Let be a finite simple directed graph embedded in a two-dimensional compact manifold M such that the faces, the connected components of the complement consists of simply connected polygons. Let denote the face set. For every let denote the number of direction changes of adjacent edges to the vertex x and for every $lteex x \in F$ let denote the number of direction changes of the edges surrounding the face. Define now the index on $V \cup F$. The Euler characteristic of is defined as the Euler characteristic of which is $\chi(\Gamma) = |V|-|E|+|F|$. For oriented surfaces , this is by Euler-Poincare equal to , where is the genus of the surface. In general, it is where are the Betti numbers (first described by Johann Listing who also coined the term topology).
Glass’s Theorem (1973):
Proof: Just note that for every neighboring edge pair at the boundary of a face f, we have either a sign change at the vertex or then a sign change at the face but not both. This gives which resembles the Euler handshake formula. Now now that .
It was a thesis of Kate Perkins which brought the attention to the theorem of Leon Glass. Let me explain this from a version from a statement I made for directed graphs.
If $\Gamma$ be a finite simple directed graph which is locally irrotational, meaning that there are no closed loops in any complete subgraph implying that the digraph structure defines a complete order on the vertex set of each complete subgraph. We can therefore assign to every simplex (=complete subgraph=face=face) the vertex , that is the minimum of this order. The Euler characteristic of the graph is the Euler characteristic of the Whitney complex, which is the number of even-dimensional simplices minus the number of odd-dimensional simplices. This is also , where . If these “energies” are shifted from the simplex to its vertex one gets where is the subgraph of the unit sphere (nodes directly attached to v), generated by all vertices which point towards . One can see this also as , the Euler characteristic of the set of all simplices for which . So:
Discrete Poincare-Hopf for directed graphs (2019): if is a locally irrotational, then .
This result implies the Glass theorem (Kate Perkins has reduced the Glass theorem in a different way): Step (i): triangulate each of the non-triangular faces first. This means that we introduce a new vertex v in each non-triangular face and connect it to the vertices bounding the face. As for the direction: at each vertex w which is a sink on the boundary take the direction from w to v. Otherwise let the direction point outwards. This choice assures that . and is the Glass index. Step (i) does not change the Euler characteristic of the cell complex and also does not affect the indices already assigned for the original vertices. Step (ii): the graph we have now might have have some triangles where the field loops in a circular way. Stellate (pyramid extend along a circle) also such a triangle by placing a new vertex inside it, then take all directions point outwards. Now we have an irrotational directed graph and the same Euler characteristic. Also here, the R(w) of the original vertices does not change. Glass theorem follows from the discrete Poincare-Hopf for directed graphs which are discrete manifolds.
The Glass theorem is formulated in a frame work of topological graph theory, which is a branch of graph theory where graphs are geometrically realized part of a continuum space M. It is possible to remove the assumption of an ambient manifold M by using the concept of a discrete CW complex , where is the face set. These are the 0-dimensional, 1-dimensional and 2-dimensional cells in this construct. Now, without mentioning any ambient manifold and within combinatorics, one just has three set of finite sets and face maps from to E$ and face maps from to V$. The assumption is that the Barycentric refinement of this cell complex is a (discrete) 2-manifold, a graph which has the property that every unit sphere is a 1-manifold, a circular graph. The digraph structure as well as an orientation on , now gives us a rule which assigns to each edge a vertex and to each face a vertex. Moving the energies $\omega(x)$ from the cells to the vertex along now defines an index and obviously, the total energy (the Euler characteristic) does not change. Every discrete CW complex is also and so part of a very important geometric category of objects. It is category which is a finite geometric topos. Topoi are the nicest objects in mathematics because one can do with them anything we want like taking products, like taking quotients.
Lets get a bit crazy: Glass’s theorem shows that the digraph structure in 2-dimensions itself determines the Delta set structure from the digraph alone. What about higher dimensions (this in other words was asked by Kate Perkins)? One of the lessons which history has shown us is that imprecise definitions and notions can produce headaches for centuries. One of these problematic notions was the notion of polyhedron. Euler implicitly assumed some convexity condition when talking about polyhedra and so assumed that polyhedra are cell complexes of 2-spheres (and not for example Kepler solids where the Euler formula fails). Much of the rigorous mathematical literature about polyhedra deals with convex polyhedra. The topologists polyhedra as geometric realizations of a cell complex and especially for simplicial complexes. A very general definition is to see a polyhedron as a geometric realization of a topos element, an element in the category of Delta sets. This is a strong enough category for many things because it has a cohomology and a simple structure of the form , where are the sets of dimension . Unlike for simplicial complexes, where the cardinality of defines its dimenion, the dimension assumption is in the case of Delta sets put more generally. The Euler characteristic of a Delta set is and could by Euler-Poincare also be expressed cohomology as , where are the Betti numbers (which are defined for a Delta set because we have a notion of exterior derivative and so a differntial complex. As usual, the heat flow proof proves the Euler-Poincare theorem also in this generality). Let be a rule which assigns to every an element such that is a lower dimensional part of . the face maps define incidence in a similar way as subset defines incidence in the simplicial complex case). Let the unit sphere of defined as , where is the star of , a subset of that is open. Let be the subset of for which . Define . More generally, let be a function on (like ), then . Poincare-Hopf looks the same also in this general frame work of energized Delta sets.
