Spin structures on 3-manifolds via arbitrary triangulations
aa r X i v : . [ m a t h . G T ] A p r Spin structures on 3-manifoldsvia arbitrary triangulations
Riccardo
Benedetti
Carlo
Petronio
April 16, 2013
Abstract
Let M be an oriented compact 3-manifold and let T be a (loose) triangula-tion of M , with ideal vertices at the components of ∂M and possibly internalvertices. We show that any spin structure s on M can be encoded by extracombinatorial structures on T . We then analyze how to change these extrastructures on T , and T itself, without changing s , thereby getting a com-binatorial realization, in the usual “objects/moves” sense, of the set of allpairs ( M, s ). Our moves have a local nature, except one, that has a globalflavour but is explicitly described anyway. We also provide an alternativeapproach where the global move is replaced by simultaneous local ones.MSC (2010): 57R15 (primary); 57N10, 57M20 (secondary).
Combinatorial presentations of 3-dimensional topological categories, such as thedescription of closed oriented 3-manifolds via surgery on framed links in S , andmany more, are among the main themes of geometric topology, and in particularhave proved crucial for the theory of quantum invariants, initiated in [16] and [18].A combinatorial presentation of the set of all pairs ( M, s ), with M a closedoriented 3-manifold and s a spin structure on M , was already contained in [5].This presentation was realized by selecting the (loose) triangulations of M havingonly one vertex and supporting a ∆- complex structure (see [8]), also called a branching . The viewpoint adopted in [5] was actually that of special spines,equivalent to that of triangulations via duality (see Matveev [14] and below). Forthe special spine dual to a triangulation, a branching is precisely a structure of oriented branched surface (see Williams [20]), and this structure was used in [5]to define a trivialization of the tangent bundle of M along the 1-skeleton of thespine, whence a spin structure on M , using constructions already proposed byIshii [9] and Christy [7].The construction just described easily extends to pairs ( M, s ) with M a com-pact oriented 3-manifold with non-empty boundary and s a spin structure on M ,1sing branchable triangulations of M with ideal vertices at the components of ∂M , and possibly internal internal vertices. This approach however suffers fromthe drawback that not all triangulations of M are branchable: for instance, thecanonical triangulation by two regular hyperbolic ideal tetrahedra of the hyper-bolic one-cusped manifold called the “figure-eight-knot-sister” is not branchable.On one hand, one easily sees that any triangulation of M has branchable sub-divisions ( e.g. , take a regular subdivision and define a branching by choosing atotal ordering of the vertices). On the other hand, in many circumstances one isinterested in sticking to a given triangulation of M , or to consider the class of allvertex-efficient triangulations of M (namely, the purely ideal triangulations fornon-empty ∂M , and the 1-vertex triangulations for closed M ).Recently, generalized versions of the notion of branching (see the definitionsbelow), with the nice property of existing on every triangulation, have emergedas useful devices to deal with simplicial formulas defined over triangulationsequipped with solutions of Thurston’s PSL(2 , C ) consistency equations (or vari-ations of them [11, 12]). For instance, motivated by his work in progress on theentropy of solutions of the homogeneous PSL(2 , R ) Thurston equations, Luo in-troduced the notion of Z / Z -taut structure on a triangulation, and it turns outthat a certain notion of weak branching , widely employed below together with theunderlying notion of pre-branching , easily allows to show that every triangulationadmits Z / Z -taut structures (see Remark 1.2). As another example, the same no-tions of weak and pre-branching were exploited in [1] to extend the constructionof quantum hyperbolic invariants [2, 3] to an arbitrary hyperbolic one-cuspedmanifold, over a canonical Zariski-open set of the geometric component of itscharacter variety.In several instances Luo [10] suggested that a combinatorial encoding of spinstructures based on arbitrary triangulations might be of use for the constructionof spin-refined invariants obtained from simplicial formulas as those mentionedin the previous paragraph. In this note we provide such a presentation, using thenotion of weak branching already alluded to.The results established in this paper provide an “objects/moves” combina-torial presentation of the set of all pairs ( M, s ), with M a compact oriented3-manifold and s a spin structure on M , in the following sense: • Given any (loose) triangulation T of M , with ideal vertices at the com-ponents of ∂M and possibly internal vertices, and any s , we encode s bydecorating T with certain extra combinatorial structures; • We exhibit combinatorial moves on decorated triangulations relating to eachother any two that encode the same (
M, s ).2e note that all our moves are explicitly described, but one of them has an in-trinsically global nature. On the other hand, in the second part of the paper wewill show that this move can actually be replaced, in a suitable sense, by a com-bination of local ones. This last result is subtle and technically quite demanding,it is based on some non-trivial algebraic constructions, and it unveils unexpectedcoherence properties of the graphic calculus we use to encode weakly branchedtriangulations.A first application of the technology developed in the present note appearsin [1], where our results are used to solve a sign indeterminacy in the phaseanomaly of the quantum hyperbolic invariants (see Remark 2.9). We also notethat adapting the arguments of [5, Chapter 8], the results of this article canbe used to provide an effective construction of the Roberts spin-refined Turaev-Viro invariants [17], and of the related Blanchet spin-refined Reshetikhin-Turaevinvariants [6] of the double of a manifold.
In this section we state some results that provide in terms of arbitrary triangu-lations a combinatorial encoding of spin structures on oriented 3-manifolds. Thegeometric construction underlying this encoding actually employs certain objectscalled special spines , and will be fully described in Sections 2 and 3. As a matterof fact, triangulations and special spines are equivalent to each other via duality,but perhaps the majority of topologists is more familiar with the language oftriangulations, which is why we are anticipating our statements in this section.
In this note M will always be a connected, compact and oriented 3-manifold, withor without boundary. We also assume that ∂M has no S component (otherwisewe canonically cap it with D ). We begin with several definitions. A triangulation of M is the datum T of • a finite number of oriented abstract tetrahedra, and • an orientation-reversing simplicial pairing of the 2-faces of these tetrahedrasuch that the space obtained by first gluing the tetrahedra along the pairings andthen removing open stars of the vertices is orientation-preservingly homeomorphicto M with some punctures (open balls removed). Any number of punctures,including zero, is allowed (but a closed M must be punctured at leats once).3 v v v v v v v Figure 1:
Left: a branched tetrahedron of index +1 and one of index −
1. Right: a weakbranching compatible with a pre-branching.
A A AA A A ‘ ‘ ‘‘ ‘ ‘‘ ‘ ‘
B B BB B B ‘ ‘ ‘‘ ‘ ‘‘ ‘ ‘
C C CC C C ‘ ‘ ‘‘ ‘ ‘‘ ‘ ‘
Figure 2:
The three types of face-pairing in a weakly branched triangulation. A branching on an abstract oriented tetrahedron ∆ is an orientation of itsedges such that no face of ∆ is a cycle. Equivalently, one vertex of ∆ should be asource and one should be a sink, as illustrated in Fig. 1-left. Note that the figureshows the only two possible branched tetrahedra up to oriented isomorphism.They are characterized by an index ±
1, to define which one denotes by v j thevertex of ∆ towards which j edges of ∆ point, and one checks whether the ordering( v , v , v , v ) defines the orientation of ∆ or not. Each face of a branched abstracttetrahedron is endowed with the prevailing orientation induced by its edges.A pre-branching on a triangulation T is an orientation ω of the edges of thegluing graph Γ of T (a 4-valent graph) such at each vertex two edges are incomingand two are outgoing. Given such an ω , a weak branching b compatible with ω isthe choice of an abstract branching for each tetrahedron in T such that Γ withits orientation ω is positively transversal to each face of each tetrahedron in T ,as in Fig. 1-right. Note that for such a b when two faces are glued in T either allthree edge orientations are matched or only one is, and in both the glued facesit is one of the prevailing two, as in Fig. 2 (the labels ∅ , +1 , − .2 Spin structure from a weak branching and a 1-chain All the constructions and results of the rest of this section will be explained andproved in Sections 2 and 3 in the dual context of special spines. Let a triangulation T with pre-branching ω and compatible weak branching b be given. We will nowdefine a chain α ( P, ω, b ) = P e α ( e ) · e ∈ C ( T ; Z / Z ), where e runs over all edgesof e . The value of α ( e ) is the sum of a fixed initial contribution 1 plus certaincontributions of two different types; both contribution types are computed in thegroup G = (cid:0) · Z (cid:1) / Z , but for each of them the sum is in Z / Z ; here comes thedescription of the two types: • Endow e with an arbitrary orientation and in the abstract tetrahedra of T consider the collection of all the edges projecting to e and of type v v or v v ; for each such abstract edge e e take a contribution + or − dependingon whether the projection from e e to e preserves or reverses the orientation; • Consider all the face-gluings as in Fig. 2 in which e is involved (with mul-tiplicity) and take a contribution depending as follows on the type t of thegluing and on the position of e within it: ⊲ t = ∅ , regardless of the position of e ; ⊲ t = ± e is matched by the gluing; ⊲ ∓ if t = ± e is not matched by the gluing. Proposition 1.1. α ( P, ω, b ) is a coboundary, and to every β ∈ C ( T ; Z / Z ) such that ∂β = α ( P, ω, b ) there corresponds a spin structure s (cid:0) T , ω, b, β (cid:1) on M .Moreover s (cid:0) T , ω, b, β (cid:1) = s (cid:0) T , ω, b, β (cid:1) if and only if β + β is in H ( T ; Z / Z ) . Remark 1.2.
