Characteristic Topological Invariants

We prove $k$-point Green function formulas

for the $m$’th characteristic of $A\subset G$

where $w(x)=(-1)^{{\rm dim}(x)}$ and where $G$ is a finite abstract simplicial complex. The stars $U(x)=\{y,x\subset y\}$ define a topological base of a finite non-Hausdorff topology on $G$. For two arbitrary open sets $U,V$, the valuation formula

holds for all $m\geq 1$. This makes the higher characteristics $w_{m}$ topological invariants: they agree for homeomorphic complexes. It also makes more general energized versions of $w_{m}$ sheaf ready. For manifolds, $w_{m}(M)=w_{1}(M)$.

We use short hand $w_{m}(A)=\sum_{X\in A^{m},\bigcap X\in A}w(X)$, summing over $X=(x_{1},\dots,x_{m})\in A^{m}$ with $\bigcap X\in A$, meaning $\bigcap_{j=1}^{m}x_{j}\in A$ and $w(X)=\prod_{j=1}^{m}(-1)^{{\rm dim}(x_{j})}$. The Euler characteristic is $w_{1}(A)$ and the Wu characteristic is $w_{2}(A)$. $k$-point Green function maps a k-point configuration $X\in G^{k}$ to $g_{m}(X)=w(X)w_{m}(U(X))$ with $U(X)=\bigcap_{j=1}^{k}U(x_{j}))$. The energy formula $w_{m}(G)=\sum_{X\in G^{k}}g_{m}(X)$ involves open sets in $G$ and so is sheaf theoretic. Unlike in $w_{m}(G)$, there is no $\bigcap X\in G$ assumption in the Green part. The closure $B(x)$ of a star $U(x)$ is called the unit ball. Its boundary $S(x)$ is the unit sphere also known as link. The local valuation formula

holds for all stars $U(x)$, but in general not for arbitrary open sets $U$ with with $B(X)=\overline{U(X)}$ and $S(X)=B(X)\setminus U(X)=\delta U(X)$.

The special one-particle case $k=1$ gives for all $m\geq 1$ a Gauss-Bonnet or Poincaré-Hopf type formula

which reduces the computation of $w_{m}(G)$ to local expressions. We can think of $g_{m}(x)$ as a “curvature” or interpret it as an “index”. Note that the Gauss-Bonnet type formula for open sets does not generalize if $G$ is replaced by an open set $U$. It only works if $G$ is replaced by a smaller simplicial complex. The unit balls $B(x)=\overline{U(x)}$ for example are by themselves again simplicial complexes and we also have $w_{m}(G)=\sum_{x\in G}w(x)w_{m}(B(x))$.

For all complexes $G$, the sphere formula

holds. We have seen that in [7, 10] for $m=1$. It follows from the local valuation formula and the ball formula $w_{m}(G)=\sum_{x\in G}w(x)w_{m}(B(x))$. This is can be useful for large complexes, where we can use the ball formula again for computing $w_{m}(B(x))$, especially if many $B(x)$ have the same type or are known. The sphere formula for example proves that if $G$ is a complex for which the characteristic $w_{m}$ of all spheres $S(x)$ is constant $c\neq 0$ then $w_{m}(G)=0$. This applies for odd-dimensional manifolds or more generally for odd-dimensional Dehn-Sommerville d-manifolds $\mathcal{X}_{d}$ [4] recursively defined by the property $\chi(G)=1+(-1)^{d}$ and that all unit spheres satisfy $S(x)\in\mathcal{X}_{d-1}$. The induction starts with $\mathcal{X}_{-1}=\{0\}$, where the void $0=\{\}$ is the empty complex, which is also the $(-1)$-sphere.

