## The Hamiltonian Manifold Theorem

We prove that connected combinatorial manifolds of positive dimension define finite simple graphs which are Hamiltonian.

## Connection Duality

A simplicial complex G defines the connection matrix L which is L(x,y)=1 if and only if x and y intersect. The dual matrix is K(x,y)=1 if and only if x and y do not intersect. It is the adjacency matrix of the dual connection graph.

## Combinatorial Alexander Duality

The beautiful Alexander duality theorem for finite abstract simplicial complexes.

## Interaction cohomology Example

We compute the quadratic interaction cohomology in the simplest case.

## The dunce hat and Lusternik-Schnirelmann

The interaction cohomology of the dunce hat is computed. We then comment on the discrete Lusternik-Schnirelmann theorem.

## The Hydrogen Relation

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.

## Green Star Formula

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.

The classical potential $V(x,y) = 1/|x-y|$ has infinite range which violently clashes with relativity. Solving this problem had required a completely new theory: GR. It remains also a fundamental problem still in general relativity: a Gedanken experiment in which the particles in the sun suddenly transition to particles without mass shows this. [This is forbidden by energy conservation but energy … ….