[Update, January 5, 2018: In “listening to the cohomology of graphs”, the Green star formula is mentioned.]

It is now almost a year old, the struggle to find a general formula for the Green function values , where is the connection matrix of an abstract finite simplicial complex $G$. Remember that $L(x,y)=1$ of the simplices and $y$ intersect and if $x$ and $y$ do not intersect. In the “Gauss Bonnet connection” we got the diagonal entries. We then found more “Green function values”. And November 2017 there was an other attempt in that quest for a Green function formula. The quest is over. The formula uses the “star” St(x) of a simplex x, which is the set of all which contain . The star makes up the vertices of the unstable sphere in the Barycentric refinement of . Define x+y as the symmetric difference of the sets x and y. This is standard notation as together with the intersection, we have a Boolean ring of sets.

Green Star Formula: . |

In particular, the Euler characteristic of the star of is the Green diagonal entry: .

[**Remark added January 30, 2018:** One could also write , which is a notation we will probably use more. Also, about notation, it make sense to see the star St(x) as the unstable manifold so that one also can look at . The matrix

is nothing else than a **signed version of the connection matrix**. It is orthogonally conjugated to the connection matrix . It is nice that the inverse of is the Green’s function

.

Thinking in terms of dynamical systems, one also would have to look at “heteroclinic points”, intersections of stable and unstable manifolds. But this is not that interesting as is for and equal to if . All of these intersection values of stable and unstable manifolds can be both interpreted as **curvatures** or as **energy values**. It is natural to think of the matrix entries as curvatures as summing up these matrix values always gives a combinatorial invariant, either Euler characteristic or Wu characteristic. [A combinatorial invariant is a quantity which is invariant under Barycentric subdivision (a definition, I found first mentioned in a paper of Bott from 1952. It is an elegant notion as it bypasses topology.)] By the way, also the set of curvature values should now be combinatorial invariants. We had only established that for the diagonal entries of the Green’s function.] As seen here both Euler and Wu characteristic have curvatures also located on the zero-dimensional part of the simplicial complex (just shove things down to zero dimensions!). These are then more familiar curvatures. In the discrete, for graphs, the curvature has already been used by Heesch in the context of graph colorings almost a hundred years ago. The Gauss-Bonnet-Chern formula for graphs (covered here more than 6 years ago) is the Euler chararacteristic case. The general formula (mentioned in the introduction of that article) has already been mentioned earlier by Norman Levitt a reference I had only found later. I call it in the universality paper the “Gauss-Bonnet-Chern-Levitt” formula. But as pointed out in the Wu paper, the formula is very general and leads to Gauss-Bonnet results for any linear or multi-linear valuation. Wu characteristics are examples of multi-linear valuations.

Gauss Bonnet interpretation of Energy:Summing up intersection values of unstable manifolds gives Euler characteristic (Energy theorem) Summing up intersection values of stable manifolds gives the Wu characteristic (this is the definition) Summing up intersection values of stable and unstable manifolds gives the Euler characteristic (this is the definition) |

End of remark added. ]

I call it the “Green star formula” because the Green function entries are related to the stars. The theme picture for this section shows our Lemon tree at home, which carries now 10 small green lemmons which are currently our “stars”. The tree had some trouble during the late fall in the rather erratic Massachusetts weather but now happily spends the winter in our warm living room. We take great care of it and even pollinated the flowers by hand using a toothbrush as there are no bees … so much about lemon tree sex … Note that unless the analogue stable star $St^-(x) = \{ y | y \subset x \}$, the star $St(x) = St^+(x) = \{ y \in G | x \subset y \}$ is not a simplicial complex in general. It is a set of sets, all right, but not every subset is there. This was actually the reason, why we looked far to far away most of the time: we were looking for formulas involving sub-simplicial complexes of $G$. It often almost worked, but not quite and our previous blog entries illustrate this. The formula actually makes one believe in the beauty of mathematics as it is a simple formula. To illustrate this, we can write

$$ L(x,y) = \chi(St^-(x) \cap St^-(y)) $$

showing that the matrix entries of $L$ are related to the stable part of simplices (which are simplicial complexes) while the Green function entries refer to the unstable part of simplices (which are not simplicial complexes in general). The Euler characteristic of a set $A$ of subsets of $G$ of course is still defined as $\chi(A) = \sum_{x \in A} \omega(x)$, where $\omega(x)=(-1)^{{\rm dim}(x)}$ with ${\rm dim}(x)=|x|-1$.

The Green star formula is of course proven with Cramer by deleting column x and row y and computing the determinant. It can be done by induction by removing an other simplex z and show that when adding that simplex both sides get multiplied with the same factor $(1-\chi(S(z)))$ by the multiplicative PoincarĂ©-Hopf result which already proved the unimodularity theorem. One has still to check some initial seeds.

Apropos: Sajal Kumar Mukherjee and Sudip Bera recently found a new elegant proof of the unimodularity theorem by seeing directly what happens if a new vertex is added to the simplicial complex. (My own proof has added single cells). Mukherjee and Bera actually found the proof I had been looking for in vain for 9 months in 2016 (interrupted a bit by some experiments in number theory). My own proof of the unimodularity theorem can be found here.

Let us explain how the above formula ties in with the diagonal entry formula $g(x,x) = 1-\chi(S(x))$. We have seen that in general, the unit sphere S(x) is a Zykov sum of the stable and unstable part $S(x) = S^-(x) + S^+(x)$ which is sort of a hyperbolicity. It is explained in the proof of the energy theorem telling that $\sum_{x} \sum_{y} g(x,y)$ of all potential energies between two simplices is the Euler characteristic of $G$. This hyperbolic splitting produced a formula for the “genus” values of the stable and unstable parts: $g(x,x) = 1-\chi(S(x)) = (1-\chi(S^+(x))) (1-\chi(S^-(x)) = (1-\chi(S^+(x)) \omega(x)$. The Green Star formula tells in the case x=y that $g(x,x)= \chi(St(x))$: which is quite memorable:

The self interaction energy of a simplex is the Euler characteristic of its star.

Generate[A_]:=Delete[Union[Sort[Flatten[Map[Subsets,A],1]]],1] R[n_,m_]:=Module[{A={},X=Range[n],k},Do[k:=1+Random[Integer,n-1]; A=Append[A,Union[RandomChoice[X,k]]],{m}];Generate[A]]; G=R[7,20];n=Length[G];SQ=SubsetQ; OmegaComplex[x_]:=-(-1)^Length[x]; EulerChi[GG_]:=Total[Map[OmegaComplex,GG]]; S[x_]:=Module[{u={}},Do[v=G[[k]];If[SQ[v,x],u=Append[u,v]],{k,n}];u]; SymmetricDifference[a_,b_]:=Union[Complement[a, b],Complement[b, a]]; K=Table[SymmetricDifference[G[[k]],G[[l]]],{k,n},{l,n}]; H=Table[Intersection[S[G[[k]]],S[G[[l]]]],{k,n},{l,n}]; h=Table[(-1)^Length[K[[k,l]]] EulerChi[H[[k,l]]],{k,n},{l,n}]; L=Table[If[DisjointQ[G[[k]],G[[l]]],0,1],{k,n},{l,n}]; h.L==IdentityMatrix[n]