Unlike manifolds, Dehn-Sommerville spaces $\mathcal{X}=\bigcup\mathcal{X}$ define a monoid under the join operation $G+H=G\cup H\cup\{xy,x\in G,y\in H\}$. It contains the sphere monoid $\mathcal{S}$ of all spheres in which the $(-1)$ sphere $0$ is the zero element. The join of a $p$-spheres with a $q$-sphere is a $(p+q+1)$-sphere using induction and $S_{G+H}(x)=S_{G}(x)+H$ for $x\in G$ or $S_{G+H}(x)=G+H(x)$ for $x\in G$. It is natural to look at generalized $d$-manifolds which are complexes for which all unit spheres are in $\mathcal{X}_{d-1}$. This is a larger class than $d$-manifolds but which like manifolds is no sub-monoid of the monoid $(\mathcal{G},+)$ of all complexes.

We have proven the energy formula in the case $m=1,k=2$ already as $\chi(G)=\sum_{x,y}g(x,y)$, where

are the matrix entries of the inverse of the connection matrix $L(x,y)=1$ if $x\cap y\neq\emptyset$ and $L(x,y)=0$ else. In the case $m=1,k=1$, we have seen the super trace expression $\chi(G)=w_{1}(G)=\sum_{x\in G}w(x)\chi(U(x))$ involving $g(x,x)=\chi(U(x))$. We then generalized the energy formula to energized complexes, where $w(x)$ was replaced by an arbitrary ring-valued function $h(x)$. We had explored such energized complexes also in the case when $h$ was division algebra valued.

The energy theorem can more generally be seen as a linear transformation which turns an interaction energy function $h_{m}:G^{m}\to R$ on intersecting points to a Green function $g_{m}:G^{k}\to R$ on k-tuples of open stars. The parameters $m$ and $k$ can be arbitrary positive integers. The energy

then defines $k$-point potential values

or $X\in G^{k}$. We then still have for any $m,k\geq 1$ the relation

Of course, for a general choice of the interaction energy $h$, there is no topological invariance. Indeed, already requiring the weaker combinatorial property of being invariant under Barycentric refinements forces $h$ to be of the form $h(x)=w(x)$.

The more general picture with energized local interaction $h_{m}$ explains, why a positive semi-definite $h$ produces positive semi-definite $g$: if $h_{m}$ is zero except for some fixed $X_{0}$, where $h_{m}(X_{0})=1$, then $g_{m}:X\in G^{k}\to g_{m}(X)$ is positive semi definite form. Convex combinations of positive semi definite forms $h_{m}$ in $m$ variables are mapped into convex combinations of positive semi definite forms $g_{m}$ in $k$ variables. In the case $m=1,k=2$ for example, where the Green function matrix $g(x,y)$ is the inverse of the connection matrix $L(x,y)$, we have a duality between positive definite quadratic forms. In this case, $L$ and $g$ are even isospectral. They gave isospectral positive definite integral quadratic forms. One can write $L(x,y)=w_{m}(V(x)\cap V(y))$ with $V(x)=\overline{\{x\}}$. For $|h(x)|=1,k=2,m=1$ we had them inverses of each other. [5, 8, 6].

As for references, we should add that the subject has a large cultural background starting with the Max Dehn and Poul Heegaard [2] who first defined abstract simplicial complexes in 1907. Together with finite topological spaces as first considered by Pavel Alexandrov [1] in 1937, this produces a powerful framework. We pointed out some reasons in [10, 9] why we do not want to look at geometric realizations. An example is that the geometric realization of the double suspension of a Poincaré homology sphere is homeomorphic to a 5-sphere even so the combinatorial complexes are obviously not homeomorphic in a discrete sense. To remain in the finite topology allows to avoid difficulties from the continuum.

