\in A$ . It is easy to see thatof the group$Q$

with values in

$A$

$c$

is a 2-cocycle. If we change the map , then the cocycle changes
$s$

$c$

only by a coboundary so that the cohomology class $[c]\in H^{2}(Q;A)$ is well defined.

This is the definition of the Euler class of central extensions. Now $assun_{1}e$ that

$Q$

is

a free abelian group (as in our case). Then we have $H^{2}(Q;A)\cong Hom(H_{2}(Q;Z), A)$

13

142
and $H_{2}(Q;Z)\cong\Lambda^{2}Q$ . For any element $f\wedge g\in\Lambda^{2}Q(f, g\in Q)$ , we have$\chi(f\wedge g)=c(f, g)-c(g, f)$$=[s(f), s(g)]\in A$

because $s(fg)=s(gf)$ . With these preparations we compute the Euler class$H^{2}(\Lambda^{3}H;{\rm Im}\tau_{3})$. Let $\xi=a\wedge b\wedge c$ and $\eta=d\wedge e\wedge f$ be any two elements of$\Lambda^{3}H(a, b, c, d, e, f\in H)$$H_{2}(\Lambda^{3}H;Z)$. To evaluate the value of

we choose elements$\varphi,\psi\in \mathcal{I}_{g’ 1}$

It is easy to deduce from our identification of$Hom(H, \Lambda^{2}H)$$\chi\in$$\chi$

on the cycle

such that$\Lambda^{3}H$(see \S 2) that the homomorphism$\xi\wedge\eta\in\Lambda^{2}(\Lambda^{3}H)\cong$$\tau_{2}(\varphi)=\xi$

andas a submodule of.

$\Lambda^{2}H\otimes H\cong$$\tau_{2}(\varphi):Harrow \mathcal{L}_{2}=\Lambda^{2}H$$\tau_{2}(\varphi)(u)=(u\cdot a)b\wedge c+(u\cdot b)c\wedge a+(u\cdot c)a\wedge b(u\in H)$$\tau_{2}(\psi)=\eta$is given by.

Similarly we have$\tau_{2}(\psi)(u)=(u\cdot d)e\wedge f+(u\cdot e)f\wedge d+(u\cdot f)d\wedge e$

Hence if we apply Proposition 3.4 and Proposition 3.5 to

14$\varphi,$.

$\psi\in \mathcal{I}_{g’ 1}=\mathcal{M}(2)$

, we

143
obtain$\tau_{3}([\varphi, \psi])=\varphi\{\tau_{2}(\psi)(u)\}-\psi\{\tau_{2}(\varphi)(u)\}$$=(u\cdot d)\{(f\cdot a)[e, [b, c]]+(f\cdot b)[e, [c, a]]+(f\cdot c)[e, [a, b]]$$+(e\cdot a)[[b, c],$ $f$] $+(e\cdot b)[[c, a],$ $f$ ] $+(e\cdot c)[[a, b],$ $f$ ]}$+(u\cdot e)\{(d\cdot a)[f, [b, c]]+(d\cdot b)[f, [c, a]]+(d\cdot c)[f, [a, b]]$]$+(f\cdot a)[[b, c],$ $d$ $+(f\cdot b)[[c, a],d]+(f\cdot c)[[a, b],$ $d$]}$+(u\cdot f)\{(e\cdot a)[d, [b, c]]+(e\cdot b)[d, [c, a]]+(e\cdot c)[d, [a, b]]$$+(d\cdot a)[[b, c],$ $e$] $+(d\cdot b)[[c, a],$ ] $+(d\cdot c)[[a, b],$ ]}
$e$

$e$

$-(u\cdot a)\{(c\cdot d)[b, [e, f]]+(c\cdot e)[b, [f, d]]+(c\cdot f)[b, [d, e]]$$+(b\cdot d)[[e, f],$ $c$] $+(b\cdot e)[[f, d],$ ] $+(b\cdot f)[[d, e],$ ]}
$c$

$c$

$-(u\cdot b)\{(a\cdot d)[c, [e, f]]+(a\cdot e)[c, [f, d]]+(a\cdot f)[c, [d, e]]$$+(c\cdot d)[[e, f],$ $a$] $+(c\cdot e)[[f, d],$ ] $+(c\cdot f)[[d, e],$ ]}
$a$

$a$

$-(u\cdot c)\{(b\cdot d)[a, [e, f]]+(b\cdot e)[a, [f, d]]+(b\cdot f)[a, [d, e]]$]] $+(a\cdot f)[[d, e],$ ]}.

$+(a\cdot d)[[e, f],$ $b$ $+(a\cdot e)[[f, d],$ $b$

Now by the identification of$t=b\wedge crightarrow e\wedge f\in\overline{T}$$\overline{T}$as a submodule of$b$$Hom(H, \mathcal{L}_{3})$(see \S 2), the element

can be written as$t(u)=(u\cdot b)[c, [e, f]]-(u\cdot c)[b, [e, f]]$$-(u\cdot e)[[b, c],$ $f$] $+(u\cdot f)[[b, c],$ ].
$e$

It is now a routine matter to check that the formula given in the theorem is the$co$rrect one. This completes the proof.
Here is an example which will clarify the effectiveness of Theorem 3.1.

Example 3.6. We consider a compact surface of genus two with one boundary
component as illustrated in Figure 1, where we fix a symplectic basis
15$x_{1},$ $x_{2},$ $y_{1},$$y_{2}$of

144
the homology group. Also let$\varphi,$$\psi\in \mathcal{I}_{2,1}$

and$\zeta\in \mathcal{K}_{2,1}$

be the elements defined there,

where $the+(resp. -)$ sign means that we take the right handed (resp. left handed)

Dehn twist on the corresponding simple closed curve.

$($$\psi$$\varphi$

Figure 116

145
We can show by a direct computation that$[\varphi, \psi]=\zeta$. Now we compute (cf.

$[7,14])$A$\tau_{2}(\varphi)=x_{1}$$y_{1}\wedge x_{2}$$\tau_{2}(\psi)=x_{1}\wedge y_{1}\wedge y_{2}$$\tau_{3}(\zeta)=-(x_{1}\wedge y_{1})^{\otimes 2}+(x_{2}\wedge y_{2})^{\otimes 2}-(x_{1}\wedge y_{1}+x_{2}\wedge y_{2})^{\otimes 2}$$=-x_{1}\wedge y_{1}rightarrow x_{1}\wedge y_{1}-x_{1}\wedge y_{1}rightarrow x_{2}\wedge y_{2}$.

