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Combinatorial Invariants from Four

arXiv:hep-th/9303110v1 19 Mar 1993Combinatorial Invariants from FourDimensional Lattice ModelsDanny Birmingham 1Universiteit van Amsterdam, Instituut voor Theoretische Fysica,1018 XE Amsterdam, The NetherlandsMark Rakowski2Yale University, Center for Theoretical Physics,New Haven, CT 06511, USAAbstractWe study the subdivision properties of certain lattice gauge theo-ries based on the groups Z2 and Z3, in four dimensions. The Boltz-mann weights are shown to be invariant under all type (k, l) subdivi-sion moves, at certain discrete values of the coupling parameter.

Thepartition function then provides a combinatorial invariant of the un-derlying simplicial complex, at least when there is no boundary. Wealso show how an extra phase factor arises when comparing Boltzmannweights under the Alexander moves, where the boundary undergoessubdivision.ITFA-93-06 / YCTP-P6-93March 19931Supported by Stichting voor Fundamenteel Onderzoek der Materie (FOM)Email: Dannyb@phys.uva.nl2Email: Rakowski@yalph2.bitnet

1IntroductionThe application of quantum field theory to the study of certain problems intopology has been a very fruitful one; we refer the reader to [1] for a generalreview of this subject. To state things quite simply, it has been possible tocompute a variety of topological invariants as correlation functions in specialquantum field theories.While many of these applications employ contin-uum field theory techniques, the lack of a precise formulation of quantumfield theory is a serious handicap when ones goal is ultimately to provideself-contained rigorous argument.

These difficulties might be sidestepped ifdiscrete lattice models can be employed in ways which avoid the continuumlimit.In [2], we presented a class of lattice gauge theories which enjoyed somenovel properties under lattice subdivision. The models were defined on atriangulated 4-manifold with boundary, in terms of compact Wilson variables.While we were motivated by some discrete structures which had a formalresemblance to the pure Chern-Simons theory [3, 4], once the model is definedno further reference to any continuum theory need be made.

We found that itwas possible to define models based on the gauge groups Z2 and Z3 in whichthe Boltzmann weight was invariant under type 4 Alexander subdivision, atcertain discrete values of the coupling parameter. These observations cameas a result of computer studies, and we were able to present some exactcalculations using Mathematica [5].Here, we will present analytic proofs of the subdivision properties of themodels defined in [2].We will show that the Boltzmann weight of thesetheories is invariant under all type (k, l) subdivisions [6] of the underlyingsimplicial complex.

This provides one with a partition function which is acombinatorial invariant of that complex, at least in the absence of a bound-ary.We also show how the Boltzmann weights behave under Alexandersubdivision [7], where the boundary itself is subdivided. We find that thereis a phase factor associated with these general subdivision moves at the levelof the Boltzmann weights.Other topological lattice models have been formulated previously [8, 9,10], and we should say a word about them here.

While these theories arealso formulated in terms of a triangulation, and an analysis of subdivision1

properties has been given, we are not aware of any connection with themodels considered in this paper. In particular, the Boltzmann weights ofthese other models are assembled from the 6j symbols, and their extensions.In [11, 12], certain Chern-Simons type theories were constructed for finitegroups.

It is unclear whether a relation exists between those models and ourfour dimensional theory defined on a manifold with boundary.The next section begins with a review of simplicial complexes and latticegauge theory, followed by a definition of the models we consider. An overviewof subdivision moves is also given.

We then present our main results regarding(k, l) subdivision invariance, and their proof. Alexander subdivision is thenexamined in these models, and we remark on some issues that arise withdifferent groups.

We close then with a few comments.2Definition of the ModelsTo begin, let us recall the basic elements of the theory of simplicial complexes;we refer to [13], which will be the source for our definitions and notation, forfurther details.Let {v0, · · · , vn} be a geometrically independent set of points in someambient euclidean space RN. An n-simplex spanned by this set of vertices,is the set of points x of RN which satisfies,x=nXi=0ti vi , with(1)1=nXi=0ti , and ti ≥0 for all i .Pictorially, these can be regarded as points, line segments, triangles, andtetrahedrons for n equals zero through three respectively.

A simplex whichis spanned by any subset of the vertices is called a face of the original simplex.An orientation of a simplex is a choice of ordering of its vertices, and we let[v0, · · ·, vn](2)denote the oriented simplex with the orientation class given by the orderingv0 · · · vn.2

A simplicial complex K in RN is a collection of simplices which are gluedtogether under two restrictions. Any face of a simplex in K is also in K,and the intersection of any two simplices in K must be a face of each ofthem.