The mathematics of Dehn,Heegard or Alexandrov is not only mathematically simpler as it is only combinatorics of finite sets of sets only, we can also branch off into area, where the continuum is awkward. For example, we can look at the cohomology of open sets in a simplicial complex. This is richer than the cohomology of closed sets. Since the Betti vector of a single set $\{x\}$ (which by definition is an open set $\{x\}$ with locally maximal $x$ in a simplicial complex) is the basis vectors $e_{|x|}$, where $|x|$ is the cardinality of the set, we can realize any Betti vector with open sets. We would not know how to realize a given Betti vector with a simplicial complex. In the continuum, the cohomology of an open set is usually only considered in the sense of a limit of cohomologies of compact subsets. Even the homotopy is different. An open interval $\{\{1,2\}\}$ for example has Euler characteristic $-1$ and is not homotopic to a single point $\{\{1\}\}$ which has Euler characteristic $1$. These two spaces should not be considered homotopic, for any sensible definition of homotopy. But we can identify $A=\{\{1,2\}\}$ for example with $B=\{\{2\},\{1,2\},\{2,3\}\}$ which is also an open interval. The open sets $A,B$ are homeomorphic. Their boundaries $\delta A=\overline{A}\setminus A=\{\{1\},\{2\}\}$ and $\delta B=\overline{A}\setminus B=\{\{1\},\{2\}\}$ are both $0$-spheres. Unlike for general finite set of sets (sometimes called multi-graphs), the cohomology and topology works well for both open and closed sets. Almost everything we describe here fails for general set of sets which are neither closed nor open.

A finite abstract simplicial complex $G$ is a finite set of sets, closed under the operation of taking non-empty subsets. $G$ carries a finite topology $\mathcal{O}$, in which the set of all stars $U(x)=\{y,x\subset y\}$ is the topological base. If $V=\bigcup_{x}x$ is the vertex set, the collection of vertex stars $U(x)$ with $x=\{v\}$ with $v\in V$ are a topological subbase which when closed under intersection produces the topological base and when closed under intersection and union produces the topology. The closed sets in the topology are the sub-simplicial complexes. If $x\subset y$, then $U(y)\subset U(x)$. A map $f:G\to H$ between simplicial complexes is continuous if $f^{-1}(U)$ is open in $G$ if $U$ is open in $H$. A simplicial map is a map from $V(G)$ to $V(H)$ that lifts to an order preserving map $G\to H$. Simplicial maps are continuous; however not all continuous maps are simplicial maps. An example is a constant map onto a simplex $c$ of positive dimension.

The topology on $G$ is Kolmogorov (T0) but neither Fréchet (T1), nor Hausdorff (T2). As any finite topology, it is Alexandroff, meaning that there are smallest neighborhoods $U(x)$ of every $x\in G$. These are the stars $U(x)=\{y\in G,x\subset y\}$. The topology is Zariski type because the closed sets in the topology agreeing with sub-simplicial complexes of $G$. In the case when $G$ is the Whitney complex of a graph $(V,E)$, formed by the vertex sets of complete subgraphs, then subgraphs define closed sets. Every $G$ again defines a graph, where $\Gamma=(V,E)=\{(x,y),x\subset y\;,{\rm or}\;,\;y\subset x\}$. The Whitney complex $G$ of this graph is the Barycentric refinement $G_{1}$ of $G$. The topology has the desired connectivity properties of $G$. The just described graph obtained from $G$ has the same connectivity properties than the topology of $G$. The topology induced from the geodesic distance of the graph would produce the discrete topology on $G$ and render $G$ completely disconnected. The non-Hausdorff property is inevitable. In the case when $G$ comes from a graph $\Gamma$, the graph $\Gamma$ is the Čech nerve of the topological subbase of vertex stars. The Čech nerve of the topological base of all stars is $\Gamma_{1}$, the Barycentric refinement of $\Gamma$.