Let b be a weak branching compatible with a pre-branching ω on a triangulation T of a manifold M . If in each abstract tetrahedron of T wechoose the pair of opposite edges of types v v and v v with respect to b , thenthe choice actually depends on ω only, not on b . Moreover one sees that for alledges e of T in M there is always an even number of abstract edges of types v v or v v projecting to e (this corresponds to the fact that the contributionsto α ( P, ω, b ) of the first type described above are in Z / Z , and it is established inProposition 2.7 below). It follows that, giving sign − v v and v v , and sign +1 to the other edges, we get a Z / Z -taut structure on T , as mentioned in the introduction. The next results provide the combinatorial encoding of spin structures announcedin the title of the paper. From now on all chains β ∈ C ( T ; Z / Z ) will be viewed5 Figure 3:
Moves preserving the pre-branching and the associated spin structure. In bothmoves the “1” means that 1 must be added to the coefficient in β of the triangle to which “1” isattached; note that in both moves it is the only one whose three edges all retain their orientationunder the move. Figure 4:
A circuit γ in the gluing graph that in each tetrahedron visits faces sharing the edge v v . The gluing encoded by an edge of γ need not match edges of type v v to each other. up to 2-boundaries, without explicit mention. Proposition 1.3. s (cid:0) T , ω, b , β (cid:1) = s (cid:0) T , ω, b , β (cid:1) if and only if ( b , β ) and ( b , β ) are related by the moves of Fig. 3 (and their compositions and inverses). Proposition 1.4. s (cid:0) T , ω , b , β (cid:1) = s (cid:0) T , ω , b , β (cid:1) if and only if ( ω , b , β ) and ( ω , b , β ) are related by the moves of Proposition 1.3 and additional moves (cid:0) T , ω, b, β (cid:1) (cid:16) T , ω ′ , b ′ , β ′ (cid:17) described as follows: • In the gluing graph of T (which is oriented by ω ) take an oriented simplecircuit γ such that, for each tetrahedron it visits, the two faces it visits sharethe edge v v with respect to b , as in Fig. 4; • Define ω ′ by reversing γ , define b ′ by reversing each edge v v in each tetra-hedron visited by γ , and define β ′ by adding to the coefficient of each faceof T visited by γ and incident to tetrahedra of distinct indices. Proposition 1.5. s (cid:0) T , ω , b , β (cid:1) = s (cid:0) T , ω , b , β (cid:1) if and only if the quadru-ples (cid:0) T , ω , b , β (cid:1) and (cid:0) T , ω , b , β (cid:1) are related by the moves of Propositions 1.3and 1.4 and those shown in Figg. 5 and 6. Remark 1.6.
In this result one can avoid the move of Fig. 6 if T and T havethe same number of internal vertices and both consist of at least two tetrahedra.6 Figure 5:
Moves preserving the spin structure. Note that in the central move the coefficients1 are given to one internal and to one external face; coefficients 0 are never shown.
A A A AB B B BC C C CV V ‘ ‘ ‘ ‘ ‘‘‘‘ ‘‘
Figure 6:
A move increasing by one the number of punctures and preserving the spin structure.The coefficients of
ABV , ACV and
BCV in the 2-chain after the move are 0, 0 and 1. Spin structures from weakly branched spines
We will now explain how the spin structure s (cid:0) T , ω, b, β (cid:1) mentioned in the previ-ous section is constructed. As announced, this employs the viewpoint of specialspines, which is dual to that of triangulations.To a triangulation T of M we can associate the dual special spine P of M minus some balls, as suggested in Fig. 7. The polyhedron P is a compact 2-dimensional one onto which M minus some balls collapses. Every point of P has a neighbourhood homeomorphic to the cone over a circle, or over a circlewith a diameter (in which case the point is said to belong to a singular edge ), orover a circle with three radii (in which case the point is called a singular vertex ,and the neighbourhood itself is called a butterfly ). Moreover P has vertices, itssingular set S ( P ) is a 4-valent graph (actually, it is the gluing graph of T ) andthe components of P minus S ( P ), that we call regions , are homeomorphic to opendiscs. Any such P is called a special polyhedron , and it is known that there canexist at most one thickening of P , namely a punctured manifold M collapsing onto P , in which case P dually defines a triangulation of M . Moreover one can addto P an easy extra combinatorial structure, called a screw-orientation (see [4])ensuring that P is thickenable and that its thickening is oriented. A screw-orientation for P is an orientation of each edge e of P and a cyclic ordering of thethree germs of regions incident to e , up to simultaneous reversal of both, withobvious compatibility at vertices. All the special polyhedra we will consider willbe embedded in an oriented 3-manifold or locally embedded in 3-space, and westipulate from now on that the screw-orientation will always be the induced one,which allows us to avoid discussing screw-orientation and orientation altogether. If an oriented tetrahedron ∆ is branched, one can endow each wing of the dualbutterfly Y with the orientation such that the edge of ∆ dual to the wing ispositively transversal to the wing. (Note that the ambient orientation is usedhere.) One can moreover smoothen Y along its singular set so that the positivetransversal directions to the wings match, as shown in Fig. 8, where we show thebutterflies dual to the branched tetrahedra of Fig. 1-left. We can further definealong the singular set of Y two vector fields ν (the positive transversal to thewings) and µ (the descending vector field), and an orientation of the 4 singularedges of the butterfly, as shown in Fig. 9. Note that the orientation of an edge e of a butterfly is always given by the wedge of ν and µ along e , and it is theprevailing orientation of the three induced by the wings incident to e .8igure 7: Duality between a tetrahedron and a butterfly (the regular neighbourhood of avertex in a special spine).
R R
23 23
R R
12 02 R R R
02 12
R R
03 03
R R
01 01 R Figure 8:
Smoothing of a butterfly carried by a branching of its dual tetrahedron ∆. Here R ij denotes the wing of the butterfly dual to the edge v i v j of ∆. Figure 9:
The fields ν (vertical) and µ (horizontal) along the singular set of a smooth butterfly,and the orientation of its edges. .2 Weakly branched triangulations and theinduced frame along the dual -skeleton Let us fix in this subsection a triangulation T of an oriented manifold M andthe special spine P dual to T . If T carries a global branching , namely if eachtetrahedron in T is endowed with a branching so that all face-pairings match theedge orientations, then the frame ( ν, µ ) extends to S ( P ), as in Fig. 10 below.However, a global branching does not always exist, and we explain here how thestructure of weak branching still allows to globally define a frame along S ( P ). Remark 2.1.
We will call frame on a subset X of M a pair of linearly indepen-dent sections defined on X of the tangent bundle T M of M ; since M is oriented,this uniquely induces a trivialization of T M on X .Let us then take a pre-branching ω of T , viewed as an orientation of S ( P )with two incoming and two outgoing edges at each vertex, and a weak branch-ing b compatible with ω . For an edge e of P the following three possibilities(corresponding to those in Fig. 2) occur: • e can be a branched edge (type ∅ ), namely one along which the branchingsdefined at the ends are compatible, as in Fig. 10; the same figure showshow to (obviously) extend the frame ( ν, µ ) along such an e ; • If e is not branched there is only one region A incident to e lying on thetwo-fold side (namely, to the left of e ) at both ends of e , and we say that: – e is a positive unbranched edge (type +1) if A is under at the beginningof e and over at the end of e , as in Fig. 11-top/left; – e is a negative unbranched edge (type −
1) if A is over at the beginningof e and under at the end of e , as in Fig. 11-top/right.In both cases we can again coherently define ν along e , by letting thetransverse orientation of A prevail on the other two, and accordingly define µ , as illustrated in Fig. 11-bottom.For a technical but important reason to a spine P with pre-branching ω andcompatible weak branching b we actually associate a frame ϕ = ( ν, µ ) that isobtained from the above-described ( ν, µ ) by adding to µ a full rotation around ν along each unbranched edge of P , as shown in Fig. 12. We summarize the mainpoints of our construction in the following: Definition 2.2.
Let T be a triangulation of a compact oriented 3-manifold M ,and let P be the dual spine of M minus some balls . A pre-branching on P is10igure 10: A branched edge and the extension of ( ν, µ ) along it. A AB C B CA AB BC C
Figure 11:
Top-left: a positive unbranched edge. Top-right: a negative one. Bottom: thecorresponding extensions of ( ν, µ ). A AB BC C
Figure 12:
The field µ obtained by adding a full rotation to µ along each (positive or negative)unbranched edge of P . Planar structure of index +1 (left) or − N . an orientation ω of its edges such that at each vertex two germs of edges areincoming and two are outgoing. A weak branching on T compatible with ω is achoice b of a branching for each tetrahedron of T , such that b induces ω at eachvertex of P according to Fig. 9. The frame ϕ ( P, ω, b ) = ( ν, µ ) defined along S ( P )is given by the pair ( ν, µ ) at the vertices of P as in Fig. 9, with extension ( ν, µ )along the edges of P as in Figg. 10 (branched edges) and 11 (unbranched edges),and correction from ( ν, µ ) to ( ν, µ ) along the unbranched edges as in Fig. 12. Remark 2.3.
For every triangulation T the dual spine P always admits somepre-branching ω . Given ω , for a compatible weak branching b there are 4 inde-pendent choices at each tetrahedron of T . The frame ϕ ( P, ω, b ) is well-definedup to homotopy on S ( P ). In this subsection we introduce a convenient graphic encoding for weakly branchedtriangulations that we will later use to prove (the dual version of) Proposition 1.1.Let N be the set of finite 4-valent graphs Γ with the following extra structures: • Each edge of Γ is oriented and bears a colour ∅ , +1 or − • At each vertex of Γ a planar structure as in Fig. 13 left/right is given.Let T be a weakly branched triangulation of an oriented 3-manifold M , andlet P be the dual spine of M minus some balls. We can turn S ( P ) into a graphΓ( T ) ∈ N by associating to a branched tetrahedron of T as Fig. 1-left (or toa smoothed vertex of the dual spine P as in Fig. 9) a vertex as in Fig. 13, andgiving colour ∅ , +1 , − ∈ N we can associate a weakly branched triangulation T (Γ) of an orientedmanifold M . Some examples of how to explicitly construct the spine P dual to T (Γ) along the edges of Γ are illustrated in Fig. 14. (Recall that P is determined12 +1-1 Figure 14:
Reconstruction of a weakly branched spine from a graph in N . +1 +1= =-1 +10 -1 Figure 15:
To each e Γ ∈ e N one can uniquely associate a weakly branched special spine, alsogiven by the graph Γ ∈ N obtained from e Γ by fusing the edges through valence-2 vertices. by the attaching circles of its regions to S ( P ), which is what we show in Fig. 14-centre, and that the screw-orientation of P is induced by the local embedding in3-space, shown in Fig. 14-right.) We summarize our construction as follows: Proposition 2.4.