The picture here is that of a collection of simplices glued togetherunder the above restriction. We will think of a spacetime manifold as beingapproximated by a certain simplicial complex.One defines the boundary operator ∂p on σ = [v0, · · · , vp] by:∂p σ =pXi=0(−1)i [v0, · · ·, ˆvi, · · · , vp] ,(3)where the ‘hat’ indicates a vertex which has been omitted.

It is easy to showthat the composition of boundary operators is zero.In a Wilson formulation of lattice gauge theory, the basic dynamical vari-ables are given by group valued maps on the 1-simplices (denoted [a, b]) withthe rule that Uba = U−1ab . A configuration of the system is then specified bya collection of group elements, one for each “link”.

One has, in addition, agauge group associated to each of the vertices, and the action of that groupon the link variables is defined by,Uab →ga Uab g−1b,(4)where ga is a group element associated with the vertex a. This group actionis also called a gauge transformation.Given a compact gauge group G, together with an invariant measure, onecan define a theory with partition functionZ =YαZdUα exp[β S(U)] ,(5)where the action functional S of the theory is taken to be a gauge invariantfunction of the link variables defined above, and the index α indicates theset of independent 1-simplices in the simplicial complex K. In the case of adiscrete gauge group, the group integration (whose volume we normalize tounity) is a discrete sum,ZdU →1|G|XU,(6)3

where |G| denotes the order of the group. One can also define correlationfunctions of the link variables,< Uγ1 · · · Uγp >=YαZdUα Uγ1 · · · Uγp exp[β S(U)] .

(7)It should be emphasized that, in general, all of these quantities depend notonly on the coupling parameter β, but also on the simplicial complex K.A central role in the construction of lattice gauge theory actions is playedby the holonomy. Let Uabc = Uab Ubc Uca be the holonomy based at the firstvertex a, around the triangle determined by a, b and c, and traversed in theorder from left to right .We take the action of our theory to be given by,S =X(U −U−1) ⋆(U −U−1) ,(8)where U is the above holonomy combination, and the sum here is over allelementary 4-simplices in the simplicial complex.

A matrix trace is also tobe included for the case of non-Abelian Lie groups. The ⋆-product [14], to berecalled presently, is designed to capture some of the properties of the wedgeproduct of differential forms.

In a continuum limit, the quantity U −U−1becomes proportional to the curvature of a connection, and (8) then goes overto the Chern form. We mention this from a purely motivational standpoint;we will not make any use of continuum theories in this paper.

Let us nowrecall the definition of the star product [14].The star product is a variant of the usual cup product of cochains whichachieves graded commutativity at the expense of associativity. Let cr andcs be two maps from the set of oriented r- and s-simplices respectively, intoa group, and let < cr, [v0, · · · , vr] > represent the evaluation of this map onthe particular r-simplex [v0, · · · , vr].

In our applications, we have been usingthe notation Uabc for the quantity < U, [abc] >. Denote by P, one of the(r + s + 1)!

permutations of the set of vertices {v0, · · · , vr+s}, which spansome (r + s)-simplex, and by Pvi the value of that permutation on vi. Thestar product of cr and cs is defined by,< cr ⋆cs, [v0, · · ·, vr+s] > =(9)1(r + s + 1)!XP(−1)|P | < cr, [Pv0, · · · , Pvr] > · < cs, [Pvr, · · ·, Pvr+s] > ,4

when the order v0 · · · vr+s is in the equivalence class of the orientation ofthe simplex [v0, · · ·, vr+s] (this determines the overall sign of the product),and where the sum is over all permutations of the vertices.The actualnumber of independent terms in this sum is given by the number of ways onecan partition the set of vertices into two parts which contain one vertex incommon, and an easy counting yields(r + s + 1)!r! s!.

(10)As we have seen, the holonomy U is a group valued map on 2-simplices,and therefore the above action is naturally defined on a 4-simplex.Oneshould note that the quantity (U −U−1)abc enjoys the property of antisym-metry in its last two indices; this is a simple consequence of the relationUabc = U−1abc. When the group is Abelian, one has, moreover, antisymmetryin all three indices.

The star product in our action, when evaluated on a given4-simplex, leads generically to 5! terms, however, the symmetries present in(8) reduce that number to 15 distinct combinations.

