数理解析研究所講究録
第 1991 巻 2016 年 164-174

164

ON EQUIVARIANT PERTURBATIVE INVARIANTS IN 3-DIMENSIONBY MORSE THEORYTADAYUKI WATANABE

1. INTRODUCTION
Aound 1992, Axelrod-Singer and Kontsevich independently developed the method to
obtain (mathcmatical) topological invariants of 3-manifolds by perturbative expansionof Witten’s path integral (Chern-Simons perturbation theory, [1, 5The invariant is
a series of terms corresponding to Feynman diagrams such that each term is given by
integration over the configuration space of a 3-manifold. This is known to be very strong,for example, the expansion around the trivial connection dominates all -valued Ohtsuki
finite type invariants for integral homology 3-spheres ([7]). In this note, we explain about
our attempt to construct ‘equivariant invariant’ of 3-manifolds with the first Betti number$\mathbb{Q}$1,

Around 2008, Ohtsuki constructed an equivariant refinement of the LMO invariantl for
3-manifolds with the first Betti number 1 ([12, 13 which pioneered a new direction of
pcrturbative invariants of 3-manifolds. Inspired by Ohtsuki’s work, Lescop constructed
an equivariant refinement of Chern-Simons perturbation theory for 3-manifolds with thefirst Betti number 1 for the 2-loop graphs by using a method similar to March\’e ([9, 11
Lescop’s construction is as follows.
Let $M$ be a closed 3-manifold with. The equivariant configuration space$Conf_{K_{2}}(M)$ is defined as the set of tuples$x_{2}\in M$ , satisfying the following,$H_{1}(M)=\mathbb{Z}$$(x_{1}, x_{2}, \gamma)$$x_{1},$

conditions.
(1)(2) is the relative bordism class of paths $c:[0, 1]arrow M$ that go fromto
The natural map $Conf_{K_{2}}(M)arrow Conf_{2}(M)=M\cross M\backslash \triangle_{M}$ that forgets is an infinite
cyclic covering. Instead of removing the diagonalin the definition of $Conf_{2}(M)$ ,consider the blowing-up along, namely replacingwith its normal sphere bundle,
to obtain a compactification
of $Conf_{2}(M)$ . Similarly, by blowing-up along thepreimage ofin the space of tuples
satisfying only (2) above, we obtain the‘closure’of $Conf_{K_{2}}(M)$ .
Lescop defined an invariant of 3-manifolds $M$ with $b_{1}(M)=1$ by an equivariant intersection theory in. The principal term of it is given by the equivariant triple$Q,$
intersection
for a fundamenta14-chain$x_{1}\neq x_{2}.$$\gamma$$x_{1}$$x_{2}.$$\gamma$$\Delta_{M}$$\Delta_{M}$$\Delta_{M}$$\overline{Conf}_{2}(M)$$(x_{I}, x_{2}, \gamma)$$\Delta_{M}$$\overline{C\circ nf}_{K_{2}}(M)$$\overline{Conf}_{K_{2}}(M)$$\langle Q,$$Q\rangle_{\mathbb{Z}}$$Q\in C_{4}(\overline{C\circ nf}_{K_{2}}(M);\mathbb{Q})\otimes_{\mathbb{Q}[t,t^{-1}]}\mathbb{Q}(t)$which satisfies a certain boundary condition (equivariant propagator)2.
Received December 30, 2015.
ILMO invariant is defined combinatorially by using Kontsevich’s link invariant and is known to beuniversal among finite type invariants of homology 3-spheres$2_{The}$Poincar\’e dual ofgenerates.
The meaning of the word ‘propagator’ here diﬀers from the usual one.
$[Q]\in H_{4}(\overline{C\circ nf}_{2}(M),\partial C\circ nf_{2}(M);\mathbb{Q}(t))$$H^{2}(\overline{Conf}_{2}(M);\mathbb{Q}(t))\cong \mathbb{Q}(t)$165
Lescop proved the existence of an equivariant propagator by means of homology theoetic arguments. We developed a notion o -paths’ (we previously called ‘AL-paths’) inand gave an explicit equivariant propagator by the natural
a surface bundle $M$ overmap from the moduli space of -paths to configuration space ([16]). By using the equivariant propagator, we construct an invariant of fiberwise Morse functions on $M$ ([17]). The
construction of the invariant can be applied to a construction of a perturbative isotopy
invariant of knots in $M$ , which is useful for the study of finite type invariants of knots in$M$ ([18]).
$f^{(}Z$$1^{\cdot}$$S^{i}$$Z$

2. MODULI SPACE OF$Z$-PATHSWe define the moduli space of -paths and its closure’.3$\grave{}$$Z$

2.1. -path. Let $M$ be an oriented closed 3-manifold. Assume that $M$ admits a structuremap :is a fiberwiseof an oriented fiber bundle $\kappa:Marrow S^{1}$ . We say that aforeach $s\in S^{1}$
Morsc
is
:
if
the
restriction
Morse function
く )$s\in S^{1}$ ,forms a(known to exist for every ). The totality of the critical points of
1-submanifold of $M$ (closed braid) and we call each component of the 1-submanifold acritical locus. Let be the gradient of along the fibers, namely, the one whose restriction
denote the union of all critical loci of . For
is grad f. Let
to each fiber overa critical locus of a fiberwise Morse function the descending/ascending manifold aredefined respectively by$Z$$C^{\infty}$$Marrow \mathbb{R}$$f$$\kappa^{-1}(s)arrow \mathbb{R}$$f_{\theta}=f|_{\kappa^{-1}}$$s$$f_{s},$$\kappa$

