数理解析研究所講究録
第 1571 巻 2007 年 82-86

82

PRE-BLOCH INVARIANT FOR 3-MANIFOLD WITH HIGHER
GENUS BOUNDARY(KABAYA Yuichi)
Department of Mathematics, Tokyo Institute of Technology

東京工業大学理工学研究科数学専攻・

蒲谷祐一

・ABSTRACT. This is an exposition of the author’s paper [3].

1. INTRODUCTION

In [5], Neumann and Yang defined the Bloch invariant for oriented hyperbolic
3-manifold with finite volume. The Bloch invariant is defined on the Bloch groupand has intimate relation with volume of the manifold and Chern-Simons invariant.
In this exposition, we generalize the Bloch invariants for infinite volume hyperbolic 3-manifolds. Uike finite volume hyperbolic manifolds, this invariant is notinvariant of a hyperbolic manifold. In this case, we essentiany need a boundarycondition. The boundary condition is given by pants decomposition.
2. DEFINITION OF THE BLOCH INVARIANT
The pre-Bloch groupis the quotient of the free abelian group generated
by $\mathbb{C}-\{0,1\}$ factored, by the relation:(2.1)$\mathcal{P}(\mathbb{C})$$[x]-[y]+[ \frac{y}{x}]-[\frac{1-x^{-1}}{1-y-1}]+[\frac{1-x}{1-y}]=0$.

the class which contains . Then we define a map, $[z]\mapsto 2(z\wedge z(1-z))$ ,:where we regard C’ as an abelian group by multiplication. For example, $xy\wedge zz=$

We denote

$[z]$

$z$

$\lambda$$x\wedge zz+y\wedge zz$$\mathcal{P}(\mathbb{C})arrow \mathbb{C}^{*}\wedge z\mathbb{C}^{*}$.

Theorem 2.1 (Bloch-Wigner, Dupon-Sah [2]).
(2.2)C’) represents i-th homology of group!))$f_{1}arrow\mathcal{P}(\mathbb{C})f_{2}arrow\lambda$$0arrow \mathbb{Q}/\mathbb{Z}arrow
H_{3}(PSL(2,\mathbb{C}),$$\mathbb{Z}$is exact. $(H_{i}(PSL(2, \mathbb{C}),$$\mathbb{Z}$$\bigwedge_{\mathbb{Z}}\mathbb{C}^{*}arrow H_{2}(PSL(2,\mathbb{C}),\mathbb{Z})f_{8}arrow 0$

In the above exact sequence, is defined by the composition. is defned by $[g_{1}|g_{2}|g_{3}]\mapsto[[z : g_{1}z : g_{1}g_{2}z : g_{1}g_{2}g_{3}z]]$ where$f_{1}$$H_{3}(PSL(2, \mathbb{C}),\mathbb{Z})$$[g_{1}|g_{2}|g_{8}]$$\mathbb{Q}/\mathbb{Z}\subset H_{3}(\mathbb{C}", \mathbb{Z})arrow$$f_{2}$

is the bar notation and$[z_{0} :z_{1} :

z_{2} :

z_{3}]= \frac{(z_{2}-z_{1})(z_{3}-z_{0})}{(z_{2}-z_{0})(z_{3}-z_{1})}f_{3}$composition C’C’
We define the Bloch group by$\bigwedge_{\mathbb{Z}}$$\cong H_{2}(\mathbb{C}^{*}, \mathbb{Z})arrow H_{2}(PSL(2, \mathbb{C}),$$\mathbb{Z}$is the).

$\mathcal{B}(\mathbb{C})=Ker(\mathcal{P}(\mathbb{C})arrow \mathbb{C}^{*}\bigwedge_{\mathbb{Z}}\mathbb{C}^{n})$.

