# Quantum Hyperbolic State Sum Invariants of 3-Manifolds

###### Abstract

Any triple , where is a compact closed
oriented 3-manifold, is a link in and is a flat principal
-bundle over ( is the Borel subgroup of upper triangular
matrices of ), can be encoded by suitable
distinguished and decorated
triangulations . For each , for each
odd integer , one defines a state sum , based on
the
Faddeev-Kashaev quantum dilogarithm at , such that
is a well-defined complex valued invariant.
The purely topological, conjectural invariants proposed earlier
by Kashaev
correspond to the special case of the trivial flat bundle.
Moreover, we extend the definition of these invariants to the case of
flat bundles on with not necessarily trivial holonomy
along the meridians of the link’s components, and also to -manifolds
endowed with a -flat bundle and with *arbitrary* non-spherical
parametrized boundary components. As a matter of fact the distinguished
and decorated triangulations are strongly reminiscent of
the way one represents the classical refined scissors congruence
class , belonging to the extended Bloch group,
of any given finite volume hyperbolic 3-manifold by
using any hyperbolic ideal triangulation of . We point out
some remarkable specializations of the invariants; among these,
the so called Seifert-type invariants, when : these seem to be
good candidates in order to fully reconstruct the Jones polynomials in the
main stream of quantum hyperbolic invariants.
Finally, we try to set our results against the heuristic backgroud of the
Euclidean analytic continuation
of (2+1) quantum gravity with negative cosmological constant,
regarded as a gauge theory with the non-compact group
as gauge group.

Laboratoire E. Picard, CNRS UMR 5580, Université Toulouse III, F-31062 TOULOUSE

Email:

Dipartimento di Matematica, Università di Pisa, Via F. Buonarroti, 2, I-56127 PISA

Email:

*Keywords: quantum dilogarithm, hyperbolic 3-manifolds, state sum invariants.*

## 1 Introduction

In a series of papers [1, 2, 3], Kashaev proposed a conjectural infinite family , being an odd positive integer, of complex valued topological invariants of links in any oriented compact closed 3-manifold , based on the theory of quantum dilogarithms at an ’th-root of unity [4, 5]. These invariants should be computed, in a purely 3-dimensional context, as a state sum , supported by an arbitrary distinguished and decorated triangulation of , say, if any. “Distinguished” means that is triangulated by a Hamiltonian subcomplex of ; the rather complicated decoration shall be specified later. The main ingredients of the state sum are the quantum-dilogarithm -symbols suitably associated to the tetrahedra of . On the “quantum” side, the state sum is similar to the Turaev-Viro one [6, 7]. On the “classical” side, it relates to the computation of the volume of a hyperbolic 3-manifold by the sum of the volumes of the ideal tetrahedra of any of its ideal triangulations.

Beside a somewhat neglected existence problem of such distinguished and decorated triangulations of , a main question left unsettled is just the invariance of when varies. On the other hand, Kashaev proved the invariance of under certain “moves” on distinguished and decorated triangulations, which gives some evidence for the conjectured topological invariance.

In [2] he also defined a family of topological invariants for links in using the solutions of the Yang-Baxter equation (i.e. the ) derived from the pentagon identity satisfied by the quantum-dilogarithm 6j-symbols [5]. Then he argued that these invariants coincide with the previous ones in the special case when . More recently, Murakami-Murakami [8] have shown that actually equals a specific coloured Jones invariant, getting, by the way, another proof that it is a well-defined invariant for links in .

Having reformulated Kashaev’s invariants (of links in ) within the main stream of Jones polynomials has been an important achievement, but it also has the negative consequence of putting aside the original purely 3-dimensional set-up (for links in an arbitrary ), willingly forgetting the complicated and somewhat mysterious decorations.

On the contrary, in our opinion, Kashaev’s 3-dimensional set-up deserved to be deeper understood and developed. As a by-product, we shall see in §7 that there are good reasons to believe that it could contain a consistent part of an “exact solution” of the Euclidean analytic continuation of (2+1) quantum gravity with negative cosmological constant, regarded as a gauge theory with the non-compact gauge group and an action of Chern-Simons type. In fact, the so called Volume Conjecture [9] about the asymptotic behaviour of when perfectly agrees, for what concerns the “real part”, with the expected “classical limit” of this theory [10, p.77]. In particular, recall that hyperbolic 3-manifolds are the pure-gravity (i.e. empty) Euclidean “classical” solutions.

The initial aim of the present paper was to work out a proof that the original 3-dimensional Kashaev set-up actually produces topological invariants. In fact, after having made it more “flexible” (as is demanded by technical and also conceptual reasons), and having better understood the decoration, we have finally obtained the following more general results that we state here in a somewhat informal way.

