We prove that any discrete surface has an Eulerian edge refinement. For a 2-disk, an Eulerian edge refinement is possible if and only if the boundary length is divisible by 3
Anatole Katok, 1944-2018
We prove that connected combinatorial manifolds of positive dimension define finite simple graphs which are Hamiltonian.
About the origin of the definitino of shellability.
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.
The beautiful Alexander duality theorem for finite abstract simplicial complexes.
We compute the quadratic interaction cohomology in the simplest case.
The interaction cohomology of the dunce hat is computed. We then comment on the discrete Lusternik-Schnirelmann theorem.
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.