Random Čech complexes on Riemannian manifolds

In this paper we study the homology of a random ?ech complex generated by a homogeneous Poisson process in a compact Riemannian manifold M. In particular, we focus on the phase transition for “homological connectivity” where the homology of the complex becomes isomorphic to that of M. The results presented in this paper are an important generalization of 7 , from the flat torus to general compact Riemannian manifolds. In addition to proving the statements related to homological connectivity, the methods we develop in this paper can be used as a framework for translating results for random geometric graphs and complexes from the Euclidean setting into the more general Riemannian one.     

On the homology of the Harmonic Archipelago

We calculate the singular homology and Čech cohomology groups of the Harmonic Archipelago. As a corollary, we prove that this space is not homotopy equivalent to the Griffiths space. This is interesting in view of Eda’s proof that the first singular homology groups of these spaces are isomorphic.     

Infinite-dimensional homology and multibump solutions

We start by introducing a Čech homology with compact supports which we then use in order to construct an infinite-dimensional homology theory. Next we show that under appropriate conditions on the nonlinearity there exists a ground state solution for a semilinear Schr?dinger equation with strongly indefinite linear part. To this solution there corresponds a nontrivial critical group, defined in terms of the infinite-dimensional homology mentioned above. Finally, we employ this fact in order to construct solutions of multibump type. Although our main purpose is to survey certain homological methods in critical point theory, we also include some new results. Dedicated to Felix Browder on the occasion of his 80th birthday     

Persistence stability for geometric complexes

In this paper we study the properties of the homology of different geometric filtered complexes (such as Vietoris–Rips, ?ech and witness complexes) built on top of totally bounded metric spaces. Using recent developments in the theory of topological persistence, we provide simple and natural proofs of the stability of the persistent homology of such complexes with respect to the Gromov–Hausdorff distance. We also exhibit a few noteworthy properties of the homology of the Rips and ?ech complexes built on top of compact spaces.     

N -Fold ?ech Derived Functors and Generalised Hopf Type Formulas

The notion of n-fold Čech derived functors is introduced and studied. This is illustrated using the n-fold Čech derived functors of the nilization functors Z_k. This gives a new purely algebraic method for the investigation of the Brown–Ellis generalised Hopf formula for the higher integral group homology and for its further generalisation. The paper ends with an application to algebraic K-theory. (Received: December 2004)     

Local n-connectivity of decomposition spaces

A corollary of the main result of this paper is the following Theorem. Suppose f: X → Y is a closed surjection of metrizable spaces whose point inverses are LC^{n + 1}-divisors (n ? 1). If Y is complete and f is homology n-stable, then Y is LC^{n + 1}provided X is LC^{n + 1}.Intuitively, f is homology n-stable if the ?ech homology groups of its point inverses are locally constant up to dimension n. In addition, we obtain sufficient conditions for the Freudenthal compactification to be LCⁿ.     

On the homology of compact spaces by using non-standard methods

It is proved that strong and ?ech homology theories coincide for compact pairs and coefficients in an equationally compact group or in the non-standard group for any abelian group G. It is also proved that McCord homology has compact supports to a reasonable degree. Furthermore, the equality of the homology classes of point embeddings into the 2-adic solenoid is studied using non-standard methods.     

Cosimplicial DGLAs in Deformation Theory

We identify ?ech cocycles in nonabelian (formal) group cohomology with Maurer–Cartan elements in a suitable L _∞-algebra. Applications to deformation theory are described.     

On a problem of Čech

Assuming CH (but we have stronger results) we partially solve a problem posed by E. ?ech. Kuratowski gave the axioms which a topological closure operator, ?, must satisfy. If we do not ask that ? be idempotent (i.e. that for all $\text{X,(?(X))=?(X)}$, then ? is known as a closure operator. E. ?ech asked if there is a nontrivial closure operator which is onto, that is, for which in some sense every subset of our ‘space’ is ‘closed’. We build such functions. The problem is also well motivated when presented in purely set theoretic terms.     

Random Geometric Complexes

We study the expected topological properties of Čech and Vietoris–Rips complexes built on random points in ℝ ^d . We find higher-dimensional analogues of known results for connectivity and component counts for random geometric graphs. However, higher homology H _k is not monotone when k>0.     

Čech Type Approach to Computing Homology of Maps

A new approach to algorithmic computation of the homology of spaces and maps is presented. The key point of the approach is a change in the representation of sets. The proposed representation is based on a combinatorial variant of the Čech homology and the Nerve Theorem. In many situations, this change of the representation of the input may help in bypassing the problems with the complexity of the standard homology algorithms by reducing the size of necessary input. We show that the approach is particularly advantageous in the case of homology map algorithms.     