Notations. We denote by a compact closed oriented 3-manifold, is a link in , is a tubular neighbourhood of in , and . So has toral boundary components, where is the number of components of ; we denote them by , . Let be any compact oriented 3-manifold with non-empty boundary consisting of toral components , and any compact oriented 3-manifold with non-empty boundary , so that each of its connected components has genus .

Given , denotes a system of
essential, that is non contractible, simple closed curves, one on
each . is the Borel subgroup of of upper
triangular matrices, and denotes either an equivalence class of
flat principal -bundles on (resp. ), i.e. a bundle
endowed with a flat connection, or, equivalently, an element of
(resp. ),
acting by inner automorphisms. One usually calls
the *character variety of for *.

Finally, we denote by a *symplectic*
parametrization
of (see Section 5 for the precise definition).
The triples and are regarded up to
the natural equivalence relation induced by orientation preserving
homeomorphisms.

We first consider triples .

###### Theorem 1.1

A family of complex valued invariants , being any odd positive integer, is defined. The triple can be encoded by suitably decorated “distinguished” triangulations , and can be computed as a state sum whose value is invariant when varies. The original Kashaev’s purely topological invariants correspond to the case when , each is a meridian of and equals the class of the trivial flat bundle, or, equivalently, equals the constant representation .

###### Remark 1.2

There are two ways to look at , whence at these invariants.

(1) Let be the closed manifold obtained by Dehn filling of each boundary component of , along the curves of . If denotes the link of surgery “cores”, then becomes a family of meridians of the ’s, and is a flat bundle on with, in general, a non-trivial holonomy along these meridians. In classical gravity an analogous situation arises in presence of closed lines of massive particles of a universe (see for instance [11]).

(2) We are actually considering with parametrized boundary. More precisely, fix a base solid torus with meridian . Then each boundary component of is parametrized by an oriented diffeomorphism such that ; is well-defined up to automorphisms of which extend to the whole .

The symplectic parametrizations of surfaces cited above are defined in the spirit of Remark 1.2 (2), starting with base handlebodies of genus with fixed systems of meridians and so on. Then we also have:

###### Theorem 1.3

One can define a family of complex valued invariants , being any odd positive integer, which specializes to the previous one when is a union of tori.

Our final set-up shall be slightly different from the original Kashaev one. So we prefer to reach it step by step.

In §4.1-4.2, we shall prove Theorem 1.1 in a particular case: , each is a meridian of and is defined on the whole of , i.e. is trivial along the ’s; that is we actually construct invariants . Note that the original Kashaev invariants already are specializations of them (with trivial). To treat this case we shall need a milder modification of the original Kashaev set-up. The related existence problem of distinguished and decorated triangulations is solved in §2-3. Some basic properties of these invariants are settled in §4.3: “projective invariance”, and “duality” (which in particular describes the behaviour of the invariants when the orientation of changes).

Next, we treat the case when does not necessarily extend to the whole of . This is done in §5 through a trick, which consists roughly in cutting up pieces of a distinguished triangulation of , turning it into a decorated cell decomposition of with tetrahedron-like building blocks, which may be used to again define our previous state sums. Thanks to our definition of parametrized surface , there is a natural way to recover the set-up used in the proof of Theorem 1.1 from triples , thus obtaining Theorem 1.3.

In §6, we consider some remarkable
specializations of the invariants ; among these
there are the so called *Seifert-type* invariants of links in
-homology spheres. For , they seem to be good candidates
in order to fully recover the coloured Jones polynomials in terms of the
quantum hyperbolic state sum invariants (generalizing [8]), and to
give them a new interpretation.
One finds a preliminary discussion at the end of §6,
our work on this matter being in progress (cf. [12]).

We stress a qualitative contribution of the present paper in understanding the nature of the distinguished and decorated triangulations: it turns out that they are strongly reminiscent (up to a sort of reduction mod ) of the way one represents the classical refined scissors congruence class (belonging to the extended Bloch group ) of any given finite volume hyperbolic 3-manifold , by using any of its hyperbolic ideal triagulations [13, 14]. It is known that the classical Rogers dilogarithm lifts, with its fundamental five-terms identity, to an analytic function defined on , and that

where denotes the Chern-Simons invariant [14, 15]. Moreover one knows that the -dimensional quantum dilogarithm 6j-symbols and their pentagonal relations recover respectively, in the limit , the classical Euler dilogarithm and the Rogers five-terms identity (see [4, 5] and, for a detailed account, [16]). All this should indicate the right conceptual framework supporting the so called volume conjecture and natural generalizations, also involving the Chern-Simons invariant; roughly speaking one expects that:

Qualitative Hyperbolic -Conjecture. When admits some “hyperbolization” (for instance: is a complete hyperbolic manifold or is hyperbolic and is a geodesic link, as it arises from hyperbolic Dehn surgery), then one can recover by means of the asymptotic behaviour for of the invariants .

