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

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.

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 strong ring is a category of geometric objects G which are disjoint unions of products of

simplicial complexes. Each has a Dirac operator D and a connection operator L. Both are related in

various ways to topology.