Neumann and Yang introduced the Bloch invariant by the $follow\dot{i}g$ way. An.
with each vertex at infinity
ideal tetrahedron is a $g\infty desic$ 3-simplex on
An oriented ideal tetrahedron with prescribed edge is parametrized by a complex$\mathbb{H}^{3}$$\mathbb{C}P^{1}$

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number. Let $M$ be an oriented complete finite volume hyperbolic manifold. Let$M=\Delta(z_{1})\cup\cdots\cup\triangle(z_{n})$ be an ideal triangulation (we omit
the precise definitionof ideal triangulation. For example, Thurston’s hyperbolic Dehn
produces

surgeryn

an ideal triangulation. See detail in [5].) Then we define$\beta(M)=\sum_{\nu=1}[z_{\nu}]$

very definition depends on the triangulation and only states that
but Neumann and Yang showed that. The$\beta(M)\in \mathcal{P}(\mathbb{C})$

Theorem 2.2 (Neumann-Yang [5]). $\beta(M)$ is an invariant of $M$ and$\beta(M)\in \mathcal{B}(\mathbb{C})$,.

If $M$ is closed, the discrete faithful representation of $\pi_{1}(M)$ induces : $H_{3}(M)arrow$$H_{8}(PSL(2, \mathbb{C}))$ and $\beta(M)$ is equal to $f_{2}((\rho_{0})_{*}([M]))$ where $[M]$ is the fundamental
cycle of $M$ .
Let $M$ be a compact 3-manifold with torus boundary and be a representation$\pi_{1}(M)$. Neumann and Yang also defined an invariantoftoin. If corresponds to the holonomy of hyperbolic Dehn filled manifold, thenis equal to the Bloch invariant of the Dehn filled manifold (in paxticular. Let. But in general
be a settakes value in. Let and be the derivatives of holonomies alongof generators of. Then we haveand$\rho 0$$\rho$$PSL(2, \mathbb{C})$$\mathcal{P}(\mathbb{C})$$\beta(M,\rho)$$\rho$$\beta(M, \rho)$$\beta(M, \rho)$$\beta(M, \rho)\in \mathcal{B}(\mathbb{C}))$$H_{1}(\partial M)$$(\mathcal{L},\mathcal{M})$$\mathcal{P}(\mathbb{C})$$M_{0}$$L_{0}$$\mathcal{L}$$\mathcal{M}$Theorem 2.3 (Neumann [4]).
$\lambda(\beta(M, \rho))=L_{0}\bigwedge_{Z}M_{0}$.

can be expressed in terms ofThis theorem states that a difference from$M$
the representation of the boundary of .
They also definedfor more general manifold. Let $M$ be a compact
and be a representation of $\pi_{1}(M)$ to $PSL(2, \mathbb{C})$ .
3-manifold with boundaryif the restriction of to $\partial M=S$ has a fixed
They defined an invariant
point i.e. the restriction of to the boundary is reducible. If the boundary has
genus more than 1, this as sumption is too strong. In fact the set of reducible
surface representations has strictly smaller dimension than the set of all the surface
representations.
$\mathcal{B}(\mathbb{C})$$\beta(M, \rho)$$S$$\rho$$\beta(M, \rho)$$\rho$$\rho$

3. PANTS DECOMPOSITION AND IDEAL TRIANGULATION

Let $M$ be acompact oriented 3-manifold. For simplicity, we assume that $S=\partial M$
is aconnected surface of genus $g>1$ . The pants decomposition of is amaximal
set of distinct isotopy classes of disjoint simple closed curves. The number of curvesof is $3g-3$ . Let be orientations of the curves of C. The pair $(C, 0)$ definae
ideal triangulation of as foUows. $S-Ci_{8}$ aset of pair8 of pants (3-holed spheres).
Then each pair of prts $P$ admits ideal tritgulation by two ideal tritgle8 so thatthe ideal vertices of ideal triangles spinning around the boundaries of the pair ofprts to the direction . Let
be an ideal triangulation which is coincide with
the ideal triangulation of the boundary given by $(C, 0)$ . We can imagine easily thissituation if we attach truncated ideal 3-simplices each other. buncated vertice8
make 2-dimensional triangles. When we attach 3-simplices along their faces to eachother, then these triangles are attached to each other. These tritgles form annuli
which are the boundaries of aneighborhood of pants curves C. For each curve, we denote these rnuli by $A_{k}(k=1, \ldots, 3g-3)$ . We can use more general
ideal tritgulation, but in this paper we use this definition for simplicity.
$C$

