Updates on the Dehn paper

[Update June 28: A bit unrelated: Positive curvature and Bosons, which is one of the maybe more treacherous alleys in the saga of positive curvature. In any case, even if it could be a dead end, it is a passage not been walked before to link the known bosons for fundamental forces with the known even dimensional positive curvature manifolds. Still, the picture is a falsifiable scientific pursuit: it can be challenged by finding one single further positive curvature manifold beyond W²⁴ or one single other Boson carrying forces. (The axion could be a candidate). This worked beautifully in Keplers Harmonices Mundi suggestion, where a single new planet killed the idea for good. What is nice that we do not only have a correspondence between Force bosons and even-dimensional positive curvature manifolds but also a correspondce between carriers with mass (gauge bosons and Higgs) and manifolds for which some Betti numbers are larger than 1. It is an epic conjecture of Ziller that all even-dimensional positive curvature manifolds have Betti numbers bounded above by 2.]

See also Chopping up Riemannian manifolds in the quantum calculus blog.
Here are some updates (mini blog) on A Dehn type invariant for Riemannian manifolds The first version of the ArXiv paper had contained mistakes and was updated corrections. Here is a local version [PDF] (last updated May 29, 2020) . The following chronological with the latest things on top as usual on blog rolls.

Some slides, I wanted to post for a while already. Here they are, before I move on to other subjects:
[5/31/2020] I made more computations with frame dependence. The frame dependence happens even in product situations like S² x S² or S³ x S³ where I had previously thought that gamma is the 4 or 0. Turning the orthogonal frame so that it is no more compatible with the product structure however changes also gamma. The updated PDF replaces the Arxiv version. The entire paper essentially has been reduced now to just example computations and also motivations for the discrete, especially about energized simplicial complexes. It is a bit thin, but that's also good. It will keep me hungry. At least, the case of Poincare-Hopf on Riemannian polytopes has remained solid as well as the integrated case of index expectation Gauss-Bonnet results on Riemannian manifolds with boundary. But this was not really much new, just much clearer now.

At the end of that document, I just added an other connection to discrete research like this theorem from 2013 which shows that the fixed point set of an automorphism T of a simplicial complex G which preserves the orientation of all simplices (this assures that the index i_T(x) of a simplex is equal to w(x)=(-1)^dim(x) and so X(F) = L(G,T), the Lefshetz number, which is the super trace of the induced linear action on cohomology). The reason for getting T(x)=x not changing orientation in the discrete is because it comes from a circle action in the continuum which is homotopic to the identity. Now, in the case when T can be deformed to the identity (of course a true circle action is not possible in the discrete), but in the case when T is embedded in a circle action, one has that L(M,T) is equal to the Euler characteristic X(M), as T must then act trivially on cohomology. And the sum of the indices of the fixed point set is by definition the Euler characteristic of F. Yesterday, while running, I wondered whether if an isometric circle action exists on a positive curvature manifold, whether the fixed point set must necessarily be a manifold. All examples, I could think of have shown manifolds as fixed point sets (which could also be zero-dimensional manifolds, discrete sets of points). Looking this up later at home showed that this is indeed the case: it is a theorem of Conner and Kobayashi (independent work) from the 1950ies. The circle action does not even have to consist of isometries: if a smooth circle action exists on a manifold, then the fixed point set is a smaller dimensional compact manifold with the same Euler characteristic. The Euler characteristic part is what one expects from Lefshetz fixed point theory (even so the fixed point set is not a discrete set of points but a manifold). Now, also in all cases I could think off during running like spheres the fixed point set F is a manifold with even co-dimension. Of course, it is still a positive curvature manifold, if M had positive curvature. One can therefore ask:

Is it true that the fixed point set of an isometric circle action on an even dimensional positive curvature manifold has even co-dimension?

