Euler Complexes

Finite Spaces and Applications to the Euler Characteristic

The aim of this paper is to introduce finite spaces and their simplicial complexes. Next, we give an application to combinatorics in the form of a relation between the Euler characteristic and the Möbius function. We begin by giving an overview of finite topological spaces. We introduce beat points and weak homotopy equivalences. Then, we show that finite spaces are weak homotopy equivalent to ...

Determinants of Perfect Complexes and Euler Characteristics in Relative K0-groups

We study the K0 and K1-groups of exact and triangulated categories of perfect complexes, and we apply the results to show how determinant functors on triangulated categories can be used for the construction of Euler characteristics in relative algebraic K0-groups.

Multi-resolution Cell Complexes Based on Homology-Preserving Euler Operators

We have proposed a complete set of basis Euler operators for updating cell complexes in arbitrary dimensions, which can be classified as homology-preserving and homology-modifying. Here, we define the effect of homology-preserving operators on the incidence graph representation of cell complexes. Based on these operators, we build a multiresolution model for cell complexes represented in the fo...

2 00 5 Euler complexes and geometry of modular varieties

Matching and Independence Complexes Related to Small Grids

The topology of the matching complex for the 2×n grid graph is mysterious. We describe a discrete Morse matching for a family of independence complexes Ind(∆n ) that include these matching complexes. Using this matching, we determine the dimensions of the chain spaces for the resulting Morse complexes and derive bounds on the location of non-trivial homology groups for certain Ind(∆n ). Further...

un 2 01 6 MATCHING AND INDEPENDENCE COMPLEXES RELATED TO SMALL GRIDS

The topology of the matching complex for the 2 × n grid graph is mysterious. We describe a discrete Morse matching for a family of independence complexes Ind(∆mn ) that include these matching complexes. Using this matching, we determine the dimensions of the chain spaces for the resulting Morse complexes and derive bounds on the location of non-trivial homology groups for certain Ind(∆mn ). Fur...