$C$

$0$

$S$

$0$

$\gamma_{k}\in C$

$T$

$S$

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4. REPRESENTATION OF

AND PRE-BLOCH INVARIANT$\pi_{1}(M)$and be as in the last section. Let be a representation of $\pi_{1}(M)$ to. We assume that restriction of to each pair of pants (3-holed sphere)
is irreducible and the holonomies around boundaries of pair of pants are hyperbolic. Then $(C, 0, \rho)$ determines developing map of uniquelyelements ofup to conjugation as follows. Let $P$ be a pair of pants of $S-C$ . Let
be boundary curves of . Let
be elements ofwhich go aroundby the assumption. Put. Then
have distinct fixed points inso that ideal vertices are at fixed points of. Then we canideal triangle indevelop this ideal triangle by the action of. Do this construction for eachpair of pants, we get a developing map of for . By putting the ideal triangle atanother position, we can conjugate the representation .
, we extend it to the developing map
After constructing developing map of$M$
map
bydeveloping
defines a complex parameter for each ideal
using . Theoftetrahedron. We denote such complex number by . Then we define
Let $M,$$C$$0$$\rho$$PSL(2, \mathbb{C})$$\rho$$PSL(2, \mathbb{C})$$S$$\gamma_{1},$$P$$\gamma_{1},$$\gamma_{2},$$g_{1},$$g_{2},$$\gamma_{3}$$\mathbb{C}P^{1}$$\rho(g_{i})$$\gamma_{3}$$\gamma_{2},$$\pi_{1}(P)$$g_{3}$$\mathbb{H}^{3}$$\rho(g_{i})$$\rho(\pi_{1}(P))$$S$$\rho$$\rho$$\partial M$$\rho$$z_{\nu}$

Definltion 4.1.

$\beta(M, \rho, C, 0)=\sum_{\dot{j}=1}^{n}[z_{i}]\in \mathcal{P}(\mathbb{C})$.

Proposition 4.2. $\beta(M, \rho, C, 0)$ only depends on $M,$$\rho_{f}C,$$0_{f}$of triangulation of $M$ .

and not on the choice

5. SOME PROPERTIES OF $\beta(M, \rho, C, 0)$
As in [6], we define the edge relation for each l-simplex ofto$\partial M$$T$which is not facing:$R_{i}= \pm\prod_{\nu=1}^{n}z_{\nu}^{r_{\nu}’}(1-z_{\nu})^{r_{\nu}’’}\dot{\cdot},=1$$(i=1,2, \ldots,n-3(g-1))$ .

, we define a complex number by multiFor l-simplex of which is facing toplication the complex numbers of edges which are adjacent to the l-simplex:$T$$\partial M$,$B_{i}= \pm\prod_{\nu=1}^{n}z_{\nu}^{b_{\nu}’}(1-z_{\nu})^{b_{\nu}’’}\cdot$$(i=1,2, \ldots, 6(g-1))$.

Because the number of pairs of pants on $S-C$ is $2g-2$ , the number of boundary
is not equal to 1. We call such a l-simplex
l-simplices are $6g-6$ . Unhike
measures how bent two ideal triangles at the boundary
boundary 1-simplex.
l-simplex.
. Take a path ofso that is a generator ofon
Take a loop
. Then we can define
represents a generator ofso thatcomplexparameters
as torus boundary casea complex number by multiplication(see, for example, [6].)$R_{i},$$B_{i}$$B_{i}$$(A_{k}, \partial A_{k})$$H_{1}(A_{k}, \mathbb{Z})$$h_{k}$$A_{k}$$h_{k}$$w_{k}$$H_{1}(A_{k}, \partial A_{k}, \mathbb{Z})$$w_{k}$$H_{k}= \pm\prod_{\nu\approx 1}^{n}z_{i}^{h_{k,\nu}’}(1-z_{i})^{h_{k,\nu}’’}$,$W_{k}= \pm\prod_{\nu=1}^{n}z_{i}^{w_{k,\nu}’}(1-z_{i})^{w_{k,\nu}’’}$$(k=1, \ldots, 3(g-1))$.