On the other hand, by Theorem 3.1 we have$\chi(x_{1}\wedge y_{1}\wedge x_{2}, x_{1}\wedge y_{1}\wedge y_{2})$$=-y_{1}\wedge x_{2}rightarrow y_{2}\wedge x_{1}+x_{2}\wedge x_{1}rightarrow y_{1}\wedge y_{2}-x_{1}\wedge y_{1}rightarrow x_{1}\wedge y_{1}$$\equiv-x_{1}\Lambda y_{1}rightarrow x_{1}\wedge y_{1}-x_{1}\wedge y_{1}rightarrow x_{2}\wedge y_{2}$as elements ofcontained in$\overline{T}$$x_{1}\wedge y_{1}rightarrow x_{2}$A $y_{2}-y_{1}$ A$x_{2}rightarrow y_{2}\wedge x_{1}+x_{2}\wedge x_{1}rightarrow y_{1}\wedge y_{2}$$\chi$

has a meaning with values in$T$(not just incorrect answer. For the genus two case we can modify
with values inis. This checks Theorem 3.1 in this case. Observe here that although$T_{0}$

the Euler class, because$T$), it does not give theto obtain the correct one

by adding a constant term, but here we omit it.

4. The mapping$\lambda_{f}$:$\mathcal{I}_{g}arrow Z$

Recall that in \S 2 we have defined a mapping$\lambda(M)(\varphi\in \mathcal{I}_{g})$$\chi$$\overline{T}$where$M_{\varphi}$$\lambda_{f}$:$\mathcal{I}_{g}arrow Z$

by$\lambda_{f}(\varphi)=\lambda(M_{\varphi})-$is the homology 3-sphere obtained by cutting a given

oriented homology 3-sphere $M$ along an embedded oriented surface $f(\Sigma_{g})\subset M$

and then reglueing the resulting two pieces
the restriction of$\lambda_{f}$to the subgroup$\mathcal{K}_{g}$17$V_{+}$

and$V_{-}$

by the map

is a homomorphism,$\lambda_{f}$$\varphi$. Althoughitself is not a146homomorphism (except the case $g=2$ ). The purpose of this section is to determine
completely the deviation of the mapping$\lambda_{f}$from the additivity in the case where $f$

is a Heegaard embedding (see Remark 4.7 for the general case). In order to formulate
the result, we define a mapping

an embedded disc$D^{2}$$D^{2}\subset\Sigma_{g}$:$\delta_{f}$$\mathcal{I}_{g}\cross \mathcal{I}_{g}arrow Z$

as f\‘ollows. First as in \S 2 we fix

and consider the Torelli group. Then we have the homomorphism$\tau_{2}$:$\mathcal{I}_{g’ 1}arrow\Lambda^{3}H$kernel of the homomorphism $H=H_{1}(\Sigma_{g}; Z)arrow H_{1}$ (the mappingand$H_{y}$

each of which has maximal rank

symplectic basis$i=1,$$\cdots$$x_{1},$ $\cdots$,$\Sigma_{g}$relative to$H_{x}$

for theZ) which is induced from
$H_{y}$

for the kernel pf theZ). Then since $M$ is assumed to be a homology sphere,

it is easy to see that $H$ is decomposed as a direct sum$H_{x}$$V_{-};$-and similarly we denote$V_{+};$of. Next we denote$\Sigma_{g}arrow f(\Sigma_{g})\subset V$

homomorphism $Harrow H_{1}$ ($\mathcal{I}_{g’ 1}$$x_{g},$ $y_{1},$$\cdot’$. ,$g$$H_{x}\oplus H_{y}$of isotropic subspaces. It follows that we can choose aof $H$ such that $x_{i}\in H_{x}$ and$y_{g}$$y_{i}\in H_{y}$for all, . With these preparations, we have$g$Definition 4.1. We deffie a mapping$\delta_{f}$as follows. For any two elements$re$spect to the natural urjection
$s$

basis

$x_{1},$$\cdots$$\mathcal{I}_{g}\cross \mathcal{I}_{g}arrow Z$$\psi\in \mathcal{I}_{g}$$\mathcal{I}_{g’ 1}arrow \mathcal{I}_{g}$, choose any of their lifts

terms$\tau_{2}(\tilde{\psi})=\sum_{i<j<k}b_{ijk}x_{i}\wedge x_{j}\wedge x_{k}+other$terms$\Lambda^{3}H$$\overline{\varphi},\overline{\psi}\in \mathcal{I}_{g’ 1}$with. Write$\tau_{2}(\overline{\varphi})=\sum_{i<j<k}a_{ijk}y_{i}\wedge y_{j}\wedge y_{k}+other$in terms of the basis of$x_{i}\wedge y_{j}\wedge y_{k}(j<k)$$\varphi,$:: $x_{i}\wedge x_{j}\wedge x_{k}(i<j<k),$ $x;\wedge x_{j}\wedge y_{k}(i<j)$ ,, and $y_{i}\wedge y_{j}\wedge y_{k}(i<j<k)$ which is indu $ced$ from the symplectic$x_{g},$ $y_{1},$$\cdots y_{g}$of $H$ chosen above. Then we set$\delta_{f}(\varphi,\psi)=\sum_{i<j<k}a_{ijk}b_{ijk}$

18

.

147
Lemma 4.2. The a\’oove defnition is well defined. Namely the$value\delta_{f}(\varphi, \psi)$does

not depend on th various choices made.
$e$

Proof. We have to prove that the value is independent of the(i) choices of the lifts

and$\overline{\varphi},\overline{\psi}\in \mathcal{I}_{g1}$(ii) choice of the symplectic basis$x_{1}$, –,First we consider (i). Recall that$\pi_{1}(T_{1}\Sigma_{g})$

[7],

where

$T_{1}\Sigma_{g}$value$\delta_{f}(\varphi, \psi)$$y_{g}$.
is naturally isomorphic to$Ker(\mathcal{I}_{g’ 1}arrow \mathcal{I}_{g})$is the unit tangent bundle of$\tau_{2}(Ker(\mathcal{I}_{g’ 1}arrow \mathcal{I}_{g}))\subset\Lambda^{3}H$$x_{1}\wedge y_{1}+\cdots+x_{g}\wedge y_{g}$$x_{g},$ $y_{1},$ $\cdots$. Also according to Johnson$\Sigma_{g}$can be identified as $\{\omega_{0}\wedge u;u\in H\}$ where$\omega_{0}=$is the “symplectic class” of $H$ . It follows immediately that theis independent of the choices of$\tilde{\varphi}$and$\overline{\psi}$.

Next we consider (ii). Since we are only considering those symplectic basis$x_{1},$ $\cdots$$x_{g},$ $y_{1},$$\cdots$,$y_{g}$

such that$x_{i}\in H_{x}$andfor all , any two such bases are$i$$y_{i}\in H_{y}$related by a matrix of the form$(\begin{array}{ll}A OO {}^{t}A^{-1}\end{array})\in Sp(2g;Z)$for some $A\in GL(g;Z)$ . As is well known $GL(g;Z)$ is generated by the following

matrices$m$( )$\cdot$$R=(\begin{array}{llll}-1 1 \ddots 1\end{array})$,$T_{\ell m}=\ell$

.

$1$

.

.

$(p\neq m)$.

