Tag: <span>simplicial complex</span>

Fusion Inequality for Quadratic Cohomology

While linear cohomology deals with functions on simplices, quadratic cohomology deals with functions on pairs of simplices that intersect. Linear cohomology is to Euler characteristic what quadratic cohomology is to Wu characteristic $w(G) = \sum_{x,y, x \cap y \in G} w(x) w(y)$. If the simplicial complex is split into a …

A multi-particle energy theorem

A finite abstraact simplicial complex or a finite simple graph comes with a natural finite topological space. Some quantities like the Euler characteristic or the higher Wu characteristics are all topological invariants. One can also reformulate the Lefschetz fixed point theorem for continuous maps on finite topological spaces.

Isospectral Simplicial Complexes

One can not hear a complex! After some hope that some kind of algebraic miracle allows to recover the complex from the spectrum (for example by looking for the minimal polynomial which an eigenvalue has and expecting that the factorization reflects some order structure in the abstract simplicial complex), I …

Hearing the shape of a simplicial complex

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?