Elements in $G$ also known as simplices or faces or simply called sets. $G$ is is a set of sets of $V$ and the topology is a set of sets in $\mathcal{O}$. The closure $\overline{A}$ of an arbitrary set $A\subset G$ is the smallest closed set in $G$ containing $A$. We need to distinguish three different things: (i) the set $x\in G$ as an element or point of $G$, (ii) the subset $A(x)=\{x\}\subset G$ and (iii) its closure $K(x)=\overline{\{x\}}=\{y,y\subset x\}\subset G$, which is the simplicial complex generated by $\{x\}$. The set $A(x)$ is open only if $x$ is a locally maximal simplex (meaning not contained in any larger simplex) and closed only if $x$ has dimension $0$. In general, the set $A(x)=\{x\}\subset G$ is neither open nor closed but $x$ always defines two natural sets, the open set $U(x)$ and the closed set $K(x)$. The open set $U(x)$ is the smallest open set containing $x$, the closed set $K(x)$ is the smallest closed set containing $x$. $U(x)$ is the star and $K(x)$ the core. The closure $B(x)$ of $U(x)$ is the unit ball, and its boundary $S(x)=B(x)\setminus U(x)$ of $U(x)$ is the unit sphere. The dimension ${\rm dim}(x)$ of $x\in G$ is defined as $|x|-1$, where $|x|$ is the cardinality of $x$. We write $X=(x_{1},\dots,x_{k})\in G^{k}$ to address a $k$-tuple of points $x_{j}\in G$, and define $w(x)=(-1)^{{\rm dim}(x)}$ and $w(X)=\prod_{j=1}^{k}w(x_{j})$. We write shorthand $\bigcap X$ for $\bigcap_{j=1}^{k}x_{j}$. Similarly, we write $\bigcup X$ for $\bigcup_{j=1}^{k}x_{j}$. We usually do not care about the order of the elements in $X$ but allow that the same element appears multiple times. The configuration $X=(x,x,x,\dots,x)$ for example is a $k$-point configuration in which all points are the same.

We often look at functions $h:G\to R$, where $R$ is some algebraic object like a ring. It should have an additive structure that is commutative (with respect to addition). It could be $\mathbb{Z}$ or a finite Abelian group for example, it can also have more structure like a vector space over a field or an operator algebra. Here we look at rings like $R=\mathbb{Z}$ but the multiplicative structure of the ring does not really enter. It could be a division algebra like the quaternions for example, where the multiplication is not commutative. As we have expressions like $w(x)w_{m}(U(x))$ in our main result, a multiplication with $1$ or $-1$. If $R$ is an additive group, then $-r$ denotes the additive inverse of $r$ in the group. As the theme is part of a finitist approach to mathematics, we prefer finite objects. Even in the case $r=\mathbb{Z}$, the range of a function $f:G\to R$ is always finite so that we still deal with finite objects, despite the fact that there is no a priori cap on the size of $R$. Having a ring rather than only an additive group has the advantage that on can also look at objects like determinants (or in the non-commutative ring case Dieudonné determinants) which can play a role in the vicinity of what we do here. Having a ring is a familiar frame work in other parts of mathematics.

Rather than have a fixed ring $R$ it is possible to attach a ring $R(x)$ to every open set $R(U)$. The ring $R(x)$ is called the ring of sections. To fix the relations, one needs restriction maps. Because the open sets $U(x)$ are minimal, the ring $R(x)$ can be called the stalk at $x$. Its elements are the germs at $x$. Since $x\subset y$ implies $U(y)\subset U(x)$, a sheaf is determined by giving restriction maps $r(x,y):U(x)\to U(y)$ satisfying pre-sheaf properties $r(x,x)=Id$ and $r(y,z)\circ r(x,y)=r(x,z)$ if $x\subset y\subset z$. Having all the germs $R(x)$ and the transition maps fixed, one already has the existence (called gluing) and uniqueness (called locality) which are necessary to have a sheaf and not only a pre-sheaf. An example is $w(x)=(-1)^{{\rm dim}(x)}$ with restriction maps $r(x,y)=w(x)w(y)$ for $x\subset y$. There is a unique extension of $w$ to all open sets $w(U)=\prod_{x\in U}w(x)$. In analogy to $\chi(U)=w_{1}(U)=\sum_{x\in U}w(x)$, we have called this the Fermi characteristic $\phi(U)$ of $U$ [7]. It is equal to ${\rm det}(L)$ with the connection matrix $L(x,y)=\chi(K(x)\cap K(y))$, the inverse of the Green function matrix $g(x,y)=w(x)w(y)\chi(U(x)\cap U(y))$.