$1$

Now it is easy to check the relevant invariance under the basis change correspond19

148
ing to the matrix$x_{1}’,$$\cdots$,$x_{g}’,$ $y_{1}’,$$\cdots$$R$. We check it for the matrix$y_{g}’$$\tau_{tm}$. The new symplectic basis

which is obtained from the old one by applying the matrix$T_{fm}$is expressed as$x_{k}’=x_{k}(k\neq m),$ $x_{m}’=x_{f}+x_{m}$$y_{k}’=y_{k}(k\neq P),$ $y_{f}’=y_{f}-y_{m}$Hence if we represent the given two elements

new basis as$x_{k}’+$$\tau_{2}(\overline{\varphi})=\sum a_{ijk}’y_{i}’$A$y_{j}’$$\tau_{2}(\overline{\varphi})$

and

.

and

$\tau_{2}(\overline{\psi})$A $y_{k}’+other$ terms andwith respect to the$\tau_{2}(\overline{\psi})=\sum b_{ijk}’x_{i}’\wedge x_{j}’\wedge$other terms, then $a_{ijk}’=a_{ijk}$ unless one of the indices $i,j,$$k$

is equal to $m$ andthe other two are different from . Similarly $b_{ijk}’=b_{ijk}$ unless one of $i,j,$$\ell$

to

$p$

$k$

is equal

and the other two are different from $m$ . Moreover we have$a_{tjk}’=a_{fjk}+a_{mjk}(j, k\neq P, m)$$b_{mjk}’=b_{mjk}-b_{\ell jk}(j, k\neq P, m)$.

It follows that $a_{1jk}’b_{fjk}’+a_{mjk}’b_{mjk}’=a_{tjk}b_{ljk}+a_{mjk}b_{mjk}$ . Hence the sum$\sum a_{ijk}b_{ijk}$

is invariant under (ii). This completes the proof.
Now we are ready to state the main theorem of this section.

Theorem 4.3. Let $M$ be an oriented homology 3-sphere and let

a Heegaar embedding. Also let$d$$\lambda(M_{\varphi})-\lambda(M)(\varphi\in \mathcal{I}_{g})$cutting along$f(\Sigma_{g})$where$\lambda_{f}$:$\mathcal{I}_{g}arrow Z$$f$:$\Sigma_{g}arrow M$

be the mapp $ing$ defi $ned$ as$\lambda_{f}(\varphi)=$is the A omology sphere obtained from $M$ by$M_{\varphi}$. and then $p$ asting back by the map$\varphi\in \mathcal{I}_{g}$. Then we have$\lambda_{f}(\varphi\psi)=\lambda_{f}(\varphi)+\lambda_{f}(\psi)+2\delta_{f}(\varphi, \psi)$

for all$\varphi,$$\psi\in \mathcal{I}_{g}$.

Corollary 4.4. (i) The mapping$\lambda_{f}$be:$\mathcal{I}_{g}arrow Z$

20

is a Aomomorphism for $g=2$ .

149(ii) If we deffie$h$$\overline{\lambda}_{f}$:$\mathcal{I}_{g}arrow Z/2$

to be the $mod 2$ reduction of$\lambda_{f}$, then$\overline{\lambda}_{f}$is aomomorphism for all .
$g$

Remark 4.5. As is well known the $mod 2$ reduction of the Casson invariant is equal

to the Rohlin invariant. Hence statement (ii) in the above corollary is due to Birman-

Craggs [2] (see also Remark 4.7).
Before proving Theorem 4.3, we prepare a few preliminary results.

Proposition 4.5. Let $M$ be an oriented homology 3-sphere and let

an embedding. Then for any two elements$\varphi,\psi\in \mathcal{I}_{g}$$f$:$\Sigma_{g}arrow M$be, we bave$\lambda_{f}([\varphi, \psi])=2\delta_{f}(\varphi, \psi)-2\delta_{f}(\psi, \varphi)$

Proof. Fix an embedded discgroup$\mathcal{I}_{g’ 1}$

rather than$\mathcal{I}_{g}$$D^{2}\subset\Sigma_{g}$and we consider everything at the level of the. Now first of all we claim that the following two assertions

hold:(i) the value(ii)$\lambda_{f}([\varphi, \psi])$depends only on$\tau_{2}(\varphi)$and$\tau_{2}(\psi)\in\Lambda^{3}H$$\lambda_{f}([\varphi_{1}\cdots\varphi_{s}, \psi_{1}\cdots\psi_{t}])=\sum_{i=1}^{s}\sum_{j=1}^{t}\lambda_{f}([\varphi_{i}, \psi_{j}])$In view of the existence of the exact sequence
we have only to prove that

we havefor any. Hence we obtain

Proposition 2.1 and Corollary 2.3. Similarly we have$\varphi_{i},$$\varphi,\psi\in \mathcal{I}_{g’ 1}$$\psi_{j}\in \mathcal{I}_{g’ 1}$

for any$\varphi_{1},$.

.

$1arrow \mathcal{K}_{g’ 1}arrow \mathcal{I}_{g’ 1}arrow\Lambda^{3}Harrow 1$$\lambda_{f}([\varphi, \psi])=\lambda_{f}([\varphi\varphi_{1}, \psi\psi_{1}])$$[\varphi\varphi_{1}, \psi]=\varphi[\varphi_{1}, \psi]\varphi^{-1}[\varphi, \psi]$

for any$\psi_{1}\in \mathcal{K}_{g1}$

,

. But

$\lambda_{f}([\varphi\varphi_{1}, \psi])=\lambda_{f}([\varphi, \psi])$by$\lambda_{f}([\varphi\varphi_{1}, \psi\psi_{1}])=\lambda_{f}([\varphi\varphi_{1}, \psi])$

proving (i). To prove (ii), we first assume that $s=2$ and $t=1$ . Then we have$[\varphi_{1}\varphi_{2}, \psi]=\varphi_{1}[\varphi_{2}, \psi]\varphi_{1}^{-1}[\varphi_{1}, \psi]$and hence$\lambda_{f}([\varphi_{1}\varphi_{2}, \psi])=\lambda_{f}([\varphi_{1}, \psi])+\lambda_{f}([\varphi_{2}, \psi])$by Proposition 2.1 and Corollary 2.3 again. The general case follows from this by aneasy inductive argument.