Nevertheless, some substantial facts are not yet well understood, and this matter needs further investigations (see also §3.3, §6 and §7).

Finally in §7 we try to set our results against the heuristic backgroud of the Euclidean analytic continuation of (2+1) quantum gravity with negative cosmological constant, regarded as a gauge theory with the non-compact group as gauge group, as defined by Witten [10]. We shall develop a few speculations about a path integral interpretation of our state sums and their relation with the absolute torsions of Farber-Turaev [17].

The aim of §8 is to provide the reader a precise statement of all the facts we use concerning our “quantum data”, allowing a substantially self-contained reading. We shall mostly insist on the geometric interpretation of the basic properties, showing how the decorations we use in this paper do arise. All the statements given without proof in the Appendix are carefully treated in [16].

## 2 Distinguished singular triangulations of

Let us recall some facts about standard spines of 3-manifolds and their dual ideal triangulations [18, 19]. A reference is [20], but one finds also a clear discussion about this material in [6] (note that sometimes the terminologies do not agree). Using the Hauptvermutung, we will freely intermingle the differentiable, piecewise linear and topological viewpoints for -dimensional manifolds.

Consider a tetrahedron and let be the interior of the 2-skeleton of the dual cell-decomposition. A simple polyhedron is a finite 2-dimensional polyhedron such that each point of has a neighbourhood which can be embedded into . A simple polyhedron is standard (in [6] one uses the term cellular) if all the components of the natural stratification of given by singularity are open cells. Depending on the dimension, we call these components vertices, edges and regions of .

Every compact 3-manifold (which for simplicity we assume connected) with non-empty boundary has standard spines, that is standard polyhedra embedded in Int such that collapses onto (i.e. is a regular neighbourhood of ). Standard spines of oriented 3-manifolds are characterized among standard polyhedra by the property of carrying an orientation, that is a suitable “screw-orientation” along the edges [20, Def. 2.1.1]. Such an oriented 3-manifold can be reconstructed (up to orientation preserving homeomorphisms) from any of its oriented standard spines. From now on we assume that is oriented.

A singular triangulation of a polyhedron is a triangulation in a weak sense, namely self-adjacencies and multiple adjacencies are allowed. For any as above, let us denote by the polyhedron obtained by collapsing each component of to a point. An ideal triangulation of is a singular triangulation of such that the vertices of are precisely the points of which correspond to the components of .

For any ideal triangulation of , the 2-skeleton
of the *dual* cell-decomposition of is a standard spine
of . This procedure can be reversed, so that we can associate
to each standard spine of an ideal triangulation of
such that . Thus standard spines and ideal triangulations
are dual equivalent viewpoints which we will freely intermingle.
Note that, by removing small neigbourhoods of the vertices of ,
any ideal triangulation leads to a cell-decomposition of by
truncated tetrahedra which induces a singular triangulation of the
boundary of .

###### Remark 2.1

For the following facts, a reference is [21, Ch. E]. The name “ideal triangulation” is inspired by the geometric triangulations of a non compact hyperbolic 3-manifold with finite volume (for example, the complement of a hyperbolic link in ) by ideal hyperbolic tetrahedra (possibly partially flat). It is a standard result of Epstein-Penner [22] that such triangulations exist. A geometric ideal triangulation can be regarded as a special topological ideal triangulation admitting suitable decorations by the moduli of the corresponding hyperbolic ideal tetrahedra. Then, the volume of the 3-manifold can be expressed as the sum of the volumes of the ideal tetrahedra of any of its geometric triangulations, and it can be computed in terms of the moduli via the Bloch-Wigner dilogarithm function [23]. Of course the volume, which by Mostow’s rigidity theorem is a topological invariant, does not depend on the specific geometric ideal triangulation used to compute it. In a sense, the quantum hyperbolic state sums we are concerned with can be considered as “quantum” deformations of this “classical” hyperbolic situation, which make sense for arbitrary 3-manifolds.

Consider now our closed 3-manifold . For any let , that is the manifold with spherical boundary components obtained by removing disjoint open balls from . Clearly and any ideal triangulation of is a singular triangulation of ; moreover all singular triangulations of are obtained in this way. We shall adopt the following terminology.

###### Definition 2.2

A singular triangulation of is simply called a triangulation. Ordinary triangulations (where neither self-adjacencies nor multi-adjacencies are allowed) are said to be regular. An almost-regular triangulation of is a triangulation having the same vertices as a regular one. This is equivalent to saying that there exists a regular triangulation of with the same number of vertices. Given a triangulation , , , shall denote the number of vertices, edges, faces, tetrahedra of .

The main advantage in using singular triangulations (standard spines) instead of only ordinary triangulations consists of the fact that there exists a finite set of moves which are sufficient in order to connect (by means of finite sequences of these moves) singular triangulations (standard spines) of the same manifold. On the contrary, if we pretend to keep ordinary triangulations at each steps, we are forced to consider an infinite set of moves (see [6]).