$3(g-1)$ .
depends only on thea twist parameter of for $k=1,$
We call
represents the square of eigenvalue of the holonomy
andhomology class of. On the other hand is not $wen$-defined. If we deformaround the curve.
changes toacross the boundary l-simplex , thenthe endpoint of$W_{k}$$\gamma_{k}$$h_{k}$$\gamma_{k}\in C$$w_{k}$$H_{k}$$\ldots,$$H_{k}$$W_{k}$$i$$W_{k}$$W_{k}B_{i}$85Letbe a boundary l-simplex of a pair of pants P.
are intersect
andwith common pants curve . Then we can observe thatmeasures how bent$P$
ideal triangles onaround . In fact we can show $B_{i}B_{j}=H_{k}$ . Since
hastwo adjacent pair of pants, we denote the other one by $P’$ . Then $P$ ‘ has twoboundary l-simplex which intersect with . We denote them by and $j’$ . By
above observation we have a relation $B_{i}B_{j}=H_{k}=B_{i’}B_{j’}$ . So $(B_{1}, \ldots, B_{6g-6})$
is essentially $(6g-6)-(3g-3)=3g-3$ dimensional object. Moreover we canrepresent
in terms of.
$(H_{k}, W_{k})$ reminds us theFenchel-Nielsen coordinate. Fenchel-Nielsen coordinate
define a coordinate of Teichm\"uller space by length and twist. A length is welldefined for given hyperbolic surface, on the other hand twist is not determined bygiven hyperbolic surface.
We have a version of Theorem 2.3 for higher genus boundary case.
$e_{i},$$e_{j}$$e_{i}$$e_{j}$$B_{i}B_{j}$$\gamma_{k}$$\gamma_{k}$$\gamma_{k}$$i’$$\gamma_{k}$$H_{k}’ s$$B_{i}$

Theorem 5.1.
$\lambda(\beta(M, \rho, C_{0}))=\sum_{k=1}^{3(g-1)}H_{k}\wedge zW_{k}$.

We remark thatis not well defined as we mentioned, but the right hand sideof the above equation is well-defined.
$W_{k}$

6. VOLUMES OF REPRESENTATIONS
By using invariance of $\beta(M, \rho,C, 0)$ , we can define a volume of a representation.

We define $Li_{2}(z)=- \int_{0}^{\infty}\frac{\log(1-t)}{t}dt$ and$D(z)={\rm Im} Li_{2}(z)+\log|z|arg(1-z)(z\in \mathbb{C}-\{0,1\})$.

For an ideal simplex with complex parameter of ideal simplex, we can describe
its volume by $D(z)$ . $D$ satisfies five term relation,$z$$D(x)-D(y)+D(y/x)-D( \frac{1-x^{-1}}{1-y-1})+D(\frac{1-x}{1-y})=0$.

. We can define volume of $\beta(M, \rho, C, 0)$So we have a homomorphism $D$ :by $D(\beta(M, \rho, C, 0))$ . We next consider the variation of volume in deformation space.
parametrized bytoConsider smooth family of representations of. Then the derivative of volume is$\mathcal{P}(\mathbb{C})arrow \bm{R}$$\pi_{1}(M)$$PSL(2,\mathbb{C})$$t$$\frac{dVo1}{dt}=-\frac{1}{2}\sum_{k=1}^{3(g-1)}(\log|W_{k}|\frac{d\arg(H_{k})}{dt}-\log|H_{k}|\frac{d\arg(W_{k})}{dt})$.

This formula shows that the derivative of volume is written by in terms of therepresentations of the boundary. We remark that Bonahon proved a variation
formula of volume for geometrically finite hyperbolic manifolds (see Theorem 3 of[1]). Bonahon’ theorem shows the variation of volume bounded by pleated surface
ation. In our case, the geodesic larnination
with fixed pleated locus geodesic
is given by ideal triangulation by pants decomposition.
$lam\dot{i}$86
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. (1998), no. 1, 25-58.
[2] J.L Dupont, H Sah, Scissors congruences II, J. Pure and Appl. Algebra 25 (1982) 159-195[3] Y. Kabaya, Pre-Bloch invariants of 3-manifolds with boundary, to appear.
[4] W.D Neumann, Combinatorics of triangulations and the Chern-Simons invariant for hyperbolic 3-manifolds, from: “Topology 90, Proceedings of the Research Semester in Low
Topology at Ohio State”, Walter de Gruyter Verlag, Berlin-New York (1992)$Dimensional243-272$[5] W.D. Neumann, J Yang, Blochof hyperbolic S-manifolds, Duke Math. J. 96 (1999)
29-59[6] W.D.Neumann, D.Zagier, Volumes of hyperbolic 3-manifolds, Topology 24 (1985), 307-332.
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