# Algebraic K-Theory and Algebraic Number Theory by Stein M.R., Dennis R.K. (eds.)

By Stein M.R., Dennis R.K. (eds.)

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**Example text**

If X is the 2-skeleton of the dual of a triangulation of M, we let a* E adm{aX) be the colouring of ax corresponding to the admissible colouring a E aT. 1 x (a*) with proof as in ref. 1. 37 1T(O) ,. B 1T(3) i Figure 6. Correspondence between an internal O-stratum of X and a. tetrahedron. defines the numbering. 38 11" E S4 Figure 7. Correspondence between a 2-simplex of aT and a O-stratum of f. the numbering. c, M and 13 Before proving the invariance of nx(a) we quote a lemma whose proof can be found in ref.

The group of invertible elements of K is denoted by K*. The element 'Iii is called conjugate of w. 3. N : adm( J3) -+ N*, (A, B, C) 1-+ NAB C is a map. The number multiplicity of the admissible triple (A, B, C). 4. W : J -+ K*, A AEJ. 1-+ WA NAB C is called is a map. The element w~ E K* is called dimension of 5. adm(J3 x N*) is the subset of J3 (A,B,Cii) E adm(J3 x N*) X N* defined as follows: iff (A,B,C) E adm(J3), i $ N AB C • 6. adm(J6 x (N*)4) is the subset of J6 x (N*)4 defined as follows: (A,B,C,D,E,Fii,j,k,l) E adm(J6 x (N*)4) iff (A,B,Cii), (C,D,Eij), (B,D,Fik), (A,F,Eil) E adm(J3 x N°).

It turns out that the Schwinger-Dyson equation takes the same form for any value of q when the surfaces involved are non-degenerate. Since the amplitude of the boundary operator in the Boulatov model is expressed a sum over triangulations in the interior of the boundary surface weighted with AFq depending on q, the independence of the Schwinger-Dyson equation on q suggests that we can restrict the topology of T in the interior of the surface without spoiling the equation. Indeed we find that, associated to an arbitrary closed orient able three-dimensional manifold, one can construct a solution to the equation.