Let us recall two elementary moves on triangulations (spines) that we shall use throughout the paper; see Fig. 1.

The move. Replace the triangulation of a portion of made by the union of tetrahedra with a common 2-face by the triangulation made by tetrahedra with a new common edge which connect the two vertices opposite to .

The move. Add a new vertex in the interior of a tetrahedron of and make from it the cone over the triangulated boundary of . The dual spine of the triangulation thus obtained is a spine of , where is an open ball in the interior of .

The and moves can be easily reformulated in dual terms (see for instance [20, p.15], where they are called “MP-moves”; in Fig. 2, we show their branched versions). Standard spines of the same with at least two vertices (which, of course, is a painless requirement) may always be connected by the (dual) move and its inverse. In particular, any almost-regular triangulation of can be obtained from a regular one via a finite sequence of or moves.

In order to handle singular triangulations of a closed manifold , we also need a move which allows us to vary the number of vertices. Although this is not the shortest way (the so-called bubble move makes a hole in any by introducing only two more vertices, see Proposition 8.11), the move (and its inverse) shall be convenient for our purposes. Let us describe the dual spine of . Consider the vertex of (which is dual to ) contained in . We are removing a ball around of . The boundary of this ball is a portion of and the natural cell decomposition of induces on it a cell decomposition which is isomorphic to the “bi-dual” tetrahedron of ; that is it is isomorphic to the cell decomposition of dual to the standard triangulation.

For technical reasons we shall need a further move which we recall in Fig. 1 in terms of standard spines and that we denote by move. It is also known as lune move and is somewhat similar to the second Reidemeister move on link diagrams. Note that the inverse of the lune move is not always admissible because one could lose the standardness property when using it. However we shall need only the “positive” lune move.

The following technical result due to Makovetskii [24] shall be necessary.

###### Proposition 2.3

Let and be standard spines of . Then there exists a spine of such that can be obtained from both and via a finite sequence of and moves (we stress that they are all positive moves).

###### Definition 2.4

A distinguished triangulation of has the property that is Hamiltonian, that is at each vertex of there are exactly two “germs” of edges of . Of course, the two germs could belong to the same edge of .

It is convenient to give a slightly different description of the distinguished singular triangulations in terms of spines.

###### Definition 2.5

Let be as before. Let be any finite family of disjoint simple closed curves on . We say that is a quasi-standard spine of relative to if:

(i) is a simple polyhedron with boundary consisting of circles. These circles bound (unilaterally) annular regions of . The other regions are cells.

(ii) is properly embedded in and transversely intersects at .

(iii) is is a spine of .

Note that annular regions is a simple spine of .

###### Lemma 2.6

Let and be as before. Quasi-standard spines of relative to do exist.

Proof. Let be any standard spine of . Consider a normal retraction . Recall that is the mapping cilynder of ; for each region of , ; for each edge , a “tripode”; for each vertev , a “quadripode”. We can assume that is in “general position” with respect to , so that the mapping cylinder of the restriction of to is a simple spine of relative to (with the obvious meaning of the words); possibly after doing some moves we then obtain a quasi-standard Q.

###### Definition 2.7

Consider and formed by the union of parallel copies of the meridian of the component of , . A spine of adapted to of type is a quasi-standard spine of relative to such an .

It is clear that Proposition 2.3 extends to the “adapted” setting. In particular, we have:

###### Proposition 2.8

Let and be quasi-standard spines of relative to and of type . Then there exists a spine of relative to and of type such that can be obtained from both and via a finite sequence of and moves, and at each step we still have spines of adapted to and of type .

###### Remark 2.9

If is a spine of adapted to as before, then by filling each boundary component of by a 2-disk, we get a standard spine of , , and the dual triangulation of naturally contains as a Hamiltonian subcomplex; that is we have obtained a distinguished triangulation of . Vice-versa, starting from any , by removing an open disk in the dual region to each edge of , we pass from to a spine of adapted to , of some type. So they are equivalent viewpoints.

As an immediate corollary we have:

###### Corollary 2.10

For any there exist distinguished triangulations of with vertices.

Let us analyze the moves on distinguished triangulations. The and moves specialize to moves between distinguished triangulations as follows.

Let us denote by any such a move. If is a distinguished triangulation of , we want to get moves . In the case there is nothing to do because is still Hamiltonian. In the case, we assume that an edge of lies in the boundary of the involved tetrahedron ; lies in the boundary of a unique 2-face of containing the new vertex. Then we get the Hamiltonian just by replacing by the other two edges of . We have also:

###### Lemma 2.11

Let be a distinguished triangulation of and a move. Then it can be completed to a move .