If that is answered positively, then a positive curvature 6-manifold admitting a circle action must have positive Euler characteristic, the reason being that the Hopf conjectures hold in 4 dimensions by Gauss-Bonnet-Chern and in 2 dimensions by Gauss-Bonnet and because by a result of Berger, if the fixed point set is zero dimensional, it is non-empty (and so also has positive Euler characteristic). On the other hand if there should exist a counter example to the question, then the fixed point set as an odd-dimensional manifold would have zero Euler characteristic and so the Hopf sign-conjecture would be false. So, the above question is actually equivalent to the Hopf sign conjecture for manifolds with an isometric circle action on positive curvature manifolds. This has been answered very affirmatively for 4-manifolds by work of Hsiang and Kleiner who even gave the list of possible manifolds in 4 dimensions (spheres or real or complex projective planes). Now, one can even be bolder and ask

Is it true that the fixed point set F of an isometric circle action on an even dimensional positive curvature manifold has even co-dimension and that the manifold F again admits an isometric circle action?

The intuition is that one could ``speed up" (or renormalize) the circle action near the fixed point set F so that in the limit, when approaching F, we do not end up with a stationary motion but with an induced flow. That would be very strong as it would in general prove the Hopf conjecture under the rather mild symmetry condition of admitting a Killing vector field. There is a whole industry out there proving such statements under more symmetry. An example is this result of Amann and Kennard.

The ``fixed point story about circle actions" leads a bit away from the original question about the nature of the functional gamma(M), but at one point, it appeared attractive that if M admits a circle action that gamma(M) is so close to Euler characteristic that it forces positive curvature of M. The functional gamma had been considered just because of the Hopf conjectures. It now appears however that estimating the difference between gamma(M) and chi(M) leads to quite serious analytic business.
[5/30/2020] I did this morning more computations also with SU(3) and chose different basis. Indeed, also there the sum depends on the basis. Since the summation over 8! = 40320 terms is a bit tedious, here is a nice trick to get it down to 2520=8!/16. There is the obvious symmetry for sectional curvature K(i,j)=K(j,i). So, permutations like (2,1,4,3,5,6,8,7) or (1,2,3,4,5,6,7,8) produce the same products of sectional curvatures. To get quickly a list of representations of the equivalence classes, we partition a permutation into groups of 2, sort each of the pairs, flatten the list again then take the union. This can be done in Mathematica as follows
```
f[p_]:=Flatten[Map[Sort,Partition[p,2]]];
P=Union[Map[f,Permutations[Range[8]]]]
```
This speeds up the computation by a factor 16 and hopefully allow to compute an averaged gamma(SU(3)). A factor 16 appears not much but whether you have to wait a minute for a result or 16 minutes really matters! Eventually, we want to integrate over a circle (fortunately, we do not have to integrate over the entire orthogonal group O(8) which would be just painful as this is a 28 dimensional group and even computer algebra systems do not like that.). The fact that we only need to average over a 1-dimensional circle to make the curvature coordinate independent is now the new content of Lemma 1 in the latest version. I secretly still hope that the actual averaged gamma value is an integer. In order to make the rotation in SU(3), we rotate two basis elements in the adjoint representation. We can not rotate the Gell-Mann matrices themselves as this is in the complex picture and not an orthogonal transformation in the 8 dimensional real Lie group su(3) of SU(3).

[5/29/2020] More bad news: also the coordinate independence can fail. Jason de Vito computed with CP² with standard Fubini-Study metric (example 4.10 in my text). I had there some mixed results too but assumed having made mistakes. Here is what Jason got: disregarding the constant, for 4-manifolds, we can by symmetry look at 8( K₁₂ K₃₄ + K₁₃ K₂₄ + K₁₄ K₂₃). For the basis (1,0),(i,0),(0,1),(0,i), then the do Carmo formula gives K₁₂ = K₃₄ = 4, while all other sectional curvatures K_ij = 1 so that the sum is 144 while for 1/sqrt(2) (1,1), (i,0), 1/sqrt(2) (1,-1), (0,i), one has K₁₃ = K₂₃ = 1, with all other K_ij = 5/2 giving 108. I found in my research notes from March the following note and code for the complex projective plane " The next example is still incomplete. We count 8*16+16 = 144 sectional curvature and not 8*4+16*4=192 which requires a different volume". While writing this I had been still under the impression that the integral should add up to Euler characteristic and assumed of having made a mistake. I should have looked at this more then.

