Tomasz Kaczynski

Homology computation by reduction of chain complexes

Abstract A new algorithm for computing the homology module of a finitely generated chain complex is given. It is based on local one-step reductions of the size of the initial chain complex and it has a clear geometrical interpretation. The complexity of the algorithm is discussed in special cases.

Coreduction Homology Algorithm for Regular CW-Complexes

In this paper we present a new algorithm for computing the homology of regular CW-complexes. This algorithm is based on the coreduction algorithm due to Mrozek and Batko and consists essentially of a geometric preprocessing algorithm for the standard chain complex generated by a CW-complex. By employing the concept of S-complexes the original chain complex can—in all known practical cases—be reduced to a significantly smaller S-complex with isomorphic homology, which can then be computed using standard methods. Furthermore, we demonstrate that in the context of non-uniform cubical grids this method significantly improves currently available algorithms based on uniform cubical grids.

AN ALGORITHMIC APPROACH TO THE CONSTRUCTION OF HOMOMORPHISMS INDUCED BY MAPS IN HOMOLOGY

This paper is devoted to giving the theoretical background for an algorithm for computing homomorphisms induced by maps in homology. The principal idea is to insert the graph of a given continuous map f into ag raph of a multi-valued representable map F. The multi-valued representable maps have well developed continuity properties and admit a nite coding that per- mits treating them by combinatorial methods. We provide the construction of the homomorphism F induced by F such that F = f. The presented construction does not require subsequent barycentric subdivisions and simpli- cial approximations of f. The main motivation for this paper comes from the project of computing the Conley Index for discrete dynamical systems.

Stable index pairs for discrete dynamical systems

A new shorter proof of the existence of index pairs for discrete dynamical systems is given. Moreover, the index pairs defined in that proof are stable with respect to small perturbations of the generating map. The existence of stable index pairs was previously known in the case of diffeomorphisms and flows generated by smooth vector fields but it was an open question in the general discrete case. Received by the editors May 23, 1995; revised November 28, 1995. The first author was supported by a grant from NSERC of Canada. The second author was partially supported by KBN grant no. 0449/P3/94/06. AMS subject classification: Primary: 54H20; secondary: 54C60, 34C35. c Canadian Mathematical Society 1997. 448

Reducing complexes in multidimensional persistent homology theory

Formans discrete Morse theory appeared to be useful for providing filtration-preserving reductions of complexes in the study of persistent homology. So far, the algorithms computing discrete Morse matchings have only been used for one-dimensional filtrations. This paper is perhaps the first attempt in the direction of extending such algorithms to multidimensional filtrations. An initial framework related to Morse matchings for the multidimensional setting is proposed, and a matching algorithm given by King, Knudson, and Mramor is extended in this direction. The correctness of the algorithm is proved, and its complexity analyzed. The algorithm is used for establishing a reduction of a simplicial complex to a smaller but not necessarily optimal cellular complex. First experiments with filtrations of triangular meshes are presented.

Comparison of persistent homologies for vector functions: From continuous to discrete and back

The theory of multidimensional persistent homology was initially developed in the discrete setting, and involved the study of simplicial complexes filtered through an ordering of the simplices. Later, stability properties of multidimensional persistence have been proved to hold when topological spaces are filtered by continuous functions, i.e. for continuous data. This paper aims to provide a bridge between the continuous setting, where stability properties hold, and the discrete setting, where actual computations are carried out. More precisely, a stability preserving method is developed to compare the rank invariants of vector functions obtained from discrete data. These advances confirm that multidimensional persistent homology is an appropriate tool for shape comparison in computer vision and computer graphics applications. The results are supported by numerical tests.

A local Hopf bifurcation theorem for a certain class of implicit differential equations

The local Hopf Bifurcation theorem is extended to implicit differential equations in R, of the form x — f(x,x, a), which are not solvable for the variable x. The proof uses the S -degree of convex-valued mappings. An example of an implicit differential equation in R to which the presented theorem applies is provided. Introduction. There is a considerable amount of research that has been devoted to the existence of continua of periodic solutions from an equilibrium of a parametrised dynamical system, i.e. the so called Hopf Bifurcation Problem. In his original work, Hopf (cf [7]) proved the result on bifurcating periodic orbits under very strong and restrictive assumptions such as: the analyticity of the function, simple characteristic root transversally crossing the imaginary axis, and no other imaginary characteristic roots. Even though his result allowed to deal with stability properties and other characteristics of bifurcating orbits, there was need for a more general type of bifurcation theorems for a diversity of problems arising from applications. We refer to the book by Marsden and McCracken [12] for a more detailed discussion of the history and background of the problem. Many techniques were developed and used for the study of Hopf bifurcation problems in more general settings. We could mention such methods as Lyapounov-Schmidt, center manifold, Fuller index, regular approximation, decomposition theory, etc. We refer the reader to [2] for details and an extensive bibliography. The concept of S-degree recently developed in [3] and [6] has been used to provide purely topological proofs of local and global Hopf bifurcation theorems for certain classes of differential equations. In particular, [5] proved the existence of a bifurcation of nonconstant periodic solutions from a trivial solution of the functional differential equation

Geometric Construction of a Coboundary of a Cycle

A new method of computing the homomorphism induced by a continuous map in homology presented in [1] and [2] relies on computing coboundaries of cycles. This paper is devoted to a precise geometric construction of a coboundary of a given cycle in a prescribed rectangle and a description of the associated algorithm.

Conley index for set-valued maps: from theory to computation

Recent results on the Conley index theory for discrete multi-valued dynamical systems with their consequences for the computation of the index for representable maps are recapitulated. The terminology is simplified with respect to previous presentations, some superfluous hypotheses are abandoned and some conclusions are proved in a simpler way.

Algorithmic Construction of Acyclic Partial Matchings for Multidimensional Persistence

Given a simplicial complex and a vector-valued function on its vertices, we present an algorithmic construction of an acyclic partial matching on the cells of the complex. This construction is used to build a reduced filtered complex with the same multidimensional persistent homology as of the original one filtered by the sublevel sets of the function. A number of numerical experiments show a substantial rate of reduction in the number of cells achieved by the algorithm.