Spectral Sequences on Combinatorial Simplicial Complexes

Vertex Decomposable Simplicial Complexes Associated to Path Graphs

Introduction Vertex decomposability of a simplicial complex is a combinatorial topological concept which is related to the algebraic properties of the Stanley-Reisner ring of the simplicial complex. This notion was first defined by Provan and Billera in 1980 for k-decomposable pure complexes which is known as vertex decomposable when . Later Bjorner and Wachs extended this concept to non-pure ...

On f-Vectors and Relative Homology

We find strong necessary conditions on the f -vectors, Betti sequences, and relative Betti sequence of a pair of simplicial complexes. We also present an example showing that these conditions are not sufficient. If only the difference between two Betti sequences is specified, and not the individual Betti sequences, then the characterization is complete, and the characterization of all pairs of ...

Isoperimetric inequalities in simplicial complexes

In graph theory there are intimate connections between the expansion properties of a graph and the spectrum of its Laplacian. In this paper we define a notion of combinatorial expansion for simplicial complexes of general dimension, and prove that similar connections exist between the combinatorial expansion of a complex, and the spectrum of the high dimensional Laplacian defined by Eckmann. In...

On Isomorphism of Simplicial Complexes and Their Related Algebras

High Dimensional Combinatorial Random Walks and Colorful Expansion

Random walks on expander graphs have been extensively studied. We define a high order combinatorial random walk on high dimensional simplicial complexes. This walk moves at random between neighboring i-dimensional faces (e.g. edges) of the complex, where two i-dimensional faces are considered neighbors if they share a common (i + 1)-dimensional face (e.g. a triangle). We prove that if the links...