Proof. This is clear if we think in dual terms. If is a spine adapted to and is the move, the dual regions in to the edges in “persist” in so we find .

Finally we can solve half of the existence problem mentioned in the introduction.

###### Proposition 2.12

There exist almost-regular distinguished triangulations of .

Proof. It is enough to remark that any distinguished triangulation, which exists by Corollary 2.10, can be made almost-regular after a finite number of moves.

## 3 Decorations

In this section we have to properly define, and possibly better understand, the decorations of distinguished triangulations of .

### 3.1 Branchings

Let be a standard spine of and consider as usual the dual ideal triangulation . A branching of is a system of orientations on the edges of such that each “abstract” tetrahedron of has one source and one sink on its 1-skeleton. This is equivalent to saying that, for any 2-face of , the edge-orientations do not induce an orientation of the boundary of .

In dual terms, a branching is a system of orientations on the regions of such that for each edge of we have the same induced orientation only twice. In particular, note that each edge of has an induced orientation.

In the original set-up of [1] one used regular triangulations of , with a given total ordering on the set of vertices; in fact the role of the ordering is just to define a branching via the natural rule: “on each edge, go from the smaller vertex towards the bigger one”. Note that one could not exclude, a priori, that after some negative moves, one eventually reaches veritable singular triangulations, for which such a total ordering no longer induces a branching.

Branchings, mostly in terms of spines, have been widely studied in [20] (see also [25]). They are rich structures: a branching of allows us to give the spine the extra structure of an embedded and oriented (hence normally oriented) branched surface in Int(); by the way, this also justifies the name. Moreover a branched carries a suitable positively transverse combing of .

We recall here part of their combinatorial content.
A branching allows to define an orientation on *any* cell of ,
not only on the edges. Indeed, consider any “abstract” tetrahedron
. For each vertex of consider the number of incoming
-oriented edges in the 1-skeleton. This gives us an
ordering of the vertices which
reproduces the
branching on , according to the former rule.
This gives us a base vertex
and an ordered triple of edges emanating from , whence an orientation
of .
Note that this orientation may or may not agree with the orientation of ;
in the first case
we say that is of index , and it is of index otherwise.

To orient 2-faces we work in a similar way on the boundary of each “abstract” 2-face . We get an ordering , a base vertex , and finally an orientation of . This 2-face orientation can be described in another equivalent way. Let us consider the 1-cochain such that for each -oriented edge. Then there is a unique way to orient any 2-face such that the coboundary .

The corresponding dual orientation on the edges of is just the induced orientation mentioned in §2.

Branching’s existence and transit. This matter is
carefully
analyzed in [20]. In general,
a given ideal triangulation of could admit no branching,
but there exist branched ideal triangulations of any . More precisely,
given any
system
of edge-orientations on and any move , a *transit*
is given by any system of edge-orientations on
*which agrees with on the “common” edges*.
In [20, Th. 3.4.9] one proves:

###### Proposition 3.1

For any there exists a finite sequence of transits such that the final is actually branched.

If is a branched spine and is either a or a move, then it can be completed (sometimes in a unique way, sometimes in two ways) to a branched transit . On the contrary, it could happen that a or inverse move is not “branchable” at all (see [20, Ch.3]). However we shall only use the “positive” moves. Note that otherwise it could stop any attempt to prove the invariance of the quantum hyperbolic state sums, via any argument of “move-invariance”. In Fig. 2 and Fig. 3 we show the branched transits.

In fact the list is not complete, but one can see in the figures all the essentially different behaviours, and easily complete the list by applying evident symmetries. Following [20] we can distinguish two quite different kinds of branched transits: the sliding moves which actually preserve the combing, and the bumping moves which eventually change it. For the moves there is a similar behaviour (see Fig. 4).

However, we shall not exploit this difference in the present paper. Note that the middle sliding move in Fig. 2 corresponds dually to the triangulation move shown in Fig. 5.

Given a distinguished triangulation of , the first component of is just a branching of . If is either a , or move, we have again some branched transit ; we have already described it in the first two cases. In the last case there are several ways to take which agree with on the edges “already” present in . Anyone of these ways is a possible transit.

By combining the existence of distinguished triangulations with the above proposition we have:

###### Lemma 3.2

There exist almost-regular branched distinguished triangulations of .

###### Remark 3.3

Branching’s rôle. To understand the rôle of the branching, let us go back to the “classical” hyperbolic ideal triangulations (see Remark 2.1). A hyperbolic ideal tetrahedron with ordered vertices on the Riemann sphere is determined up to congruence by the cross ratio belongs to the upper half plane of . Changing the ordering by even permutations produces the cyclically ordered triple , which is the actual modular triple of . Changing the orientation corresponds to considering the complex conjugate triple. . If the tetrahedron is positively oriented

A branching on the ideal triangulation allows one to choose, in a somewhat globally coherent way, for each , a complex valued in the corresponding triple. In the usual decoration of the edges of by means of the elements of the triple, is associated to the first edge emanating from the base vertex , and so on, respecting the orders of the triple and of the edges.

