EMIS/ELibM Electronic Journals

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These pages are not updated anymore. They reflect the state of 20 August 2005. For the current production of this journal, please refer to http://www.math.psu.edu/era/.


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\begin{document}
\setcounter{page}{72}
\title[Invariants from triangulations of hyperbolic 3-manifolds]
{Invariants from triangulations of
hyperbolic 3-manifolds}
\author{Walter D. Neumann} 
\address{Department of Mathematics\\The University of
Melbourne\\Carlton, Vic 3052\\Australia}
\email{neumann@maths.mu.oz.au} 
\author{Jun Yang} 
\address{Department of Mathematics\\Duke University\\Durham NC 27707}
\email{yang@math.duke.edu}
\subjclass{57M50,30F40;19E99,22E40,57R20}
\date{May 5, 1995\inRevisedForm July 19, 1995}

\renewcommand{\currentissue}{2}
\renewcommand{\currentvolume}{1}

\begin{abstract}
For any finite volume hyperbolic 3-manifold $M$ we use ideal
triangulation to define an invariant $\beta(M)$ in the Bloch group
$\B(\C)$.  It actually lies in the subgroup of $\B(\C)$ determined by
the invariant trace field of $M$.  The Chern-Simons invariant of $M$
is determined modulo rationals by $\beta(M)$.  This implies
rationality and --- assuming the Ramakrishnan conjecture ---
irrationality results for Chern Simons invariants.
\end{abstract}
\maketitle

\section{Main Results}

The {\em pre-Bloch group} $\P(k)$ of a field $k$ is the quotient of
the free $\Z$-module $\Z(k-\{0,1\})$ by all instances of the following
relations:
\begin{gather}
[x]-[y]+[\frac yx]-[\frac{1-y^{-1}}{1-x^{-1}}]+[\frac{1-y}{1-x}]=0\label{5term}\\
[x]=[1-\frac 1x]=[\frac 1{1-x}]=-[\frac1x]=-[\frac x{x-1}]=-[1-x]\label{invsim}
\end{gather}
The {\em Bloch group $\B(k)$} is the kernel of the map
$$\mu\colon\P(k)\to k^*\wedge_\Z k^*\quad\text{given by}\quad
\mu([z])=2(z\wedge(1-z)).
$$
(There are several variants of this definition in the literature. 
Dupont and Sah \cite{dupont-sah} show that the various definitions
differ at most by torsion and that they agree with each other for
algebraically closed fields. See also the discussion in
\cite{neumann-yang-cs}.)

By results of Borel, Bloch, and Suslin \cite{borel,bloch,suslin} (see
Theorem \ref{borel}) the
Bloch group $\B(k)$ of a number field is isomorphic modulo torsion to
$\Z^{r_2}$, where $r_2$ is the number of complex embeddings of $k$ (a
complex embedding is an embedding $k\to\C$ with image not in
$\R$). Thus $\B(k)\otimes\Q\cong\Q^{r_2}$.  Moreover, if a specific
complex embedding is chosen, that is, $k$ is given as a subfield of
$\C$, then the induced map $\B(k)\otimes\Q\to\B(\C)\otimes \Q$ is
injective, so we may write $\B(k)\otimes\Q\subset\B(\C)\otimes \Q$.

Let $M=\H^3/\Gamma$ be an oriented complete hyperbolic manifold of
finite volume (briefly just ``hyperbolic 3-manifold'' from now on).
The {\em invariant trace field} $k(M)=k(\Gamma)$ is the field
generated over $\Q$ by squares of traces of elements of $\Gamma$. It
is the smallest field among trace fields of finite index subgroups of
$\Gamma$ (\cite{reid}, see also \cite{neumann-reid1}).  It is a number
field and comes with a specific embedding in $\C$.

