Euler Characteristic for metric spaces

Euler characteristic super sums energy of space. It subtracts the number of odd dimensional parts from the number of even dimensional parts. It can not be defined for an arbitrary metric space in general. Bouquets of spheres like the Hawaiian rings (am infinite union of circles) or the set $\{0\} \cup \{ 1/n, n \geq 1 \}$ (an infinite union of zero dimensional spheres) show that compactness and finite dimension is not enough. Super summing is magic. Euler characteristic is invariant under Barycentric refinements. It can be computed by computing the dimension of haromnic forms, it been be computed by transporting the energy along a vector field to critical points, where it is stuck locally as an index. By distributing the energy using the Laplacian one gets curvature and can be computed by summing this curvature up.

Both in the case of C^1 manifolds as well as finite simplicial complexes, one can compute Euler characteristic in three different ways: in an algebraic way using kernels of Laplacians, in a combinatorial way by super summing simplicial elements in space, or in an analytic way by adding up indices of a function. For topological manifolds there are limitations in general. The combinatorial path through triangulations is out because not all topological manifolds can be triangulated.There are ways out however as we can build a CW structure or then build a homotopic equivalent thickened manifold with boundary on which a triangulation and so counting is possible. As for the analytic case, the definition of the index is not an issue if one takes the index formula $i_g(x) = 1-\chi(S(x))/2 - \chi(S_g(x))/2$ , where $S(x)$ is a a small enough sphere (topological q-manifolds can by Whitney be embedded in some Euclidean space $\latex R^{2q+1}$ and are especially metrizable) and where $S_g(x)=\{ y \in S(x), g(y)=g(x) \}$ is a level set. We only need to find a nice function $g$ for which $S_g(x)$ is a $q-2$ -manifold for which we just need again that $g$ restricted to $S(x)$ and $S_g(x)$ is nice. In the case of $C^2$ manifolds, this is no problem as almost all linear functions in the ambient space induce Morse functions on the manifold. This works also for piecewise smooth $C^2$ manifolds and so polytopes. It is also no problem for varieties, where classically, we have an issue as the index formula defines Poincare-Hopf indices at singular points. There is also no problem at all for structures that do not have a uniform dimension. We can take a 2-sphere for example and glue to it finitely many one dimensional hairs. All we need is a CW structure which is obtained by gluing recursively k-balls to already existing (k-1)-spheres.

In this presentation I muse about using the analytic approach more seriously and defining a class of metric spaces which have the property that at every point a sufficiently small $S_r(x)$ is in the class and such that the space carries a continuous function g such that for sufficiently small $r$ also the level set $S_{r,g}(x)= \{ y \in S_r(x), g(y)=g(x)\}$ is of this type. Because every topological manifold carries a CW structure, this works for topological manifolds. It also seems to work for metric spaces that are absolute neighborhood retracts because these are the metric space which admit a CW structure and allow to define a nice function g for which all except finitely many indices are non-zero.

I really like the index formula as usually, the index of vector fields refers to a tangent space structure and a notion of topological degree, which we do not have in the discrete. It makes in any situation where we have a notion of unit sphere and a notion of level set. What is nice in the discrete is that both of these notions are very easy there.

There is an other reason, why I started to think a bit more about this topic. I will again teach differential geometry and it would be really nice to teach such a course “radically different” and for example forget about any tensor calculus. Curvature can be defined very nicely as the expectation of Poincare-Hopf indices. The strategy would be to define Euler characteristic inductively with respect to dimension via Poincare-Hopf. Start with zero dimensional spaces for which it is just the cardinality, then one dimensional spaces for which the index formula reduces things to count the number of boundary points: the index at a boundary point is 1/2, the index of any other point is zero. Now we can proceed to two dimensional surfaces, where we just need to count the number of nodes on a small sphere where the function value is the same than the function value at the center. This produces index 1 at maxima and minima and index -1 at saddle points and index -2 at a monkey saddle etc. The curvature expectation on the boundary agrees with the geodesic curvature, tin the interior with Gaussian curvature and we would have a simple proof of the Gauss-Bonnet theorem in two dimensions. But it does not stop here. For an interior point of an odd dimensional manifold for example, the curvature is zero because every index is zero: $i_g(x) = 1-\chi(S(x))/2 - \chi(S_g(x))/2 = -\chi(S_g(x))//2 = 0$ because every $S_g(x)$ is odd an odd dimensional manifold which has zero Euler chaeracteristic. For even dimensional manifolds, we get the Gauss-Bonnet-Chern integrand. When taking the Nash embedding theorem for granted, the proof of Gauss-Bonnet-Chern (using the index expectation curvature) is almost a joke as it just boils down to the Poincar\’e-Hopf theorem. Getting rid of traditional differential geometric frame work would however change the subject too much and hardly be called differentialgeometry any more as we do no more differentiate!

Author: oliverknill