数理解析研究所講究録
第 1911 巻 2014 年 122-129

122

An alternative construction of$Kontsevich-Kuperberg-Thurston$ ’s universal finite typeinvariant of homology 3-spheres
Tatsuro Shimizu
Research Institute for Mathematical Sciences, Kyoto University

1

Introduction

Kontsevich-Kuperberg-Thurston invariant is one variation of M. Kontsevich’s ChernSimons perturbation theoretic invariant. G. Kuperberg and D. Thurston ([5]) gave theconstruction of the invariant based on M. Kontsevich’s idea in [4] and they showed that
this invariant is a universal finite type invariant for integral homology 3-spheres as the
LMO is.
Kontsevich-Kuperberg-Thurston invariant, denoted by $z^{KKT}$ , is a sequenceis a topological invariant of rational homology 3-spheres taking values in the finite
dimensional rational vector space.
is the quotient space divided by somerelations (called IHX, AS relations) from the vector space freely generated by oriented
Jacobi diagrams with $2n$ -vertexes. We don’t give an explicit definition of this space andJacobi diagrams (For example, see [5], [6]). In this article we treat only the case of $n=1.$
In this caseis isomorphic to the 1-dimensional vector space . So we take and fix
such an isomorphism and then we consider
asvalued invariant. It is known
equals to times the Casson-Walker invariant.
that$\{z_{n}^{KKT}\}_{n\in N}.$$z_{n}^{KKT}$$\mathcal{A}_{n}(\emptyset)$$\mathcal{A}_{n}(\emptyset)$$\mathcal{A}_{1}(\emptyset)$$\mathbb{Q}$$z_{1}^{KKT}$$z_{1}^{KKT}$

2$a\mathbb{Q}$$\frac{4}{3}$Preliminary

In this article, all homology 3-spheres are oriented, smooth and with a metric. The
assumptions “smooth”, “oriented and “with a metric” are not usual. We will use thesestructures in the construction of the invariant. The invariant is, however, independent of
the choices of these structure (i.e. topological invariant).
Let be a rational homology 3-sphere. Let $\infty\in Y$ be a base point. Take $N(\infty;Y)\subset Y$ina neighborhood ofin and let $N(\infty;S^{3})$ be a neighborhood of: $(N(\infty;Y), \infty)arrow\simeq(N(\infty;S^{3}), \infty)$
We take an orientation preserving diﬀeomorphism$Y$$\infty$$Y$$\infty$$\varphi^{\infty}$$S^{3}=\mathbb{R}^{3}\cup\{\infty\}.$123
and then we identify $N(\infty;Y)$ with $N(\infty;S^{3})$ via. So we consider
under this identification.
We introduce a compactification $C_{2}(Y)$ of the two point configuration space$\triangle=\{(x, y)|x\neq y\}$ . We denote by $B\ell(A, B)$ the real blowing-up of
along
for
submanifold in $A:B\ell(A, B)=A\backslash B\cup S\nu_{B}$ . Hereis the normal bundle of inandis the unit sphere bundle of:Let
be the blow-down map. There are three submanifolds. The over-lineofandmeans that the closure. We define$N(\infty;Y)\backslash \infty\subset \mathbb{R}^{3}$$\varphi^{\infty}$$(Y\backslash \infty)^{2}\backslash$

$B$

$A$

$B$

$S\nu_{B}$$q_{1}$$B$$\nu_{B}$$A$$\nu_{B}.$$B\ell(Y^{2}, \infty^{2})arrow Y^{2}$$\overline{q_{1}^{-1}((Y\backslash \infty)\cross\infty)},$$q_{1}^{-1}(\infty\cross(Y\backslash \infty))$$B\ell(Y^{2}, \infty^{2})$$q_{1}^{-1}(\triangle\backslash \infty^{2})$$C_{2}(Y)=Bl(B\ell(Y^{2}, \infty^{2}), \overline{q_{1}^{-1}((Y\backslash \infty)\cross\infty)}\sqcup\overline{q_{1}^{-1}(\infty\cross(Y\backslash \infty))}\sqcup\overline{q_{1}^{-1}(\triangle\backslash \infty^{2})})$.