Thurston has shown \cite{thurston2} that any hyperbolic 3-manifold $M$
has a degree one ideal triangulation (see sect.~2) by ideal simplices
$\Delta_1,\ldots, \Delta_n$.  Let $z_i\in\C$ be the parameter of the
ideal simplex $\Delta_i$ for each $i$ (cross-ratio of its four
vertices).  These $z_i$ define an element $\beta(M)=\sum_{i=1}^n[z_i]$
in the pre-Bloch group $\P(\C)$.

\begin{theorem}\label{theorem1}
This $\beta(M)$ depends only on $M$. It lies in the Bloch group
$\B(\C)\subset\P(\C)$. As an element of $\B(\C)\otimes\Q$, it lies in
the subgroup $\B(k(M))\otimes\Q$.
\end{theorem}

One of the main ingredients for this theorem --- Proposition
\ref{beta-fundamental} below --- fills a gap in the literature.
Statements have appeared in papers of at least two authors that
implicitly assume it.

In the non-compact case one can find genuine (rather than just degree
one) ideal triangulations of $M$ (see \cite{epstein-penner}) and the
simplex parameters $z_i$ then lie in the invariant trace field $k(M)$
(see \cite{neumann-reid1}). Thus $\beta(M)$ is the image of a class
$\beta_k(M):=\sum[z_i]\in \P(k(M))$.
\begin{theorem}
$\beta_k(M)$ lies in $\B(k(M))$ and is independent of triangulation.
\end{theorem}
Thus, in the non-compact case the ``$\otimes\Q$'' of Theorem
\ref{theorem1} can be deleted. We do not know if it can in the compact
case, though we can describe an explicit integer $c=c(M)\ge 0$ such
that $2^c\beta(M)$ is in the image of $\B(k(M))\to\B(\C)$.
\smallskip

The Bloch invariant $\beta(M)$ is intimately related to the volume
$\vol(M)$ and
the Chern-Simons invariant $\CS(M)$.  Chern and Simons defined the
latter invariant in (\cite{chern-simons}) for any compact
$(4n-1)$-dimensional Riemannian manifold. Meyerhoff \cite{meyerhoff}
extended the definition in the case of hyperbolic 3-manifolds to allow
noncompact ones, that is hyperbolic 3-manifolds with cusps. The
Chern-Simons invariant $\CS(M)$ of such a hyperbolic 3-manifold $M$ is
an element in $\R/\pi^2\Z$. It is called {\em rational} (also called
{\em torsion}) if it lies in $\pi^2\Q/\pi^2\Z$.

The relation with the Bloch invariant uses the ``Bloch regulator
map''
$$\rho\colon \B(\C) \longrightarrow \C/\Q,$$
defined as follows. For $z\in
\C-\{0,1\}$, define
$$
\rho(z) = \frac{\log z}{2\pi i}\wedge\frac{\log (1-z)}{2\pi i} +
1\wedge 
\frac{{\cal R}(z)}{2\pi^2},
$$
where ${\cal R}(z)$ is the ``Rogers dilogarithm function''
$${\cal R}(z)=\frac12\log(z)\log(1-z)-\int_0^z\frac{\log(1-t)}tdt.$$
  See section 4 of
\cite{dupont-sah} or \cite{hain} for details on how to interpret this
formula. 
This $\rho$ vanishes on the relations (\ref{5term}) and
(\ref{invsim}) which define $\P(\C)$ and hence $\rho$ induces a map
$$\rho\colon \P(\C)\longrightarrow\C\wedge_\Z\C.$$
This fits in a commutative diagram
$$\begin{matrix}
\P(\C)&\stackrel{\mu}{\longrightarrow}&\C^*\wedge\C^*\\
\downarrow\scriptstyle\rho&&\downarrow\scriptstyle=\\
\C\wedge\C&\stackrel{\epsilon}{\longrightarrow}&\C^*\wedge\C^*\\
\end{matrix}
$$ where $\epsilon=2(e\wedge e)$ with $e(z)=\exp(2\pi iz)$. 
The kernel of $\mu$ is $\B(\C)$ and the kernel of $\epsilon$ is
$\C/\Q$. Hence $\rho$ restricts to give the desired map
$\rho\colon\B(\C)\to\C/\Q$.  

\begin{theorem}\label{formula}
$\rho(\beta(M))=\frac1{2\pi^2}(\CS(M)+i\vol(M))\in\C/\Q$.
\end{theorem}

The volume part of this result is not hard (in part because ``volume''
is already well defined on $\P(\C)$), but the part referring to
Chern-Simons lies deeper. Modulo Proposition \ref{beta-fundamental}
below, the compact case is due to Dupont \cite{dupont1}.