The Boltzmann weightfor this theory, evaluated on the simplex [v0, v1, v2, v3, v4], can now be writtenin the form:W[v0, v1, v2, v3, v4]=B[v0, v1, v2, v3, v4] B[v0, v1, v3, v4, v2] B[v0, v1, v4, v2, v3]B[v1, v0, v2, v4, v3] B[v1, v0, v3, v2, v4] B[v1, v0, v4, v3, v2]B[v2, v0, v1, v3, v4] B[v2, v0, v3, v4, v1] B[v2, v0, v4, v1, v3]B[v3, v0, v1, v4, v2] B[v3, v0, v2, v1, v4] B[v3, v0, v4, v2, v1]B[v4, v0, v1, v2, v3] B[v4, v0, v2, v3, v1] B[v4, v0, v3, v1, v2] ,(11)where,B[v0, v1, v2, v3, v4] = exp[β (U −U−1)v0v1v2 (U −U−1)v0v3v4] . (12)One general feature worth observing is that each B factor depends on twoindependent holonomies which share a common vertex; the term “bowtie”seems appropriate to describe this structure.

In the following, our analysisshall proceed for general complex coupling β.Our main concern here is to study these models for the case of the discreteAbelian groups Zp. However, an action which depends on the combination5

(U −U−1) necessarily leads to a trivial theory for the case of Z2. One maythen wish to consider the actionS =X(U −1) ⋆(U −1) .

(13)However, for Abelian groups, the holonomy Uabc is invariant under cyclic per-mutations of the indices. In addition, for the case of Z2, we have the relationU = U−1, for all group elements.

As a result, the holonomy combination isin fact symmetric in all indices, and the action above vanishes. Neverthe-less, as shown in [2], one can simply define the Boltzmann weight for a given4-simplex to be of the form (11), withB[v0, v1, v2, v3, v4] = exp[β(U −1)v0v1v2(U −1)v0v3v4] .

(14)One can proceed and compute the partition function for these theoriesdefined on a simplicial complex K. We wish to study, however, the behaviorof this function under subdivision of the complex. Let us now recall two basesof subdivision operations which accommodate an analysis of this question.The Alexander Moves:Consider a single 4-simplex [v0, v1, v2, v3, v4].

The subdivision operations ofAlexander type can in turn be described as follows.Type 1 Alexander subdivision:[v0, v1, v2, v3, v4] →[x, v1, v2, v3, v4] + [v0, x, v2, v3, v4] ,(15)Type 2 Alexander subdivision:[v0, v1, v2, v3, v4] →[x, v1, v2, v3, v4] + [v0, x, v2, v3, v4] + [v0, v1, x, v3, v4] , (16)Type 3 Alexander subdivision:[v0, v1, v2, v3, v4]→[x, v1, v2, v3, v4] + [v0, x, v2, v3, v4] + [v0, v1, x, v3, v4]+[v0, v1, v2, x, v4] ,(17)Type 4 Alexander subdivision:[v0, v1, v2, v3, v4]→[x, v1, v2, v3, v4] + [v0, x, v2, v3, v4] + [v0, v1, x, v3, v4]+[v0, v1, v2, x, v4] + [v0, v1, v2, v3, x] . (18)6

One can picture the move of type 1 as the introduction of an additionalvertex x, which is placed at the center of the 1-simplex [v0, v1], and is thenjoined to all the remaining vertices of the 4-simplex.Moves 2 to 4 in-volve a similar construction, where x is placed at the center of the simplices[v0, v1, v2], [v0, v1, v2, v3], and finally [v0, v1, v2, v3, v4]. There is, in addition, atype 0 move which is effected by replacing a vertex of the simplicial complexby a new vertex.

This can be considered as a degenerate case, and need notconcern us in the following.According to Alexander [7], two simplicial complexes are said to be equiv-alent if and only if it is possible to transform one into the other by a sequenceof these moves. Hence, any function of K which is invariant under thesemoves yields a combinatorial invariant of the simplicial complex.The (k, l) Moves:Another basis of subdivision operations is available, known as the (k, l)moves, and these allow for a more convenient analysis.

In particular, it hasbeen shown [6] that the basis of (k, l) moves is equivalent to the Alexandermoves for the case of closed manifolds, for dimensions less than or equal tofour. In the four dimensional case under study, we have five (k, l) moves, withk = 1, · · ·, 5, and k + l = 6.