The set N of decorated graphs corresponds bijectively to theset of triples ( P, ω, b ) with P an oriented special spine, ω a pre-branching on P and b a weak branching compatible with ω . For later purpose we now need to extend the set of graphs N to some e N , byallowing 2-valent vertices besides the 4-valent (decorated) ones, and insisting thatthe edge orientations should match through the 2-valent vertices. By interpretingeach 2-valent vertex as or we can then associate as above to each element e Γ of e N a weakly branched special spine. On the other hand we can define the fusion of two edges separated by a valence-2 vertex by interpreting the set ofcolours {∅ , +1 , − } as Z / Z and postulating that colours sum up under fusion.Applying fusion as long as possible to e Γ ∈ e N we then get some Γ ∈ N . Thefollowing result can be easily verified —see Fig. 15 Proposition 2.5.
The weakly branched special spine associated to e Γ ∈ e N isindependent of the interpretation of the -valent vertices, and it coincides with AB C B CA AB BC C
Figure 16:
The frames ( ν, µ ) corresponding to +1 + 1 and to − the spine corresponding to the graph Γ ∈ N obtained from e Γ by edge-fusion. We conclude this subsection by explaining why have defined ϕ ( P, ω, b ) = ( ν, µ )not simply as ( ν, µ ), but adding instead a full twist to µ along unbranched edges: Proposition 2.6.
Take e Γ ∈ e N and let Γ ∈ N be obtained from e Γ by fusing edgesthrough valence- vertices. Then the frames ( e ν, e µ ) and ( ν, µ ) carried by e Γ and by Γ are homotopic to each other.Proof. We have to show that when we fuse two coloured edges into one the frame( ν, µ ) defined by the fusion is homotopic to the concatenation of the framesdefined by the two edges. Recall that the colour of the combination is the sum ofthe colours, and note that the conclusion is obvious when one of the edge coloursis ∅ . When the two edge colours are opposite to each other one can examineFig. 11 and see that the concatenation of the two frames ( ν, µ ) is homotopic toa constant frame; at the level of ( ν, µ ) we would have to add two full twists to µ , which amounts to nothing, and the conclusion follows. We are left to dealwith the sum of two edges with identical colour. We deal with the case +1 + 1,since − − ν, µ ) corresponding to +1 + 1 and to − ν, µ )we have to add two full twists to µ (that is, nothing) in the +1 + 1 configuration,and one full twist in the − Figure 17:
Decoration of the attaching circles of the regions near vertices and edges.
Let us now denote by α ( P, ω, b ) ∈ C ( P ; Z / Z ) the obstruction to extending ϕ ( P, ω, b ) to a frame defined on P . To define α ( P, ω, b ), note that
T M can alwaysbe trivialized as GL + (3; R ) × R on each open region R of P , and α ( P, ω, b )( R ) isthe element of π (GL + (3; R )) = Z / Z represented by the restriction of ϕ ( P, ω, b )to (a loop parallel to) ∂R . The next result shows that the chain α ( P, ω, b ) = P e α ( e ) · e ∈ C ( T ; Z / Z ) introduced in Section 1 is dual to α ( P, ω, b ), namelythat α ( e ) = α ( P, ω, b ) ( R ) if R is the region of P dual to an edge e of T . Proposition 2.7.
Given Γ ∈ N decorate the attaching circles of the regions ofthe special spine P defined by Γ as follows: • At each vertex of Γ put arrows as in Fig. 17-top; • At each edge e of Γ , if the edge colour is ∅ , put nothing, while if the edgecolour is ± put a weight on the region that lies to the left of e at both endsof e , and ∓ on the two other regions (see two examples in Fig. 17-bottom).Then α ( P, ω, b )( R ) ∈ Z / Z is computed as plus the sum of the numerical contri-butions along ∂R plus the sum of contributions from arrows, turned numerical asfollows: choose for ∂R an arbitrary orientation and give each arrow value + or − depending on whether it agrees or not with the orientation. Moreover, boththe sum of the numerical contributions and that of the contributions from arrowsturned numerical belong to Z / Z .Proof. Recall first that ϕ ( P, ω, b ) = ( ν, µ ) is obtained from ( ν, µ ) by adding afull twist to µ along the edges of P having colour ±
1. It is then sufficient toshow that the obstruction α to extending ( ν, µ ) is computed by decorating theregions of P as in Fig. 17-top near the vertices and as in Fig. 18 near the edges.Let us now pick a region R , give it some orientation, and compute α ( R ).Thanks to the orientation of R and of the ambient manifold M , for a vector at15 Figure 18:
Reduced decoration near edges, used to compute α . R R R R R R up down in out for(ward) back(ward)
Figure 19:
Positions of a vector on the boundary of an oriented region. some point of ∂R the positions shown in Fig. 19 are well-defined. We now analyzehow the positions of ν and µ change as ∂R travels near a vertex or edge of P .From Fig. 9 one sees that ( ν, µ ) does not change at a vertex except if ∂R is inone of the two positions indicated by arrows in Fig. 17-top (the sink and the sourcequadrants of the vertex); for these positions, we have 4 different possibilities, twoas follows Position of R Position of RV Sink quadrant of vertex V Source quadrant of the vertexwith ∂R oriented as the arrow in Fig. 17-top, and two more V and V withopposite orientation of ∂R ; the corresponding changes of ν and µ are∆ ν ∆ µ ∆ ν ∆ µ V up → up → up out → back → in V down → down → down in → for → out V up → up → up in → for → out V down → down → down out → back → in and this description applies whatever the index of the vertex.Turning to ∆( ν, µ ) along an edge e , of course nothing happens if e is branchedor e is unbranched but R is in position A in Fig. 11; otherwise we have 8 possi-bilities, 4 with ∂R concordant with e and R in the following positionPosition of R Position of RE B in Fig. 11-left E C in Fig. 11-left E B in Fig. 11-right E C in Fig. 11-rightand 4 more E j with ∂R discordant with e ; the corresponding ∆( ν, µ ) is16 ν ∆ µ ∆ ν ∆ µ E up → in → down out → up → in E up → in → down in → down → out E down → out → up in → down → out E down → out → up out → up → in E down → in → up in → up → out E down → in → up out → down → in E up → out → down out → down → in E up → out → down in → up → out. The value of α ( R ) will be given in π (GL + (3; R )) = Z / Z = { , } by 1 plussome contribution of each configuration V i , V i , E j , E j , but: • The V i , V i , E j , E j cannot appear in arbitrary order: only some concatena-tions are possible; • The individual V i , V i , E j , E j do not make sense in π (GL + (3; R )) but someof their concatenations do, when ( ν, µ ) is the same at the two ends of theconfiguration.The idea of the proof is then to assign to each V i , V i , E j , E j a value ± so that,whatever concatenation is possible and makes sense in π (GL + (3; R )), its geomet-rically correct value in π (GL + (3; R )) is the sum of the values of the V i , V i , E j , E j appearing in it. Turning to the details, the possible concatenations are (cid:8) V , E , E (cid:9) + (cid:8) V , E , E (cid:9) { V , E , E } + { V , E , E } (cid:8) V , E , E (cid:9) + (cid:8) V , E , E (cid:9) (cid:8) V , E , E (cid:9) + (cid:8) V , E , E (cid:9) . (1)and some concatenations that readily make sense in π (GL + (3; R )) are V + V = V + V = 1 V + V = V + V = 1 E + E = E + E = 1 E + E = E + E = 1 E + E = E + E = 0 E + E = E + E = 0 E + E = E + E = 0 E + E = E + E = 0 E + E = E + E = 1 E + E = E + E = 1;see for instance Fig. 20 for E + E = 1, where the concatenation is shown on theleft and then homotoped to 1 ∈ π (GL + (3; R )).These relations (subject to the condition that all V i , V i , E j , E j should beassigned ± as a value) are equivalent to V = V V = V E = E = − E = − E E = E = − E = − E (2)(note that the relations E = − E , E = − E , E = − E , E = − E come fromthe algebra but they are also geometrically clear). We now claim that E + V + E + V = 117igure 20: Proof that E + E = 1. Figure 21:
A concatenation giving 1 ∈ π (GL + (3; R )). Figure 22:
Alternative method to compute α ( P, ω, b ). which is proved in Fig. 21. Taking into account (2) the last condition is equivalentto any one of the following V = E = E = − V V = E = V = − E V = E = E = − V V = E = V = − E . (3)Choosing one of the relations (3) and combining it with (2) one can now computethe correct value of any possible concatenation according to (1). Let us nowchoose V = E = E = + and V = − , and note that the concatenationrules (1) imply that the total number of V , V , V , V found along ∂R is even(see also below). The desired computation rule and the last assertion of thestatement easily follow. At the end of the proof of Proposition 2.7 one can also choose V = V = E = + and E = − , which implies that α ( P, ω, b ) can be also computed by decoratingthe attaching circles of the regions as in Fig. 22. More generally, if we indicate by c i , c i the number of configurations C i , C i along ∂R , we have that α ( R ) is equalto 1 plus (+ ( v + v ) + ( v + v ) + ( e + e − e − e ) − ( e + e − e − e ))= (+ ( v + v ) + ( v + v ) − ( e + e − e − e ) + ( e + e − e − e ))= (+ ( v + v ) − ( v + v ) + ( e + e − e − e ) + ( e + e − e − e ))= ( − ( v + v ) + ( v + v ) + ( e + e − e − e ) + ( e + e − e − e ))and these expressions are recognized to be equivalent to each other because v + v + e + e + e + e v + v + e + e + e + e v + v + e + e + e + e v + v + e + e + e + e Figure 23:
Additivity of the computation of α . are all even numbers, thanks to (1). This implies that v + v + v + v is alsoeven (as noted above), and e + e + e + e + e + e + e + e is even as well(which is clear, since it counts the number of up/down switches of ν ). Remark 2.8.