$f$

$\xi$

$s\in S^{1}$$\xi$$\Sigma(\xi)$

$f$

$p$

}

$\mathscr{D}_{p}(\xi)=\{x\in M|\lim_{tarrow-\infty}\Phi_{-\xi}^{\partial}(x)\inp\}$$\mathscr{A}_{p}(\xi)=\{x\in M|\lim_{tarrow\infty}\Phi_{-\xi}^{t}(x)\epsilon p\}$

where
Let

: $Marrow M$ is the flow of:be the pullback of$-\xi.$$\Phi_{-\xi}^{t}$$\tilde{M}arrow \mathbb{R}$$\tilde{\kappa}$$\kappa$by the projection$\tilde{M}\underline{\tilde{\kappa}}$$\mathbb{R}arrow \mathbb{R}/\mathbb{Z}=S^{1}.$IR$\pi\downarrow \}$$M\underline{\kappa}S^{1}$: $\tilde{M}arrow M$ on the total space is an infinite cyclic covering. The
is a fiberwise Morse function (for a fiber bundle over ). Letfunction $\tilde{f}=fo\pi$ :denote the gradient for along the fibers. By replacing with , the critical locus, its
descending/ascending manifolds are defined similarly.
is included in a single
is horizontal if
We say that an embedding :is included in a single critical locus
fiber of and say that it is vertical if:
is descending if
embeddingvertical)(resp.
horizontal. A horizontal embedding : $[\mu, v]arrow M$ is a flow-line of if(resp.
is a positive multiple offor each $t\in(\mu, \nu)$ ,The induced map$\pi$$\tilde{M}arrow \mathbb{R}$$\mathbb{R}$$\overline{\xi}$$\tilde{f}$$\tilde{\xi}$$\xi$$[\mu, \nu]arrow\overline{M}$${\rm Im}\sigma$$\sigma$$0_{\sim^{f\tilde{f}}}.$${\rm Im}\sigma$$\tilde{\kappa}$$\sigma$$A$$\tilde{f}(\sigma(\mu))\geq f(\sigma(\nu)\rangle$$[\mu, \nu]arrow\tilde{M}$$\tilde{\xi}$$\tilde{\kappa}(\sigma(\mu\rangle)\geq\tilde{\kappa}(\sigma(v)))$$\sigma$$(-\tilde{\xi})_{\sigma(t)}.$$d \sigma_{t}(\frac{\partial}{\partial t})$Definition 2.1. Let
sequence$\gamma=$$3_{The}$$x,$

A -path frombe such thatsatisfying the following six conditions.

$y\in\overline{M}$$(\sigma_{1}, \sigma_{2}, \ldots , \sigma_{n})$$\tilde{\kappa}(x)\geq\tilde{\kappa}(y).$definition in this note diﬀers slightly from that of [16].

$Z$

$x$

to

$y$

\’is a166$-flow$-line$Z$-path :

locupartof
acritical$\{$

FIGURE 1.

$Z$$of-\tilde{\xi}$-path

are real numbers such thatis an embedding
and it is either horizontal or vertical.
(2) For each
is descending.
(3) If is horizontal, then is a flow-line of If it is vertical, then
(1) For each$i,$$[\mu_{i}, \nu_{i}]arrow\tilde{M}(\mu_{i},$$\nu_{i}$$\sigma_{i}$$\mu_{i}\leq\nu_{i})$$i,$$\sigma_{i}$$\tilde{\xi.}$$\sigma_{i}$$\mu_{i}<\nu_{i}.$$\sigma_{i}$(4)(5)
for $1\leq i<n.$(6) If is horizontal (resp. vertical) and if $i<n$ , thental).
$\sigma_{1}(\mu_{1})=x,$$\sigma_{n}(\nu_{\mathfrak{n}})=y.$$\sigma_{i}(\nu_{i})=\sigma_{i+1}(\mu_{i+1})$$\sigma_{i}$$\sigma_{i+1}$

is vertical (resp. horizon-

We say that two -paths are equivalent if they are related by piecewise reparametrizations.
We call a sequence of paths of the form
for a -path in-path in $M.$
$Z$$\pi 0\gamma=(\pi 0\sigma_{1},$$\tilde{M}a$$Z$$\pi\circ\sigma_{n}\rangle$$\gamma$$\ldots,$$Z$

Let
be the set of all equivalence classes of -paths in . This has a natural
structure of a noncompact manifold with corners. Let denote the covering translationof the covering :that induces the translation $x\mapsto x-1$ in R. This induces$(x, y)\mapsto(t^{n}x, t^{n}y)$ on
diagonal ,-actions. We denote theandquotient spaces
respectively by
and
. We
and
,-action
consider anotheron the quotient spaces, denoted by by abuse of notation, as$\tilde{M}$$Z$$\mathscr{M}_{2}^{z}(\tilde{\xi})$$t$$\tilde{M}arrow M$$\pi$$\mathbb{Z}$$(\tilde{M}\cross\tilde{M})/\mathbb{Z}$$\mathscr{M}_{2}^{z}(\tilde{\xi})/\mathbb{Z}$$\tilde{M}\cross\tilde{M}$$\mathscr{M}_{2}^{z}(\tilde{\xi})$$\gamma\mapsto t^{n}\gamma,$$\mathscr{M}_{2}^{z}(\tilde{\xi})_{Z}$$\tilde{M}\cross z^{\tilde{M}}$$t^{n}$$\mathbb{Z}$$t([x\cross y])=[x\cross ty].$