(* The curvature 2-form of M=CP^2 is in suitable coordinates
   R_{01} =-R_{23}=e^0 wedge e^1 - e^2 wedge e^3, R_{02}=-R_{31}=e^0 wedge e^2 - e^3 wedge e^1  
   R_{03} = 4 e^0 wedge e^3 + 2 e^1 wedge e^2 ,  R_{12} = 2 e^0 wedge e^3 + 4 e^1 wedge e^2     
   First write down the full curvature tensor                                                   *)
R=Table[0,{i,4},{j,4},{k,4},{m,4}];
R[[1,2,1,2]]=1; R[[1,2,3,4]]=-1; R[[3,4,1,2]]=-1; R[[3,4,3,4]]=1;
R[[1,3,1,3]]=1; R[[1,3,4,2]]=-1; R[[2,4,1,3]]=-1; R[[2,4,2,4]]=1;
R[[1,4,1,4]]=4; R[[1,4,2,3]]= 2; R[[2,3,1,4]]=2; R[[2,3,2,3]]=4;
Do[R[[j,i,k,l]]=-R[[i,j,k,l]],{k,4},{l,4},{i,4},{j,i+1,4}];
Do[R[[i,j,l,k]]=-R[[i,j,k,l]],{i,4},{j,4},{k,4},{l,k+1,4}];
Do[R[[j,i,l,k]]=R[[i,j,k,l]],{i,4},{j,i+1,4},{k,4},{l,k+1,4}];
K=Table[R[[i,j,i,j]],{i,4},{j,4}];
P=Permutations[Range[4]];
S = Table[ K[[P[[k,1]],P[[k,2]]]]*K[[P[[k,3]],P[[k,4]]]],{k,Length[P]}]
Total[S]      (* gives 144  *)

Apropos looking up old code: Here is code for the real projective plane, where of course things work well, as it is a two-dimensional surface. The surface is nicely Nash embedded in 6-dimensional space. The example had been mentioned in the preprint too. One can not realize it of course without self-intersections in 3-dimensional space. The volume of this 2-dim real projective plane is the same than the volume of the 2-sphere and the curvature is constant 1/2. This come handy when looking at other metrics on RP² x RP² for example. Unlike for S² x S², one knows that RP² x RP2 does not admit a metric with positive curvature because of Synge theorem.

(* Gauss-Bonnet for RP^2 with constant K=1/2 and V=4Pi. Use parameterization       *)
   given in Berger-Gostiaux, page 89, O. Knill, Feb, 2020                          *)
{x,y,z}={Cos[t] Sin[s], Sin[t] Sin[s], Cos[s]};
r={x,y,z};             a=2Pi; b=Pi;    (* Sphere case *)
r={x^2,y^2,z^2,Sqrt[2]y*z,Sqrt[2]*z*x,Sqrt[2]*x*y};  a=Pi; b=Pi;   (* RP^1         *)
dr={D[r,t],D[r,s]}; T=dr.Transpose[dr]; g=Simplify[T]; gi=Inverse[g];
dV=Simplify[Sqrt[Det[g]]]; M=Length[T];
P=Permutations[Range[M]]; X = {t,s};  S4=2d! (4Pi)^d/(2d)! /. d->2;
Ch[X_,g_]:=gi.Simplify[Table[D[g[[i,j]],X[[k]]]+D[g[[i,k]],X[[j]]]-
     D[g[[j,k]],X[[i]]],{i,M},{j,M},{k,M}]/2];
R[X_,g_]:=g.Module[{c=Ch[X,g]},Table[D[c[[i,j,l]],X[[k]]]-
     D[c[[i,j,k]],X[[l]]]+(c.c)[[i,k,l,j]]-(c.c)[[i,l,k,j]],{i,M},{j,M},{k,M},{l,M}]];
K=R[X,g][[1,2,1,2]]/Det[g];                   (* Simplifies to 1/2     *)
Integrate[ dV,{t,0,a},{s,0,b}]                (* Volume                *)
Integrate[ K*dV,{t,0,a},{s,0,b}]/(2Pi)        (* Euler characteristic  *)