The main reason why we have defined ϕ ( P, ω, b ) = ( ν, µ ) notas ( ν, µ ), but rather adding a full twist to µ along unbranched edges, wasto have additivity of the frames with respect to edge-fusion, as explained inProposition 2.6. Coherently with this we now have that the obstruction α ( P, ω, b )is also additive, namely it can be computed at the level of the graphs in e N , whichwould be false for α . Two examples of additivity (that again holds independentlyof the interpretation of the 2-valent vertices) are shown in Fig. 23. Remark 2.9.
Extending results of [2, 3], in [1] certain quantum hyperbolic invari-ants H N ( P ) have been constructed for a variety of patterns P , with N ≥ pattern consists of an oriented compact 3-manifold M with (possiblyempty) toric boundary, and an elaborated extra structure on M , which includesa PSL(2 , C )-character. Each invariant is computed as a state sum over a suitablydecorated weakly branched triangulation of M with some number k of punc-tures, and it is well-defined up to a phase anomaly . Namely, for N ≡ N -th root of unity, while for N ≡ N -th root of unity and a sign. And it turns our that inthe latter case the sign ambiguity can be removed by multiplying the state sumby ( − k − α ( P,ω,b )([ P ]) , where ( P, ω, b ) is the weakly branched spine dual to the20riangulation, and [ P ] ∈ C ( P ; Z / Z ) is the sum of all the regions of P . We close this section with a result that dualizes to Proposition 1.1.
Proposition 2.10.
The class of α = α ( P, ω, b ) vanishes in H ( P ; Z / Z ) . Forevery β ∈ C ( P ; Z / Z ) such that δβ = α a spin structure s ( P, ω, b, β ) is well-defined as the homotopy class of the frame ( ν, β ( µ )) on S ( P ) , where ( ν, µ ) = ϕ ( P, ω, b ) and β ( µ ) is obtained by giving a full twist to µ along all the edges e of P such that β ( e ) = 1 . Moreover s ( P, ω, b, β ) = s ( P, ω, b, β ) if and only if β + β vanishes in H ( P ; Z / Z ) .Proof. All three assertions are general topological facts. To prove the first one, let( ν, µ ) be any given spin structure on M , namely a frame on S ( P ) that extends to P and is viewed up to homotopy on S ( P ). Homotoping ( ν, µ ) we can suppose itcoincides with ( ν, µ ) at the vertices of P , so we can define β ∈ C ( P ; Z / Z ) where β ( e ) is the difference between ( ν, µ ) and ( ν, µ ) along e . Since the obstruction toextending ( ν, µ ) to a region R of P vanishes, we see that the obstruction α ( R ) toextending ( ν, µ ) to R is the the sum of β ( e ) for all the edges e of P contained in ∂R , namely δβ = α .The second assertion is now easy: if δβ = α then the obstruction to extending( ν, β ( µ )) to P vanishes.Turning to the third assertion, it is first of all evident that if v is a vertex of P and b v ∈ C ( P ; Z / Z ) is its dual then the frames on S ( P ) carried by some β with δβ = α and by β + δ b v are homotopic on S ( P ), with homotopy supported near v . Conversely, suppose β , β with δβ = δβ = α give frames (cid:0) ν (0) , µ (0) (cid:1) and (cid:0) ν (1) , µ (1) (cid:1) that are homotopic on S ( P ) via (cid:0) ν ( t ) , µ ( t ) (cid:1) t ∈ [0 , . If v is a vertex of P ,by construction (cid:0) ν (0) , µ (0) (cid:1) equals (cid:0) ν (1) , µ (1) (cid:1) at v , so we can view (cid:0) ν ( t ) , µ ( t ) (cid:1) t ∈ [0 , at v as an element γ ( v ) of π (GL + (3; R )) = Z / Z . We then have γ ∈ C ( P ; Z / Z )and β = β + δγ , whence the conclusion.Note that the previous result is coherent with the known fact that the set ofspin structures on M is an affine space over H ( P ; Z / Z ) = H ( M ; Z / Z ). We will establish in this section the dual versions of Propositions 1.3 to 1.5.From now on we will regard any β ∈ C ( P ; Z / Z ) such that δβ = α ( P, ω, b ) upto coboundaries. To discuss when two quadruples (
P, ω, b, β ) define the same21
I II -1+1 +1-1 11
III - Figure 24:
Moves that change the weak branching while preserving the pre-branching and thespin structure. Recall that ± Z / Z while 1 is a weight in Z / Z . s ( P, ω, b, β ) we can then describe right to left how the quadruple must change,and we have already dealt with the change of β .Before proceeding further we introduce a convenient graphic encoding forthe quadruples ( P, ω, b, β ). Namely we define N w as the set of all graphs Γ asin N , with the extra structure of weight in Z / Z attached to each edge of Γ. Anatural correspondence between N w and the set of all quadruples ( P, ω, b, β ), with(
P, ω, b ) as in Proposition 2.4 and β ∈ C ( P ; Z / Z ), is obtained by interpretingthe weight of an edge as the value of β on it. Note that for Γ ∈ N w the edgecolours belong to Z / Z = {∅ , +1 , − } and the weights to Z / Z = { , } , so noconfusion between colours and weights is possible. Colours ∅ and weights 0 willoften be omitted. We can similarly define e N w as the set of graphs in e N withweights in Z / Z attached to the edges, stipulating that weights sum up in Z / Z when two edges are fused together. The next result dualizes to Proposition 1.3.
Proposition 3.1.
Two graphs in N w define quadruples ( P j , ω j , b j , β j ) for j = 0 , with P = P , ω = ω and s ( P , ω , b , β ) = s ( P , ω , b , β ) if and only if theyare obtained from each other by repeated applications of the moves I and II ofFig. 24 (and their inverses, followed by the reduction from e N to N ).Proof. We must prove that the moves I and II generate all possible changes at avertex V of a weak branching compatible with a given pre-branching and, takingweights into account, that the associated spin structure is preserved. For bothindices ε = ± V there are 3 such possible changes; for ε = − I , II and III − (already shown in Fig. 24), which can be realizedas III − = I · II = II · I , with I and II the inverses of I and II , and productswritten with the move applied first on the left; for ε = +1 the 3 possible changesare given by I , II and III + = II · I = I · II .22 AB BC CC CD DE EE EB BF FF FD DA A
Figure 25:
Left: move I preserves the pre-branched spine. Right: the frame ( ν, µ ) is un-changed under move I on the region D . It is then sufficient to show that the moves I and II correctly represent onechange of weak branching and preserve the spin structure, which we will doexplicitly only for I . Ignoring the frame, the proof that I preserves the pre-branched spine is contained in Fig. 25-left. Turning to the frames, thanks toProposition 2.6, we can carry out a completely local analysis. Moreover we notethat locally before the move the frame ( ν, β ( µ )) coincides with ( ν, µ ), while afterthe move the frame ( ν, β ( µ )) is obtained from ( ν, µ ) by giving a full twist to µ along each of the 4 involved edges (three edges have colour ± ∅ and weight 1). These four twists are induced by ahomotopy, so it will be enough to show that the frames ( ν, µ ) before and afterthe move coincide up to homotopy. Showing this on a single global picture istoo complicated, so we confine ourselves to proving that ( ν, µ ) is unchanged upto homotopy separately on the boundary of each of the regions A, B, C, D, E, F of Fig. 25-left. This is very easy for all the regions except A , see for instanceFig. 25-right for D . In Fig. 26 we treat instead the case of the region A . Remark 3.2.
Let the change of weak branching on the pre-branched spine (
P, ω )in move I be given by b b ′ . The difference ∆ α = α ( P, ω, b ) + α ( P, ω, b ′ ) is thencomputed locally, and Proposition 3.1 implies that ∆ α = δ b e , with e as in Fig. 27.This fact can actually be checked directly, as in the rest of Fig. 27, since thepicture shows that ∆ α is 0 on A, B, F and 1 on
C, D, E . The next result dualizes to Proposition 1.4.23igure 26:
The frame ( ν, µ ) is unchanged up to homotopy on A . Left: before the move; right:after the move. Top: locally embedded configuration; bottom: abstract configuration. A AB BC CC CD DE EE EB BF FF FD DA Ae
A:C: E:B:D: F: -1/21 1 1-1/2 -1/2-1/2 +1/2 +1/2
Figure 27:
Variation of α with move I . +1 +1 -1+1 -1+1 -1 -1 -in-out-left-in-out-right -in-out -in-out -out-in -in-out-along Figure 28:
The edges in the circuit γ and the corresponding regions of P . Proposition 3.3.