In view of the above two assertions (i) and (ii), for any two elements

we may writethat$\lambda_{f}([\xi,\eta])$$\tau_{2}(\varphi)=\xi$

andinstead of$\tau_{2}(\psi)=\eta$$\lambda_{f}([\varphi, \psi])$

where. Also the value
21$\varphi,$$\psi\in \mathcal{I}_{g’ 1}$$\lambda_{f}([\xi, \eta])$$\xi,$$\eta\in\Lambda^{3}H$

are any elements suchis “bilinear” with respect to

150
$\xi$

and . Now we choose a symplectic basis$\eta$

is homologous to$u\in H$ ,let$u^{+}$

in -and$V$$0$$\cdots$

is homologous to$y_{i}$

be the cycle in$x_{1},$$0$of $H$ such that$x_{g},$ $y_{1},$$\cdots$

infor all . For any element$V_{+}$$y_{g}$$x_{i}$$i$-which is obtained by pushing

$V$

$u$

to the positive

direction. Then by a standard argument we have$lk(x_{i}, y_{j}^{+})=-\delta_{ij}$and$\ell k(x_{i}, x_{j}^{+})=\ell k(y_{i}, y_{j}^{+})=0$

Now we compute$\Lambda^{3}H$$\lambda_{f}([\xi, \eta])$

where

$\xi$

and

$\eta$.

run through any member of the basis of: $x_{i}\wedge x_{j}\wedge x_{k}(i<j<k),$ $x_{i}\wedge x_{j}\wedge y_{k}(i<j),$ $x_{i}\wedge y_{j}\wedge y_{k}(j<k)$ and$y_{i}\wedge y_{j}\wedge y_{k}(i<j<k)$which is as sociated to the symplectic basis of $H$ chosen above.

To determine the value$\lambda_{f}([\xi, \eta])$

we first use Theorem 3.1 to computeand$\tau_{3}([\xi, \eta])$then apply Theorem 2.2 and Theorem 2.4. We give the results in the following list
(we refer to [14] for the explicit evaluations of the invariants$\theta_{0}=\epsilon_{f}0\theta,\overline{d}$and ).
$d$In the list we denote $r=r(\xi, \eta)$ for $\chi(.\xi\wedge\eta)\in T$ which is a lift of the element$\tau_{3}([\xi, \eta])\in\overline{T}$

to

$T$

(recall that$\lambda_{f}=\frac{1}{24}d+\theta_{0}+\frac{1}{3}\overline{d}$$\chi$

has a meaning with values in$T$) and also recall that(see \S 2).

(I) The case where $\xi=x;\wedge x_{j}\wedge x_{k},$ $\eta=Xf\wedge x_{m}$ A$x_{n}$.

In this case $r=r(\xi, \eta)=0$ and hence $\theta_{0}(r)=\overline{d}(r)=0$ . Also we have $d([\xi, \eta])=$$0$. Therefore we can conclude that

(II) The case where$\lambda_{f}([\xi, \eta])=0$$\xi=x_{i}\wedge x_{j}\wedge x_{k},$$\eta=x_{f}$A.

$x_{m}\wedge y_{n}$.

In this case we have$r(\xi, \eta)=-\delta_{in}x_{j}\wedge x_{k}rightarrow x_{\ell}\wedge x_{m}-\delta_{jn}x_{k}\wedge x_{i}rightarrow x_{f}\wedge x_{m}-\delta_{kn}x_{i}\wedge x_{j}rightarrow x_{f}\wedge x_{m}$and hence$\theta_{0}(r)=\overline{d}(r)=0$(III) The case where. Also we have $d([\xi, \eta])=0$ . Therefore$\xi=x_{i}\wedge x_{j}\wedge x_{k},$$\eta=x_{1}\wedge y_{m}\wedge y_{n}$22.

$\lambda_{f}([\xi, \eta])=0$.

151
In this case we have$r(\xi, \eta)=-\delta_{im}x_{j}$A$x_{k}rightarrow y_{n}\wedge x\ell-\delta_{in}x_{j}\wedge x_{k}rightarrow x_{f}\wedge y_{m}$$-\delta_{jm}x_{k}\wedge x_{i}rightarrow y_{n}$A$x_{f}-\delta_{jn}x_{k}\wedge x_{i}rightarrow x_{f}\wedge y_{m}$$-\delta_{km}x_{i}\wedge x_{j}rightarrow y_{n}\wedge x_{f}-\delta_{kn}x_{i}\wedge x_{j}rightarrow x_{f}\wedge y_{m}$and hence$\theta_{0}(r)=\overline{d}(r)=0$(IV) The case where. Also we have $d([\xi, \eta])=0$ and we conclude$\xi=x_{i}\wedge x_{j}\wedge x_{k},$$\eta=y\ell$A$y_{m}\wedge y_{n}$$\lambda_{f}([\xi, \eta])=0$.

.

In this case we haveA$y_{\ell}-\delta_{in^{X}j}\wedge x_{k}rightarrow y_{f}\wedge y_{m}$$-\delta_{jl}x_{k}\wedge x_{i}rightarrow y_{m}\wedge y_{n}-\delta_{jm}x_{k}\wedge x_{i}rightarrow y_{n}$A$y\ell-\delta_{jn}x_{k}\wedge x;rightarrow y\ell$A$y_{m}$$-\delta_{kt}x_{i}\wedge x_{j}rightarrow y_{m}\wedge y_{n}-\delta_{km}x_{i}\wedge x_{j}rightarrow y_{n}$A$y_{f}-\delta_{kn}x_{i}\wedge x_{j}rightarrow y\ell$A$y_{m}$$r(\xi, \eta)=-\delta_{i\ell}x_{j}\wedge x_{k}rightarrow y_{m}\wedge y_{n}-\delta_{im}x_{j}\wedge x_{k}rightarrow y_{n}$Hence we have$\theta_{0}(r)=-\overline{d}(r)=-3\delta_{if}\delta_{jm}\delta_{kn}$.

(here we have used the assumptions

that $i<j<k$ and $p<m<n$ ). Also we have $d([\xi, \eta])=0$ . Therefore we obtain$\lambda_{f}([\xi, \eta])=-2\delta_{il}\delta_{jm}\delta_{kn}$.

(V) The case where $\xi=x;\wedge x_{j}\wedge y_{k},$ $\eta=x_{f}$ A$x_{m}\wedge y_{n}$.

In this case we have$r(\xi, \eta)=-\delta_{in}x_{j}\wedge y_{k}rightarrow x_{f}$A$x_{m}-\delta_{jn}y_{k}\wedge x_{i}rightarrow x_{f}\wedge x_{m}$$+\delta_{kl}x_{i}\wedge x_{j}rightarrow x_{m}\wedge y_{n}+\delta_{km}x_{i}\wedge x_{j}rightarrow y_{n}\wedge x_{l}$and hence we obtain $\theta_{0}(r)=\overline{d}(r)=0$ .
$\lambda_{f}([\xi, \eta])=0$

Also we have $d(r)=0$ and therefore.

(VI) The case where$\xi=x_{i}\wedge x_{j}\wedge y_{k},$ $\eta=x_{f}$23A$y_{m}$

A$y_{n}$.