For a general function $h:G^{m}\to R$, where we write $h(X)=h(x_{1},\cdots,x_{m})$, we can define $w_{m}(G)=\sum_{X\in G^{m},\bigcap X\in G}h(X)$ and more generally,

for $A\subset G$ for sets $A$ that are not necessarily simplicial complexes. No symmetry like that $h(X)=h(Y)$ if $X$ and $Y$ are permutations is assumed. We mostly take $h(X)=\prod_{j=1}^{m}(-1)^{{\rm dim}(x)}$ because this assures that $w_{m}(G)$ are combinatorial invariants, meaning invariant under Barycentric refinements. For $k$ arbitrary stars $U(x_{1}),\dots,U(x_{m})$, define $U(X)=\bigcap_{j=1}^{k}U(x_{j})$ and $\omega_{m}(U)=\sum_{X\in G^{m},\bigcap X\in G}w(U(X))$. The first characteristic $w_{1}(G)=\sum_{x\in G}w(x)$ is the Euler characteristic of $G$, the second characteristic $w_{2}(G)=\sum_{x,y,x\cap y\in G}w(x)w(y)$ is the Wu characteristic and the third characteristic is

While for any subsets $A,B$, we have for $m=1$ the property $w_{1}(A\cup B)=w_{1}(A)+w_{1}(B)-w_{1}(A\cap B)$, this valuation formula fails for $m>1$ for general $A,B$, even for closed $A,B$ in general. But $w_{m}(G)$ is a multi-linear valuation if extended to $w_{m}(G_{1},\dots,G_{m})$ with $G_{j}\subset G$ arbitrary subsets of $G$. Now $G_{j}\to w_{m}(G_{1},\dots,G_{m})$ with all other sets $G_{i},i\neq j$ fixed, satisfies linearity $\omega(A)+\omega(B)=\omega_{m}(A\cup B)+\omega_{m}(A\cap B)$.

The 1-point complex is defined to be contractible. If $G$ is a complex and both $G\setminus U(x)$ and $S(x)$ are contractible, then $G$ is called contractible. Two complexes $G,H$ which can morphed into each other by homotopy reductions and extensions are called homotopic. This is an equivalence relation on the space of all complexes. Homotopy does not honor dimension and so is not topological. Homotopy preserves $w_{1}(G)$ but not $w_{m}(G)$ with $m>1$. The higher characteristics are topological invariants in the sense that they are preserved under homeomorphisms, an other equivalence relation: $H$ is called a continuous image of $G$ if there exists a Barycentric refinement $G_{m}$ and a continuous map $f:G_{m}\to H$ (of course using the finite topology defined above), such that $f^{-1}(S(x))$ is homeomorphic to $S(x)$ (inductively defined as the maximal dimension of $S(x)$ is smaller than the maximal dimension of $G$) and such that for every locally maximal $x\in G$ of dimension $d$, the complex $f^{-1}(B(x))$ is a $d$-ball. $G$ and $H$ are homeomorphic, if $G$ is a continuous image of $H$ and $H$ is a continuous image of $G$. A $d$-ball is a complex of the form $G-U(x)$, where $G$ is a $d$-sphere. A $d$-sphere is a $d$-manifold $G$ such that $G-U(x)$ is contractible. A $d$-manifold $G$ is a complex such that for all $x$, the unit sphere $S(x)$ is a $(d-1)$-sphere. The empty complex is the $(-1)$-sphere.

We assume $m,k\geq 1$ are integers. Define the $m$’th order potential energy of the $k$-point configuration $X\in G^{k}$ as $g_{m}(X)=w(X)w_{m}(U(X))$. By design, it is zero if $U(X)=\bigcap U(x_{j})=\emptyset$ which is the case if at least one of the points is out of reach of the others. Even for non-intersecting $x,y$ it can happen that $U(x)\cap U(y)$ is non-empty like if $x,y$ are zero-dimensional parts of a higher dimensional simplex $z$, where $z$ is in $U(x)\cap U(y)$. This is a case where $x,y$ can not be separated by open sets. Our goal is to have for any $k\geq 1$ and any $m\geq 1$, the $m$’th characteristic can be expressed using $k$-point Green function entries $g_{m}(x_{1},\dots,x_{k})=\prod_{j=1}^{k}w(x_{j})w_{m}(\bigcap_{j=1}^{k}U(x_{j}))$. We think of this as the $m$-th potential energy of the $k$-point configuration $X=(x_{1},\dots,x_{k})$.