Starting with a topological ideal triangulation of a
3-manifold , one usually tries to make it “geometric”
(proving by the
way that is hyperbolic) by solving some suitable system of
hyperbolicity equations in complex
indeterminates, being the number of tetrahedra. Then, a branching
allows us to specify *which* system of equations, and this
last is governed by a certain global coherence.

Such a kind of global coherence shall be important to control the behaviour of our state sums up to decorated moves. On the other hand, it is not too surprising, in the spirit of the classical situation, that the value of the state sums shall not eventually depend on the branching of the decoration.

### 3.2 Full cocycles

Recall that we are actually considering a triple where is an equivalence class of flat principal -bundles on . Set .

Let be a branched distinguished triangulation of . Let denote the equivalence class of a cellular 1-cocycle on via the usual equivalence relation up to coboundaries: since a (cellular) 0-cochain is a -valued function defined on the vertices of , and are equivalent if they differ by the coboundary of some 0-cochain . This means that for any -oriented edge with ordered end-points , one has . We denote the quotient set by ; recall that it can be identified with the set of equivalence classes of flat -principal bundles on .

The second component of is a -valued full 1-cocycle on representing ; we write . For each -oriented edge of , is an upper triangular matrix; we denote by the upper-diagonal entry of this matrix. “Full” means that for each , .

###### Remark 3.4

The fullness property strictly concerns the cocycle and not its class . It only depends on whether a given class could be represented by full cocycles. Moreover, fullness does not depend on the branching . In fact we can define this notion by using any arbitrary system of edge-orientations.

We refer to §6 for more details and examples of -bundles. Here we simply recall that there are two distinguished abelian subgroups of , and this fact induces some distinguished kinds of cocycles. They are:

(1) the Cartan subgroup of diagonal matrices; it is isomorphic to the multiplicative group . The evident isomorphism maps to ;

(2) the parabolic subgroup of matrices with double eigenvalue ; it is isomorphic to the additive group . The evident isomorphism maps to .

Denote by any such subgroup. We get a map , where is endowed with the natural abelian group structure. Note that is isomorphic to the ordinary (singular or de Rham) 1-cohomology of .

A complex valued injective function defined on the set of vertices of a regular distinguished triangulation of (as in the original set-up of [1]) should be regarded as a -valued -cochain. Its coboundary is a basic example of a full cocycle representing (whence the trivial flat bundle in ).

Existence and transit of full cocycles. This is a somewhat delicate matter. Let be a triangulation of , with any edge-orientation system . Let be a 1-cocycle on . Consider any transit as before. In the or cases (and their inverses), there is a unique on which agrees with on the common edges. This defines a transit . Clearly .

For moves , there is an infinite set of possible transits such that and the cocycles agree on the common edges. Moreover, given one transit with a full cocycle, we can always turn into an equivalent full , which differs from by the coboundary of some -cochain with support consisting of the new vertex of . Hence for moves there always exists an infinite set of full transits.

Assume now that is full, and consider or moves (or their inverses). The trouble is that, in general, is no longer full. If is a branched distinguished transit and is full on , then we have a completion to a full transit only if also the final is full. Otherwise we must stop.

However almost-regular triangulations have generically a good behaviour with respect to the existence and transit of full cocycles.

###### Proposition 3.5

Let be an almost-regular triangulation of endowed with any edge-orientations system . Let be any finite sequence of or orientation-transits. Then there exists a sequence of open dense sets of full cocycles on (in the natural subspace topology of ), such that is contained in the image of via the elementary transit , and each class can be represented by cocycles in .

As an immediate corollary (use the case ) we make a further step towards a solution of the existence problem:

###### Corollary 3.6

Given a triple , then a partially decorated (even almost-regular) distinguished triangulation do exists.

Proof of the proposition. As is almost-regular, there exists a regular triangulation and a sequence of or orientation-transits. It is enough to prove the proposition for , so let us assume that is regular. The conclusion of the theorem holds for . In fact we can suitably perturb any by the coboundaries of -cochains, see the discussion above. Now we remark that each elementary cocycle-transit can be regarded as an algebraic bijective map from the space of 1-cocycles on to the space of 1-cocycles on . The set of full cocycles for which the full elementary transit fails are contained in a proper algebraic subvariety. So the conclusion follows, working by induction on .

Full cocycles rôle. Let be the full cocycle of a decoration . For each edge denote by the -entry of ; is as before. Fix a determination of the ’th-root holding for all entries of , for all . Consider the Weil algebra , which is described in details in §8. Then we can associate to each the -dimensional standard representation of characterized by the pair .