Two graphs in N w define quadruples ( P j , ω j , b j , β j ) for j = 0 , with P = P and s ( P , ω , b , β ) = s ( P , ω , b , β ) if and only if they are relatedby the moves of Proposition 3.1 plus moves of the form Γ Γ ′ , where: • Γ contains a simple oriented circuit γ that at all its vertices is an overarc; • Γ ′ is obtained from Γ by reversing the orientation of the edges in γ andadding to the weights of the edges of γ whose ends have distinct indices.Proof. Suppose that ω and ω are distinct pre-branchings on the same spine P .The union of the edges of P on which ω and ω disagree can be expressed as adisjoint union of simple circuits oriented by ω . It is then sufficient to considerthe situation of two weak branchings ω, ω ′ that differ only on a simple circuit γ oriented by ω , and then iterate the procedure. Moreover, having already de-scribed how to obtain from each other any two pairs ( b, β ) yielding the same spinstructure on a given ( P, ω ), it is now sufficient to find one specific weak branch-ing b on ( P, ω ) and one b ′ on ( P, ω ′ ) and to describe a move β β ′ such that s ( P, ω, b, β ) = s ( P, ω ′ , b ′ , β ′ ). This move will be that of the statement, implyingthe conclusion. To describe the move we note that indeed via Proposition 3.1 wecan arrange so that γ contains overarcs only in a graph Γ ∈ N giving a weakbranching b on ( P, ω ). Examining Fig. 1-left and 13 one readily sees that theweak branching b ′ obtained by reversing γ is derived from b by switching theorientation of the edge v v in the tetrahedra dual to the edges in γ . We areonly left to show that the move β β ′ such that s ( P, ω, b, β ) = s ( P, ω ′ , b ′ , β ′ )consists in adding to β the 1-cochain ∆ β given by the duals of the edges in γ having endpoints with distinct indices. We will prove this in a slightly indirectway, in the spirit of Remark 3.2, by computing ∆ α = α ( P, ω, b ) + α ( P, ω ′ , b ′ ) andshowing that ∆ α = δ (∆ β ).We begin by noting that there are 9 possible positions of a region R withrespect to an edge e of γ , as shown in Fig. 28. One can now check that thecontributions carried by e to (∆ α )( R ), depending on the indices ± / ± e , are as given in the following table (with ∂R oriented as in in Fig. 28):25 -1/2 -1/2-1/2 -1/21 1 +1 +1/2 +1/2+1/2 +1/21 1- - - Figure 29:
Three computations of the contribution of an edge e in γ to (∆ α )( R ). The colour ∅ / + 1 / − e (unchanged by the switch of γ ) is shown in the middle. On the left we see theindices of the ends of e before the switch, and the local computation of α ( R ). On the right thecomputation of α ′ ( R ) after the switch. +1 / + 1 − / − / − − / + 1 ∅ -in-out-right 0 0 1 1 ∅ -along 0 0 0 0 ∅ -in-out-left 0 0 1 1+1-in-out 0 0 1 1+1-in 1 0 1 0+1-out 1 0 0 1 − − − α )( R ) obtained by summingthe contributions given by the various edges of γ equals (mod 2) the number ofedges in γ visited by ∂R and having ends with distinct indices. If ∂R visits onlyone edge, i.e. , if it is of type in-out, the conclusion is evident from the values in thetable. Otherwise (∆ α )( R ) is the sum of only two possibly non-0 contributions,one from the edge of γ where ∂R enters and one from the edge of γ where itleaves. More precisely, as one sees from the table, there is an “in” contributiondepending only on the index of the vertex of γ where ∂R enters (contribution 1 forindex +1 and contribution 0 for index − Figure 30:
Moves on e N w preserving the associated spin structure; the edges entirely containedin the picture have colour ∅ and weight 0. only on the index of the vertex of γ where ∂R leaves (again, contribution 1 forindex +1 and contribution 0 for index − α )( R )has the desired value, and the proof is complete. The next result dualizes to Proposition 1.5.
Proposition 3.4.
Two graphs in N w define quadruples ( P j , ω j , b j , β j ) for j = 0 , with s ( P , ω , b , β ) = s ( P , ω , b , β ) if and only if they are obtained from eachother by the moves of Propositions 3.1 and 3.3 and those shown in Fig. 30.Proof. Two special polyhedra are spines (in the punctured sense) of the samemanifold without boundary spheres if and only if they are related by bubble and2-3 moves [13, 15]. It is then sufficient to prove the following: • Using the moves I and II any edge e with distinct ends V , V of a graphin N w can be transformed into one to which a move in Fig. 30 applies; • At the level of spines the moves in Fig. 30 translate the bubble and the 2-3move, and at the level of quadruples (
P, ω, b, β ) represented by graphs in N w the associated spin structure is unchanged under these moves.With ε being the index of a vertex, the following steps establish the first assertion:27. If e is an underpass at some V j , apply to each such V j the move III ε ( V j ) (with III + the analogue of III − for a vertex of index +1); this allows toassume that e is an overpass at V and V ;2. If the colour of e is now +1, act as follows:(a) If ε ( V ) = ε ( V ) = −
1, apply I to V and II to V ;(b) If ε ( V ) = − ε ( V ) = +1, apply II to V ;(c) If ε ( V ) = +1, apply I to V ;3. If the colour of e is now −
1, act as follows:(a) If ε ( V ) = ε ( V ) = +1, apply I to V and II to V ;(b) If ε ( V ) = +1 and ε ( V ) = −
1, apply II to V ;(c) If ε ( V ) = −
1, apply I to V ;4. The colour of e is now ∅ , and we want to exclude the case ε ( V ) = +1 and ε ( V ) = −
1, for which we apply I to V and II to V ;5. Up to coboundaries we turn the weight of e to 0.For the second assertion, once again we start by an indirect argument in the spiritof Remark 3.2, showing that the weights appearing in the moves compensate forthe variation of the obstruction α ∈ C ( P ; Z / Z ), which is done in Fig. 31. Amore direct arguments for the move of Fig. 30-top/left is carried out in Fig. 32;for the other moves the argument follows from [5]. In this section we show that the global move of Proposition 3.3 can be replaced,in a suitable sense, by a simultaneous combination of local ones.
We introduce now a set A of decorated graphs via which we can encode an arbi-trarily branched triangulation, namely a triangulation in which each tetrahedronis endowed with a branching, without any compatibility whatsoever. Each vertexof a graph Γ in A will be given a planar structure as in Fig. 13, which correspondsto giving the dual tetrahedron a branching. Note that each edge of Γ then has anorientation defined at each of its ends. We are left to choose colours for the edges28 B CDA AB BC CDDE EF FG GH HI I N
Figure 31:
Proof that under the moves of Fig. 30 the change in the obstruction α is com-pensated by the weights on the edges. For the two moves on the right this is easy: on all theregions that survive α keeps the same value, and on the newborn region it has value 0. Forthe top-left move ∆ α is 1 on A, B, C and 0 on D , while for the bottom-left move ∆ α is 0 on A, B, C, E, F, G and 1 on
D, H, I, N , and indeed for both cases these values are given by theweights in the moves.
Figure 32:
Top: the branched bubble move. Middle: frames carrying the same spin structurebefore and after the move. Bottom: the frame carried by the spine after the move, that becomesthe previous one taking into account the weight 1 appearing in the move of Fig. 30-left. Figure 33:
Labels for the germs of region near a (branched) vertex. of Γ in order to encode the face-pairings, or equivalently the attaching circles to S ( P ) = Γ of the regions of the dual spine P . To do so we note that dual to agerm e of edge of S ( P ) at some vertex there is a branched triangle. We can nowlabel by 0 , , e . We show in Fig. 33-left(in a cross-section) this abstract labeling rule, and in Fig. 33-right its concreteconsequences. One can now easily check the following: Lemma 4.1.
Let e be an edge of Γ and let n ( e ) be the number of regions incidentto e having the same label at both ends of e . • If the two ends of e are consistently oriented then n ( e ) = 0 or n ( e ) = 3 ; • If the two ends of e are inconsistently oriented then n ( e ) = 1 . This implies that we can give an edge e of Γ the following colours in S (seeFig. 34 for some examples): • If the two ends of e are consistently oriented colour σ ∈ S +3 = {∅ , (0 1 2) , (0 2 1) } if region j at the first end of e is matched to region σ ( j ) at the second end; • If the two ends of e are inconsistently oriented, colour τ ∈ S − = { (0 1) , (0 2) , (1 2) } if region j at one end is matched to region τ ( j ) at the other end. Remark 4.2.
A graph Γ in A defines a weakly branched triangulation if andonly if all the edges are consistently oriented. In this case Γ is converted into agraph in N representing the same weakly branched triangulation by the colour-replacements (0 1 2) +1 and (0 2 1)
7→ − (0 1 2) (0 1)(0 2 1) (1 2)(0 2)
Figure 34:
Meaning of the edge colours for a graph in A . s s t t t tsss ts st tt s -1 -1 Figure 35:
How to fuse together two edges of a graph in e A . From now on we will call even (respectively, odd ) an edge of a graph in A with colour in S +3 (respectively, in S − ), or, equivalently, with consistently(respectively, inconsistently) oriented ends. As we did for N , to define moves on A it is convenient to enlarge it to some e A byallowing valence-2 vertices; edges are again decorated by an orientation at each oftheir ends and a colour (in S +3 if the orientations match, in S − if they do not),but we also insist that orientations should match across the valence-2 vertices.Note that if we choose for each 2-valent vertex of e Γ ∈ e A an interpretation asor as we can associate to e Γ an arbitrarily branched special spine.We now define a projection e A → A by illustrating in Fig. 35 how fuse togethertwo edges sharing a valence-2 vertex; note that σ, σ , σ ∈ S +3 and τ, τ , τ ∈ S − ;moreover σ − ◦ τ = τ ◦ σ and τ ◦ σ − = σ ◦ τ , which gives alternative ways ofexpressing the fusion rules. We have the following: Proposition 4.3.
The fusion rules of Fig. 35 are associative, so each graph e Γ ∈ e A defines a unique Γ ∈ A . Moreover the arbitrarily branched spine associated (1 2) =(0 1) (0 1 2) Figure 36:
To each e Γ ∈ e A one can uniquely associate an arbitrarily branched special spine,also given by the graph Γ ∈ A obtained from e Γ by fusing the edges through valence-2 vertices. (1 2)(1 2)
Figure 37:
A move on e A . to e Γ is well-defined regardless of the interpretation of the valence- vertices asor , and it coincides with the arbitrarily branched spine associated to Γ . The first assertion of this result follows from the second one, that can beestablished with some patience; see some examples in Fig. 36.
Let us consider the move on graphs in e A described in Fig. 37-left. In Fig. 37-right we show that the move preserves the spine (or triangulation) encoded bythe graph, while of course changing the arbitrary branching. The following result(that will also follow from the rest of this section) is not difficult to show: Proposition 4.4.