We denote by : $C_{2}(Y)arrow Y^{2}$ the composition of blow-down maps. $C_{2}(Y)$ is a closed 6manifold with boundary and corner. It is known that there is a natural smooth structure
on $C_{2}(Y)$ ([6]).
$q$3The original construction of Kontsevich-Kuperberg-Thurston
invariantTake
be unit vectors. Let :framing such thatcoincides with the standard trivialization of,
To define the invariant, we will construct 4-cycles
and$(C_{2}(Y), \partial C_{2}(Y))$ by using,andWe first construct 3-dimensional submanifolds
Let: $C_{2}(S^{3})arrow S^{2}$ be the extended map of$a_{1},$$a_{2},$$a_{3}\in S^{2}\subset \mathbb{R}^{3}$$\tau$$T(Y\backslash \infty)arrow\underline{\simeq}(Y\backslash \infty)\cross \mathbb{R}^{3}$$\tau|_{N(\infty;Y)\backslash \infty}$$W_{1}(\tau)$$W_{2}(\tau)$be a$T\mathbb{R}^{3}=\mathbb{R}^{3}\cross \mathbb{R}^{3}.$$W_{3}(\tau)$of$W_{3}^{\partial}(\tau)$

of$\tau.$$W_{1}^{\partial}(\tau)$$p_{S^{3}}$$W_{2}^{\partial}(\tau)$$\mathbb{R}^{3}\cross \mathbb{R}^{3}\backslash \trianglearrow S^{2},$

.
. Let

$\partial C_{2}(Y)$$(x, y) \mapsto\frac{y-x}{\Vert y-x\Vert}$

124$p_{0}:\partial C_{2}(Y)\backslash q^{-1}(\triangle\backslash \infty^{2})arrow S^{2}$be the smooth map defined as follows:$p_{0}|_{q^{-1}(\infty^{2})}=p_{S^{3}}|_{q^{-1}(\infty^{2})}:q^{-1}(\infty^{2})arrow S^{2},$$p_{0}|_{q^{-1}((Y\backslash \infty)\cross\infty)}$:$p_{0}|_{q^{-1}(\infty\cross(Y\backslash \infty))}$: $q^{-1}(\infty\cross(Y\backslash \infty))=ST_{\infty}Y\cross(Y\backslash \infty)arrow ST_{\infty}Y=S^{2}.$$q^{-1}(Y\backslash \infty)\cross\infty=(Y\backslash \infty)\cross ST_{\infty}Yarrow ST_{\infty}Y=S^{2}arrow-1S^{2}$andHere the maps
are projection
andis an involution defined by $x\mapsto-x$ . Thenandare 3-dimensional submanifolds of. There is a canonical bundle
isomorphism. We define:
as follows:$ST_{\infty}Y\cross(Y\backslash \infty)arrow ST_{\infty}Y,$$ST_{\infty}Y\cross(Y\backslash \infty)arrow ST_{\infty}Y$$S^{2}arrow-1S^{2}$$p_{0}^{-1}(a_{1}),p_{0}^{-1}(a_{2})$$p_{0}^{-1}(a_{3})$$\partial C_{2}(Y)\backslash q^{-1}(\triangle\backslash \infty^{2})$$p(\tau)$$\nu_{\Delta\backslash \infty^{2}}\cong ST(Y\backslash \infty)$$q^{-1}(\triangle\backslash \infty^{2})arrow S^{2}$$q^{-1}(\triangle\backslash \infty^{2})=Sv_{\Delta\backslash \infty^{2}}\cong ST(Y\backslash \infty)\cong \mathcal{T}(Y\backslash \infty)\cross S^{2}arrow S^{2}.$Here the last map is the projection.