152
In this case we have$r(\xi, \eta)=-\delta_{im}x_{j}\wedge y_{k}rightarrow y_{n}\wedge x_{f}-\delta_{in}x_{j}\wedge y_{k}rightarrow x_{f}\wedge y_{m}$$-\delta_{jm}y_{k}\wedge x_{i}rightarrow y_{n}\wedge x_{f}-\delta_{jn}y_{k}\wedge x_{i}rightarrow x_{f}\wedge y_{m}$$+\delta_{k\ell}x_{i}\wedge x_{j}rightarrow y_{m}\wedge y_{n}$and hence$\theta_{0}(r)=\delta_{kf}\delta_{im}\delta_{jn}$. Recalling the conditions $i<j$ and $m<n$ , we compute$\overline{d}(r)=\delta_{im}\delta_{jk}\delta_{nt}-\delta_{in}\delta_{jk}\delta_{1m}-\delta_{jm}\delta_{ik}\delta_{tn}$$+\delta_{jn}\delta_{ik}\delta_{tm}-3\delta_{im}\delta_{jn}\delta_{k\ell}$$d(r)=8(\delta\delta\delta-\delta\delta_{n}\delta_{im}-\delta_{ik}\delta_{lm}\delta_{jn}+\delta_{ik}\delta_{nf}\delta_{jm})$Hence we can conclude that$\lambda_{f}([\xi, \eta])=0$.

In the above list we omit four other cases like$\xi=y_{i}\wedge y_{j}\wedge y_{k}$andbecause of a symmetry in the above computations with respect to$x$

can easily read off from the above list that the only possible pairs ofzero$\lambda_{f}$is$(x_{i}\wedge x_{j}\wedge x_{k}, y_{i}\wedge y_{j}\wedge y_{k})$and.

$(y_{i}\wedge y_{j}\wedge y_{k}, x_{i}\wedge x_{j}\wedge x_{k})$$\eta=y\ell\wedge y_{m}\wedge y_{n}$and . Now we$y$$(\xi, \eta)$with non-and in these cases

the values are-2 and 2 respectively. Therefore we can conclude that$\lambda_{f}([\varphi, \psi])=2\delta_{f}(\varphi, \psi)-2\delta_{f}(\psi, \varphi)$.

This completes the proof of Proposition 4.5.

Lemma 4.6. Let$\Sigma_{g}$. Let$\Sigma_{g}$$\mathcal{M}_{g1}$and let$V$be a handlebody of genus

be the mapping class $gro$ up of$\mathcal{N}_{g,1}$

be the subgro up of

be extended to diffeomorphisms ofLet$H_{y}$$\mathcal{M}_{g’ 1}$$V$$\Sigma_{g}$$g$and fix any difFeomorphism$\partial V\cong$

relati $ve$ to an embedded disc$D^{2}\subset$

consisting of all th elemen$e$(under the above identffication$ts$which an$c$$\partial V=\Sigma_{g}$).

be the kernel of the A omomorphism $H=H_{1}(\Sigma_{g} ; Z)arrow H_{1}(V;Z)$ , whichis induced from the inclu sion $\Sigma_{g}=\partial Varrow V$ , and choose any symplectic
24

$b$

asis

153
$x_{1},$$\cdots x_{g},$ $y_{1},$$W_{y}$

ofand$\Lambda^{3}H$$\cdots y_{g}ofH$

such that$y_{i}\in H_{y}$for all $i=1,$ $\cdots g$ . Defin the subgroup$e$

to be the one generated by the elements of the forms$y_{i}\wedge y_{j}\wedge y_{k}$$x_{i}\wedge x_{j}\wedge y_{k},$ $x_{i}\wedge y_{j}\wedge y_{k}$. Then we bave$\tau_{2}(\mathcal{I}_{g’ 1}\cap N_{g,1})=W_{y}$.

We omit the proof of the above lemma because it follows from Lemma 2.5 of[14] by an easy argument.

Proof of Theorem 4.3. By the assumption that$f$:$\Sigma_{g}arrow M$

is a Heegaard em-

bedding, both of the pieces $V+andV$-are handlebodies of genus . Now as before$g$choose a symplectic basis

in$V_{-}$

and$y_{i}$$x_{1},$$\cdots x_{g},$ $y_{1},$is homologous toand choose any lifts$0$$\tilde{\varphi},\overline{\psi}\in \mathcal{I}_{g’ 1}$$\cdots$,$y_{g}$

of $H$ such that$x_{i}$

is homologous toin $V+for$ all . Also fix an embedded disc$i$of the given elements$\varphi,$$\psi\in \mathcal{I}_{g}$$0$$D^{2}\subset\Sigma_{g}$. Now write$\tau_{2}(\tilde{\varphi})=\sum_{i<j<k}a_{ijk}y_{i}\wedge y_{j}\wedge y_{k}+\xi_{x}$$\tau_{2}(\tilde{\psi})=\sum_{i<j<k}b_{ijk}x_{i}\wedge x_{j}\wedge x_{k}+\eta_{y}$with respect to the basis of$\Lambda^{3}H$

which is associated to the symplectic basis of $H$ cho-sen above. If we apply Lemma 4.6 to the pairthere exists two elementsof$V_{+}$(here we identify$\varphi+,$ $\psi+\in \mathcal{I}_{g1}$$f(\Sigma_{9})$extends to a diffeomorphism of
the pair (V-,$f(\Sigma_{g})$with$V_{+}$$V_{-}$$\varphi_{2}=\varphi_{-\varphi\varphi+}^{\sim}$

and

and

$\tau_{2}(\varphi_{+})=-\sum a_{ijk}y_{i}\wedge y_{j}\wedge y_{k}$$\tau_{2}(\psi_{+})=-\eta_{y}$extends to a diffeomorphism of -andmorphism of) and) implies the existence of elements$V$tends to diffeomorphisms of$V_{+}$$\varphi_{-},$$\psi_{-}\in \mathcal{I}_{g’ 1}$(iv)$\psi_{-}$$V_{-}$), we have25$\psi_{+}$. Now setsuch that (iii)$\varphi_{-}$

extends to a diffeo-$\varphi_{1}=\varphi-\overline{\varphi},$Then since $\varphi+and\psi_{+}$ (resp.

(resp.

(ii). The same argument applied to$\tau_{2}(\varphi_{-})=-\xi_{x}$$\tau_{2}(\psi_{-})=-\sum b_{ijk}x_{i}\wedge x_{j}\wedge x_{k}$$\psi_{2}=\psi_{-}\overline{\psi}\psi+\cdot$, then we can conclude thatsuch that (i) $\varphi+extends$ to a diffeomorphism$\partial V_{+}$

and$(V_{+}, f(\Sigma_{g}))$,$\psi_{-}$) ex-$M_{\varphi_{2}}=M_{\varphi}$

and$\varphi_{-}$$M_{\varphi_{1}\psi_{1}}=M_{\varphi}\psi,$$\psi_{1}=\tilde{\psi}\psi+$

and

154
$M\psi_{2}=M\psi$Hence$\lambda_{f}(\varphi_{1}\psi_{1})=\lambda_{f}(\varphi\psi)$$\lambda_{f}(\varphi_{2})=\lambda_{f}(\varphi)$$\lambda_{f}(\psi_{2})=\lambda_{f}(\psi)$

Also observe that both of the elements$\tau_{2}(\psi_{2})=0$. Similarly we have$\varphi_{2}$and$\psi_{2}$.

belong to$M_{\varphi_{1}\psi_{1}}=M_{\psi_{-}\varphi_{1}\psi_{1}\varphi+}$$\mathcal{K}_{g’ 1}$

because$\tau_{2}(\varphi_{2})=$and hence.