It suffices to consider the first three cases; the(4, 2) and (5, 1) moves are inverse to the (2, 4) and (1, 5) moves, respectively.The (1, 5) move:[v0, v1, v2, v3, v4]→[x, v1, v2, v3, v4] + [v0, x, v2, v3, v4] + [v0, v1, x, v3, v4]+[v0, v1, v2, x, v4] + [v0, v1, v2, v3, x] ,(19)The (2, 4) move:[v0, v1, v2, x, v3]+[v0, v1, v2, v3, y] →[v0, v1, v2, x, y] + [v0, v2, v3, x, y]+[v0, v1, v3, y, x] + [v1, v2, v3, y, x] ,(20)The (3, 3) move:[v0, v1, v2, x, y]+[v0, v1, v2, y, z] + [v0, v1, v2, z, x] →[x, y, z, v0, v1]+[x, y, z, v1, v2] + [x, y, z, v2, v0] . (21)A simple observation is that the (1, 5) move is identical to the type 4Alexander subdivision.

One notes that the simplices on the left hand side7

of the (2, 4) move share a common 3-simplex [v0, v1, v2, v3], while those onthe right have a common 1-simplex [x, y]. For the case of the (3, 3) move,the 2-simplex [v0, v1, v2] is common to the left hand side, with [x, y, z] beingcommon to the right.

Furthermore, one can verify that the boundary of the4-simplex remains unchanged as a result of these operations.Previous Results:Before proceeding with the general analysis, we recall the results obtained forthe 4-disk and 4-sphere, for the groups Z2 and Z3. A complete calculation ofthe partition function for arbitrary complex coupling β was presented in [2].The central observation was that subdivision invariant points were presentin both models.

Indeed, it was explicitly checked that the partition functionsof the 4-disk remained invariant under all Alexander moves, at these specialpoints.It was further shown that the results for S4 were invariant withrespect to Alexander type 4 subdivision. For each of the models studied, anatural scale factor was present, and we denote this by s(2) = exp[4β] ands(3) = exp[−3β] for the case of Z2 and Z3, respectively.

The correspondingsubdivision invariant points are then given when s(2)2 = 1, and s(3)3 = 1.A point worth mentioning is that the Boltzmann weights themselves aregroup valued at these special points.Let us quickly summarize some ofthose results.Beginning with the Z2 theory, we found the partition function on the4-disk to be given byZ(s(2)) = 124(9 + 7s(2)) ,(22)when s(2)2 = 1. The two roots of unity, +1 and −1 yield the values 1 and1/8 respectively.

For the case of the 4-sphere S4, we find that the partitionfunction assumes a value of Z = 1, when s(2)2 = 1.Turning now to the Z3 theory, we found the partition function on the4-disk to be:Z(s(3)) = 134(29 + 26(s(3) + s(3)−1)) ,(23)when s(3)3 = 1. The trivial subdivision invariant point s(3) = 1 yields avalue Z = 1, while the other two cube roots of unity give a value Z = 1/27.Again, for the case of S4, one finds a value of Z = 1 when s(3)3 = 1.8

These results were obtained through the use of Mathematica [5] to eval-uate the partition functions. While we could show through exhaustive com-puter checks that these models had interesting subdivision invariant points, aclear analytic understanding of these special properties was generally lacking.This we supply in the following sections.3Main ResultsHaving laid the foundational material in the preceding sections, we can nowstate and prove the main results.The aim of this section is to establishthe behavior of the Boltzmann weights under all (k, l) type subdivisions.

Inorder to treat both the Z2 and Z3 models uniformly, it is expedient to let Xdenote the combinations U −1 and U −U−1, respectively, for those models.It will further be convenient to let B[0, 1, 2, 3, 4] represent the expressionB[v0, v1, v2, v3, v4]; using subscripts to keep track of vertices should cause noconfusion. We begin with a lemma.Lemma: The Boltzmann weights for a given vertex ordering satisfy theconditions,B[0, 1, 2, 3, 4] B[0, 1, 2, 4, 5] B[0, 1, 2, 5, 3] = exp[β Xv0v1v2Xv3v4v5] ,(24)B[0, 1, 2, 3, 4] B[1, 2, 0, 4, 3] = exp[β Xv0v1v2Xv0v1v4] exp[−β Xv0v1v2Xv0v1v3] ,(25)at the points s(2)2 = 1 and s(3)3 = 1, in the Z2 and Z3 theories respectively.Consider first the Z2 case.One notices immediately that the relation(24) is trivially satisfied for Uv0v1v2 = 1, so we only need to consider the caseUv0v1v2 = −1.