[5/29/2020] We have to explore that if M is the product of an odd dimensional space manifold N and a time manifold (like T=T¹ the circle, or T=[a,b], an interval) and the metric in N is time independent, then the functional γ(M) is equal to χ(M) if the basis is chosen with one vector in T. This should also holds in the case of manifold with two different boundaries as the boundary is then consists of two disjoint unions of odd-dimensional N. What we need is that the metric is time independent. In that case this should even work for cobordisms.
[5/29/2020] Here is a bit of connection to combinatorics (which is actually our main focus). The Riemannian explorations were an excursion (definitely out of my comfort zone). But there are relations. The fact that we have non-removable curvature at the intersections leads to an abstract combinatorial problem for a geometric realization of a finite abstract simplicial complex G with maximal simplices generating the chambers M_k in M. If f is Morse on M inducing Morse functions on M_k we can look at h(x) = ∑_{m in x} i_f(m) = 1 with Morse indices i_f(m) in {-1,1}. This leads to the matrix L(x,y) counting the number of simplices in x ∩ y, a positive definite quadratic form in SL(n,Z) of simplectic nature as L is conjugated to its inverse g, the Green function which is again integer-valued and for which ∑_x,y g(x,y) = |G| is the number of simplices in G. For a global Morse function f, we can count the indices using exclusion-inclusion. This leads to h(x)=w(x)=(-1)^dim(x) and ∑_x h(x) = X(G)=X(M) is the Euler characteristic of G a or M. If f_k are Morse on M_k and independent of each other, then the gluing does not work and h(x) becomes an arbitrary integer-valued function on the simplicial complex. There is still a connection matrix L which satisfies det(L) = ∏_x h(x) and ∑_x,y g(x,y) = ∑_x h(x). The fact that γ(M) is not metric-dependent brings the functional γ closer energized simplicial complexes, where the topological h(x)=w(x) leading to Euler characteristic X(M) is replaced by a more general h(x) for which γ(M) = ∑_x h(x) differs from X(M). Averaging over probability spaces of Morse functions then renders h(x) real-valued for which the abstract combinatorial results still work. A combinatorics person can forget completely about any motivation or any relation with Riemannian geometry and just use the finite combinatorial or linear algebra frame work. Nice is for example that the number of positive eigenvalues of L are the number of simplices with positive energy h(x) and the number of negative eigenvalues of L the number of simplices with negative energy. This is just linear algebra for finite matrices and finite combinatorics. No Riemannian geometry is at all involved. Still, the world of Riemannian geometry is beautiful and can inspire combinatorics, not at least also because in Riemannian geometry there is lots of very successful physics which is physics that is confirmed by experiments.