Our main theorem tells that the $m$’th characteristic is the total energy over all possible $k$’point configurations.

A second major point is the sphere formula for $S(X)=\delta U(X)=B(x)\setminus U(x)$.

This means that the total energy of all unit spheres of even configurations is the same than the total energy of all unit spheres of odd configurations.

Let us add already a remark which however leads to an other story to which we hope to be able to write more about in the future: the theorem also works for the dual spheres $\hat{X}=S(x_{1})\cap\cdots\cap S(x_{k})$ which is different from $S(X)=\delta(U(x_{1})\cap\cdots\cap U(x_{k}))$.

The dual spheres $\hat{X}$ play an important role in other places like in graph coloring. In the case when $G$ is a $d$-manifold and if all points in $X$ are adjacent in the metric in which two points $x,y\in S(x)$ have distance $1$, then $\hat{X}$ is always a $(d-k)$-sphere. If $k=1$ then $\hat{X}=S(x)$, which is in the manifold case a $(d-1)$-sphere. For $k=2$, and $x,y\in S(x)$ we have $\hat{(x,y)}=S(x)\cap S(y)$ is a $(d-2)$-sphere. We have looked at this earlier in the context of graphs, where $X=(v_{1},\dots,v_{k})$ are the vertices of a complete graph $K_{k}$ and so is associated to a vertex $x$ in the Barycentric refined graph. If $G$ is a $d$-manifold, then by definition $\hat{X}=\hat{x}$ is a $(d-k)$-sphere. We have seen this as a duality because $\bigcap_{v\in x}S(v)=\hat{x}$ and $\bigcap_{v\in\hat{x}}S(v)=x$. This can be seen as a duality between $(k-1)$-spheres (the boundary sphere of a simplex) and $(d-k)$-spheres. Shifting $k$, there is a duality between $k$-spheres and $(d+1-k)$-spheres.

Still dwelling on that, moving cohomology from simplices to spheres could allow then to see Poincaré duality more elegantly, avoiding concepts like CW complexes which are necessary already in elementary setups like that the cube is the dual of the octahedron. When the cube is seen as a 2-sphere, it by definition is not a simplicial complex but only a more general CW complex if one wants to understand it as dual to the octahedron. So, if one wants to avoid the continuum (as we do), one usually goes to CW complexes. Poincaré himself already struggled quite a bit with the difficulty that the dual of a simplicial complex is not a simplicial complex. We will see the duality in the context of $\delta$-sets which generalize simplicial complexes too but more naturally than CW complexes. $\delta$ sets are more general than simplicial sets, a construct which has a bit more structure than $\delta$ sets. But simplicial sets as a subclass of $\delta$ sets less accessible. Entire articles have been written just to explain the definition.

Here is an other remark, maybe more for the mathematical physics minded: motivated by the Fock picture in which $G^{k}$ are considered as $k$-particle configurations, we could sum up the theorem over $k$ and get for example

If we would replace $w(x)$ with $h(x)=(-1)^{{\rm dim}(x)}/2$ and set $h(X)=\prod_{j}h(x_{j})$ and still use $U(X)=\bigcap_{j}U(x_{j})$, if $X=(x_{1},\dots,x_{k})$ and setting $g_{m}(X)=h(X)w_{m}(U(X))$, this would allow to write $w_{m}(G)=\sum_{X}g_{m}(X)$ where $X$ runs over all particle configurations ranging from single particles $k=1$ to pair interactions $k=2$, triple interactions $k=3$ etc. The total energy of space is then the sum over all interaction energies overall possible particle configurations $X$. If we think of $g_{m}(X)$ as the potential energy of the particle configuration $X$, then $w_{m}(G)$ is the sum over all possible potential energies which particle configurations which can be realized in $G$.