This system of representations satisfies the following properties:

(1) Consider any -oriented “abstract” 2-face of . Starting from the base vertex , according to the orientation, we find a cycle of edges . The first two are positively -oriented. The last one has the negative orientation. Then the cocycle condition on implies that

* is, up to isomorphism, the unique irreducible summand of the
representation .*

(2) Consider, in particular, the 2-face opposite to the base vertex , in any tetrahedron of . For each of , let us denote by the opposite edge in . Then the -components of the representations satisfy the following Fermat relation:

(3) The same conclusions hold for any other branching on , as they only depend on the cocycle condition.

After a full transit (once also the charge-transit shall be ruled out), we observe that:

The system of representations varies exactly in the way one needs in order to apply the fundamental algebraic identities (the pentagon, orthogonality, and bubble relations of the Appendix) satisfied by the quantum-dilogarithm c-6j-symbols.

One could ask if systems of representations verifying the above properties are more general than the one obtained starting from full cocycles. In fact, setting and one obtains a full cocycle . In a sense the system can be considered as a sort of reduction mod of . In order to study any “classical” limit, when , of our state sums invariants, it seems quite appropriate to consider the reductions of the same cocycle.

### 3.3 Charges

The “classical” source of the charges in the decorations clearly emerges from Neumann’s work on the Cheeger-Chern-Simons classes of hyperbolic -manifolds and scissors congruences of hyperbolic polytopes. Since there is a wide literature on this subject (see the references in [13]), we shall only report a few details.

Refined Scissors Congruence.
From the work of [23] and [26], we know that the
volume of any oriented hyperbolic
3-manifold has a deep analytic relationship with
another geometric invariant, the Chern-Simons invariant . Recall that is a -valued invariant defined for any oriented compact Riemannian 3-manifold, and that its definition can also be extended, with value in , to *non-compact* complete and finite volume hyperbolic 3-manifolds [27]. Consider then

It turns out that (for any finite volume ), actually depends on a weaker sub-structure of the full hyperbolic structure, called the refined scissors congruence class and denoted by [13, 14]; it is orientation sensitive and takes values in the extended Bloch group . If is non-compact, this class may be represented with the help of the usual geometric ideal triangulations ; if is compact, we assume here that it is obtained by hyperbolic Dehn surgery on some non-compact one, so that we have some deformed geometric ideal triangulations of , where is a finite set of simple closed geodesics in . Then, by abuse of notations, in both cases we shall denote these special decompositions of by and call them “ideal triangulations of ”.

For the sake of clarity, let us present the construction of . It relies heavily on the functional properties of the classical Rogers dilogarithm, defined on by:

and in particular on its five-term identity, which reads:

(1) |

One then starts with a four components non connected covering of :

It can be regarded as the Riemann surface for the collection of all branches of the functions on , . If is obtained by splitting along the rays and , is a suitable identification space from : namely, . The map

is well-defined on , and it gives an identification between and the set of triples of the form

with

and for some determination of the function, where is a modular triple for an ideal tetrahedron ; in other words we adjust its dihedral angles by means of multiples of so that the resulting angle sum is zero. Such a triple is called a combinatorial flattening of the ideal tetrahedron. So can be regarded as the set of these combinatorial flattenings, for all ideal tetrahedra of the hyperbolic space .

The Rogers dilogarithm lifts to an analytic function:

and it extends to the free
-module . Moreover, one may
lift the classical five-term identity satisfied by the Rogers
dilogarithm to the function , provided that we pass to some
quotient of . There is such a natural
maximal one , which is called the
*extended pre-Bloch group*. In this way, we can turn into a homomorphism.

Take our finite volume oriented hyperbolic 3-manifold ; for any ideal triangulation of , one can define

where depends on the orientation of . When is non-compact just represents the (refined) scissors congruence class of . In the compact case, how to explicitly represent is a subtler fact; anyway the above is adequate to represent a “classical” counterpart of our state sums, which in general depend also on the link and not only on the ambient manifold.

###### Remark 3.7

Consider the map:

It lifts the Dehn invariant of the classical scissors congruence [13]. Then is called the extended Bloch group, and one can show that or are in .

The classes or may be seen as representatives of the fundamental class of in the (discrete) homology group ([14, 15])

which itself maps surjectively onto [14] (see the definition in Remark 3.7). The relations in express the move for ideal hyperbolic tetrahedra endowed with combinatorial flattenings, and then they give the independence of from .

The relations in allow to have a global control on the values of the combinatorial flattenings, and this is made easier by the presence of a branching of . For instance, note that a, let us say, -flattening of a presupposes a choice of opposite edges; changing this choice turns it into a or flattening.