[5/28/2020] A rewrite [PDF] has started. It will soon be ready to replace the Arxiv version. [Update: has replaced] The example of the metric on T⁴ considered by Cliff also allows to compute the Gauss-Bonnet-Chern integrand. It is interesting because I so far do not know examples in the literature, where the integrand is non-constant and can be explicitly computed. It is a sum of 24² terms. In the case of g=dx²+dy²+exp(2u(x,y)) dt² + exp(-2u(x,y)) ds² the GBC integrand is -cos(x) cos(y)/(2π²), which as it should integrates up to 0, the Euler characteristic of the 4-torus. Actually, one can make the computation for general u(x,y) and have GBC=-det(d²(u)/(2π²), which is up to a scale the discriminant of the function u(t,s), the determinant of the Hessian, which is familiar in multi-variable calculus. This is cute as one knows in the case of a surface z = u(x,y), that at critical points the discriminant D is the Gauss curvature of the surface at that point. This also shows that if one wants to have on the 4-torus a given Gauss-Bonnet-Chern integrand K(x,y,t,s) = k(x,y), then one has to solve the Monge-Ampere partial differential equation det d² u(x,y) = k(x,y) to get a metric which produces this curvature. The Monge-Ampere PDE is here even a bit simpler than in the case of a two dimensional surface z=u(x,y), where the Gauss curvature is det( d² u)/(1+|du|²)².
[5/27/2020] In the paper, I made the assumption that averaging the product curvature over all frames, produces a curvature which is asymptotically rotationally symmetric at the vertices of the Riemannian polyhedra. This is not true in general. There can be small discrepancies and they do not go to zero fast enough when the grid size goes to zero. The interior is fine as it is a Riemann sum. But while the errors go to zero if the triangulation is refined, the number of vertices increases. It appears that the error with diameter r can be O(r^2d) but we expect V/r^2d polytopes, if the volume is V. So, the gluing curvature does not necessarily go to zero. Looking at a fixed triangulation should however allow to make statements like: if the metric does not fluctuate too much and the curvature is positive, then the functional γ(M) is close enough to Euler characteristic or ``if the metric is smooth enough" and the ``sectional curvatures are large enough", then the Euler characteristic is positive. Also for the sphere theorems, one had needed some pinching condition additionally to the positive curvature condition to establish that M is a sphere. The analytic problem is then, assume the 2d-manifold M has sectional curvatures larger or equal than some positive δ and the derivatives of the metric g are uniformly smaller than some ε, then M has positive Euler characteristic. An other way to rephrase this, if M has curvature sign e and is close enough to a constant curvature space (space form), then M has Euler characteristic with positive &chi(M) e^d < Of course, this has to be formulated more precisely. Still, it should indicate that looking at &gamma(M); is not necessarily a waste of time.