Any two arbitrary branchings on the same triangulations arerelated by compositions of the moves I and II (ignoring weights), that of Fig. 37,and their inverses. Since for a single tetrahedron there are 24 different branchings, this result32
Figure 38:
The inverse of the move of Fig. 37 and one generated by those in Fig. 37 and 24. means that at each vertex using the moves I and II and that of Fig. 37 one cancreate all 24 possible configurations. See for instance Fig. 38. We define e A w as the set of graphs in e A with weights attached to the edges. Theweight of an edge e is given by an internal orientation and by a numerical weight in the group G = (cid:0) · Z (cid:1) / Z , with the following restrictions: • If e is even then its internal orientation matches those at its ends (so it isnot shown in the pictures) and the numerical weight is 0 or 1; • If e is odd the numerical weight is ± .Note that there is a natural inclusion e N w ⊂ e A w The numerical part of a systemof weights will be viewed up to 1-coboundaries with values in Z / Z , namely thenumerical weights of all 4 edges incident to a vertex can simultaneously changeby 1. We next define the weighted fusion rules of Fig. 39. Remark 4.5.
The fusion rules do not cover the case of two odd edges with in-ternal orientations both opposite to the external orientation after fusion, becausethis case will never occur for us. For the fusion of two odd edges with discordantinternal orientations, we note that a , a are ± , so a − a = a − a in Z / Z .The following fact, proved in Fig. 40, must be taken into account: Proposition 4.6.
The weighted fusion rules of Fig. 39 are not associative. w w + w + w + a + a + a + a - a - wwaaaaa sttt ttt ttt tttss w w aawwaaa aawwaaa sstt ss ssttttttss
11 1111111 1111 222 222 111222 2222222 -1-1 -1-1
Figure 39:
Edge fusion rules for graphs in e A w . a a + a + + a + - a - a t t (t(tttt t t )t )tt t tt a a aaaa a aa -1-1 Figure 40:
The fusion rules for graphs in e A w are not associative. In this example both theinternal orientation and the numerical weight ± depend on the order in which fusions areperformed; note however that ( τ ◦ τ ) − ◦ τ = ( τ ◦ τ ) − ◦ τ , coherently with the fact that thefusion rules for unweighted graphs are associative. We now introduce certain moves on N w , to define which we also use e A w . To beginwe call elementary move on e A w one of I, I, II, II, M, M from Figg. 41 and 42.The pictures contain more moves whose rˆole will be explained soon.
Remark 4.7. • In the symbols denoting the moves, overlining and subscriptsare used to indicate the type of index transition ±
7→ ± • The moves I and II are those of Fig. 24 and at the level of N w , namelyunder the associative fusion rules for e N w , we have I = I − and II = II − ;moreover III − = I · II = II · I and III + = I · II = II · I ; • In the product of two moves that to the left applies first; not all productsmake sense.We now establish some results concerning relations between moves:
Proposition 4.8.
Consider a vertex as in Fig. 13, apply to it one of the followingcombination of weighted moves, and locally apply near the vertex the weightedfusion rules of Fig. 39; then the result is the same as indicated: M · M = id − , N · N = id − , M · M = id + , N · N = id + , III III - (021) (021)(021) (021)(012) (012)1 1 III III + (021) (012)(012) (021)1 1 Figure 41:
Moves on e A w derived from those on e N w . (0 1) (0 1)(0 1) (0 1)(1 2) (1 2)(1 2) (1 2)-1/2 +1/2+1/2 -1/2 M M -1/2 -1/2+1/2 +1/2
N N
Figure 42:
More moves on weighted graphs. M M = Figure 43:
Proof that M · M = id − under local application of weighted fusion. III − · M = N · III + , III + · M = N · III − , M · N = N · M , M · N = N · M. Remark 4.9.
Since the fusion rules in e A w are not associative, these equalities do not imply that at the level A w we have the relations M = M − , N = N − , N = III − · M · III + but these relations do make sense and hold in a restricted context, see below.The proofs of some of the equalities in Proposition 4.8 are given in Figg. 43to 45; they all crucially use the weighted fusion rules of Fig. 39 and the conventionthat weights are viewed up to Z / Z -coboundaries; the other proofs are similar.We now call weighted move on a vertex as in Fig. 13 any sequence of elemen-tary weighted moves (not followed by any fusion). We first have the following: Proposition 4.10.
Take Γ ∈ N w and apply to each of its vertices a weightedmove to get e Γ ∈ e A . Suppose that by applying (in some order) the weighted fusionrules of Fig. 39 one gets Θ ∈ N w . Then the system of weights on Θ is well-definedindependently of the order of application of the weighted fusion rules.Proof. The statement contains the implicit claim that the rules of Fig. 39 sufficeto go from e Γ to some Θ, namely that no situation as in Remark 4.5 occurs.We prove the proposition ignoring the colours in S , because we already knowthat fusion is associative at the S level. We concentrate on a single edge of Γand we imagine e is initially drawn in front of us with orientation from left toright and weight w ∈ { , } . Replacing external orientations of edges by letters36 III - (0 1) (0 1)(0 1) (0 1)-1/2 -1/2 +1/2+1/2 +1/2 -1/2 N (1 2)(1 2) -1/2+1/2 M (021) (012)1 1(021) (012)(012) (021)11 III + (012)(021) = (021) (012)(02) (02)= = Figure 44:
Proof that
III − · M = N · III + under local application of weighted fusion. (0 1) (0 1)(1 2)(1 2) -1/2+1/2 M N -1/2+1/2
N M (0 1) (0 1)(0 1) (0 1)(1 2) (1 2)(1 2) (1 2)-1/2 +1/2+1/2 -1/2+1/2 -1/2-1/2 +1/2=
Figure 45:
Proof that M · N = N · M under local application of weighted fusion. /ℓ and internal orientations by R/L we then have in Γ an initial edge rrw thatgets replaced in e Γ by a concatenation e e of edges of the form rru , ℓℓu , rRℓa , rLℓa , ℓRra , ℓLra . A careful examination of the elementary weighted moves actually shows that thepossibilities for e e are only as follows: (cid:16)(cid:0) rru (cid:1) ∗ · rRℓa · (cid:0) ℓℓu (cid:1) ∗ · ℓLra (cid:17) ∗ · (cid:0) rru (cid:1) ∗ · rrw · (cid:0) rru (cid:1) ∗ · (cid:16) rLℓa · (cid:0) ℓℓu (cid:1) ∗ · ℓRra · (cid:0) rru (cid:1) ∗ (cid:17) ∗ (4) (cid:0) ℓℓu (cid:1) ∗ · ℓLra · (4) · rLℓa · (cid:0) ℓℓu (cid:1) ∗ (5)where y ∗ means any number (including 0) of repetitions of a string y , and theweight u (respectively, a ) can have a different value in { , } (respectively, ± )each time it appears. It is then clear that we never get any of the adjacencies rLℓa · ℓLra or ℓRra · rRℓa not contemplated by the weighted fusion rules of Fig. 39.Moreover these rules can be expressed as rru · rru = rru + u , rru · rDℓa = rDℓu + a , ℓDra · rru = ℓDra + u , ℓℓu · ℓℓu = ℓℓu + u , ℓℓu · ℓDra = ℓDru + a , rDℓa · ℓℓu = rDℓa + u , ℓLra · rLℓa = ℓℓa + a , rRℓa · ℓRra = rra + a , ℓLra · rRℓa = ℓℓa − a , ℓRra · rLℓa = ℓℓa − a , rRℓa · ℓLra = rra − a , rLℓa · ℓRra = rra − a . We must show that by applying them as long as possible to (4) or (5) we get awell-defined result. Note first that each edge ℓℓu or rru can be ignored; in fact, itscontribution is independent of the time it is involved in weighted fusions, because: • On the internal orientation it acts identically to the right and to the left; • Its numerical weight is in { , } , so it is insensitive to later sign change.We then have to deal with concatenations of the form rRℓa · ℓLrb · . . . rRℓa k · ℓLrb k · rLℓd h · ℓRrc h · . . . rLℓd · ℓRrc , (6) ℓLrb · rRℓa · ℓLrb · . . . rRℓa k · ℓLrb k · rLℓd h · ℓRrc h · . . . rLℓd · ℓRrc · rLℓd (7)but we also consider the following (that arise starting from an edge ℓℓw in Γ): ℓLra · rRℓb · . . . ℓLra k · rRℓb k · ℓRrd h · rLℓc h · . . . ℓRrd · rLℓc , (8)38 Rℓb · ℓLra · rRℓb · . . . ℓLra k · rRℓb k · ℓRrd h · rLℓc h · . . . ℓRrd · rLℓc · ℓRrd . (9)We now claim that, regardless of the order in which the weighted fusion rules areapplied, the numerical edge weight on the final result is P i =1 ,...,k a i − P i =1 ,...,k b i + P j =1 ,...,h c j − P j =1 ,...,h d j for (6) and (8) , P i =1 ,...,k a i − P i =0 ,...,k b i + P j =1 ,...,h c j − P j =0 ,...,h d j for (7) and (9) . The claim of course implies the conclusion, and we can prove it by induction onhalf the length of the concatenation. The base step of the induction is with length0 in cases (6) and (8), so it is empty, and with length 2 in cases (7) and (9), so itfollows directly from the weighted fusion rules (remember that − b − d = b + d because both b and d are ± ). For the inductive step we must analyze whathappens by applying one weighted fusion to one of (6)-(9). In all four cases wecan distinguish between the “central” fusion ℓLrb k · rLℓd h → ℓℓb k + d h or rRℓb k · ℓRrd h → rrb k + d h and any “lateral” fusion. Dealing with lateral fusions is easier, and we make itexplicit only for case (6) and for a fusion performed to the left of the centre; thisfusion will be rRℓa t · ℓLrb t → rra t − b t or ℓLrb t · rRℓa t +1 → ℓℓb t − a t +1 = ℓℓa t +1 − b t ; then we can forgetthe fused edge (remembering that its weight must be added to the final one) sowe are led to a shorter concatenation of type (6); the inductive assumption theneasily implies the conclusion.Turning to the central fusion, in case (6) forgetting the fused edge we get theshorter concatenation of type (9) rRℓa · ℓLrb · rRℓa · . . . ℓLrb k − · rRℓa k · ℓRrc h · rLℓd h − · . . . ℓRrc · rLℓd · ℓRrc whence, by the inductive assumption, independently of the order, a final weight b k + d h + P i =1 ,...,k − b i − P i =1 ,...,k a i + P j =1 ,...,h − d j − P j =1 ,...,h c j = P i =1 ,...,k a i − P i =1 ,...,k b i + P j =1 ,...,h c j − P j =1 ,...,h d j (10)as desired. The central fusion in (7) gives the type (8) concatenation ℓLrb · rRℓa · . . . ℓLrb k − · rRℓa k · ℓRrc h · rLℓd h − · . . . ℓRrc · rLℓd whence final weight precisely as in (10), as desired. The central fusion in cases (8)and (9) is similarly reduced to the inductive assumption in cases (7) and (6)respectively. 39 orollary 4.11. If a sequence of elementary weighted moves is applied to avertex as in Fig. 13 and the weighted fusions are applied as long as possible to theedges generated by these moves, the result is independent of the order of fusions.Proof.