Phase transitions of the Moran process and algorithmic consequences

The Moran process is a random process that models the spread of genetic mutations through graphs. On connected graphs, the process eventually reaches “fixation,” where all vertices are mutants, or “extinction,” where none are. Our main result is an almost‐tight upper bound on expected absorption time. For all ?>0, we show that the expected absorption time on an n‐vertex graph is o(n^3+?). Specifically, it is at most , and there is a family of graphs where it is Ω(n³). In proving this, we establish a phase transition in the probability of fixation, depending on the mutants' fitness r. We show that no similar phase transition occurs for digraphs, where it is already known that the expected absorption time can be exponential. Finally, we give an improved fully polynomial randomized approximation scheme (FPRAS) for approximating the probability of fixation. On degree‐bounded graphs where some basic properties are given, its running time is independent of the number of vertices.     

Generalization of the concept of continuous function and homeomorphism

Summary We give here a generalization of the concept of continuous function and homeomorphism which seems to be useful in some questions of topology. For instance, applications are given here to separation theorems and to Čech homology theory. This research was supported by the Air Force Office of Scientific Research.     

Integer Cech cohomology of a class of n‐dimensional substitutions

A class of non‐periodic tilings in n‐dimensions is considered. They are based on one‐dimensional substitution tilings that force the border, a property preserved in the construction for higher dimensions. This fact allows to compute the integer?ech cohomology of the tiling spaces in an efficient way. Several examples are analyzed, some of them with PV numbers as inflation factors, and they have finitely or infinitely generated torsion‐free cohomologies. Copyright © 2010 John Wiley & Sons, Ltd.     

Fibrations with Hopfian properties

This study explores the homotopy-theoretic meeting-point of topics in differential topology, combinatorial group theory and algebraicK-theory. The first two are due to H. Hopf and date from around 1930. The third arose in the author’s characterisation of plus-constructive fibrations. LetF ( _→ ^ί )E →B be a fibration such thati induces an isomorphism of homology with trivial integer coefficients; what is the effect ofi on fundamental groups? In particular, when one passes to hypoabelianisations by factoring out perfect radicals, doesi induce an epimorphism? Numerous conditions are determined which force an affirmative answer. On the other hand, negative examples of a non-finitary nature are also provided. This leaves the question open in the finitely generated case, where it forms a homological version of the dual to Hopf’s original, famous question in group theory.     

Growth of maps, distortion in groups and symplectic geometry

In the present paper we study two sequences of real numbers associated to a symplectic diffeomorphism:?• The uniform norm of the differential of its n-th iteration;?• The word length of its n-th iteration, where we assume that our diffeomorphism lies in a finitely generated group of symplectic diffeomorphisms.?We find lower bounds for the growth rates of these sequences in a number of situations. These bounds depend on the symplectic geometry of the manifold rather than on the specific choice of a diffeomorphism. They are obtained by using recent results of Schwarz on Floer homology. As an application, we prove non-existence of certain non-linear symplectic representations for finitely generated groups. Oblatum 6-XII-2001 & 19-VI-2002?Published online: 5 September 2002 RID="*" ID="*"Supported by the Israel Science Foundation founded by the Israel Academy of Sciences and Humanities.     

Expected invariants of simplicial complexes obtained from random point samples

In this paper, we study the expectation values of topological invariants of the Vietoris–Rips complex and ?ech complex for a finite set of sample points on a Riemannian manifold. We show that the Betti number and Euler characteristic of the complexes are Lipschitz functions of the scale parameter and that there is an interval such that the Betti curve converges to the Betti number of the underlying manifold.

     

UNIFORM VIETORIS-BEGLE THEOREMS

Abstract

In this paper we generalize the well-known Vietoris-Begle theorem to uniform spaces. We formulate two uniform versions: one for the ?ech cohomology based on all finite uniform coverings and one for the ?ech cohomology based on all uniform coverings.     

Phase transition for dielectric breakdown model

In this paper, the phase transition for DBM when ν varies is investigated by using real-space renormalization-group method. The result demonstrates that there are phase transitions for almost all the value of ν, and we find a new result that the larger ν is, the larger the value of phase transition point q_c is.     

Supercomplete topological spaces

Supercomplete topological spaces and other variants of supercompleteness are defined in this paper. The main idea is to give different characterizations of Čech complete spaces. In particular it is shown that for paracompacta (and so for metrizable spaces) and topological groups the two notions of supercompleteness and Čech completeness coincide.