For a gradient along the fiber for a fiberwise Morse function , let denote the nonsingular vector field $\xi+$ grad on $M$ . Let: $Marrow ST(M)(ST$ denotes the unit tangent
bundle) be the section given by
$f$

$\xi$

$\hat{\xi}$$\kappa$$s_{\hat{\xi}}$$-\hat{\xi}/\Vert\hat{\xi}\Vert.$

Theorem 2.2 ([16]). Let be an oriented connected closed surface and let $M$ be themapping torus of an orientation preserving diﬀeomorphism. Let:
be the preimage of the diagonalof $M\cross M.$$\Sigma$$\Sigmaarrow\Sigma$$\tilde{\Delta}_{M}\subset$$\varphi$$\tilde{M}\cross z^{\tilde{M}}$$\Delta_{M}$(1) There a naturalthat is a countable union of comof
pact manifolds with $CO7ners.$(2) Suppose that induces an $isomo7phismH_{1}(M)/$ Torsion. Let
be the map that assigns the endpoints. Let
be the$tclosure^{f}\overline{\mathscr{M}}_{2}^{z}(\tilde{\xi})_{Z}$$i\mathcal{S}$$\mathscr{M}_{2}^{Z}(\tilde{\xi})_{Z}$$\cong H_{1}(S^{1})$$\kappa$$\overline{M}\cross z^{\overline{M}}$blow-up of$\overline{b}:\overline{\mathscr{M}}_{2}^{z}(\tilde{\xi})_{Z}arrow$$B\ell_{\overline{b}^{-1}(\tilde{\Delta}_{M}\rangle}(\overline{\mathscr{M}}_{2}^{z}(\tilde{\xi})_{Z})$$\overline{\mathscr{M}}_{2}^{z}(\tilde{\xi})_{Z}$along$\overline{b}^{-1}(\tilde{\Delta}_{M})$. Then induces a map$B\ell_{\overline{b}^{-1}(\tilde{\Delta}_{M})}(\overline{\mathscr{M}}_{2}^{z}(\tilde{\xi})_{Z})arrow\overline{C\circ nf}_{K_{2}}(M)$167
that represents a 4-dimensionat
following identity in$\mathbb{Q}(t)$-chain$Q(\tilde{\xi})$of$\overline{Conf}_{K_{2}}(M)$$H_{3}(\partial\overline{Conf}_{K_{2}}(M);\mathbb{Q})\otimes_{\mathbb{Q}[t_{2}t^{-1}]}\mathbb{Q}(t)$. Moreover, theholds.

$[ \partial Q(\tilde{\xi})]=[s_{\hat{\zeta}}(M)]+\frac{t\zeta_{\varphi}’}{\zeta_{\varphi}}[ST(M)|_{K}],$is the Lefschetz zeta function for and $K$ is a knot such that
where
. Furthermore, there is a product $P(t)$ of cyis the positive generator of$P(t)\Delta(M)Q(\tilde{\xi})$
is the-chainis aclotomic polynomials such that$M$).
polynomial
Alexanderof$\kappa_{*}([K])$$\zeta_{\varphi}$$\varphi$$H_{1}(S^{1})$$\mathbb{Q}[t, t^{-1}]$

2.2. Closure of the moduli space of -paths. We define the$Z$paths in$\tilde{M}$$(\triangle(M)$$space_{\fbox{Error::0x0000}}\mathscr{K}_{2}(\tilde{ \xi})$

of horizontalby$\mathscr{M}_{2}(\tilde{\xi})=\{(x, y)\in\overline{M}\cross\overline{M};\tilde{\kappa}(x)=\tilde{\kappa}(y)$,$y=\Phi_{-\tilde{\xi}}^{t}(x\rangle$

for somc $t>0$ }.

denote the inclusion map. For a continuous parameter $s\in S$
and if the parameter is at mostbysuch as real numbers, we denote the sumLet$b:\mathscr{M}_{2}(\tilde{\xi})arrow\tilde{M}\cross\tilde{M}$$\int_{s\in S}V_{s}$$\bigcup_{s\epsilon s}V_{s}$countable, then we denote it by$\sum_{s\in S}V_{s}$the intersection
For a genericmanifold. There is a free -action on$\tilde{\xi,}$

or $V_{81}+V_{s}2+\cdots$ etc.

$\mathscr{D}_{p}(\tilde{\xi})\cap \mathscr{A}_{q}(\tilde{\xi})_{-}is$$\mathbb{R}$$\mathscr{D}_{p}(\tilde{\xi})\cap \mathscr{A}_{q}(\xi)$transversal and hence is a smooth. We putby$x\mapsto\Phi_{-\tilde{\xi}}^{T}(x)(T\in \mathbb{R}\rangle$$\mathscr{M}_{pq}(\tilde{\xi})=(\mathscr{D}_{p}(\tilde{\xi})\cap \mathscr{A}_{q}(\tilde{\xi}))/\mathbb{R}.$and the extension :Proposition 2.3. There is a natural closureofdenotesof such that for a generic the following holdthe diagonal for any set ).
$\overline{b}$$\overline{\mathscr{M}}_{2}(\tilde{\xi})$$\overline{\mathscr{M}}_{2}(\tilde{\xi})arrow\overline{M}\cross\overline{M}$$\tilde{\xi}$$b$$\mathscr{M}_{2}(\tilde{\xi)}$$(\Delta_{S}\subseteq S\cross S$