$\lambda_{f}(\varphi_{1}\psi_{1})=\lambda_{f}(\psi_{-}\varphi_{1}\psi_{1}\varphi_{+})$On the other hand we have$\psi_{-\varphi_{1}\psi_{1}\varphi_{+}=\psi_{-}\varphi_{2}\psi_{-}^{-1}[\psi_{-},\varphi_{+}^{-1}]\varphi_{+}^{-1}\psi_{2}\varphi+}$and therefore$\lambda_{f}(\psi_{-}\varphi_{1}\psi_{1}\varphi_{+})=\lambda_{f}(\psi_{-}\varphi_{2}\psi_{-}^{-1})+\lambda_{f}([\psi_{-}, \varphi_{+}^{-1}])+\lambda_{f}(\varphi_{+}^{-1}\psi_{2}\varphi_{+})$by Proposition 2.1.
$\lambda_{f}(\psi_{2})$But we haveand$\lambda_{f}(\psi_{-}\varphi_{2}\psi_{-}^{-1})=\lambda_{f}(\varphi_{2})$by Corollary 2.3,(i). If we combine the above equations, we see that$\lambda_{f}(\varphi\psi)=\lambda_{f}(\varphi)+\lambda_{f}(\psi)+\lambda_{f}([\psi_{-}, \varphi_{+}^{-1}])$

Here recall thatHence$\lambda_{f}(\varphi_{+}^{-1}\psi_{2}\varphi_{+})=$$\tau_{2}(\psi_{-})=-\sum b_{ijk}x_{i}\wedge x_{j}\wedge x_{k}$and$\tau_{2}(\varphi+)=-\sum a_{ijk}y_{i}\wedge y_{j}\wedge y_{k}$$\lambda_{f}([\psi_{-}, \varphi_{+}^{-1})=\lambda_{f}([\varphi+, \psi_{-}])=\sum a_{ijk}b_{ijk}=2\delta_{f}(\varphi, \psi)$the definition of$\delta_{f}$. Therefore$\lambda_{f}(\varphi\psi)=\lambda_{f}(\varphi)+\lambda_{f}(\psi)+2\delta_{f}(\varphi, \psi)$

26

.
.

by Proposition 4.5 and155
as required. This completes the proof.
Remark 4.7. In the general case where the embedding$f$:$\Sigma_{g}arrow M$

is not Heegaard,

the formula of Theorem 4.3 seems to be not true in general anymore. However we

can still conclude that the mapping$\lambda_{f}$$\overline{\lambda}_{f}$:$\mathcal{I}_{g}arrow Z/2$

, which is the $mod 2$ reduction of, is a homomorphism. This follows from the fact that we can modify the embedded

surface$f(\Sigma_{g})$to a Heegaard one by adding l-handles from both sides.

5. Determination of$H^{1}(\mathcal{K}_{g}; Z)^{f\Lambda_{g}}$Recall that we denote$\mathcal{K}_{g}$for the subgroup of the mapping class group

generated by all the Dehn twists on separating simple closed curves on
purpose of this section is to determineof all homomorphisms
and$\varphi\in \mathcal{M}_{g}$$r$:$\mathcal{K}_{g}arrow Z$$H^{1}(\mathcal{K}_{g} ; Z)^{\Lambda 4_{g}}$$\Sigma_{g}$$\mathcal{M}_{g}$. Thewhich is the group consistingsuch that $r(\psi)=r(\varphi\psi\varphi^{-1})$ for all$\psi\in \mathcal{K}_{g}$. Namely we will determine all the integer valued additive invariants

for elements of$\mathcal{K}_{g}$which are invariant under the natural action ofdetermine $bhe$ related grouphomomorphisms$\mathcal{K}_{g’ 1}arrow Z$a homomorphism$H^{1}(\mathcal{K}_{g’ 1} ; Z)^{\mathcal{M}_{g1}}$

which is the set of all. We first consider the group$d:\mathcal{K}_{g1}arrow Z$$\mathcal{K}_{g’ 1}$$\mathcal{M}_{g}$. We also$\mathcal{M}_{g’ 1}$-invariant. Recall that we have

which is the “core” of the Casson invariant from thepoint of view of our approach (see \S 2 and [14] for details). It has the property that for
any BSCC mapofof genus , we have $d(\varphi)=4h(h-1)$ . It follows that$h$$\varphi$$H^{1}(\mathcal{K}_{g’ 1} ; Z)^{\Lambda 4_{g’ 1}}$

. Next we construct another element

making use of Johnson’s homomorphism$\tau_{3,}:\mathcal{K}_{g’ 1}arrow\overline{T}$$d^{\overline{\prime}}((a\wedge b)^{\otimes 2})=-3(a\cdot b)^{2}$27

is an element$d’\in H^{1}(\mathcal{K}_{g’ 1} ; Z)^{\mathcal{M}_{g1}}$(see \S 2 and [14]).

Proposition 5.1. The following two types of correspondences

(i)

$d$

by

156
(ii) $\overline{d}’(a\wedge brightarrow c\wedge d)=-4(a\cdot b)(c\cdot d)-2(a\cdot c)(b\cdot d)+2(a\cdot d)(b\cdot c)$define a well-defined $Sp(2g;Z)$ -invariant homomorphi $sm$ d’ : $Tarrow Z$ . Moreover it istrivial on $Ker(Tarrow\overline{T})$ so that it defines a homomorphism$d^{\urcorner}$:$\overline{T}arrow Z$(we use thesame letter).

Definition 5.2. We define a homomorphism d’ :$\mathcal{K}_{g’ 1}arrow Z$$\mathcal{K}_{g1}arrow^{\tau_{3}}\overline{T}arrow^{d^{\overline}\prime}$

It is easy to see that d’ is$\mathcal{M}_{g’ 1}$

to be the compositionZ.

-invariant so that it is contained in$H^{1}$ $(\mathcal{K}_{g’ 1} ; Z)^{J\Lambda_{g1}}$.

Proof of Proposition 5.1. First of all it is easy to check that the two formulae (i)and (ii) are consistent with the relation$a\wedge brightarrow a\wedge b=2(a\wedge b)^{\otimes 2}$. Next observe thattype (ii) correspondence is linear with respect to any variables and also it is skew
symmetric with respect to

$a,$

$b$

and

$c,$

$d$

respectively. Moreover the value remains to

be unchanged if we interchange the two pairs $(a, b)$ and $(c, d)$ . It follows that thecorrespondence is well-defined. That the resulting homomorphism

invariant follows directly from the definition. Finally we check that$Ker(Tarrow\overline{T})$$\overline{d}$$\overline{d}$is $Sp(2g;Z)-$is trivial on. Recall that $Ker(Tarrow\overline{T})$ is generated by the elements of the form

$a\wedge brightarrow c\wedge d-a\wedge crightarrow b\wedge d+a\wedge drightarrow b\wedge c$.