This theorem has various consequences for rationality and
irrationality of the Chern-Simons invariant.  For example, we show in
\cite{neumann-yang-cs}:
\begin{theorem}\label{CMthm}
The Chern-Simons invariant $\CS(M)$ of a hyperbolic 3-manifold is
rational if the invariant trace field $k(M)$ of $M$, as a subfield of
$\C$, is an imaginary quadratic extension of a totally real field
(briefly, ``$k$ is CM-embedded in $\C$'').
\end{theorem}
On the other hand we have the following irrationality conjecture
for Chern-Simons invariant. We use $\overline k$ to mean the complex
conjugate of the subfield $k\subset\C$.
\begin{conjecture}\label{conj} If the 
 invariant trace field $k=k(M)$ satisfies $k\cap\overline
k\subset\R$ then $\CS(M)$ is irrational.  In particular, $\CS(M)$ is
irrational if $k(M)$ has odd degree over $\Q$.
\end{conjecture}

We show in \cite{neumann-yang-cs} that this conjecture would follow from
a conjecture of Ramakrishnan \cite{ramakrishnan}:
\begin{conjecture} For any
number field $k$ the Bloch map $\rho$ restricted to
$\B(k)\otimes\Q$ is injective.
\end{conjecture}

A number field $k$ occurs as the invariant trace field of an {\em
arithmetic} hyperbolic 3-manifold if and only if it has just one
complex place (cf.~e.g., \cite{neumann-reid1}). It then either
satisfies $k=\overline k$ and is CM-embedded or it satisfies
$k\cap\overline k\subset\R$. Thus, in the arithmetic case Theorem
\ref{CMthm} and Conjecture \ref{conj} would say that rationality or
irrationality of the Chern-Simons invariant is completely determined
by whether or not $k=\overline k$.

Although no example of irrationality of the Chern-Simons invariant of
a hyperbolic 3-manifold has been proved, there is a lot of numerical
evidence for the above conjecture.  A similar comment applies to
volume.  We say more about computational aspects in the final section
of this announcement. We also describe there a generalization of
$\beta(M)$ to an invariant of a homomorphism
$\Gamma=\pi_1(M)\to\PGL(2,\C)$.  In \cite{neumann-yang} we define an
analogous invariant in any dimension, but its significance is not
clear at this time.

\smallskip
\noindent{\it Added May 1995}.  A.B. Goncharov has kindly shared with
us his manuscript \cite{goncharov} in which he defines an invariant in
$K_{2n-1}(\overline\Q)\otimes\Q$ for any hyperbolic $(2n-1)$-manifold
of finite volume.

\section{Background}
%
\subsection{Ideal simplices and degree one ideal triangulations}
\label{ideal triang}
%
We shall denote the standard compactification of $\H^3$ by $\overline
\H^3 = \H^3\cup\CP^1$. An ideal simplex $\Delta$ with vertices
$z_1,z_2,z_3,z_4\in\CP^1$ is determined up to congruence by the cross
ratio
$$z=[z_1:z_2:z_3:z_4]=\frac{(z_3-z_2)(z_4-z_1)}{(z_3-z_1)(z_4-z_2)}.$$
This $z$ lies in the upper half plane of $\C$ if the orientation
induced by the given ordering of the vertices agrees with the
orientation of $\H^3$. Permuting the vertices by an even (i.e.,
orientation preserving) permutation replaces $z$ by one of
$$
z,\quad 1-\frac 1z, \quad\text{or}\quad \frac 1{1-z},
$$
while an odd permutation replaces $z$ by
$$
\frac 1z, \quad\frac z{z-1},\quad\text{or}\quad 1-z.
$$
We will also allow degenerate ideal simplices where the vertices lie
in a plane, so the parameter $z$ is real.  However, we always require
that the vertices are distinct.  Thus the parameter $z$ of the simplex
lies in $\C-\{0,1\}$ and every such $z$ corresponds to an ideal
simplex.