$S$

(1)
is a manifold with corners.
.
(2) induces a diﬀeomorphism Int
corresponds to broken flow-lines(3) The codimension stratum ofthat are broken times at critical points. The codimension stratum offor $r\geq 1$ is canonically diﬀeomorphic to$\overline{\mathscr{M}}_{2}(\tilde{\xi})-\overline{b}^{-1}(\Delta_{\tilde{M}}\rangle$$\overline{\mathscr{M}}_{2}(\tilde{\xi})arrow \mathscr{M}_{2}(\tilde{\xi})$$\overline{b}$$\overline{\mathscr{M}}_{2}(\tilde{\xi})-\overline{b}^{-1}(\Delta_{\tilde{M}})$$r$$\overline{\mathscr{M}}_{2}(\tilde{\xi})-$

$r$

$r$

$\overline{b}^{-1}(\triangle_{\tilde{M}})$$\{\begin{array}{ll}\prime_{s\in \mathbb{R}}\sum_{q_{1\in\Sigma}(\tilde{\xi})}\mathscr{A}_{qx}(\tilde{\xi_{s}})\cross \mathscr{D}_{qx}(\tilde{\xi_{s}})-\sum_{qz\in\Sigma(\tilde{\xi})}\Delta_{q_{1}} (r=1)\int_{s\epsilon R}\sum_{r^{disti}q_{1},..,qnct}\mathscr{A}_{q\iota}(\tilde{\xi_{s}})\cross \mathscr{M}_{q_{1}q_{2}}(\tilde{\xi_{s}})q_{1.’\cdot\cdot\backslash ,Qr\in\Sigma(\tilde{\epsilon})}\cross\cdots\cross \mathscr{M}_{q_{r}} \cross \mathscr{D}_{q,}(\tilde{\xi_{s}}) (r\geq 2)\end{array}$

The formula for the codimension
as follows.

$r$stratum $(r\geq 2)$ in Proposition 2.3 can be rewritten

168
Here, if$\Sigma(\tilde{\xi})=\{p_{1},p_{2}, .

.

.

,p_{N}\}$, then$X(s)=(\mathscr{A}_{p_{1}}(\tilde{\xi_{s}})\mathscr{A}_{p_{2}}(\tilde{\xi_{s}}) \cdots \mathscr{A}_{PN}(\tilde{\xi_{\theta}}))$,$Y(s)=(\mathscr{D}_{P1}(\tilde{\xi_{s}})\mathscr{D}_{p_{2}}(\tilde{\xi_{s}}) .

,..

\mathscr{D}_{p_{N}}(\tilde{\xi_{s}}))$$\Omega(s)=(\mathscr{M}_{p_{N}p_{1}}(\tilde{\xi_{\delta}})\mathscr{M}_{p_{3}p_{1}}(\tilde{\xi_{s}})\mathscr{M}_{P2P1}(\tilde{\xi_{s}})\emptyset \mathscr{M}_{p_{N}p_{2}}(\tilde{\xi}\mathscr{M}_{p_{3}p_{1}}(\tilde{\xi}\mathscr{M}_{p_{1}p_{2},\emptyset}(\tilde{\xi} \mathscr{M}_{p_{NP3}}(\tilde{\xi_{\theta}})\mathscr{M}_{p2P3,\emptyset}(\tilde{\xi_{s}})\mathscr{M}_{P1p_{3}}(\tilde{\xi_{s}}) \mathscr{M}_{p_{3PN}}..(\tilde{\xi_{s}})\mathscr{M}_{P2r_{\emptyset}N}(\tilde{\xi_{8}})\mathscr{M}_{p_{1PN}}.(\tilde{\xi_{s}}))$and the direct product of matrices is defined by replacing multiplications and sums withdirect products and disjoint unions, respectively.
Proposition 2.4. Let be a critical locus of and let. For a generic , the following are satisfied.
$\tilde{\xi}$$p$$\overline{b}^{-1}(\overline{M}\cross p)$$\overline{\mathscr{D}}_{p}(\tilde{\xi})=\overline{b}^{-1}(p\cross M$$\overline{\mathscr{A}}_{p}(\tilde{\xi})=$$\tilde{\xi}$(1)(resp.
) is a manifold with corners.
.
(resp. Int(2) induces a diﬀeomorphism Int...
(resp. X(3) The codimension stratum offor $r\geq 1$ is canonically diﬀeomorphic to$\overline{\mathscr{D}}_{p}(\tilde{\xi})$$\overline{\mathscr{A}}_{p}(\tilde{\xi})$$\overline{b}$$\overline{\mathscr{A}}_{p}(\tilde{\xi})arrow \mathscr{A}_{p}(\tilde{\xi}\rangle)$$\overline{\mathscr{D}}_{p}(\tilde{\xi})arrow \mathscr{D}_{p}(\tilde{\xi})$$t\overline{Y}=(\overline{\mathscr{D}}_{p_{1}}(\tilde{\xi})$$r$$(\overline{\mathscr{A}}_{p_{1}}(\tilde{\xi})\overline{\mathscr{A}}_{p_{2}}(\tilde{\xi}) .

.

.

$\overline{\mathscr{D}}_{p_{N}}(\tilde{\xi})\rangle$$\overline{\mathscr{D}}_{p_{2}}(\tilde{\xi})$$=$\overline{\mathscr{A}}_{p_{N}}(\tilde{\xi})))$(resp.