Anyway, the point here is that part of this global coherence is expressed in terms of relations which must be satisfied by the integral components of the flattenings , which then give, by definition, an integral charge on . These relations are strongly reminiscent of “the” system of hyperbolicity equations (for instance the one specified by the branching) which is satisfied by the arguments of the modular triples of the ideal triangulation. The charge of a decoration shall be, essentially, the reduction mod of an integral charge. In §7 we shall return on these scissors congruence classes and on their relationship with the -invariant.

Integral charges. We shall now give the formal definition of the integral charges in our own setting. It is a straightforward adaptation of the integral charges of the previous paragraph.

Let be a distiguished triangulation of . With the notations of Definitions 2.5 and 2.7, let us assume for simplicity that the associated simple spine of is in fact a standard one, as in the proof of Lemma 2.6. The following discussion could be adapted also to the case when is merely simple; anyway, we could also add the standardness assumption to our set-up without any substantial modification in all our arguments.

We know that the truncated tetrahedra of induce a triangulation of . Let be an oriented simple closed curve on in general position with respect to . We say that has no back-tracking with respect to if it never departs a triangle of across the same edge by which it entered. Thus each time passes through a triangle, it selects the vertex between the entering and departing edges, and gives it a sign , according as it goes past this vertex positively or negatively with respect to the boundary orientation.

Let be a simple closed curve in in general position with respect to the ideal triangulation . We say that has no back-tracking with respect to if it never departs a tetrahedron of across the same 2-face by which it entered. Thus each time passes through a tetrahedron, it selects the edge between the entering and departing faces.

Fix in each component of one of the meridians in the boundary of , say, and a simple closed curve in general position with respect to , which intersects transversely at one point. Orient and . We can assume that these curves have no back-tracking with respect to . It is clear that any class in may be represented in the set of isotopy classes of curves without back-tracking with respect to and generated by and . In the same manner, any class in may be represented up to isotopy by a curve in without back-tracking with respect to .

Denote by the set of all edges of all “abstract” tetrahedra of ; there is a natural map .

###### Definition 3.8

An integral charge on is a map

which satisfies the following properties:

(1) For each 2-face of any abstract with edges ,

for each ,

for each ,

(2) Let be a curve on which has no back-tracking with respect to . Each time enters a triangle of , associates in a natural way an integer to the selected vertex; multiply this integer by the sign of the vertex and take the sum of these signed integers. Then for every ,

(3) Let be any curve which has no back-tracking with respect to . Each time enters a tetrahedron of , associates in a natural way an integer to the selected edge. Take the sum of these integers. Then, for each ,

###### Definition 3.9

The third component of any decoration , called the charge, is of the form:

where (with the notations of the Appendix), and is an integral charge on .

###### Remark 3.10

By conditions (1), satisfies, in particular, the charge requirements of the quantum data, see Proposition 8.5. By the way, note the importance of being odd in the present definition. Conditions (2) and (3) are in fact purely homological; in particular, as already said, their validity does not depend on the particular choice of and on each component of .

Charge’s existence and transit. The existence of integral charges is obtained by just rephrasing the proof of Theorem 2.4.(i) (that is of Lemma 6.1) in [28]. The only modification is in considering as an ideal triangulation of (see Remark 2.9); if we set the sum of the charges around the edges of to be equal to , the existence follows via the same arguments.

Also Theorem 2.4.(ii) [28] shall be important in order to prove the state sum charge-invariance. Let us first describe qualitatively this result. Let and be respectively the number of vertices and edges of ; an easy computation with the Euler characteristic shows that there are exactly tetrahedra in (see Proposition 4.4). The first condition in Definition 3.8 (1) says that there are only two independent charges on the edges of each abstract , and given a branching on there is a preferred such ordered pair . Set (then ). Then, there is a canonical way to write down an integral charge on as a vector in with first components , and then . Theorem 2.4. (ii) in [28] says that integral charges lie in a specific sublattice of :

###### Proposition 3.11

There exist determined , , such that, given any integral charge , all the other integral charges are of the form

where the second addendum is an arbitrary integral linear combination of the .

The vectors have the following form. For each tetrahedron glued along a specific , define and as the coefficients in , when written in terms of . Then is the vector in with first components , and then . For instance, in the situation described on the right of Fig. 5 (where the ordering of the tetrahedra is induced by the ordering of the vertices), we easily see that .

Next, we describe the transit of integral charges. The transit of -charges shall be obtained by reduction mod . Let be a move. Let be an integral charge on , and the edge that appears. Consider the two “abstract” tetrahedra of involved in the move. They determine a subset of . Denote by the restriction of to . Let be an integral charge on . Consider the three “abstract” tetrahedra of involved in the move. So we have and with the clear meaning of the symbols. Denote by the complement of in and the restriction of . Do similarly for and . Clearly and can be naturally confused. The following lemma is the key point of the charge-invariance.

###### Lemma-Definition 3.12

We have a charge-transit if:

(1) For each “common edge” of and ,

(2) and agree on