By the argument showing Proposition 4.10 we must prove that concate-nations of the form rRℓa · ℓLrb · . . . rRℓa k · ℓLrb k , ℓLra · rRℓb · . . . ℓLra k · rRℓb k ℓLrb · rRℓa · ℓLrb · . . . rRℓa k · ℓLrb k , rRℓb · ℓLra · rRℓb · . . . ℓLra k · rRℓb k give a well-defined result. By induction on the length one can indeed see that thefirst two give P i =1 ,...,k b i − P i =1 ,...,k a i and the last two give P i =0 ,...,k b i − P i =1 ,...,k a i .The two previous results imply that: • We can define a weighted move at a vertex as in Fig. 13 as a sequence ofelementary weighted moves followed by weighted fusion; • If we apply to a graph in N w some weighted moves and after weighted fusionwe get another graph in N w , the weights on this last graph are well-defined. Proposition 4.12.
Two weighted moves at a vertex that coincide as unweightedmoves also coincide as weighted moves.Proof.
We will prove the result for moves turning a vertex of index − −
1, the general case following by pre-composition with move I and/or post-composition with move I . The 12 moves described form a groupΠ − which is intrinsically isomorphic to the alternating group S +4 on 4 objects.This isomorphism is made explicit with the choice of generators and the resultingpresentation as follows: α = III − , β = I · M , Π − = (cid:10) α, β | α , β , ( α · β ) (cid:11) (with moves and relations understood without weights). To conclude it is thensufficient to show that the three relations hold also in a weighted sense. For α this was already implicit above and very easy anyway; the other two weightedrelations are established in Figg. 46 and 47We are eventually ready to establish our main result of this section: Theorem 4.13.
Two graphs in N w represent the same spine of some manifold M and the same spin structure s on M if and only if they can be obtained fromeach other by a combination of the moves I, II, M, I , II, M and weighted fusion.
01) (01)(01) (01)(01) (01) (01)(012)(012) (012)(021) (021)(012) (012) (012)(012)(01) (01)(01) (01)(01)(01) (01)(021)1 11 111 11 I b b (1 2)(1 2) +1/2+1/2 +1/2+1/2 +1/2+1/2+1/2 +1/2-1/2 M = = -1/2 -1/2-1/2 -1/2-1/2-1/2 -1/2(012)(012) (021)1 b = = Figure 46:
Computation of β = I · M and proof that β = id − in a weighted sense. (021)(021)(021) (021) (021) (012)(012)(012) (012)(012) (12)(12)(12) (12) (12) II ab (012)(012)(021)1 (1 2) (1 2)+1/2-1/2 M = +1/2+1/2+1/2 +1/2 +1/2 +1/2+1/2+1/2 +1/2+1/2 (02)(02)(02) (02)(02)(012)(012)(021)1ab (02) (02)(021) (021)(12) (12)+1/2 +1/2+1/2 +1/2(012) (012) = ab = Figure 47:
Computation of α · β = III − · I · M = II · I · I · M = II · M and proof that( α · β ) = id − in a weighted sense. M MM M ss ss ss ss ww w ww w ww +1= == = -1-1
Figure 48:
Generation of the circuit move via the moves
M, M and weighted fusion.
Proof.
Suppose that Γ , Γ ∈ N w represent the same ( M, s ). Then they arerelated bymoves
I, II, I, II and circuit moves. To get the desired conclusion it isthen enough to show that the moves
M, M generate the circuit move, which isdone in Fig. 48 for an edge of a circuit with first end of index − Proposition 4.14.
Suppose that in Γ ∈ N w there are some (possibly intersecting)oriented circuits γ , . . . , γ n , and that each γ j is either an undercircuit (an overpassat all its vertices) or an overcircuit (an underpass at all its vertices). Thenthe spin structure defined by Γ is also defined by the graph obtained from Γ byreversing the orientation of each edge e of γ ∪ . . . ∪ γ n and adding to the weightof e if the ends if e have different indices. Proposition 4.15. If Γ ∈ N w then using the moves I, II, I , II at the vertices of Γ , followed by fusion, one can get Θ ∈ N w such that each edge of Θ is either anoverpass at both its ends or an underpass at both its ends.Proof. To begin we note that given a vertex V of Γ and the choice of two germsof edges of Γ at V having consistent orientation through V , the moves I, II, I, II allow to put the two chosen germs of edges in the overpass position at v . It isthen enough to show that we can attach labels o (over) and u (under) to thegerms of edges of Γ at vertices, so that: • For each edge the labels at its ends are the same; • At each vertex the germs having the same label have consistent orientation.One such labeling will be termed good , and the coming argument proving itsexistence is due to Federico Petronio. We choose a vertex V of Γ and attach anylabel to any of the germs of edge at V . Then we propagate the labeling along apath in Γ by applying alternatively the following rules until V is reached again:42 u uu uu uu uuu uu uu uuu uoo oo oo oo oooo oo oV VV V Figure 49:
Extension of the labeling in case of initial label u on an outgoing germ at V . • If an end of an edge has a label, give the other end the same label; • If at a vertex an incoming (respectively, outgoing) germ has a label, givethe other incoming (respectively, outgoing) germ the other label.Note that the propagation path need not be simple, but at each vertex visitedtwice the labeling is good —see Fig. 49-left. When V is reached again we haveone of the situations in Fig. 49-right; in the top one we proceed by applying thesecond rule, and eventually get back to V again with a good labeling; in thebottom one we proceed with an arbitrary choice of the label, but once more weget back to V with a good labeling. We can now similarly start from some othervertex, until all the germs of edges at vertices are labeled.Back to the proof of Theorem 4.13, suppose that Γ ∈ N w is obtained fromΓ ∈ N w by a combination of weighted moves I, II, M, I , II, M and weightedfusion. Let ∆ be the union of the edges of Γ having a different orientation inΓ . By Proposition 4.15 we can find weighted moves generated by I, II turningΓ into Γ ∈ N w in which ∆ appears as a union of overcircuits and undercircuits.Note that Γ carries the same spin structure as Γ by Proposition 3.1. Withpictures similar to Fig. 48 one can now see that the multiple circuit moves ofProposition 4.14 are generated by the moves M, M , N, N , D − = M · N = N · M , D + = M · N = N · M (the move D − is shown in Fig. 45, and D + is obtainedsimilarly).This shows that we can find a combination of the moves I, II, M, I , II, M that,after weighted fusion, turn Γ into some Γ carrying the same spin structure as Γ and the same pre-branching as Γ . Proposition 3.1 then implies that via moves43 , II, I, II we can turn Γ into some e Γ carrying the same spin structure as Γ and different from Γ possibly only for the weights. We then have a sequence ofweighted moves I, II, M, I, II, M that under weighted fusion giveΓ −→ Γ −→ Γ −→ Γ −→ e Γ and that ignoring weights give the identity of Γ (namely, they give the identityat every vertex of Γ ). Proposition 4.12 then implies that e Γ coincides with Γ also as a weighted graph (up to coboundaries). This shows that Γ carries thesame spin structure as Γ . Even if this is not strictly necessary for our main results, we provide here twomethods for the computation of the obstruction α ( P, ω, b ) carried by a graph e Γ ∈ e A that after fusion becomes a graph in Θ ∈ N defining a triple ( P, ω, b ). The firstmethod is general, direct and easy; the second one only applies to a e Γ resultingfrom the application to some Γ ∈ N of the moves of Proposition 4.10 (ignoring thenumerical weights but using internal orientations), and it is more complicated,but it also shows that some non-trivial algebra underlies the computation.
First method . Take e Γ ∈ e A that after fusion gives Γ ∈ N representing ( P, ω, b ).We claim that α ( P, ω, b ) can be computed from e Γ by considering on the boundaryof each region of P some numerical contributions in G = (cid:0) · Z (cid:1) / Z and somearrows, as in Proposition 2.7. Contributions from vertices and from even edgesare the same as in Proposition 2.7, while those from an odd edge e are describedas follows (with the regions labeled 0,1,2 as in Fig. 33 and contributions 0 notmentioned): e τ = (0 1) τ = (0 2) τ = (1 2) t regions 0 and 1 get + all regions get 1 regions 1 and 2 get − t regions 0 and 1 get − all regions get 1 regions 1 and 2 get + The proof that this recipe works follows from the fact that the contributionscombine consistently under fusion, which is shown on examples in Fig. 50.
Second method . We begin with an apparently unrelated algebraic result. Forany set G we consider the right action of S on G given by( g , g , g ) · η = (cid:0) g η (0) , g η (1) , g η (2) (cid:1) . We check that indeed this is a right action on an example: (cid:0) ( g , g , g ) · (0 1) (cid:1) · (1 2) = ( g , g , g ) · (1 2) = ( g , g , g )( g , g , g ) · (cid:0) (0 1) ◦ (1 2) (cid:1) = ( g , g , g ) · (0 1 2) = ( g , g , g ) .