[5/27/2020] Cliff Taubes sent me an example of a metric on a 4-torus for which the integral is not zero. So, the invariance result is not true. The example is a metric g=dx²+dy²+exp(2u) dt² + exp(-2u) ds² with u=cos(x) + cos(y), where the integral (with the constants given in the paper) is π². Here is the computation of Cliff Taubes [PDF]. Of course, the paper will have to be modified accordingly and to be seen what can be salvaged. Here is my Mathematica code for Cliff's computation. The example shows also that the discrepancy between the integral and the Euler characteristic can become arbitrary large and arbitrary small in general. Here is mathematica code. I had already in 1995 for this course had students verify some computations in Riemannian geometry using Mathematica (see page 97 of that document).
(* Example of Cliff Taubes, Mathematica by Oliver Knill, 5/27/2020 *) f=Cos[t]+Cos[s]; (* f=Cos[t +s]; *) S4=2d! (4Pi)^d/(2d)! /. d->2; g={{1,0,0,0}, {0,1,0,0}, {0,0,Exp[2f],0}, {0,0,0,Exp[-2f]}}; M=Length[g]; dV=Simplify[Sqrt[Det[g]]]; P=Permutations[Range[M]]; X = {t,s,u,v}; gi=Inverse[g]; g0=g; X0=X; Ch[X_,g_]:=Simplify[gi.Table[D[g[[i,j]],X[[k]]]+D[g[[i,k]],X[[j]]]-D[g[[j,k]],X[[i]]],{i,M},{j,M},{k,M}]/2] R[X_,g_]:=g.Module[{c=Ch[X,g]},Table[D[c[[i,j,l]],X[[k]]]-D[c[[i,j,k]],X[[l]]]+(c.c)[[i,k,l,j]]-(c.c)[[i,l,k,j]], {i,M},{j,M},{k,M},{l,M}]]; S[X_,g_]:=Module[{R1,R2,K}, R1=R[X,g]; R2=Table[R1[[i,j,k,l]],{i,M},{j,M},{k,M},{l,M}]; K[i_,j_]:=If[i==j,0,Simplify[R2[[i,j,i,j]]/(g[[i,i]]*g[[j,j]]-g[[i,j]]^2)]]; Table[K[i,j],{i,M},{j,M}]]; U=S[X,g]; k=Sum[ U[[P[[k,1]],P[[k,2]]]]*U[[P[[k,3]],P[[k,4]]]],{k,M!}] c[d_]:=(d!)^(-1)*(4Pi)^(-d); Integrate[k*dV,{t,0,2Pi},{s,0,2Pi},{u,0,Pi},{v,0,2Pi}]*c[2] (* this is Pi^2 *) (* It becomes -pi^2/2 for f = Cos[t+s] *)
Just if you are interested, here is an example of a deformed 4-sphere, where the integral is exactly 2. It kind of illustrates how complicated the curvature expressions can become. And it also illustrates why I had become confident that γ is χ. Of course, we will have to investigate for which metrics the integral γ is equal to the Euler characteristic χ.
(* special 4 ellipsoid computation, Mathematica Oliver Knill, February 19, 2020 *) r={3Cos[t],Sin[t]*Cos[s],Sin[t]*Sin[s]*Cos[u],Sin[t]*Sin[s]*Sin[u]*Cos[v],Sin[t]*Sin[s]*Sin[u]*Sin[v]}; dr={D[r,t],D[r,s],D[r,u],D[r,v]}; T=dr.Transpose[dr]; M=Length[T]; dV=Simplify[Sqrt[Det[T]]]; M=Length[T]; P=Permutations[Range[M]]; X = {t,s,u,v}; S4=2d! (4Pi)^d/(2d)! /. d->2; g = Simplify[T]; gi=Inverse[g]; g0=g; X0=X; Ch[X_,g_]:=Simplify[gi.Table[D[g[[i,j]],X[[k]]]+D[g[[i,k]],X[[j]]]-D[g[[j,k]],X[[i]]],{i,M},{j,M},{k,M}]/2] R[X_,g_]:=g.Module[{c=Ch[X,g]},Table[D[c[[i,j,l]],X[[k]]]-D[c[[i,j,k]],X[[l]]]+(c.c)[[i,k,l,j]]-(c.c)[[i,l,k,j]], {i,M},{j,M},{k,M},{l,M}]]; S[X_,g_]:=Module[{R1,R2,K}, R1=R[X,g]; R2=Table[R1[[i,j,k,l]],{i,M},{j,M},{k,M},{l,M}]; K[i_,j_]:=If[i==j,0,Simplify[R2[[i,j,i,j]]/(g[[i,i]]*g[[j,j]]-g[[i,j]]^2)]]; Table[K[i,j],{i,M},{j,M}]]; U=S[X,g]; k=Sum[ U[[P[[k,1]],P[[k,2]]]]*U[[P[[k,3]],P[[k,4]]]],{k,M!}]; c[d_]:=(d!)^(-1) (4Pi)^(-d); NIntegrate[ k*dV,{t,0,Pi},{s,0,Pi},{u,0,Pi},{v,0,2Pi}]*c[2]
And also, if you are curious, here is the computation for SU(3). First the structure constants are computed in the adjoint representation. I use two computations. One using a formula which appears in a paper of Milnor of 1976. The computation needs some time. This is an 8-dimensional manifold and the formula has to compute 8! = 40320 curvature expression terms. Note that the Gauss-Bonnet-Chern integrand (Euler form) has 40320² = 1625702400 terms which would take much longer to compute.
(* computing the curvatures for the 8-manifold SU(3), Oliver Knill March 21/2020 *) (* Start with Gell-Mann Matrices *) A1={{0,1,0},{1,0,0},{0,0,0}}; A2={{0,-I,0},{I,0,0},{0,0,0}}; A3={{1,0,0},{0,-1,0},{0,0,0}}; A4={{0,0,1},{0,0,0},{1,0,0}}; A5={{0,0,-I},{0,0,0},{I,0,0}}; A6={{0,0,0},{0,0,1},{0,1,0}}; A7={{0,0,0},{0,0,-I},{0,I,0}}; A8={{1,0,0},{0,1,0},{0,0,-2}}/Sqrt[3]; AA={A1,A2,A3,A4,A5,A6,A7,A8}; g[i_,j_]:=Tr[AA[[i]].Transpose[Conjugate[AA[[j]]]]]; Table[g[i,j],{i,8},{j,8}] //MatrixForm M[i_,j_]:=First[({x1,x2,x3,x4,x5,x6,x7,x8} /. Solve[AA[[i]].AA[[j]]-AA[[j]].AA[[i]]==x1*A1+x2*A2+x3*A3+x4*A4+x5*A5+x6*A6+x7*A7+x8*A8,{x1,x2,x3,x4,x5,x6,x7,x8}])/(2I)]; (* SU(3) structure constants, see Milnor 1976 *) EE=Table[Table[M[i,j],{i,8}],{j,8}]; A[i_,j_,k_]:=EE[[k]][[i,j]]; K[i_,j_]:=Sum[(1/2) A[i,j,k](-A[i,j,k]+A[j,k,i]+A[k,i,j])- (1/4) (A[i,j,k]-A[j,k,i]+A[k,i,j])(A[i,j,k]+A[j,k,i]-A[k,i,j])- (A[k,i,i]*A[k,j,j]),{k,8}]; MatrixForm[Table[K[i,j],{i,8},{j,8}]] (* Other computation with 7.3 Milnor *) g[i_,j_]:=-Tr[EE[[i]].EE[[j]]]/3; Table[g[i,j],{i,8},{j,8}]; //MatrixForm R[i_,j_,k_,l_]:=Tr[(EE[[i]].EE[[j]]-EE[[j]].EE[[i]]).(EE[[k]].EE[[l]]-EE[[l]].EE[[k]])]/4; K[i_,j_]:=-Tr[(EE[[i]].EE[[j]]-EE[[j]].EE[[i]]).(EE[[i]].EE[[j]]-EE[[j]].EE[[i]])]/12; MatrixForm[Table[K[i,j],{i,8},{j,8}]] (* Gives the same as above *) P=Permutations[Range[8]]; T=0; PP=Length[P]; Do[If[Mod[m,1000]==0,Print[N[T/m]]]; T=T+K[P[[m,1]],P[[m,2]]]*K[P[[m,3]],P[[m,4]]]*K[P[[m,5]],P[[m,6]]]*K[P[[m,7]],P[[m,8]]], {m,Length[P]}]; c[d_]:=(d!)^(-1) (4Pi)^(-d); T (* Gives 351/64, this s multiplied by c[4] and volume *) (351/64)*c[4]*Pi^5 =117 Pi/2^17