But a direct computation shows that the value of the homomorphism$\overline{d}$on thiselement is . This completes the proof.
$0$

Proposition 5.3. Let$\varphi\in \mathcal{K}_{g’ 1}$

be a BSCC map of genus . Then we have $d’(\varphi)=$$h$$h(2h+1)$ .

Proof. Letelement$\varphi$$\omega$be the bounding simple closed curve on. Choose a symplectic basis$u_{1},$$\Sigma_{g}\backslash D^{2}$$\cdots u_{h},$ $v_{1},$ $\cdots v_{h}$28correspondint to theof the homology of the

157
subsurface which$\omega$bounds. Then by Proposition 1.1 of [14], we have$\tau_{3}(\varphi)=-(u_{1}\wedge v_{1}+\cdots+u_{h}\wedge v_{h})^{\otimes 2}$.

On the other hand, a direct computation shows that$d^{\urcorner}((u_{1}\wedge v_{1}+\cdots+u_{h}\wedge v_{h})^{\otimes 2})=-h(2h+1)$.

Hence $d’(\varphi)=h(2h+1)$ , completing the proof.
Theorem 5.4. The groupis a free abeli an group of rank two and$H^{1}$ $(\mathcal{K}_{g’ 1} ; Z)^{\Lambda t_{g’ 1}}$over the rationals it is genera $ted$ by

$d$

and

$d’$

.

Proof. It is easy to see that two BSCC maps of the same genus are conjugate eachother in$\mathcal{M}_{g’ 1}$. On the other hand, according to Johnson [5],all BSCC maps of genus one and two. Hence the rank of
two. Since we have already constructed two elements$\mathcal{K}_{g’ 1}$

is generated by
is at most$H^{1}$ $(\mathcal{K}_{g1} ; Z)^{\Lambda 4_{g1}}$$d,$ $d’$of this group which areclearly linearly independent, we are done.
Remark 5.5. We can restate the above theorem as follows. Namely a function $f(h)$ of$h$, where$h$stands for the genera of BSCC maps inhomomorphism
Remark 5.6.

$\mathcal{K}_{g’ 1}arrow Z$$\mathcal{K}_{g’ 1}$, extends to an

if and only if $f(h)=ph+qh^{2}$ for some

Although the two homomorphisms$d$, d’ :$p,$ $q\in$\mathcal{K}_{g’ 1}arrow Z$-invariant$\mathcal{M}_{g’ 1}$Z$.

are seemingly

similar each other, the essential meaning of them are completely different. Moreprecisely, for each element$\Sigma_{g}$$\varphi\in \mathcal{K}_{g’ 1}$

let us consider the manifold$W_{\varphi}$-bundle over the circle with its monodromy diffeomorphism equal toassumption that$\varphi\in \mathcal{K}_{g’ 1}$, the homology group of$W_{\varphi}$

which is a$\varphi$. By theis the same as that of$S^{1}\cross\Sigma_{g}$.

But higher order intersectional properties may be different. In fact as is explainedin Johnson [8], his homomorphisms$\tau_{k}$can be interpreted as the invariants whichmeasure higher order Massey products of the manifolds liked’ is a quotient of$\tau_{3}$, we can conclude that
29

$d’$

$W_{\varphi}$. Since our invariant

can be read off from a certain higher

158
order Massey product of$W_{\varphi}$. In particular it is a local invariant in the sense that it

is computable by means of integrations of forms. On the contrary we have recently$d$can be interpreted as the Hirzebruch’s signature defect (see$W_{\varphi}$with respect to a certain canonical framing of its tangent

found that the invariant[4]) of the 3-manifold

bundle (see [16] for details). If we recall here that the signature defect is closely
related with the -invariant of Atiyah-Patodi-Singer [1] which is by no means a local$\eta$

invariant, we may say that the invariant
At least we can say that

$d$

$d$

should be an essentially global invariant.

is a much more deep invariant than

$d’$

.

Now we consider the case of closed surfaces.
Theorem 5.7. The group$d_{0}$$H^{1}(\mathcal{K}_{g}; Z)^{1\cdot 1_{g}}$can be defined as follows. Let$\omega$is isomorphic to$Z$

whose rational gen erator

be a $sep$ arating simple closed curve onthat the gen era of the two compact urfaces which are obtain $ed$ by cutting$s$$\omega$onar $eh$ and $(g-h)$ . Then we have $d_{0}(D.)=h(g-h)$ where$\omega$$D_{\omega}$$\Sigma_{g}$$\Sigma_{g}$su $ch$alon$g$is the Dehn twist.

Proof. Since the natural homomorphism$H^{1}(\mathcal{K}_{g} ; Z)^{\Lambda 4_{9}}arrow H^{1}(\mathcal{K}_{g’ 1} ; Z)^{\mathcal{M}_{g1}}$

is cearly

injective, in view of Theorem 5.4 it is enough to prove that there is one and only
one relation between the two elements$d,$$d’\in H^{1}$ $(\mathcal{K}_{g’ 1} ; Z)^{\Lambda 4_{9}}$ “on$Ker(\mathcal{K}_{g’ 1}arrow \mathcal{K}_{g})$.

More precisely, since we have
$12h(g-h)=4(g-1)\cdot h(2h+1)-(2g+1)\cdot 4h(h-1)$

and since $d(\varphi)=4h(h-1)$ and $d’(\varphi)=h(2h+1)$ for any BSCC map$\varphi\in \mathcal{K}_{g’ 1}$ofgenus , we have only to prove that the equality$h$$(2g+1)d(\varphi)=4(g-1)d’(\varphi)$holds for any$\varphi\in Ker(\mathcal{K}_{g’ 1}arrow \mathcal{K}_{g})$as was recalled in \S 4,

and that$Ker(\mathcal{I}_{g’ 1}arrow \mathcal{I}_{g})$$d$is non trivial on$Ker(\mathcal{K}_{g’ 1}arrow \mathcal{K}_{9})$is naturally isomorphic to30$\pi_{1}(T_{1}\Sigma_{g})$. Nowand also