21 1 1 1 1 2 1 1 111101101101011100111 00 2 0 2 2 0 2 2 222022220020220022220 12 0 2 0 0 1 0 0 000210012212102211002 +1/2 +1/2+1/2 +1/2 +1/2-1/2-1/2 +1/2+1/2+1/2+1/2+1/2+1/2 1 -1/2 -1/2
Figure 50:
Associativity of the computation of α on a graph in e A . If G is an Abelian group of course we have (cid:0) ( g , g , g ) + ( h , h , h ) (cid:1) · η = ( g , g , g ) · η + ( h , h , h ) · η so we can define the semidirect product S ∐ G as S × G with operation( η, ( g , g , g )) · ( θ, ( h , h , h )) = ( η ◦ θ, ( g , g , g ) · θ + ( h , h , h ) . We now specialize our choice to = (cid:0) · Z (cid:1) / Z and we establish the following: Proposition 4.16.
Define s : S → G by s ( ∅ ) = (0 , , s ((0 1 2)) = (cid:0) − , − , (cid:1) s ((0 1)) = (cid:0) − , + , (cid:1) s ((0 2)) = (1 , , s ((0 2 1)) = (cid:0) , + , + (cid:1) s ((1 2)) = (cid:0) , − , + (cid:1) . Then
Ψ : S → S ∐ G given by Ψ( η ) = ( η, s ( η )) is a group homomorphism.Proof. If x = (0 1) and y = (1 2) we have the presentation of S given by (cid:10) x, y | x , y , ( x · y ) (cid:11) with (0 1 2) = x · y , (0 2 1) = y · x , (0 2) = x · y · x . The proposition will then be aconsequence of the relationsΨ( x ) = Ψ( y ) = (Ψ( x ) · Ψ( y )) = ( ∅ , (0 , , , Ψ((0 2)) = Ψ( x ) · Ψ( y ) · Ψ( x )45((0 1 2)) = Ψ( x ) · Ψ( y ) , Ψ((0 2 1)) = Ψ( y ) · Ψ( x ) . We start withΨ( x ) = (cid:0) (0 1) , (cid:0) − , + , (cid:1)(cid:1) · (cid:0) (0 1) , (cid:0) − , + , (cid:1)(cid:1) = (cid:0) (0 1) ◦ (0 1) , (cid:0) + , − , (cid:1) + (cid:0) − , + , (cid:1)(cid:1) = ( ∅ , (0 , , . The computation of Ψ( y ) is similar. Before checking that Ψ( x ) · Ψ( y ) has van-ishing cube we compute it, checking it is Ψ((0 1 2)):Ψ( x ) · Ψ( y ) = (cid:0) (0 1) , (cid:0) − , + , (cid:1)(cid:1) · (cid:0) (1 2) (cid:0) , − , + (cid:1)(cid:1) = (cid:0) (0 1) ◦ (1 2) , (cid:0) − , , + (cid:1) + (cid:0) , − , + (cid:1)(cid:1) = (cid:0) (0 1 2) , (cid:0) − , − , (cid:1)(cid:1) . And now we conclude:(Ψ((0 1 2))) = (cid:0) (0 1 2) , (cid:0) − , − , (cid:1)(cid:1) = (cid:0) (0 1 2) ◦ (0 1 2) , (cid:0) − , , − (cid:1) + (cid:0) − , − , (cid:1)(cid:1) · Ψ((0 1 2))= (cid:0) (0 2 1) , (cid:0) , + , + (cid:1)(cid:1) · (cid:0) (0 1 2) , (cid:0) − , − , (cid:1)(cid:1) = (cid:0) (0 2 1) ◦ (0 1 2) , (cid:0) + , + , (cid:1) + (cid:0) − , − , (cid:1)(cid:1) = ( ∅ , (0 , , y ) · Ψ( x ) = (cid:0) (1 2) , (cid:0) , − , + (cid:1)(cid:1) · (cid:0) (0 1) , (cid:0) − , + , (cid:1)(cid:1) = (cid:0) (1 2) ◦ (0 1) , (cid:0) − , , + (cid:1) + (cid:0) − , + , (cid:1)(cid:1) = (cid:0) (0 2 1) , (cid:0) , + , + (cid:1)(cid:1) ;Ψ( x ) · Ψ( y ) · Ψ( x ) = (cid:0) (0 1) , (cid:0) − , + , (cid:1)(cid:1) · (cid:0) (0 2 1) , (cid:0) , + , + (cid:1)(cid:1) = (cid:0) (0 1) ◦ (0 2 1) , (cid:0) , − , + (cid:1) + (cid:0) , + , + (cid:1)(cid:1) = ((0 2) , (1 , , . Remark 4.17.
The previous result remains true, with the same proof, if thevalues on s on the transpositions are redefined as s ((0 1)) = (cid:0) + , − , (cid:1) , s ((1 2)) = (cid:0) , + , − (cid:1) , s ((0 2)) = (0 , , . Let us then turn to the computation of the obstruction α ( P, ω, b ). We startfrom Γ ∈ N , we apply to it some of the moves of Proposition 4.10 (but neglectingthe numerical weight) and we call e Γ the result. Next, we assume that applyingfusion to e Γ we get Θ ∈ N defining (
P, ω, b ). Note that every edge of e Γ carriesan internal orientation (that for an even edge we stipulate to be the the same as46
102 102201201 102102 -1/2-1/2-1/2 -1/2+1/2+1/2 1 1= -1/2-1/2(0 2) (0 1 2) (0 1)
102 102201201 102102 =-1/2+1/2 +1/2+1/21 11 0(0 1) (0 2) (0 1 2) (0 2 1)
102 102102201 102102
Figure 51:
Examples of computation of α with the second method. the orientations at the ends). Let us concentrate on an edge e of Θ, that in e Γ(before fusion) will be subdivided into several edges. Since in Θ ∈ N the edge e is oriented, we can speak of a global orientation of e (that coincides with theinternal orientations of the two extremal subedges of e ). Now note that eachsubedge e ′ of e brings three portions of strands of attaching circles of P to S ( P ),and that these strands are numbered 0 , , e ′ according to theorientation of these. The recipe for the computation of α ( P, ω, b ) now uses themap s of Proposition 4.16, and goes at follows: • Let η ∈ S be the permutation attached to e ′ , and define ( h , h , h ) tobe s ( η ) if the internal orientation of e ′ is consistent with the global one,otherwise define ( h , h , h ) as s (cid:0) η − (cid:1) ; • At the first end of e ′ with respect to the global orientation, attach to thestrands 0 , , h , h , h .A formal proof that summing the contributions of the various e ′ one gets theedge contributions to α ( P, ω, b ) as in Proposition 2.7 employs Proposition 4.16,but we confine ourselves here to some examples only, see Fig. 51 and 52.47
102 102102 201 102102
11 11 +1/2+1/2+1/2-1/2-1/2-1/2 =(1 2) (0 2 1) (0 2) (0 1 2)
120 120021 201 102 102
111 1+1/2 -1/2-1/2 -1/2-1/2-1/2 =(0 1) (0 2 1) (0 1) (0 2 1)
102 102021 201 120 120
11 +1/2+1/2 +1/2+1/2 -1/2 -1/2-1/2 -1/2 =
Figure 52:
More examples of computation of α with the second method. References [1]
S. Baseilhac, R. Benedetti , Analytic families of quantum hyperbolic in-variants and their asymptotical behaviour, I , arXiv:1212.4261v1, new versionin preparation.[2]
S. Baseilhac, R. Benedetti , Classical and quantum dilogarithmic in-variants of flat PSL (2 , C ) -bundles over -manifolds , Geom. Topol. (2005),493-569.[3] S. Baseilhac, R. Benedetti , Quantum hyperbolic invariants of -manifolds with PSL (2 , C ) -characters , Topology (2004), 1373-1423.[4] R. Benedetti, C. Petronio , A finite graphic calculus for -manifolds ,Manuscripta Math. (1995), 291-310.[5] R. Benedetti, C. Petronio , “Branched Standard Spines of 3-Manifolds,”Lecture Notes in Mathematics Vol. 1653, Springer-Verlag, Berlin, 1997.[6]
C. Blanchet , Invariants on three-manifolds with spin stucture , Comm.Math. Helv. (1992), 406-427.[7] J. Christy , Branched surfaces and attractors. I. Dynamic branched sur-faces , Trans. Amer. Math. Soc. (1993), 759-784.488]
A. Hatcher , “Algebraic Topology,” Cambridge Univ. Press, Cambridge,2002.[9]
I. Ishii , Flows and spines , Tokyo J. Math. (1986), 505-525.[10] F. Luo , Private communications , 2011-2013.[11]
F. Luo, J.-M. Schlenker , Volume maximization and the extended hyper-bolic space , Proc. Amer. Math. Soc. (2012), 1053-1068.[12]
F. Luo, S. Tillmann, T. Yang , Thurston’s spinning construction andsolutions to the hyperbolic gluing equations for closed hyperbolic -manifolds ,Proc. Amer. Math. Soc. (2013), 335-350.[13] S. V. Matveev , Transformations of special spines and the Zeeman conjec-ture , Math. USSR-Izv. (1988), 423-434.[14] S. V. Matveev , Complexity theory of three-dimensional manifolds , ActaAppl. Math. (1990), 101-130.[15] R. Piergallini , Standard moves for standard polyhedra and spines , Suppl.Rend. Circ. Mat. Palermo (II) (1988), 391-414.[16] N. Reshetikhin, V. G. Turaev , Invariants of -manifolds via link poly-nomials and quantum groups , Invent. Math. (1991), 547-597.[17] J. Roberts , Refined state-sum invariants of - and -manifolds , In: “Ge-ometric topology, 1993. Georgia international topology conference, August2–13, 1993, Athens, GA, USA” (W. H. Kazez, ed.), American MathematicalSociety, Providence, RI (1997), 217-234.[18] V. G. Turaev, O. Y. Viro , State sum invariants of -manifolds and quan-tum j -symbols , Topology (1992), 865-902.[19] V. G. Turaev “Quantum invariants of knots and 3-manifolds,” de GruyterStudies in Mathematics, Vol. 18, Berlin, 1994.[20]
R. F. Williams , Expanding attractors , Inst. Hautes ´Etudes Sci. Publ. Math.43