[5/27/2020] The early feedback pointed to the subject of invariant theory in differential geometry. It is a subject which has started in the 1920ies with Herman Weyl who gave the general structure which such invariants must have. These early structure theorems are mentioned in a 100 page monograph of Bott from 1975 who refers to Weyl's book "The Classical groups: their invariants and representations" from 1939 (1946 version). Such invariant theory appeared in Spyros proof of the Deser-Schwimmer conjecture. Weyl gives in his book a bit of history of invariant theory. It started with Cayley who passed from determinants to to more general invariants. He mentions Cayley's 1846 paper "Memoire sur les Hyperdeterminants" as the birth certificate of invariant theory. His work was taken up by Sylvester and others, then also connections from number theory came in, like Gauss work in the arithmetic of binary quadratic forms, the relations with modular functions, automorphic functions, crystallography and representation theory which started with Frobenius around 1900. Of course there are relations and motivations also from physics as one is interested in quantities which are invariant under symmetries. Symmetries are the bread and butter in physics and reducing the possibility of invariants is pivotal also from a computational point of view. At the moment, there is especially a clash with a conjecture of Singer, proven by Gilkey in 1975 that implies that any local invariant given by integrating formulas involving derivatives of the metric must be the Euler characteristic. This obviously disagrees with the Dehn-type invariant in the case M=SU(3). In short: if my paper would be correct, then this disagrees with that conjecture of Singer. The most obvious conclusion is that I'm wrong. My proof is so simple however, that it is hard to imagine for me yet where this happened. I hope to clear this up. On the other hand the subject of invariant theory is huge and might be beyond my skills.
[5/26/2020] Jeffrey Case also informed me about the work (part I) and (part II) of Spyros Alexakis on the decomposition of global conformal integral invariants which extends the Gilkey result to the more narrow class of conformal re-scalings g -> exp(2 f(x)) g of the metrics, where f is a smooth function. Spyros proves a part of the Deser-Schwimmer conjecture (which originated from particle physics): a curvature invariant K of weight (-n) that is invariant under conformal changes and only depends on the tensor elements themselves, then there is a scalar conformal integral invariant W of weight -n that locally only depends on the Weyl tensor, so that K(g) = W(g) + c GBC(g). The work is exhibited by Charles Fefferman here in this video from 2009. The Gilkey result is mentioned in II as Theorem 2. I currently do not see a collision between the theorems of Alexakis but there is still a collision with Gilkeys result in the SU(3) case. [5/27/2019: again since gamma is no invariant, there is no need to look further]
[5/26/2020] Jeffrey Case pointed out that Gilkey proved in 1973 that the only diffeomorphism invariants among compact Riemannian manifolds which are obtained by integrating local formulas in the derivatives of the metric are multiples of the Euler characteristic" and that this had been a conjecture of Singer, stated in the paper as a personal communication of Singer. The Gilkey theorem therefore implies that the Dehn invariant should be the Euler characteristic. My analysis only agrees with this for any manifold we can cut into rectangular pieces and that this is especially true locally and true for orthotope manifolds. My paper claims a counter example to the Singer conjecture stating that any metric independent invariant of the form Y(M) = ∫_M P(x) dV(x), where P depends only locally on the curvature tensor must be the Euler characteristic. There is no contradiction with that statement for most manifolds but SU(3) is a counter example. I claim that there is on M=SU(3) a curvature based invariant which is metric independent but which is not the Euler characteristic. There would be no contradiction of the Gilkey theorem were true for orthotope Riemannian manifolds only. If deformations of the metric make K_GBC-K = div(F) for some vector fields F it could be possible that such vector fields can not all be extended to a global vector field on M. [Update 5/27: this is all mute now. There is no contradiction to the Gilkey theorem because gamma(M) actually is not a metric invariant.]
[5/26/2020] Martin Kerin pointed out that he would have expected SU(3) to be orthotope as it allows for a 2-torus action. Yes, indeed as a sphere bundle over a sphere, one would expect to have an orthotope cutting in each fiber and overlay it with an orthotop cutting of the base. A consequence of my claim is that one can not glue all these orthotope cuttings of the fibres to a global partition. I don't have intuition yet why we can not partition SU(3) into little orthotope pieces. As one can read on this page, there are two non-trivial S³ bundles over S⁵ and SU(3) is the non-trivial one. Andre Henriques and Bruce Wetbury both point out in that mathoverflow entry that S^5 = |z₁|^2+|z₂|^2+|z₃|^2=1 has a natural transitive action by SU(3) and that the stabiliser at any point is SU(2). [Update 6/27: since Gamma is not an invariant of the metric, the claim that there was no orthotope partition of SU(3) is not valid. I don't yet know an explicit partition]