159
we have an exact sequence

naturally isomorphic to$1arrow \mathcal{K}_{g’ 1}arrow \mathcal{I}_{g’ 1}arrow\Lambda^{3}Harrow 1$$Ker(\tau_{2} :\pi_{1}(T_{1}Z_{g})arrow\Lambda^{3}H)$. Hence$Ker(\mathcal{K}_{g’ 1}arrow \mathcal{K}_{g})$is. On the other hand we have acentral extension$0arrow Zarrow\pi_{1}(T_{1}\Sigma_{9})arrow\pi_{1}(\Sigma_{g})arrow 1$

where the center$Z$is generated by the elementa simple closed curve on$\tau_{2}(\zeta)=0$

a homomorphism
for any$\tau_{2}$$\gamma\in\pi_{1}(\Sigma_{g})$:restricted to$\tau_{2}$$\pi_{1}(\Sigma_{g})arrow\Lambda^{3}H$, where$[\gamma]\in H$$\pi_{1}(T_{1}\Sigma_{g})$$\zeta$and the elements of the form

is essentially equivalent to(we use the same letter) andis the homology class of

symplectic class (see [7]). We can now conclude thatby, which is the Dehn twist on

which is parallel to the boundary (see [14]). Since$\Sigma_{9}\backslash IntD^{2}$, the homomorphism$\zeta\in \mathcal{K}_{g’ 1}$$[\alpha, \beta]\in \mathcal{K}_{g1}$with$\gamma$and$\tau_{2}(\gamma)=[\gamma]\wedge\omega_{0}$$\omega_{0}\in\Lambda^{2}H$$Ker(\mathcal{K}_{g’ 1}arrow \mathcal{K}_{g})$$\alpha,$is theis generated$\beta\in\pi_{1}(T_{1}\Sigma_{g})\subset \mathcal{I}_{g1}$. Now

we check that the required relation on these generators.
Since $d(\zeta)=4g(g-1)$ and $d^{t}(\zeta)=g(2g+1)$ (see Proposition 5.3), the reuired
relation $(2g+1)d=4(g-1)d’$ certainly holds on . Next we consider the element of$\zeta$the formclasses$\varphi=[\alpha, \beta]\in \mathcal{K}_{g’ 1}$$[\alpha],$$[\beta]\in H$

with$\alpha,$$\beta\in\pi_{1}(T_{1}\Sigma_{9})$. Write the corresponding homologyas$[ \alpha]=\sum_{*}(a_{i}x_{i}+b_{i}y_{i})$$[ \beta]=\sum_{i}(cx_{i}+d_{i}y_{i})w^{i}$where$x_{1},$$\cdots x_{g},$ $y_{1},$ $\cdots y_{g}$5.1,(iv) to the element

is a symplectic basis of $H$ . If we apply Proposition$\varphi=[\alpha, \beta]$

we obtain$d(\varphi)=8(g-1)^{2}[\alpha]\cdot[\beta]$$=8(g-1)^{2} \sum(a_{i}d_{i}-b_{i}c_{i})$Next we compute$d’(\varphi)$. We have.

$\tau_{2}(\alpha)=\sum_{i}(a_{i}x_{i}+b_{i}y_{i})\wedge\omega_{0}$

$31$

and

$\tau_{2}(\beta)=\sum_{i}(c_{i}x_{i}+$$16U$$d_{i}y_{i})\wedge\omega_{0}$where $\omega_{0}=x_{1}\wedge y_{1}+\cdots+x_{g}\wedge y_{g}$ . Therefore$\tau_{3}(\varphi)=\chi(\tau_{2}(\alpha)\wedge\tau_{2}(\beta))$$=_{\mu} \sum_{i,j,k,l}\{a_{i}c_{k}\chi$($\xi_{ij}$A$\xi_{kf}$)$+a_{i}d_{k}\chi(\xi_{ij}\wedge\eta_{kt})$$+b_{i}c_{k}\chi(\eta_{ij}\wedge\xi_{kl})+b_{i}d_{kx(\eta_{ij}}$

where$\xi_{ij}=x_{i}\wedge x_{j}\wedge y_{j},$$\xi_{kI}=x_{k}\wedge x_{f}\wedge y_{f},$A$\eta_{ij}=y_{i}\wedge x_{j}\wedge y_{j}$$\eta_{kf}$

)}

and

$\eta_{kI}=y_{k}\wedge x_{f}\wedge y_{f}$By Theorem 3.1 we have$\chi(\xi_{ij}\wedge\xi_{k\ell})=-\delta_{il}x_{j}\wedge y_{j}rightarrow x_{k}\wedge x_{f}-\delta_{jty_{j}\wedge x_{i}}rightarrow x_{k}\wedge x_{f}$$+\delta_{jk}x_{i}\wedge x_{j}rightarrow x_{f}\wedge y_{f}$and hence

$d’$

( (
$\chi$$\xi_{ij}$A$\xi_{kf})$) $=0$ . Similarly we have$\chi(\xi_{ij}\wedge\eta_{kl})=-\delta_{ik}x_{j}\wedge y_{j}rightarrow x_{f}\wedge y_{f}-\delta_{it}x_{j}\wedge y_{j}rightarrow y_{k}\wedge x_{f}$$-\delta_{jk}y_{j}\wedge x_{i}rightarrow x_{f}\wedge y_{f}-\delta_{j\ell}y_{j}\wedge x_{i}rightarrow y_{k}\wedge x_{f}$$+S_{J^{f}}x_{i}\wedge x_{j}rightarrow y_{f}\wedge y_{k}$and therefore$d’(\chi(\xi_{ij}\wedge\eta_{k\ell}))=-\delta_{ik}(-4-2\delta_{jt})-\delta_{it}(4\delta_{kl}+2\delta_{jk}\delta_{J^{f}})$$-\delta_{jk}(4\delta_{ij}+2\delta_{j\ell}\delta_{il})-\delta_{jl}(-4\delta_{ji}\delta_{kf}-2\delta_{j\ell}\delta_{ik})$$+\delta_{jt}(-2\delta_{i\ell}\delta_{jk}+2\delta_{ik}\delta_{jt})$.

A direct computation shows that$\sum_{k,l}d’(\chi(\xi_{ij}\wedge\eta_{kl}))=\delta_{ik}2(g-1)(2g+1)$

32

.

.

161
By an obvious anti-symmetry of the above computations with respect to$x$and , we$y$obtainand$\sum_{k,\ell}d’(\chi(\eta_{ij}\wedge\xi_{kI}))=-\delta_{ik}2(g-1)(2g+1)$$d’(\chi(\eta_{ij}\wedge\eta_{kl}))=0$.

We can now conclude that$d’( \varphi)=2(g-1)(2g+1)\sum(a_{i}d_{i}-b_{i}c_{i})$.

Hence $(2g+1)d(\varphi)=4(g-1)d’(\varphi)$ as required. This completes the proof.

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[11] S. Morita, Characteristic classes of surface bundles, Invent. Math. 90 (1987),

551-577.
[12] S. Morita, Families of Jacobian manifolds and characteristic classes of surface

bundles I, Ann. Inst. Fourier 39 (1989), -.
[13] S. Morita, Families of Jacobian manifolds and characteristic classes of surface

bundles II, Math. Proc. Camb. Phil. Soc. 105 (1989), 79-101.
[14] S. Morita, Casson’s invariant for homology 3-spheres and characteristic classesof surface bundles I, Topology 28 (1989), 305-323.
[15] S. Morita, Casson’s invariant for homology 3-spheres and characteristic classesof surface bundles II, in preparation.
[16] S. Morita, Signature defects of certain framed 3-manifolds and the Casson in-variant, in preparation.

Department of Mathematics
Faculty of Science

Tokyo Institute of Technology
Tokyo, Japan

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