Here are special cases, written out in more detail. First of all, lets write down the definitions of Euler characteristic

and Wu characteristic

i) For $m=1$, we have expressions for the Euler characteristic:
$w_{1}(G)=\sum_{x}w(x)w_{1}(U(x))=\sum_{x}g_{1}(x)$
$w_{1}(G)=\sum_{x,y}w(x)w(y)w_{1}(U(x)\cap U(y))=\sum_{x,y}g_{1}(x,y)$
$w_{1}(G)=\sum_{x,y,z}w(x)w(y)w(z)w_{1}(U(x)\cap U(y)\cap U(z))=\sum_{x,y,z}g_{1}(x,y,z)$
$w_{1}(G)=\sum_{x,y,z,w}g_{1}(x,y,z,w)$
The first two expressions for $k=1$ and $k=2$ have appeared in [7]. The inverse of the matrix $g_{1}(x,y)$ is $L(x,y)=1$ if $x=y$ and $L(x,y)=0$ if $x\neq y$.
ii) Next come expressions for the Wu characteristic:
$w_{2}(G)=\sum_{x}w(x)w_{2}(U(x))=\sum_{x}g_{2}(x)$
$w_{2}(G)=\sum_{x,y\in G}w(x)w(y)w_{2}(U(x)\cap U(y))=\sum_{x,y}g_{2}(x,y)$
$w_{2}(G)=\sum_{x,y,z\in G}w(x)w(y)w(z)w_{2}(U(x)\cap U(y)\cap U(z))=\sum_{x,y}g_{2}(x,y,z)$
$w_{2}(G)=\sum_{x,y,z,w}g_{2}(x,y,z,w)$
iii) And here are expressions for the third characteristic
$w_{3}(G)=\sum_{x}w(x)w_{3}(U(x))=\sum_{x}g_{3}(x)$
$w_{3}(G)=\sum_{x,y}w(x)w(y)w_{3}(U(x)\cap U(y))=\sum_{x,y}g_{3}(x,y)$
$w_{3}(G)=\sum_{x,y,z}w(x)w(y)w(z)w_{3}(U(x)\cap U(y))\cap U(z))=\sum_{x,y,z}g_{3}(x,y,z)$
$w_{3}(G)=\sum_{x,y,z,w}g_{3}(x,y,z,w)$

Remarks: 1) The theorem works if $h(X)=w(X)$ is replaced by any $R$-valued interaction function $h:G^{k}\to R$. We would for example set $w_{2}(A)=\sum_{x,y,x\cap y\in A}h(x,y)$ and get this equal to $\sum_{x,y\in G^{2}}g_{2}(x,y)$ with

or equal to $\sum_{x\in G}g_{1}(x)$ with $g_{1}(x)=w(x)w_{2}(U(x))$.
2) The energy theorems in the case of $k=1$ are of Gauss-Bonnet type because we attach a fixed curvature $g_{m}(x)$ to a point. We can actually interpret this also as a Poincaré-Hopf theorem. There is some duality here as $U(x)$ can be seen dual to $K(x)$.
3) The energy formula reduces the time for the computation of the characteristic substantially. Especially for $k=1$, where we have only to compute the $m$’th characteristic of the $n=|G|$ stars $U(x)$.
4) Instead of self-interactions of all $G^{j}$, we could take open sets $G_{j}\subset G$ and get expressions

For example, $w_{m}(A,B,C)=\sum_{x\in A,y\in B,z\in C}g_{m}(x,y,z)$, where

Think of this as the total m’th characteristic energy of the three sets $A,B,C$.
5) We have seen that in order to prove the theorems, it is helpful to energize more generally $h(x_{1},\dots,x_{m})$ and set $w_{m}(G)=\sum_{X\in G^{m},X>0}h(X)$. The Green function procedure which maps functions $h:G^{k}\to R$ to functions $g:G^{m}\to R$ with $g_{m}(X)=w(X)w_{m}(U(X))$ is linear. It is therefore enough to verify the claim for basis elements, where h(x) is zero except at $h(x_{1},\dots,x_{k})=1$.