Poincare-Hopf for Delta sets: For any energized Delta set .
This is not yet the most general version of the phenomenon of distributing stuff from higher dimensional to lower dimensional parts. If we take a probability space of rules , we can take the expectation of the indices and call it curvature. The Poincare-Hopf theorem becomes now a Gauss-Bonnet theorem. A very special case is if one takes a uniform measure on all possible rules. In that case one has a natural curvature which in the continuum goes to the Gauss curvature in two dimensions (defined by Gauss) and in higher dimensions to the Gauss-Bonnet-Chern curvature (if we deal with a discrete even dimensional manifold). In a combinatorial setting, discrete curvature was first considered by Victor Eberhard in 1891 and is the graph case (using modern notation) ,where are the number of dimensional simplices that contain v. In the case of a 2-dimensional manifold (a triangulation of a 2-dimensional smooth manifold), one has where is the vertex degree of v. This curvature was heavily used in the graph theory literature, when studying the 4 color theorem. In an icosahedron for example it is constant 1/6 and with 12 vertices the total curvuature is 12. In the octahedron case, it is 1/3 and the 6 vertices again produce total curvature 2 as it should be. If we look at the cube, then its Whitney complex is 1 dimensional (there are no triangles). The curvature is 1-deg/2=-1/2 at each vertex and the total curvature is -4. This reflects the fact that we have not counted the 6 faces. If one wants to look at the cube as a discrete representation of a sphere, one has either to stellate the 6 missing faces and get |V|=8+6=14, |E|=12+6*4=36, |F|=6*4=24 and so |V|-|E|+|F|=2 or then see the cube as a discrete CW complex (a special Delta set) and directly get the Euler characteristic 6-12+8=2 as Maurolico or Descartes have seen.
Gauss-Bonnet theorem for Delta sets: .
And if that is not yet general enough, consider this a static version of a dynamical setup, where we have an endomorphism on the topos. (the version vbefore applies then when T is the identity map). Now, the Euler characteristic (the case when the morphism is the identity map) becomes the Lefshetz number . The analog of the Euler-Poincar\’e theorem becomes the Brouwer-Lefschetz theorem which can be proven in the same way than the Euler-Poincare theorem using the heat flow using the McKean-Singer super symmetry, a result I once dubbed to be the correct discrete replacement for what one calls elliptic regularity in the continuum. A very special case is the Brouwer fixed point theorem which is the situation when the cohomology of the Delta set is trivial, meaning that the Lefschetz number is one, forcing to have at least one fixed point.
Brouwer-Lefschetz theorem for Delta sets: where is the index of a fixed point.
Lets get even more crazy. If one combines the Poincare-Hopf idea (relating algebra with topology using the heat flow) with the Poincare-Hopf resp Gauss-Bonnet idea one gets the most general version which still works perfectly well in the case of Delta sets. I once dared to call it a discrete Atiyah-Singer result because if you look closely this is what it is if one also frees up how to define the differential complex. One can on a Delta set also define other differential complex structures. An example where this more general set-up appears is when we look at Wu characteristic and Wu cohomology (which is an example of a higher order differential complex). The super trace induced of the endomorphism on the cohomology is then a generalized Lefschetz number and just the trace induced on the even dimensional part minus the trace on the odd dimensional part. On the other side, the sum of the indices of fixed parts of can be pushed (by a rule or then by averaging in a natural way) to the zero-dimensional part. The sum of all these curvatures is then what one could call an analytic index. So, for any endomorphism T of a Delta set one has a general rule which relates an algebraic cohomological part (expressed as traces of matrices) with an analytic index part (expressed as a sum of curvatures).
Atihay-Singer for Delta sets: .
There are just two simple ideas involved: A) on the analytic side, there is the Poincare-Hopf deformation idea to push basic energy numbers like of constituents of space modeled by the Delta set to the lowest dimensional part (this push can happen through gradient fields, through vector fields, through giving an order on simplices). B) on the algebraic side, there is the heat flow deformation idea to use the heat flow to push the energy to harmonic forms (cohomology). To finish this blasphemy of very tough mathematics with a bit more polemic, one should add that these two ideas A,B are probably simpler than some of the ideas involved to prove basic geometric results in Euclid’s elements. Enough insults. Lets finish up this post.