Definition 3.1. For $i=1$ , 2, 3,$W_{i}^{\partial}(\tau)=p_{0}^{-1}(a_{i})\cup p(\tau)^{-1}(a_{i})$We remark thatsumption of$(C_{2}(Y), \partial C_{2}(Y))$ .

is a compact 3-manifold without boundary owing to the asrepresents a cyclel of. We next extend it to a cycle of$W_{i}^{\partial}(\tau)$$W_{i}^{\partial}(\tau)$$\tau.$$\partial C_{2}(Y)$Lemma 3.2. There exists a 4-cycle
for $i=1$ , 2, 3.

Proof. Since$Y$.

$W_{i}(\tau)$of$(C_{2}(Y), \partial C_{2}(Y))$

such that$\partial W_{i}(\tau)=W_{i}^{\partial}(\tau)$

is a rational homology 3-sphere, the boundary map
is an isomorphism.

$H_{4}(C_{2}(Y), \partial C_{2}(Y);\mathbb{Q})arrow H_{3}(\partial C_{2}(Y);\mathbb{Q})$We take$W_{i}(\tau)$,$\square$as above.

Remark 3.3. This 4-cycle or the Poincar\’e dual of this 4-cycle is called a propagator.
For generic,, the intersection
andoriented -dimensional manifold. So we can count it with sign.
$W_{1}(\tau)$$W_{2}(\tau)$$W_{1}(\tau)\cap W_{2}(\tau)\cap W_{3}(\tau)$$W_{3}(\tau)$

is a compact$0$

Theorem 3.4 (Kuperberg and Thurston [5]). For generic$a_{i}$

and$W_{i}(\tau)$,$z_{1}^{KKT}(Y)= \#(W_{1}(\tau)\cap W_{2}(\tau)\cap W_{3}(\tau))+\frac{1}{4}\sigma(\tau)$

is a topological invariant of Y. In particular, $z_{1}^{KKT}(Y)$ is independent of the choice ofHeresatisfying
Let$\sigma(\tau)\in \mathbb{Z}$We definebe a framing ofis the signature defect of defined as follows. Let. Thenis a framing of $Y.$be the signature defect2 of it andbe the signature defect of$\tau_{S^{3}}|_{S^{3}\backslash N(\infty;S^{3})}=\tau_{\mathbb{R}^{3}}$$\sigma_{Y}(\tau\cup\tau_{S^{3}})$$\tau.$$S^{3}$$\tau$$\tau_{S^{3}}$$\tau\cup\tau_{S^{3}}=\tau|_{Y\backslash N(\infty,\cdot Y)}\cup\tau_{S^{3}}|_{N(\infty;S^{3})}$$\sigma(\tau)=\sigma_{Y}(\tau\cup\tau_{S^{3}})-\sigma_{S^{3}}(\tau_{S^{3}})$$\sigma_{S^{3}}(\tau_{S^{3}})$.

1In this article, all cycles are with rational coeﬃcients.
2See[1] for the definition of the signature defect of an honest framing.

$\tau_{S^{3}}.$

125
An alternative construction of Kontsevich-Kuperberg-Thurston
invariant

4

In this section, we give an alternative construction of
We also construct propagators as 4-cycles.
such thatTake
as above. Let
and be vector fields oncoincides with the constant vector field ofand transversally intersect
to the zero-section of
for $i=1$ , 2, 3.
of,andWe first construct 3-dimensional submanifolds
as in the above section. Let$z_{1}^{KKT}$$a_{1},$$a_{2},$$a_{3}\in S^{2}\subset \mathbb{R}^{3}$$\gamma_{1},$$\gamma_{2}$$Y\backslash \infty$$\gamma_{3}$$\mathbb{R}^{3}$$\gamma_{i}|_{N(\infty;Y)\backslash \infty}$$a_{i}$$\gamma_{i}$$T(Y\backslash \infty)$$W^{\partial}(\gamma_{2})$$W^{\partial}(\gamma_{1})$$\partial C_{2}(Y)$$W^{\partial}(\gamma_{3})$osure,$c_{\gamma_{i}}= \{\frac{\gamma_{i}(x)}{\Vert\gamma_{i}(x)\Vert}\in ST_{x}Y|x\inY\backslash (\infty\cup\gamma_{i}^{-1}(0))\}\subset ST(Y\backslash \infty)\ovalbox{\tt\small REJECT} 1$

for $i=1$ , 2, 3.
Lemma 4.1.