For simplicity of notation, let x = Uv0v3v4, y = Uv0v4v5, andz = Uv0v5v3. Noticing that Uv3v4v5 = x y z, our assertion is then equivalent to,1 = exp[−2 β ((x −1) + (y −1) + (z −1) −(xyz −1))] .

(26)Now, x, y, and z are independent group elements, and the image set of thefunction(x, y, z) →(x −1) + (y −1) + (z −1) −(xyz −1)(27)9

is easily seen to be {0, −4}. Recalling that s(2) = exp[4 β], one sees thenthat (26) is satisfied at s(2)2 = 1.The Z3 relation follows in the same way; here we need only check the caseUv0v1v2 = exp[±2πi/3].

The assertion (24) is then equivalent to,1 = exp[±βi√3 ((x −x−1) + (y −y−1) + (z −z−1) −(xyz −x−1y−1z−1))] ,(28)where x = Uv0v3v4, y = Uv0v4v5, and z = Uv0v5v3 are independent Z3 elements.The image set of the function,(x, y, z) →(x −x−1) + (y −y−1) + (z −z−1) −(xyz −x−1y−1z−1) , (29)is easily seen to be {0, ±3i√3}. With s(3) = exp[−3β], one then finds that(28) is satisfied at the points s(3)3 = 1.The proof of the second relation (25) is very similar, and we omit thedetails.

Our main result is the following theorem.Theorem: The full Boltzmann weights satisfy the relation,W[0, 1, 2, 3, 4] W[0, 1, 2, 4, 5] W[0, 1, 2, 5, 3] =W[0, 1, 3, 4, 5] W[1, 2, 3, 4, 5]W[2, 0, 3, 4, 5] ,(30)at the points s(2)2 = 1 and s(3)3 = 1, in the Z2 and Z3 theories respectively.The proof of this, while straightforward, is surprisingly tedious. Each ofthe W factors is itself a product of 15 factors.

One can write out all 90 Bfactors that occur in (30), and methodically use the the identities establishedin the lemma to verify the claim. In our analysis, we used the identity,B[0, 1, 2, 3, 4] B[0, 1, 2, 4, 5] B[0, 1, 2, 5, 3] =B[3, 4, 5, 0, 1] B[3, 4, 5, 1, 2]B[3, 4, 5, 2, 0] ,(31)which is a trivial consequence of (24), to eliminate all but 18 of the 90 terms.The identity (25) was then used to polish offthe remaining factors.Armed with this theorem, is is now a simple matter to understand thesubdivision properties of the Boltzmann weights under the remaining moves.Corollary: The full Boltzmann weights satisfy the following two rela-tions:W[0, 1, 2, 3, 4] W[0, 1, 2, 5, 3] =W[1, 2, 3, 4, 5] W[2, 0, 3, 4, 5] W[0, 1, 3, 4, 5]W[1, 0, 2, 4, 5] ,(32)10

W[0, 1, 2, 3, 4]=W[5, 1, 2, 3, 4] W[0, 5, 2, 3, 4] W[0, 1, 5, 3, 4] W[0, 1, 2, 5, 4]W[0, 1, 2, 3, 5] ,(33)at the points s(2)2 = 1 and s(3)3 = 1 for the groups Z2 and Z3 respectively.The proof here is a simple application of the theorem, together with thefact that,W[0, 1, 2, 3, 4]−1 = W[0, 1, 2, 4, 3](34)in our theories; this relation holds at s(2)2 = 1 in the Z2 case, and quitegenerally in the Z3 model. This means that W is actually invariant under areversal of orientation at the special points in the Z2 model, and is exchangedfor its inverse in all models based on the action (8).As a consequence of these results, one can immediately establish the factthat the partition function for these models provides a combinatorial invari-ant of an arbitrary simplicial complex, at least for the case of zero boundary.In particular, we can now assert that the results presented previously [2] forthe case of S4 do indeed correspond to a combinatorial invariant.4Behavior under the Alexander MovesThe subdivision moves introduced by Alexander, and reviewed in an earliersection, are slightly more complicated.