Let $Y$ be a CW-complex obtained by gluing together finitely many
3-simplices by identifying the 2-faces in pairs.  The complement of
the 1-skeleton is then a 3-manifold, and if this 3-manifold is
oriented we call $Y$ a {\em 3-cycle}.  In this case the complement
$Y-Y^{(0)}$ of the vertices is an oriented 3-manifold.

Suppose $M^3=\H^3/\Gamma$ is a hyperbolic manifold.  A {\em degree one
ideal triangulation of $M$} consists of a 3-cycle $Y$ plus a map
$f\colon Y-Y^{(0)}\to M$ satisfying
\begin{itemize} \item $f$ is degree one almost everywhere
in $M$; \item for each 3-simplex $S$ of $Y$ there is a map
$f_S$ of $S$ onto an ideal simplex in $\overline\H^3$, mapping
vertices to ideal vertices,  such that
$f|S-S^{(0)}\colon S-S^{(0)}\to M$ is the composition $\pi\circ
(f_S|S-S^{(0)})$, where $\pi\colon\H^3\to M$ is the projection.
\end{itemize}

Thurston shows in \cite{thurston2} that any hyperbolic 3-manifold has
degree one ideal triangulations. Such triangulations also arise ``in
practice'' (e.g., in the program SNAPPEA for exploring hyperbolic
manifolds --- \cite{weeks}) as follows.  It follows from
\cite{epstein-penner} that any non-compact $M$ has a ``genuine'' ideal
triangulation: one for which $f$ is arbitrarily closely homotopic to a
homeomorphism (\cite{epstein-penner} gives an ideal polyhedral
subdivision and some flat simplices may be needed to subdivide the
polyhedra consistently into ideal tetrahedra).  The ideal simplices
can be deformed to give degree one ideal triangulations (based on the
same 3-cycle $Y$) on almost all manifolds obtained by Dehn filling
cusps of $M$ (see e.g., \cite{neumann-zagier}).

\subsection{Bloch group} We describe the geometric background for our
definition of the Bloch group.  For $k=\C$, the relations
(\ref{invsim}) express the fact that the pre-Bloch group $\P(\C)$ may
be thought of as being generated by congruence classes of ideal
hyperbolic 3-simplices.  The convex hull of five distinct points in
the ideal boundary of $\H^3$ can be decomposed into ideal simplices in
exactly two ways: once into two ideal simplices and once into
three. The ``five term relation'' (\ref{5term}) expresses the fact
that these two decompositions represent the same element in $\P(\C)$.

As already mentioned, there are several different definitions of the
Bloch group in the literature. By \cite{dupont-sah} they differ at
most by torsion and agree with each other for algebraically closed
fields.  The version (\ref{5term}) of the five term relation we use is
the one of Suslin \cite{suslin}. Dupont and Sah use a slightly
different one first written
down by Bloch and Wigner:
$$[x]-[y]+[y/x]-[(1-y)/(1-x)]+[(1-y^{-1})/(1-x^{-1})]=0.$$ 
This is conjugate to Suslin's by the self-map $[z]\mapsto [z^{-1}]$ of
$\Z(k-\{0,1\})$.  See \cite{neumann-yang-cs} or \cite{suslin} for a
discussion of the reason for Suslin's choice.  Also, relation
(\ref{invsim}) is already implied modulo torsion by the five term
relation (\ref{5term}).  Omitting it gives the version of the Bloch
group used by Dupont and Sah \cite{dupont-sah}.

