Simplicial Perturbation Techniques and Effective Homology

2013

We study the effect of edge contractions on simplicial homology because these contractions have turned to be useful in various applications involving topology. It was observed previously that contracting edges that satisfy the so called link condition preserves homeomorphism in low dimensional complexes, and homotopy in general. But, checking the link condition involves computation in all dimensions, and hence can be costly, especially in high dimensional complexes. We define a weaker and more local condition called the p-link condition for each dimension p, and study its effect on edge contractions. We prove the following: (i) For homology groups, edges satisfying the p- and (p-1)-link conditions can be contracted without disturbing the p-dimensional homology group. (ii) For relative homology groups, the (p-1)-, and the (p-2)-link conditions suffice to guarantee that the contraction does not introduce any new class in any of the resulting relative homology groups, though some of the existing classes can be destroyed. Unfortunately, the surjection in relative homolgy groups does not guarantee that no new relative torsion is created. (iii) For torsions, edges satisfying the p-link condition alone can be contracted without creating any new relative torsion and the p-link condition cannot be avoided. The results on relative homology and relative torsion are motivated by recent results on computing optimal homologous chains, which state that such problems can be solved by linear programming if the complex has no relative torsion. Edge contractions that do not introduce new relative torsions, can safely be availed in these contexts.

Homology, Homotopy and Applications, 2003

We propose a method for calculating cohomology operations on finite simplicial complexes. Of course, there exist well-known methods for computing (co)homology groups, for example, the "reduction algorithm" consisting in reducing the matrices corresponding to the differential in each dimension to the Smith normal form, from which one can read off the (co)homology groups of the complex [Mun84], or the "incremental algorithm" for computing Betti numbers [DE93]. Nevertheless, little is known about general methods for computing cohomology operations. For any finite simplicial complex K, we give a procedure including the computation of some primary and secondary cohomology operations. This method is based on the transcription of the reduction algorithm mentioned above, in terms of a special type of algebraic homotopy equivalences, called contractions [McL75], of the (co)chain complex of K to a "minimal" (co)chain complex M (K). More concretely, whenever the ground ring is a field or the (co)homology of K is free, then M (K) is isomorphic to the (co)homology of K. Combining this contraction with the combinatorial formulae for Steenrod reduced pth powers at cochain level developed in [GR99] and [Gon00], these operations at cohomology level can be computed. Finally, a method for calculating Adem secondary cohomology operations Φ q : Ker(Sq 2 H q (K)) → H q+3 (K)/Sq 2 H q (K) is showed.

In this article we will study a generalization of the homotopy theory we know from algebraic topology. We discuss the abstract tools needed for this generalization, namely model categories and their homotopy categories. We will apply our general setting to topological spaces to find the familiar homotopy theory.

The aim of the paper is to study some of the homological groups in general and related these groups with simplicial complexes. Characterization these groups revealed the successful method to study the simplicial complex which has the following two properties: (a) each q-simplex determines (q +1) faces of dimension q-1, (b) the faces of a simplex determine the simplex and a semi-simplicial complex K is a collection of elements {f} called simplexes together with two functions. The main examples of homological groups are r-chain group, r-cycle group and r-boundary group. When we calculating the Euler characteristic of surface, we need to building a multi-surface equivalent to the original surface, therefore in this paper we achieved that the homological groups are a type of improvement for the Euler characteristic. If there is no simplex of order two (2-simplexs) in K, then B1(K) and H1(K) are equal to Z1(K). Also if K is a simplex complex, then r-chain (Cr(K) is a group. We obtained that if three points and three lines such that is triangulation of the rings and there is no simplex of order two (2-simplexs) in K, in this case the boundary homological group equal zero and H1(K)=Z1(K).

unirioja.es

In this paper, an algorithm building the eective homology version of the pushout of the simplicial morphisms f : X → Y and g : X → Z, where X, Y and Z are simplicial sets with eective homology is presented.

arXiv: History and Overview, 2018

Defined by a single axiom, finite abstract simplicial complexes belong to the simplest constructs of mathematics. We look at a a few theorems.

2001

We propose a method for calculating cohomology operations on finite simplicial complexes.

Journal of Algebraic Combinatorics, 2000

We develop an iterated homology theory for simplicial complexes. Thistheory is a variation on one due to Kalai. For ? a simplicial complex of dimension d - 1, and each r = 0, ...,d, we define rth iterated homology groups of ?. When r = 0, this corresponds to ordinary homology. If ? is a cone over ?', then when

Journal of Symbolic Computation, 2005

This paper offers an algorithmic solution to the problem of obtaining "economical" formulae for some maps in Simplicial Topology, having, in principle, a high computational cost in their evaluation. In particular, maps of this kind are used for defining cohomology operations at the cochain level. As an example, we obtain explicit combinatorial descriptions of Steenrod kth powers exclusively in terms of face operators.

In this article we give characteristic properties of the simplicial homology groups of digital images which are based on the simplicial homology groups of topological spaces in algebraic topology. We then investigate Eilenberg-Steenrod axioms for the simplicial homology groups of digital images. We state universal coefficient theorem for digital images. We conclude that the Künneth formula for simplicial homology doesn't be hold in digital images. Moreover we show that the Hurewicz Theorem need not be hold in digital images. Copyright © AJCTA, all rights reserved.