$c_{\gamma_{i}}$is a manifold with boundary and an end. Its boundary is in near$c(\gamma_{i})=c_{\gamma_{i}}\cup c_{-\gamma_{i}}$$\gamma_{i}^{-1}(0)$.

is a manifold without boundary3.

Outline of proof.
andhave same boundaries but their orientation are opposite. So
these boundaries are cancel each other.
$c_{-\gamma_{i}}$$c_{\gamma_{i}}$$\square$Definition 4.2.

$W^{\partial}(\gamma_{i})=c(\gamma_{i})\cup p_{0}^{-1}(\{a_{i}, -a_{i}\})$

We take a 4-cycle$W(\gamma_{i})$of $(C_{2}(Y), \partial C_{2}(Y))$ such that

Proposition 4.3. For generic,choice ofand$W(\gamma_{1})$$a_{i}$

and$\gamma_{i},$$\partial W(\gamma_{i})=\frac{1}{2}W^{\partial}(\gamma_{i})$$\#(W(\gamma_{1})\cap W(\gamma_{2})\cap W(\gamma_{3}))$.

is independent of the$W(\gamma_{3})4.$$W(\gamma_{2})$This proposition is proved by a homological argument similar to the argument to prove
the well-definedness of the linking number of two component links.
We next define the correction term to cancel out the influence of the choice ofand . Recall that:such thatis a framing ofis abundle. Then
We consider
as a constant vector field of trivial
constant vector filed of . Let $X$ be a compact oriented 4-manifold with $\chi(X)=0$
is the outward
and $\partial X=Y$ . Take a non-vanishing vector field
on $X$ such that
$Y=\partial
X$
in $TX$ . Then
normal vector field of. Let $T^{v}Xarrow X$ be the normal bundle of$T^{v}X$$T^{v}X|_{Y}=TY$ . Let $ST^{v}Xarrow X$ be the unit sphere bundle of. Take is a generic$T^{v}XarrowX$. Let
section ofsuch that$\gamma_{1},$$TS^{3}arrow S^{3}\cross \mathbb{R}^{3}$$\gamma_{3}$$S^{3}$$\tau_{S^{3}}|_{S^{3}\backslash N(\infty;S^{3})}=\tau_{\mathbb{R}^{3}}.$$\tau_{S^{3}}$$\mathbb{R}^{3}$$a_{i}\in \mathbb{R}^{3}$$\tau_{S^{3}}^{*}a_{i}$$S^{3}$$\eta_{X}|_{Y}$$\eta_{X}$$-$$\eta_{X}$$\beta_{i}$$\beta_{i}|_{Y}=\gamma_{i}|_{Y\backslash N(\infty,Y)}\cup\tau_{S^{3}}^{*}a_{i}|_{N(\infty,S^{3})}$losure$c_{\beta_{i}}= \{\frac{\beta_{i}(x)}{\Vert\beta_{i}(x)\Vert}\in(ST^{v}X)_{x}|x\in X\backslash \beta_{l}^{-1}(0)\} \subset ST^{v}X.$