Proposition 2.5. Let
the following fold.

$p,$ $q$

be critical loci of$\tilde{\xi}$and let$\overline{\mathscr{M}}_{pq}(\tilde{\xi})=\overline{b}^{-1}(p\cross q)$. For a generic$\tilde{\xi,}$(1)
is a manifold with comers.
(2) There is a natural diﬀeomorp hism Int(3) The codimension stratum of
diﬀeomorphic to$\overline{\mathscr{M}}_{pq}(\tilde{\xi})$$\overline{\mathscr{M}}_{pq}(\tilde{\xi})arrow \mathscr{M}_{pq}(\tilde{\xi})$$\overline{\Omega}=((1-\delta_{ij})\overline{\mathscr{M}}_{p_{i}p_{j}}(\tilde{\xi}))$

$r$

.

for

$r\geq 1$

is canonicallyfiberwise space over a space is a pair of a space and a continuous map $\phi:Earrow B.$
fiber over a point $s\in B$ is $E(s)=\phi^{-1}(s)$ ([2]). For two fiberwise spaces
is defined as the following subspace
andoverfiberwise product
$E$

$B$

$A$

$E_{1}=(E_{1}, \phi_{1})$$A$$E_{2}=(E_{2}, \phi_{2})$of$E_{1}\cross E_{2}$$B,$$E_{1}\cross E$$a$:$E_{1} \cross B2\int_{s\in B}E_{1}(s)\cross E_{2}(s)$Namely,$E_{1}\cross BE_{2}$is the pullback of$E_{1^{-}}^{\phi}4Be_{E_{2}}.$

.

169
,
: $A_{i}arrow \mathbb{R}(i=1,2, \ldots , n)$ of fiberwise spaces overFor a sequence
we define its iterated integrals as$A_{i}=(A_{i}, \phi_{i})$$\phi_{i}$$\mathbb{R},$$\int_{\mathbb{R}}A_{1}A_{2}\cdots A_{n}=\int_{s1>32>\cdots>8_{h}}A_{1}(s_{1})\cross A_{2}(s_{2})\cross\cdots\cross A_{n}(s_{n})$$=(\phi_{1}\cross\cdots\cross\phi_{n})^{-1}(\{(s_{1}, \ldots, s_{n})\in \mathbb{R}^{n}|s_{1}>\cdots>s_{n}$$\overline{\int_{R}}A_{1}A_{2}\cdots A_{n}=\int_{sz\geq s_{2}\geq\cdots\geq s_{n}}A_{1}(s_{1})\cross A_{2}(s_{2})\cross\cdots\cross A_{n}(s_{n})$$=(\phi_{1}\cross\cdots\cross\phi_{n})^{-1}\langle\{(s_{1)}\ldots, s_{n})\in \mathbb{R}^{n}|s_{1}\geq\cdots\geq s_{n}\})$For a matrix $P=(A_{\iota’j})$ of fiberwise spaces over , we define a fiber of$(A_{ij}(s))$ .
ices of fiberwise spaces over
Then iterated integrals forby similar formulas as above.
of fiberwise spaces over by
We define matrices $X,$
$Y,$

by $P(s)=$can be defined$s\in \mathbb{R}$$\mathbb{R}$$mat_{I}\cdot$$\mathbb{R}$$\mathbb{R}$$\Omega$$X= (\mathscr{A}_{P1}(\tilde{\xi})\mathscr{A}_{p_{2}}(\tilde{\xi}) .

..

\mathscr{A}_{p_{N}}(\tilde{\xi})) , Y=(\mathscr{D}_{p_{1}}(\tilde{\xi})\mathscr{D}_{p_{2}}(\tilde{\xi}) .

..

\mathscr{D}_{PN}(\tilde{\xi}))$,$\Omega=((1-\delta_{\dot{z}j})\mathscr{M}_{p_{i}p_{j}}(\tilde{\xi}))_{1\leq i,i\leq N}.$

Then the space of -paths in$\tilde{M}$$Z$

is rewritten by means of the iterated integrals as follows.

$\mathscr{M}_{2}^{z}(\tilde{\xi})=\mathscr{M}_{2}(\tilde{\xi})+\int_{\mathbb{R}}XtY+\int_{\mathbb{R}}X\Omega tY+\int_{\mathbb{R}}X\Omega\Omega tY+\cdots$

We would like to define the ‘closure’ of this space.
Lemma 2.6. For a generic , the space$\tilde{\xi}$$\overline{\int_{\mathbb{R}}}\overline{X}\overline{\Omega}\cdots\overline{\Omega}t\overline{Y}\vee$is the disjoint union of finitely$n$many manifolds }$ith$ corners, and the closure of its codimension 1 stratum is given by thefollowing formula.
$u$$\overline{\int_{\mathbb{R}}}(\partial\overline{X})\overline{\Omega}\cdots\overline{\Omega}t\overline{Y}+n\sum_{i=1}^{-}\prime_{\mathbb{R}}\overline{X}\overline{\Omega}_{\check{i-1}}\overline{\Omega}(\partial\overline{\Omega})\overline{\Omega}_{\check{n-i}}\overline{\Omega}t\overline{Y}+\overline{\int_{\mathbb{R}}}\overline{X}\overline{\Omega}\cdots\overline{\Omega}(\partial^{t}\overline{Y})\check{n}\ldots\ldots\check{n}$$+ \overline{\int_{R}}(\cross \mathbb{R}\ldots\overline{\Omega}t\overline{Y}+\sum_{i=1}^{n-1}\overline{\int_{R}}\cdots 1R.$

For $n\geq 0$ , letin Lemma 2.6.