These moves generally induce subdi-visions of the boundary of a given 4-simplex. Nevertheless, our understandingof the type (k, l) subdivision will allow us to fully analyze these other moves.Consider the type 3 Alexander move where we add the v5 vertex to thecenter of [v0, v1, v2, v3].Using subscripts once again to keep track of thevertices, this move takes the form:[0, 1, 2, 3, 4] →[5, 1, 2, 3, 4] + [0, 5, 2, 3, 4] + [0, 1, 5, 3, 4] + [0, 1, 2, 5, 4] .

(35)It is useful to note how the boundary transforms under this move; a sim-ple check reveals that the boundary component [0, 1, 2, 3] undergoes a threedimensional Alexander type 3 subdivision, namely,[0, 1, 2, 3] →[5, 1, 2, 3] + [0, 5, 2, 3] + [0, 1, 5, 3] + [0, 1, 2, 5] . (36)11

The fundamental question is how W[0, 1, 2, 3, 4] is related to the weights ofthe four 4-simplices on the right hand side of equation (35). Again, the (3, 3)identity we established in the last section proves to be the key to resolvingthis.

It is a quick exercise to show that,W[0, 1, 2, 3, 4]=W[5, 0, 1, 2, 3] ( W[5, 1, 2, 3, 4] W[0, 5, 2, 3, 4] W[0, 1, 5, 3, 4]W[0, 1, 2, 5, 4] ) . (37)The Boltzmann weight W is not invariant under this move, but it picksup what one might wish to view as a phase factor associated with addingthe v5 vertex to the center of [v0, v1, v2, v3]; the “phase” being the quantityW[5, 0, 1, 2, 3].

It is equally simple to write the corresponding “phases” as-sociated with the type 1 and 2 Alexander moves, though we won’t cataloguethem here. The type 4 move is identical to (1, 5), and we know that there isno phase factor in that case.5Z4 and BeyondOne rather immediate question about the results we have laid out in theprevious two sections would be with regard to extending them to the generalZp case.

This is not automatic, and an analysis of the group Z4 already beginsto show a departure from what happened for Z2 and Z3. If one repeats thesame analysis for a Z4 type theory defined by the action (8), one finds thatnot all fourth roots of unity yield subdivision invariant points under the (k, l)moves.

Defining the analogous scale factor to be s(4) = exp[−4β], one findsonly the two points corresponding to s(4)2 = 1. A similar situation arises forZ6; with the scale factor denoted by s(6) = exp[−3β], one finds subdivisioninvariant points when s(6)3 = 1.

For Z5, however, the entire structure ofthe theory is rather more complicated, and it is an open question as towhether one can find subdivision invariant points. Equally, we must leaveextensions to other types of groups, both discrete and continuous, for futureinvestigation.12

6Concluding RemarksHaving established the combinatorial invariance of the partition function forthe Z2 and Z3 models, perhaps the most pressing issue is to determine theprecise nature of this invariant. In particular, it is of interest to explicitlycompute the invariant for a variety of closed manifolds.

As we have seen,the Boltzmann weight is invariant under the Alexander moves, up to certain“phase” factors associated with the subdivision induced on the boundary.Our conclusion thus falls short of declaring the partition function to be acombinatorial invariant for a four dimensional manifold with boundary. Nev-ertheless, as we have seen from our computer studies, the partition functionon the 4-disk is indeed invariant under all subdivision moves of Alexandertype.

Further work is required in this arena. One might also hope that thenatural correlation functions associated with each of these models may en-joy special properties with respect to subdivision, but we leave that for thefuture.References[1] D. Birmingham, M. Blau, M. Rakowski, and G. Thompson, TopologicalField Theory, Phys.

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[7] J.W. Alexander, The Combinatorial Theory of Complexes, Ann.

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[9] H. Ooguri, Topological Lattice Models in Four Dimensions, RIMS-878preprint, May 1992. [10] L. Crane and D. Yetter, A Categorical Construction of 4D TopologicalQuantum Field Theories, Kansas State University preprint.

[11] R. Dijkgraaf and E. Witten, Topological Gauge Theories and GroupCohomology, Commun. Math.

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[12] D. Altschuler and A. Coste, Quasi-Quantum Groups, Knots, Three-Manifolds, and Topological Field Theory, Commun. Math.

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[13] J. Munkres, Elements of Algebraic Topology, Addison-Wesley, MenloPark, 1984. [14] D. Birmingham and M. Rakowski, A Star Product in Lattice GaugeTheory, Phys.

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