has two ends near
This number depends on the choice of$3_{\mathcal{C}(\gamma_{i})}$$4$$\infty.$$\gamma_{1},$$\gamma_{2}$and$\gamma_{3}.$$\gamma_{2}$126
By a similar argument as in Lemma 4.1,
is a 4-dimensional submanifold
of $ST^{v}X$ such that.
For genericandis a compact oriented -dimensional
manifold. Furthermore, this argument is extended to any closed 4-manifold whose Euler
number is zero and we can check thatis a cobordism invariant ofclosed 4-dimensional manifold whose Euler number is zero.
$c(\beta_{i})=c_{\beta_{i}}\cup c_{-\beta_{i}}$$\partial c(\beta_{i})=c(\gamma_{i}\cup\tau_{S^{3}}^{*}a_{i})$$\beta_{1},$$\beta_{2}$$c(\beta_{1})\cap c(\beta_{2})\cap c(\beta_{3})$$\beta_{3},$$0$$\#(c(\beta_{1})\cap c(\beta_{2})\cap c(\beta_{3}))$

Lemma 4.4 ([8], [7]).
of the choice of and $X.$$\tilde{I}(\gamma_{1}, \gamma_{2}, \gamma_{3})=\frac{1}{8}\#(c(\beta_{1})\cap c(\beta_{2})\cap c(\beta_{3}))-\frac{3}{4}$SignX is independent$\beta_{i}$Remark 4.5. This correction term was first defined by T. Watanabe in [8] for integral
homology 3-spheres to construct the Morse homotopy invariant. We modified his construction to extend to rational homology 3-spheres and we determined the number-$\frac{3}{4}$before the term SignX.
Theorem 4.6 ([7]).
$\tilde{z}_{1}(Y)=\#(W(\gamma_{1})\cap W(\gamma_{2})\cap W(\gamma_{3}))-\tilde{I}(\gamma_{1}, \gamma_{2}, \gamma_{3})$is a topological invariant of $Y.$

Theorem 4.7 ([7]).

Proof. Let$\tilde{z}_{1}(Y)=z_{1}^{KKT}(Y)$be a framing offield ofby considering
the definition, we have

$Y\backslash \infty$$\tau$$Y\backslash \infty$

for any rational homology 3-sphere$Y.$

as above section. Then
is a non-vanishing vector
as a constant vector filed of trivial
bundle. By. Then,$\tau^{*}a_{i}$$a_{i}\in \mathbb{R}^{3}$$\mathbb{R}^{3}$$\partial W(\tau^{*}a_{i})=\partial W_{i}(\tau)$$\#(W(\tau^{*}a_{1})\cap W(\tau^{*}a_{2})\cap W(\tau^{*}a_{3}))=\#(W_{1}(\tau)\cap W_{2}(\tau)\cap W_{3}(\tau))$Let.

be the cobordism group generated by all 3-dimensional framed manifolds
and dividing by a cobordism relation
if andonly if there exists compact framed 4-manifold $(X, T)$ such that$\Omega_{3}^{Sign=0}$$\{(Y, \tau)|\tau :TYarrow\underline{\simeq}Y\cross \mathbb{R}^{3}\}$$\bullet$Sign(X) $=0,$$\bullet$$T|_{Y}$$\sim:(Y, \tau)\sim\emptyset$

is isomorphic to the stable framing of$\tau.$

We consider
andas an invariant of framed manifold $(Y,$. Then these two invariant factor throughWe can show thatandfor a framingof. Then we havefor any and Y.
$\tilde{I}(\tau^{*}a_{1}, \tau^{*}a_{2}, \tau^{*}a_{3})$$\sigma(\tau)$$\Omega_{3}^{Sign=0}$$\tau_{S^{3}}|_{N(\infty;S^{3})})$$\mathbb{Q}\cong \mathbb{Q}$$\tau|_{Y\backslash N(\infty;Y)}\cup$$\Omega_{3}^{Sign=0}\otimes$$\tilde{I}(\tau_{00\rangle}^{*}a_{1}, \tau^{*}a_{2}\tau_{0}^{*}a_{3})=-\frac{1}{4}\sigma(\tau_{0})\neq 0$$\tau_{0}$$\tilde{I}(\tau^{*}a_{1}, \tau^{*}a_{2}, \tau^{*}a_{3})=-\frac{1}{4}\sigma(\tau)$5$S^{3}\backslash \infty$$\tau$