$S_{n}$(resp.

$T_{n}$) denote the first line (resp. the second line) of the formula

Lemma 2.7. There is a natural stratification preserving diﬀeomorphisms$\partial X\cong\overline{X}\cross \mathbb{R}\overline{\Omega}, \partialt\overline{Y}\cong\overline{\Omega}\cross \mathbb{R}t\overline{Y},$$\partial\overline{\Omega}\cong\overline{\Omega}\cross R\overline{\Omega}, \partial\overline{\mathscr{M}}_{2}(\tilde{\xi})\cong\Delta_{\tilde{M}}+\overline{X}\cross \mathbb{R}t\overline{Y}.$These induce, for $n\geq 0$ , a stratification preserving $diﬀeomo7phism$$S_{n}\cong T_{n+1}.$

Let
Lemma 2.7.

$S_{-1}\subset\partial\overline{\mathscr{M}}_{2}(\tilde{\xi})$

be the face that corresponds to$\overline{X}\cross \mathbb{R}t\overline{Y}$by the diﬀeomorphism of

170
Definition 2.8.
$\overline{\mathscr{M}}_{2}^{z}(\tilde{\xi})=[\overline{\mathscr{M}}_{2}(\tilde{\xi})+\overline{\int_{R}}\overline{X}t\overline{Y}+\overline{\int_{R}}\overline{X}\overline{\Omega}t\overline{Y}+\overline{\int_{R}}\overline{X}\overline{\Omega}\overline{\Omega}t\overline{Y}+\cdots]/\sim$

Here, for each $n\geq 0$ , we identify
by the diﬀeomorphism of Lemma 2.7.
with$(x_{1}, x_{2}, \ldots, x_{n})\mapsto(tx_{1}, tx_{2)}tx_{n})$byacts on. We put$S_{n-1}$$T_{n}$$\mathbb{Z}$$\overline{\mathscr{M}}_{2}^{z}(\tilde{\xi})$$\overline{\mathscr{M}}_{2}^{z}(\tilde{\xi})_{\mathbb{Z}}=\overline{\mathscr{M}}_{2}^{z}(\tilde{\xi})/\mathbb{Z}.$Outline of the proof of Theorem 2.2. By fixing orientations on the manifold pieces in the, the map :stratified spacea-chain of. (The proof that the coeﬃcients are rational functions is an analogue of theproof of the rationality of Novikov complexes by Pajitnov ([14, 15
By Lemmas 2.6,
2.7 and by checking the orientations on the gluing parts, it turns out that the boundaryofconcentrates on the liftof the diagonal. Hence the boundary of
consists of -paths with endpoints agree (in ) and of closed Z–paths(in $M$ ). One sees that in the homology class of the boundary of, thepart for closed -paths corresponds to the logarithmic derivative of the Lefschetz zetafunction.
$\overline{\mathscr{M}}_{2}^{z}(\tilde{\xi})_{Z}$$\overline{b}$$\overline{\mathscr{M}}_{2}^{z}(\tilde{\xi})_{Z}arrow\tilde{M}\cross z^{\tilde{M}represents}$$\mathbb{Q}(t)$$\tilde{M}\cross z^{\tilde{M}}$$\overline{\mathscr{M}}_{2}^{z}(\tilde{\xi})_{Z}$$\tilde{\Delta}_{M}$$\Delta_{M}$$\tilde{M}$$Z$$B\ell_{\overline{b}^{-1}(\tilde{\Delta}_{M})}(\overline{\mathscr{M}}_{2}^{z}(\tilde{\xi})_{Z})$$BP_{\overline{b}^{-1}(\tilde{\Delta}_{M})}(\overline{\mathscr{M}}_{2}^{z}(\tilde{\xi})_{Z})$$Z$

3. PERTURBATION THEORY FOR$\hat{\Lambda}$-COEFFICIENTSPut. Since a recipe for the perturbation theory for Lie algebra
local coeﬃcient systems is given by Axelrod-Singer, Kontsevich ([1, 5 it is expectedthat one can obtain a perturbative invariant with -coeﬃcients if there is an appropriate
propagator with -coeﬃcients. The-chain given by the moduli space of -paths canbe considered as an appropriate equivariant propagator and we use it.
$\Lambda=\mathbb{Q}[t, t^{-1}],$$\hat{\Lambda}=\mathbb{Q}(t)$$\hat{\Lambda}$$\hat{\Lambda}$$Z$$\mathbb{Q}(t)$-colored graph. We call a finite connected graph with edges oriented a graph.
vertex-orientation of a graph is an assignment of cyclic order of edges incident to eachvertex. For a vertex-oriented graph , a -coloring of is a mapping

3.1.