\begin{remark}
The above suggests that our invariant $\beta(M)$ captures a type of
``ideal scissors congruence class'' of the hyperbolic manifold $M$.
In fact, $\P(\C)$ is the ``scissors congruence group'' generated by
hyperbolic polyhedra with only ideal vertices and triangular faces
modulo the relations generated by cutting and pasting along such
faces.  From a scissors congruence point of view it is more natural
not to restrict the faces to be triangular; we then obtain the 
quotient of $\P(\C)$ by the subgroup generated by ``flat'' simplices,
that is, the quotient of $\P(\C)$ by the image of $\P(\R)$. Hilbert's 
third problem for hyperbolic geometry is often interpreted as the 
problem of evaluating the analogous group for polyhedra without ideal
vertices. Dupont and Sah \cite{dupont-sah} show that allowing only 
non-ideal vertices gives the same scissors congruence group up to
2-torsion as allowing both ideal and non-ideal vertices, and that the
resulting group is the $-1$ co-eigenspace $\P_-(\C)$ for the action
of conjugation on $\P(\C)$.  The geometric background to this is that
the group $\P(\C)$ is orientation sensitive, while the non-ideal
scissors congruence group is not. In fact, any ideal simplex $\Delta$
can be cut into three simplices which can be re-assembled to give the
mirror image of $\Delta$ by dropping a perpendicular from a vertex of
$\Delta$ to the opposite face.

The ``imaginary part'' of the Ramakrishnan conjecture (also conjectured 
in \cite{zagier}) would imply that the imaginary part
of $\rho(\beta(M))$, namely $\vol(M)$, is a complete scissors
congruence invariant up to torsion for a hyperbolic manifold.  This
might be considered a weak positive answer to Hilbert's third problem
for hyperbolic manifolds!
\end{remark}

\section{Sketch of Proofs}

The consequences of our results for rationality and irrationality of
Chern-Simons invariant follow in \cite{neumann-yang-cs} from an
analysis of the dimensions of the eigenspaces of the action of complex
conjugation on the Bloch group $\B(K)$ for a number field $K=\overline
K\subset\C$.  We will not discuss this further here.

We first discuss the compact case of Theorem \ref{theorem1}.

There is an exact sequence due to Bloch and Wigner
(cf.~\cite{dupont-sah})
\begin{equation}0\to\mu\to H_3(\PGL(2,\C)^\delta;\Z)\stackrel\sigma\longrightarrow
\B(\C)\to 0,\label{BWsequ}\end{equation} 
where $\mu\subset\C^*$ is the group of roots of unity
and the superscript $\delta$ means we are taking $\PGL(2,\C)$ with
discrete topology. 
If $M=\H^3/\Gamma$ is compact then the map
$\Gamma\to\PGL(2,\C)$ induces a map $H_3(\Gamma;\Z)\to
H_3(\PGL(2,\C)^\delta;\Z)$. But $H_3(\Gamma;\Z)=H_3(M;\Z)\equiv\Z$.
The image of a generator gives a ``fundamental class'' $[M]\in
H_3(\PGL(2,\C)^\delta;\Z)$. 

The independence of $\beta(M)$ on ideal triangulation is given by the
following proposition.

\begin{proposition}\label{beta-fundamental}$\beta(M)=\sigma([M]) \in
\B(\C)$.  
\end{proposition}

We next explain why $\beta(M)\otimes\Q$ lies in the image of
$\B(k)\otimes\Q\to\B(\C)\otimes\Q$, where $k$ is the invariant trace
field $k=k(\Gamma)$. We denote this subgroup by $\B(k)_\Q$. 

If $\Gamma$ has trace field K then one can find a quadratic extension
$K_1$ of $K$ so that the embedding $\Gamma\hookrightarrow \PGL(2,\C)$
factors up to conjugacy through $\PGL(2,K_1)$.  Now the exact sequence
(\ref{BWsequ}) holds modulo torsion for any field, in particular for
$K_1$.  We deduce that $\beta(M)\otimes\Q$ lives in $\B(K_1)_\Q$.  By
\cite{neumann-reid2} there are infinitely many different fields $K_1$
for which we can do this.  If we take two of them, say $K_1$ and
$K_2$, we see that $\beta(M)\otimes\Q$ lives in
$\B(K_1)_\Q\cap\B(K_2)_\Q$. This is $\B(K_1\cap K_2)_\Q=\B(K)_\Q$ by
Proposition 2.1 of \cite{neumann-yang-cs}.  Finally, by replacing
$\Gamma$ by a subgroup of finite index we can arrange that its trace
field is the invariant trace field $k$.  Since this just multiplies
$\beta(M)\otimes\Q$ by the index of the subgroup, it follows that
$\beta(M)\otimes\Q$ is in $\B(k)_\Q$.

