## Combinatorial Alexander Duality

The beautiful Alexander duality theorem for finite abstract simplicial complexes.

The beautiful Alexander duality theorem for finite abstract simplicial complexes.

We compute the quadratic interaction cohomology in the simplest case.

This is an other blog entry about interaction cohomology [PDF], (now on the ArXiv), a draft which just got finished over spring break. The paper had been started more than 2 years ago and got delayed when the unimodularity of the connection Laplacian took over. There was an announcement [PDF] which is now included as an appendix. [Not to appear … ….

For a one-dimensional simplicial complex, the sign less Hodge operator can be written as L-g, where g is the inverse of L. This leads to a Laplace equation shows solutions are given by a two-sided random walk.

Here is the code to compute a basis of the cohomology groups of an arbitrary simplicial complex. It takes 6 lines in mathematica without any outside libraries. The input is a simplicial complex, the out put is the basis for $H^0,H^1,H^2 etc$. The length of the code compares in complexity with computations in basic planimetric computations in a triangle (Example … ….

We found a formula of the green function entries g(x,y). Where g is the inverse of the connection matrix of a finite abstract simplicial complex. The formula involves the Euler characteristic of the intersection of the stars of the simplices x and y, hence the name.

When replacing the circle group with the dyadic group of integers, the Riemann zeta function becomes an explicit entire function for which all roots are on the imaginary axes. This is the Dyadic Riemann Hypothesis.

A simplicial complex G, a finite set of non-empty sets closed under the operation of taking finite non-empty subsets, has the Laplacian $L(x,y) = {\rm sign}(|x \cap y|)$. It is natural as it is always unimodular so that its inverse $g(x,y)$ is always integer valued. In a potential theoretical setup, the Green function values $g(x,y)$ measure a potential energy between … ….

A finite abstract simplicial complex has a natural connection Laplacian which is unimodular. The energy of the complex is the sum of the Green function entries. We see that the energy is also the number of positive eigenvalues minus the number of negative eigenvalues. One can therefore hear the Euler characteristic. Does the spectrum determine the complex?

The area of discrete geometry is a maze. There are various flavors.