$\hat{\Lambda}$$A$$\hat{\Lambda}$$\Gamma$$\Gamma$$\phi:Edges(\Gamma)arrow\hat{\Lambda}.$Definition 3.1 (Garoufalidis-Rozansky [4]).
$d_{n}( \hat{\Lambda})=\frac{span_{\mathbb{Q}}\{\Gamma:3-va1ent,2nverticae,\hat{\Lambda}-co1oredvertex-orientedgraphs\}}{AS,1HX,orientationreversa1,Linearity,Ho1onomy}$$p(t)\downarrow$$=$$\uparrow p(t^{-1})$

orientation
Reversal$p+\alpha q|=p\dagger+\alpha(q\})$Linearity

Holonomy171
3.2. Equivariant configuration spac and equivariant intersection. Let: $Marrow$$\kappa$$e(l)$$S^{1}$

be a fiber bundle$M^{\Theta} :=\{(x_{1},x_{2};\gamma_{1}, \gamma_{2},\gamma_{3})|x_{1}, x_{2}\in M,$$\gamma_{i}$: homotopy class of: $[0, 1]arrow S^{1}$ such that

$C_{?}$$c_{i}(0\rangle=\kappa(x_{1}), c_{i}(1)=\kappa(x_{2})\}.$

When $H_{1}(M)=\mathbb{Z}$ , the homotopy class of is the same thing as the relative bordism
forclass of the lift : $[0, 1]arrow M$ of . The equivariant configuration spaceis defined by$:=Bl(M^{e},$ preimage,
where $BP(X, A)$ is the blow-up of (real) manifold $X$ along a submanifold . The projection
is a -covering and we$c_{i}$$\gamma_{i}$$\overline{Conf}_{\Theta}(M)$$\tilde{c_{i}}$$c_{f}$$\Theta$$\overline{Conf}e(M)$$of \Delta_{M})$

$A$

$a$

$\overline{Conf}_{e}(M\ranglearrow\overline{Conf}_{2}(M)$$\mathbb{Z}^{3}$$hav\fbox{Error::0x0000}\pi_{0}(\overline{Conf}_{ \Theta}(M))\approx H^{1}( \Theta, \mathbb{Z})=$$[\Theta, S^{1}].$

By extending the intersections of chains by -linearity, we define the multilinear form$\hat{\Lambda}$$Q_{1}\otimes Q_{2}\otimes Q_{3}\mapsto\langle Q_{1)}Q_{2}, Q_{3}\rangle_{\Theta}\in C_{0}(\overline{Conf}_{\Theta}(M);\mathbb{Q})\otimes_{\Lambda\otimes 3}\hat{\Lambda}^{\otimes 3}$$(\Lambda^{\otimes 3}=\mathbb{Q}[t_{1}^{\pm 1}, t_{2}^{\pm 1}, t_{3}^{\pm 1}],\hat{\Lambda}^{\otimes 3}=\mathbb{Q}(t_{1})\otimes_{\mathbb{Q}}\mathbb{Q}(t_{2})\otimes_{\mathbb{Q}}\mathbb{Q}(t_{3}))$

in $Conf_{K_{2}}(M)$ .
chainsWe define the trace:$Q_{1},$ $Q_{2},$for ‘generic’ 4-dimensional$\hat{\Lambda}-$$Q_{j\}}$$Tr_{\Theta}$$\hat{\Lambda}^{\otimes 3}arrow \mathscr{A}_{1}(\hat{\Lambda})$for A colored graphs byThis induces the following map.
$rr_{r_{\Theta}}$:$H_{0}(\overline{Conf}_{8}(M);\mathbb{Q})\otimes_{\Lambda\otimes 3}\hat{\Lambda}^{\otimes 3}arrow H_{0}(\overline{Conf}_{2}(M);\mathbb{Q})\otimes_{\mathbb{Q}}\mathscr{A}_{1}(\hat{\Lambda})=\mathscr{A}_{1}(\hat{\Lambda})$Similarly, for a 3-valent graph$Q_{1}\otimes Q_{2}\otimes\cdots\otimes Q_{3n}$$Q_{2}$, . ..,$\mapsto$$\langle Q_{1},$$\mapsto$$Tr_{1^{t}}\langle Q_{1},$$Q_{2}$($r$with $2n$ vertices$Q_{i}$: codimension 2$く 3n$$\hat{\Lambda}$.

edges), we obtain-chain in$\overline{C\circ nf}_{K_{2}}(M)$or in $M$ )$Q_{3n}\rangle_{r}\in C_{0}(\overline{Conf}_{\Gamma}(M);\mathbb{Q})\otimes_{\Lambda\otimes Sn}\hat{\Lambda}^{\otimes 3n}$, . .. ,$Q_{3n}\rangle_{f’}\in H_{0}(\overline{Conf}_{2n}(M);\mathbb{Q})\otimes_{\mathbb{Q}}\mathscr{A}_{n}(\hat{\Lambda})=\mathscr{A}_{n}(\hat{\Lambda})$

.

. Let
: $Marrow S^{1}$ be an oriented surface bundle such that
Definition 3.2. Let,
gradientfor
let be thebe an oriented
erwise Morse function. We definealong the fibers $(i=1,2,$
as follows.
$\kappa_{i}\simeq\kappa$$\kappa_{i}$$f_{i}:Marrow \mathbb{R}$$4fi$$(w.r.t. \kappa_{i})$$Z_{n}$$3n\rangle$$\kappa_{i}$$\xi_{i}$$\ldots,$$Z_{n}:= \sum_{\Gamma}Tr_{\Gamma}\langle Q(\tilde{\xi_{1}})$,$Q(\tilde{\xi_{2}})$,...,$Q(\tilde{\xi_{3n}})\rangle_{\Gamma}\in \mathscr{A}_{n}(\hat{\Lambda})$The sum is over all (labeled) 3-valent graphs with $2n$ vertices.
$4$

Namely,$\mathscr{D}_{p}(\xi_{i})$is oriented for each critical locus

$p.$

.