We prove Proposition \ref{beta-fundamental} in \cite{neumann-yang} by
factoring through a relative homology group for which the relationship
between $[M]$ and $\beta(M)$ is easier to see. In the notation of
\cite{dicks-dunwoody} this relative group is
$H_3(\PGL(2,\C),\CP^1;\Z)$.  It has a natural map to $\P(\C)$. (Dupont
and Sah \cite{dupont-sah} show --- without using this notation ---
that this map is an isomorphism.)

In fact, our key lemma in both the compact and non-compact cases is 

\begin{lemma}
$H_3(\Gamma,\CP^1;\Z)$ is infinite cyclic, generated by a
``fundamental class'' $[M]$.  
The composition  $H_3(\Gamma,\CP^1;\Z)\to H_3(\PGL(2,\C),\CP^1;\Z)\to
\P(\C)$ maps $[M]$ to $\beta(M)$.
\end{lemma}

In the non-compact case the fact that $\beta(M)$ lies in $\B(\C)$ is
the relation $\sum z_i\wedge(1-z_i)=0\in\C^*\wedge\C^*$ on the simplex
parameters $z_i$.  This has been attributed to Thurston (unpublished)
by Gross \cite{gross}. It also follows easily from
\cite{neumann-zagier} (see also \cite{neumann}).  We give a
cohomological proof in \cite{neumann-yang}.  

Finally, we discuss Theorem \ref{formula}.  We have already remarked
that it is in the compact case, modulo Proposition
\ref{beta-fundamental}, essentially a result of Dupont \cite{dupont1}.
In general it follows from the simplicial formula for Chern-Simons
invariant of \cite{neumann}.  That formula was deduced in the general
case from the compact case and included an unknown constant which was
claimed there to be a rational multiple of $\pi^2$.  There was a gap
in the proof of this rationality, which is filled by Proposition
\ref{beta-fundamental} above.

\section{Final Remarks}\label{final}

\subsection{Computing the invariant $\beta(M)$}
Define the {\em Bloch-Wigner} function $D_2\colon
\C-\{0,1\}\rightarrow \R$ by (cf. \cite{bloch})  
$$D_2(z) = \mathop{Im} \ln_2(z) + \log |z|\arg(1-z),\quad z\in \C -\{0,1\}$$
where $\ln_2(z)$ is the classical dilogarithm function.  The
hyperbolic volume of an ideal tetrahedron $\Delta$ with cross ratio
$z$ is equal to $D_2(z)$. It follows that $D_2$ satisfies the
functional equations given by the relations which define $\P(\C)$, and
therefore $D_2$ induces a map
$$D_2\colon \B(\C) \longrightarrow \R,$$
by defining $D_2[z]=D_2(z)$.

Given a number field $k$ let
$\sigma_{1},\bar{\sigma}_{1},
\ldots, \sigma_{r_2},\bar{\sigma}_{r_2}$ denote the complex
embeddings of $k$. One then has a map
$$\begin{array}{rrcl}
c_2: & \B(k)& \longrightarrow & \R^{r_2}\\
     & \sum_i (n_i[z_i]) & \mapsto & (\sum_i n_i
D_2(\sigma_{1}(z_i)), \ldots, \sum_i n_i
D_2(\sigma_{r_2}(z_i))).
\end{array}
$$
The following theorem is a re-interpretation by Bloch and Suslin of a
theorem about K-groups of Borel.
\begin{theorem}\label{borel} The kernel of $c_2$ is
exactly the torsion subgroup of $\B(k)$ and the image of $c_2$ is a
maximal lattice in $\R^{r_2}$. In particular,  the rank
of $\B(k)$ is $r_2$.
\end{theorem}