An application of our construction
In this section, we give an application of our construction of $z^{KKT}.$$\square$

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5.1

Watanabe’s invariant

In the $1990s$ , K. Fukaya constructed an invariant of a pair of two local systems on a
3-manifold by using three Morse functions in [2]. Fukaya’s invariant is sum of principal
term depending on Morse functions and the correction term to cancel out the influenceof the choice of Morse functions. M. Futaki pointed out in [3] that Fukaya’s invariant
sometimes depends on the choice of Morse functions.
In 2012, T. Watanabe introduced a new type of correction term and then constructed a
topological invariant of a integral homology 3-spheres taking values in.
In this subsection, we review the degree 1-part, i.e.
-valued part, of Watanabe’s invariant with a little modification.
Take
as above. Let:
be Morse functions suchthatcoincides with the projection:
to the -direction andhas no critical points of index or 3. Let Critbe the set ofcritical points of . We assume that $ind(p_{j}^{i})=2,$ $ind(q_{j}^{i})=1$ . Let:,
be the boundary map of the Morse-Smale complex ofSince
is rational homology ball, the map is an isomorphism. Then there exists
the inverse map :,]. Letbe
1-parameterthefamily of diﬀeomorphisms associated to $-gradf_{i}$ . We denote byandthe ascending manifold of and the descending manifold of respectively.
$\mathcal{A}(\emptyset)=\Pi_{n}\mathcal{A}_{n}(\emptyset)$$A_{1}(\emptyset)\cong \mathbb{Q}$$a_{1},$$a_{2},$$a_{3}\in S^{2}\subset \mathbb{R}^{3}$$f_{1},$$f_{2},$$f_{3}$$Y\backslash \inftyarrow \mathbb{R}$$\mathbb{R}^{3}arrow \mathbb{R}$$f_{i}|_{N(\infty,Y)\backslash \infty}$$0$$(f_{i})=\{pi, \rangle p_{k_{i}}^{i}, qi, .

.

.

, q_{k_{i}}^{i}\}$$\partial^{i}$$f_{i}$$C_{1}(Y\backslash \infty;\mathbb{Q})$$f_{i}$$a_{i}$$q_{a_{i}}$$C_{2}(Y\backslash \infty;\mathbb{Q})arrow$$\partial^{i}\lceil p_{j}^{i}]=\sum_{k}\partial_{jk}^{i}[q_{k}^{i}]$$Y\backslash \infty$$f_{i}.$$\partial^{i}$$g^{i}$$C_{1}(Y\backslash \infty;\mathbb{Q})arrow C_{2}(Y\backslash \infty;\mathbb{Q})$$g^{i}[q_{j}^{i}]= \sum_{k}g_{jk}^{i}\lceil\int 2_{k}^{i}$$\{\Phi_{f_{i}}^{t}\}_{t\in \mathbb{R}}$$\mathcal{A}_{q_{k}^{j}}$$q_{k}^{i}$$\mathcal{D}_{p_{j}^{i}}$$p_{j}^{i}$

Definition 5.1.
$W(f_{i})=\{(x, y)\in(Y\backslash \infty)^{2}\backslash \triangle|\exists t\in \mathbb{R}\backslash 0, \Phi_{f_{i}}^{t}(x)=y\}$$+ \sum_{j,k}(9_{jk(\mathcal{A}_{q_{k}^{i}}}^{i}\cross \mathcal{D}_{p_{j}^{i}})-9_{jk(\mathcal{D}_{p_{j}^{i}}}^{i}\cross \mathcal{A}_{q_{k}^{i}}$$W(f_{i})$is a wighted sum of 4-manifold in$(Y\backslash \infty)^{2}.$Theorem 5.2 (Watanabe [8]).
$z_{1}^{FW}(Y)= \frac{1}{8}\#(W(f_{1})\cap W(f_{2})\cap W(f_{3}))-\tilde{I}(gradf_{1}, gradf_{2}, gradf_{3})$is a topological invariant of $Y.$

Watanabe also defined$\bullet$Is$z_{n}^{FW}$$z_{n}^{FW}(Y)\in \mathcal{A}_{n}(\emptyset)$trivial or not?