172

FIGURE 2.

$Z$-graphTheorem 3.3 ([17]).
$\hat{Z}_{n}=Z_{n}-Z_{n}^{anomaly}(\vec{\rho}_{W})\in d_{n}(\hat{\Lambda})$is an invariant of.
is a term obtained by counting aﬃnegraphs in a rank 3 vector bundle over some compact 4-manifold $W$ such that $\partial W=M.$
Here,(1) is a spin structure on $M.$
(2) $[\kappa]\in H^{1}(M)$ is the homotopy class of(3) $[f]$ is the ‘concordance class’ of an oriented fiberwise Morse function $f:Marrow \mathbb{R}.$(Oriented fiberwise Morse functions
and are concordant if there a generic
homotopy $F$ :betweenandsuch that for each birth-deathlocus, its projection to, 1 is a simple closed curve and is not nullhomotopic.)$(Z_{n}^{anomaly}(\vec{\rho}_{W})$$(M,\mathfrak{s}, [\kappa], [f])$$\mathfrak{s}$$\kappa.$$f_{0}$$M\cross[O, 1]arrow \mathbb{R}$$S^{1}\cross[0$$f_{1}$$f_{0}$$i\mathcal{S}$$f_{1}$$]$, one must show that
To get an invariant ofdoes not depend on the choiceof concordance class of oriented fiberwise Morse functions, namely, thatis invariant
under a generic homotopy of oriented fiberwise Morse functions. However, as suggestedby the definition of concordance, the topology of the moduli space of -paths may change
if there is a birth-death locus whose projection on, 1 is nullhomotopic. We guessthat the restriction of the homotopy to concordances might be too strong.
Though, this is suﬃcient to study finite type isotopy invariants of knots in a 3-manifold([18]). Thanks to the definition ofby -paths, Theorem 3.3 can be proved by astandard argument (constructing a cobordism between moduli spaces on the endpoints)
without diﬃculty.
$\hat{Z}_{n}$$(M, [\kappa])$$\hat{Z}_{n}$$Z$$S^{1}\cross[0$$\hat{Z}_{n}$

$]$

$Z$

-graph. Now we explain thatcan be defined by counting certain graphs. In
the following, we only consider the graphfor simplicity.

3.3.

$Z$

$Z_{n}$$\Gamma=\Theta$

Definition 3.4. Put$\Sigma=\kappa^{-1}(0)$. For$(a_{1}, a_{2}, a_{3})\in \mathbb{Z}^{3}$, we define$\mathscr{M}_{\Theta(a_{1},a_{2},a_{S})}^{z}(\Sigma;\xi_{1}, \xi_{2}, \xi_{3})$as the set of maps :such that(1) i-th edge is a -path for$\Thetaarrow M$

$I$

$Z$

$\xi_{i}.$(2) ( -th edge of )(count with signs)$\ThetaarrowM$
We call such a map :a -graph.
$\#$

$I$

$i$

$I$

$\cap\Sigma=a_{i}$$Z$This definition is an analogue of the flow-graphs considered in Fukaya’s Morse homotopy
theory [3]. The following lemma can be proved by a transversality argument as in [3].

173$\xi_{i}(i=1,2,3)$ , the moduli space
Lemma 3.5. For a genericis a compact oriented -dimensional manifold$\kappa_{i},$$\mathscr{M}_{\Theta(a1,a2,a3)}^{Z}(\Sigma;\xi_{1},\xi_{2_{\rangle}}\xi_{3})$$(\forall(a_{1}, a_{2}, a_{3}\rangle\in \mathbb{Z}^{3})$$0$Proposition 3.6. Choose$\kappa_{i},$$\xi_{i}(i=1,2,3)$

generically as in the Lemma. Put$F_{\Theta}:= \sum_{3(a1a2,a)\epsilon \mathbb{Z}^{3}}\#\mathscr{M}_{\Theta(a_{1},aa_{3})}^{z}2,(\Sigma;\xi_{1}, \xi_{2},\xi_{3})t_{1}^{a1}t_{2}^{a_{2}}t_{3}^{\emptyset 3}.$

Then there exist a poiynomial
cyclotomic polynomials such that

$P(t_{1}, t_{2}, t_{3})\in \mathbb{Q}[t_{1}^{\pm 1}, t_{2}^{\pm 1}, t_{3}^{\pm 1}]$and a product $C(t)\in\Lambda$ of$F_{8}= \frac{P(t_{1},t_{2},t_{3})}{C(t_{1})C(t_{2})C(t_{3})\Delta(t_{1})\Delta(t_{2})\Delta(t_{3})}$$=\langle Q(\tilde{\xi_{1}}) , Q(\tilde{\xi_{2}}) , Q(\tilde{\xi_{3}})\rangle_{\Theta}$

holds. ($\Delta(t)$is the Alexander polynomial of $M$)

Acknowledgment. I would like to thank the organizers, Professors Teruaki Kitano,
Takayuki Morifuji, Ken’ichi Ohshika and Yasushi Yamashita, of RIMS Seminar
Geometry and Algebra of low-dimensional manifolds’ for inviting me to the workshop.
This work is supported by JSPS Grant-in-Aid for Young Scientists (B) 26800041.
$Topology_{\rangle}$

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$S^{1}$$S^{1}$$S$

DEPARTMENT OF MATHEMATICS, SHIMANE UNIVERSITY-mail address: tadayuki@riko. shimane-u. ac. jp$E$$S^{1}$