Using this theorem we can compute $\beta(M)$ up to torsion by
computing its image $c_2(\beta(M))$ using the simplex parameters of an
ideal triangulation.  The program SNAPPEA works with ideal
triangulations.  However, in its current incarnation it computes the
simplex parameters numerically rather than as exact algebraic numbers.
A preliminary version of a modification of Snappea that computes to
high precision and derives exact simplex parameters was written
(mostly by Oliver Goodman) as part of an Australian Research Council
project at Melbourne University.  Using this, the element
$c_2(\beta(M))\in \R^{r_2}$ can be computed to high precision when the
invariant trace field $k=k(M)$ does not have too high degree.  Such
calculations support the predictions of the Ramakrishnan conjecture.

These calculations have also yielded interesting examples.  For
example, there exist two compact arithmetic 3-manifolds of volume
$2D_2(i)=1.83193119\dots$ with invariant trace field 
$\Q(i)$ but with different
quaternion algebras (so they are non-commensurable) such that one can
disassemble one of them into three ideal tetrahedra that can be
reassembled into the other.  The tetrahedral parameters lie in
$K:=\Q(i,\sqrt{4-2i})$, so the Bloch invariant is defined over this
field.  The examples have Chern-Simons invariants differing by
$\pi^2/12$ modulo $\pi^2$.  Thus Chern-Simons invariant is not
determined by Bloch invariant if one does not take it modulo
$\pi^2\Q$. Craig Hodgson found these examples and Alan Reid helped
compute the quaternion algebras.

\subsection{Generalization of $\beta(M)$}

\begin{theorem}
If $M=\H^3/\Gamma$ is a finite volume hyperbolic manifold and
$f\colon\Gamma\to \PGL(2,\C)$ is any homomorphism then there is a natural
invariant $\beta(f)\in\P(\C)$. If $f$ is the homomorphism
corresponding to some Dehn filling $M'$ of $M$ then
$\beta(f)=\beta(M')$.  In particular, for the discrete embedding we
have $\beta(f)=\beta(M)$.  If each cusp subgroup of $\Gamma$ has 
non-trivial elements $\gamma$ with $f(\gamma)$ parabolic (or trivial)
then $\beta(f)\in\B(\C)$.
\end{theorem}

This theorem gives a way of defining the ``volume'' of any
homomorphism $f\colon\Gamma\to \PGL(2,\C)$. The existence of such a
volume was mentioned in \cite{thurston3}. In \cite{neumann-yang} a
version of this theorem is proved for any dimension.

If $M=\H^3/\Gamma$ is compact then we can define $\beta(f)$ as the
image of the fundamental class $[M]\in H_3(\Gamma;\Z)$ under
$H_3(\Gamma;\Z)\stackrel{f_*}\exlongrightarrow
H_3(\PGL(2,\C)^\delta;\Z)\to\B(\C)$.  This clearly generalizes the
invariant $\beta(M)$, which is the value of $\beta(f)$ for the
discrete embedding.  If $M$ is non-compact it is harder to define
$\beta(f)$.  We cannot directly follow the program of the previous
section since $H_3(\Gamma,\CP^1;\Z)$ will depend on the action of
$\Gamma$ on $\CP^1$ and hence on the homomorphism $f$.  We use instead
$H_3(\Gamma,\calC;\Z)$, where $\calC$ is the union of $\Gamma/P$
as $P$ runs through a set of representatives for the conjugacy classes
of cusp subgroups of $\Gamma$.  This homology group is cyclic
generated by a class $[M]$.  There are, in general, several maps
$\calC\to\CP^1$ which are equivariant with respect to
$f\colon\Gamma\to \PGL(2,\C)$, so $f$ induces several maps
$H_3(\Gamma,\calC;\Z)\to H_3(\PGL(2,\C),\CP^1;\Z)\to \P(\C)$.  One
can define a simplicial version of $\beta(M)$ and show that it is the
image of $[M]$ under any one of these maps.  Thus the maps are all the
same and the image of $[M]$ agrees again with the simplicial version
of the invariant.



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\end{thebibliography}

\end{document}