$\bullet z_{n}^{FW}=z^{KKT}$?

and he conjectured that

128
An application of our construction to Watanabe’s invariant

5.2

Theorem 5.3. $z_{1}^{FW}(Y)=z_{1}^{KKT}(Y)$ .
be the closure of
4-cycle of $(C_{2}(Y), \partial C_{2}(Y))$ . By the definition of $W(f_{i})$ ,Proof. Let$W(f_{i})\cap((Y\backslash \infty)^{2}\backslash \triangle)$$\overline{W}(f_{i})$in $C_{2}(Y)$ . Then$\overline{W}(f_{i})$is a$\partial\overline{W}(f_{i})$$=p_{0}^{-1}( \{a_{i}, -a_{i}\})\bigcup_{\mathcal{C}}(gradf_{i})+\sum_{jk}(9_{jk(\mathcal{A}_{q_{k}^{i}}\cap \mathcal{D}_{p_{j}^{i}})-g_{jk}^{i}(\mathcal{D}_{p_{j}^{i}}\cap \mathcal{A}_{q_{k}^{i}}))}^{i}$$=p_{0}^{-1}(\{a_{i}, -a_{i}\})\cup c(gradf_{i})$$= W^{\partial}(gradf_{i})$.

Then $z_{1}^{FW}(Y)=\tilde{z}_{1}(Y)=z_{1}^{KKT}(Y)$ .

$\square$Remark 5.4. We can show that $z_{n}^{FW}(Y)=\tilde{z}_{n}(Y)=z_{n}^{KKT}(Y)$ for any $n\geq 1$ . See [7] formore detail.

References[1] M. Atiyah. On framings of 3-manifolds. Topology, $29(1):1-7$ , 1990.
[2] K. Fukaya. Morse homotopy and Chern-Simons perturbation theory. Comm. Math.
Phys., $181(1):37-90$ , 1996.

[3] M. Futaki. On Kontsevich’s configuration space integral and invariants of 3-manifolds.
Master thesis, Univ. of Tokyo, 2006.
[4] M. Kontsevich. Feynman diagrams and low-dimensional topology. In First European
Congress of Mathematics, Vol. II (Paris, 1992), volume 120 of Progr. Math., pages
97-121. Birkh\"auser, Basel, 1994.

[5] G. Kuperberg and D. P. Thurston. Perturbative 3-manifold invariants by cut-andpaste topology. ArXiv Mathematics -prints, December 1999.
$e$[6] C. Lescop. On the Kontsevich-Kuperberg-Thurston construction of a configurationspace invariant for rational homology 3-spheres. ArXiv Mathematics -prints, November 2004.
$e$[7] T. Shimizu. An invariant of rational homology 3-spheres via vector fields. ArXiv-prints, 2013.
$e$[8] T. Watanabe. Higher order generalization of Fukaya’s Morse homotopy invariant of
3-manifolds I. Invariants of homology 3-spheres. ArXiv -prints, February 2012.
$e$

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Research Institute for Mathematical Sciences
Kyoto University
Kyoto 606-8502
JAPAN-mail address: shimizu@kurims.kyoto-u.ac.jp$E$$\overline{/5_{\backslash }^{1}}g\beta*\not\cong$.

$\mathscr{X}fgg\not\in 7Rffl_{J\iota f}^{*}BJ$$\ovalbox{\tt\small REJECT}_{7}\backslash$く$\grave{7}\underline{\not\equiv}B\beta$