These are things i came across when I first started looking at the Euler characteristic, in fact, when I was looking at triangulations in particular.
The Euler characteristic generalizes to dimensions other than 2, and there are at least three noteworthy theorems involving the Euler characteristic. I’m not going to say much about them, because they, like so much else, are still outside my comfort zone. I’ll just barely tell you what they are, and leave you to chase them down if they interest you.
As we’ve seen, the Euler characteristic of a polyhedron is given by
where v, e, f are the numbers of vertices, edges, and faces. Homeomorphic polyhedra have the same Euler characteristic, and that means we can define the Euler characteristic of a topological surface as the Euler characteristic of any polyhedron which is homeomorphic to it.
This alternating-sign sum of the numbers of 0-, 1-, and 2- simplices generalizes in the obvious way: for an n-simplex, we take the sum, with alternating signs, of the numbers of k-simplices, for k <= n. As for surfaces, so for n-manifolds: this is a topological invariant, and we want to define the Euler characteristic of an n-manifold as the Euler characteristic of any k-simplex homeomorphic to it.
But we’ve been told that topological, simplicial, and differential structures on manifolds do not always coincide for n > 3. We can define the Euler characteristic for k-simplices and for manifolds with simplicial structure, but I am going to look skeptically at the Euler characteristic of topological and differentiable manifolds until I learn more about the different structures.
BTW, we can go down a dimension, too: we can define the Euler characteristic of a graph as v – e.
The Euler characteristic of an n-manifold has an extremely interesting equivalence: not only does the formula generalize from 2 to n, but it can also be generalized to a topological space X: we can write
where is the kth Betti number of X, and we assume that eventually (i.e. for k large enough) . (Lee, “Introduction to Topological Manifolds”; or Rotman, “An Introduction to Algebraic Topology”.)
But hold on just a minute. The Euler characteristic was defined for things with simplicial structure, and the Betti numbers are defined (not by me!) for topological spaces. But we know that simplicial & topological do not coincide for higher dimensions than 3, so we probably have to be rather careful here.
When all else fails, look carefully at the book. What we actually have is two definitions, a theorem, and an extension of the definition. We define the Euler characteristic of a simplicial complex, and the Betti numbers of a topological space; the theorem says that for a finite simplicial complex, the Euler characteristic is equal to the alternating sum of the Betti numbers:
Having the theorem, we then extend the definition of the Euler characteristic to any topological space: we define the Euler characteristic as the alternating sum of the Betti numbers; the theorem assures us that the two definitions of the Euler characteristic coincide on simplicial complexes.
We’re still not out of the woods. Looking in Rotman, I am reminded that there are different homology theories. Do they all give the same Betti numbers? I have no idea. (Well, I suppose they do, but I’m ready for anything.) Gee, no wonder algebraic topology is still on my list of things to study.
As an aside, in Massey’s combined book, “A Basic Course in Algebraic Topology”, I find a note which says that until about 1930, mathematicians focused on the Betti numbers (and the torsion coefficients) of homology groups, rather than on the groups themselves.
Index of a surface
Next, on this detour, we have the Poincare-Hopf index theorem, which says that the Euler characteristic of a surface is equal to the index of the surface.
As the Betti numbers came out of left field, the study of vector fields on a surface is coming out of right field. If you’ve ever been told that you can’t comb the hair on a ball, you’ve encountered one of the key theorems of the subject.
There are two ways to look at the index theorem. In the special setting of dynamical systems (i.e. differential equations), we are looking at points where the vector field vanishes (critical points, which are equilibrium points, but not necessarily stable equilibria). More specifically, we are interested in the flow pattern in the vicinity of critical points.
In this context, we break flow patterns into sectors each of which is called parabolic, hyperbolic, or elliptic; and we have a recipe (due to Bendixson) for planar systems which says that the index of a critical point (wrt the vector field) is
I = 1 + (e-h)/2,
where e and h are the numbers of elliptic and hyperbolic sectors. (The parabolic sectors don’t contribute to the index.)
The index of the surface wrt the vector field is the sum of the indices of the critical points. The Poincare-Hopf theorem says that the
index of the surface (in fact, of an n-manifold) wrt the vector field is equal to the Euler characteristic of the manifold.
(I believe that Poincare proved it for surfaces and Hopf extended it to higher dimensions. I am not sure that the definition of sectors applies in higher dimensions; that is, I am not sure how the index is defined in higher dimensions.)
This imposes significant constraints on the possible vector fields.
That’s one way to do it. My references for this approach are Perko, “Differential Equations and Dynamical Systems” and Firby & Gardiner, “Surface Topology”.
I was hoping to at least construct illustrations of the three kinds of sectors, but I’m finding it difficult and very annoying to update my version 5 drawings to version 6 of Mathematica®. (Backwards compatibility? What’s that?)
From Firby & Gardiner, I extract the following description. A sector is parabolic if all paths lead to the critical point, or all paths lead away. A sector is elliptic if all paths begin and end at the critical point. A sector is hyperbolic if all paths sweep past the critical point.
But there is another way to define and compute the index of a critical point. In fact, we can compute the index of any point wrt a curve, but if it’s not a critical point, then its index is zero; therefore we can confine our attention to critical points. This approach starts with winding numbers. Fulton, “Algebraic Topology: A First Course”, is excellent for surfaces (with multiple chapters on winding numbers and vector fields), and even suggests how to generalize to higher dimensions. Also, Sieradski, “An Introduction to Topology and Homotopy”, has a few sections, rather than chapters, on winding numbers and vector fields.
I’m beginning to look forward to tackling Fulton.
The Gauss-Bonnet theorem, as O’Neill states it (“Elementary Differential Geometry”) says that the gaussian curvature, integrated over all of a compact orientable surface, is equal to , where is the Euler characteristic of the surface.
Oh, but doesn’t that require differentiable structure?
Yes and no.
Bloch shows us the Gauss-Bonnet theorem for simplicial surfaces! It turns out we can associate curvature with the vertices of a simplicial surface, in such a way that the total curvature (the sum of the curvatures at each vertex) is equal to .
I’ll talk about this when I discuss chapter 3 of Bloch. I finished the chapter eons ago, but I haven’t tallied it up yet.