Cech Cohomology Group Research Articles

Topology of the Random Fibonacci Tiling Space

We look at the topology of the tiling space of locally random Fibonacci substitution, which is defined as a -> ba with probability p, a -> ab with probability 1-p and b -> a for 0 < p < 1. We show that its Cech cohomology group is not finitely generated, in contrast to the case where random substitutions are applied globally.

Stable Ergodic Properties of Cocycles¶over Hyperbolic Attractors

We consider a hyperbolic flow φt defined on an attracting basic set Λ. A map from the first (Cech) cohomology group of Λ into the dynamic cohomology group is constructed. This map is used to discuss the stable ergodicity and mixing of compact Lie group extensions and velocity changes of φt.

Cohomology and asymptotic stability of 1-dimensional continua

We prove that a 1-dimensional continuum carrying a flow without singular points is homeomorphic to the unit circle if its first Cech cohomology group with integer coefficients is isomorphic to ℤ. As an application of this we obtain that an asymptotically stable invariant 1-dimensional continuum of a flow on a locally compact ANR, which does not contain singular points, must be a periodic orbit.

Systems of differential equations which are competitive or cooperative: III. Competing species

For pt.II, see J. Math. Anal., vol.16, p.423 (1985). Persistent trajectories of the n-dimensional system xi=xiNi(x1, ., xn), xi>or=0, are studied under the assumptions that the system is competitive and dissipative with irreducible community matrices ( delta Ni/ delta xj). The main result is that there is a canonically defined countable (generically finite) family of disjoint invariant open (n-1) cells which attract all non-convergent persistent trajectories. These cells are Lipschitz submanifolds and are transverse to positive rays. In dimension 3 this implies that an omega limit set of a persistent orbit is either an equilibrium, a cycle bounding an invariant disc, or a one-dimensional continuum having a non-trivial first Cech cohomology group and containing an equilibrium. Thus the existence of a persistent trajectory in the three-dimensional case implies the existence of a positive equilibrium. In any dimension it is shown that if the community matrices are strictly negative then there is a closed invariant (n-1) cell which attracts every persistent trajectory. This shows that a seemingly special construction by Smale (1976) of certain competitive systems is in fact close to the general case.

K-Theory and K-Homology Relative to a II ∞ -Factor

Let X be a compact space and M be a factor of type II, acting on a separable Hilbert space. Let KM(X) denote the Grothendieck group generated by the semigroup of isomorphism classes of M-vector bundles over X, and, if X is also metric, let ExtM{(X) denote the group of equivalence classes of extensions of C (X) relative to M. We show that KM(X) is the direct sum of the even-dimensional Cech cohomology groups of X, and that Extm(X) is the direct product of the odd-dimensional Cech homology groups of X. Introduction. Recently Brown, Douglas, and Fillmore [8] have constructed a generalised homology theory called K-homology, which, in a sense made rigorous in [8], is dual to K-theory. Their construction is in terms of extensions of commutative C*-algebras by the ideal of compact operators on a separable Hilbert space. Fillmore [12] and Cho [9] have investigated the analogous construction with the compact operators replaced by the closed two-sided ideal generated by the finite projections in a factor of type II,,. They have constructed (see [9]) a generalised homology theory {Ext/} on the category of compact metric spaces, which we shall call K-homology relative to the II.-factor M. In [61 Breuer has considered a theory of vector bundles relative to M and introduced a functor KM which has topological properties like those of K-theory. We shall constructa generalised cohomology theory { KM}) (K-theory relative to M) from Breuer's functor, identify it in terms of the conventional K-functor and show that KM(X) is the direct sum of the even-dimensional real cohomology of X for any compact space X. Then we shall deduce the corresponding result for ExtM; namely that Extm (X) is the direct product of the odd-dimensional real homology of X. We mention that the results in this note all follow in standard fashion from the recent literature; our goal is merely to point out some interesting consequences of the work of Breuer [6] and Cho [9]. Along the way we provide a proof of Proposition 2, which has been stated and used by Singer in [18]. First we set up some notation. Throughout, all topological spaces will be Hausdorff, and M will be a factor of type II acting on a separable Hilbert space H. We shall denote by Pf(M) the set of finite projections of M and by dim: Pf (M) -* R+ the Murray-von Neumann dimension function of M. For Received by the editors November 21, 1977 and, in revised form, January 25, 1978. AMS (MOS) subject classifications (1970). Primary 55B20; Secondary 46L10, 47B30, 47C15. ?d American Mathematical Society 1978

Topological invariants of the maximal ideal space of a Banach algebra

The theme of this paper is as old as the concept of Banach algebra. Given the functor that assigns to a commutative Banach algebra A its maximal ideal space d, , the urge to interpret topological data concerning AA in terms of the algebraic structure of A is irresistable. The first result in this direction is due to Shilov [29]; it says that each opencompact subset of A, is the support of the Gelfand transform of a unique idempotent in A. A corollary is that A, itself is compact if and only if A has an identity. The Shilov idempotent theorem can be viewed as a characterization of the zero-dimensional Cech cohomology group H”(AA , Z). In fact, it implies immediately that N”(AA , Z) is isomorphic to the additive subgroup of A generated by the idempotents in A. An early result of Brushlinsky [9] points out that if X is compact, then the first Cech group W(X, Z) can be identified with C(X)-l,‘exp(C(X)), where for any commutative Banach algebra A with identity, A-l denotes the group of invertible elements of A and exp(A) denotes the subgroup consisting of elements of the form ea for a E A. Arens [l] and Royden [26] proved that the analogous result holds in general. That is, there is a natural isomorphism

An Algebraic Property of the Cech Cohomology Groups Which Prevents Local Connectivity and Movability

An Algebraic Property of the Cech Cohomology Groups Which Prevents Local Connectivity and Movability

Change of Velocity in Dynamical Systems

In this work we study the properties of topological dynamical systems under a positive continuous change of velocity. In §l we define a flow obtained from a flow by a positive continuous change of velocity. We then prove that the time change flow is reversible so that we can recover the original flow. In §2 we define, following Kirillov, the first cohomology group of a dynamical system. A time change flow is then seen to be related to this first cohomology group. We now prove that there exists a group homomorphism between the first Cech cohomology group with integer coefficients and the first cohomology group of a compact dynamical system with coefficients in the reals. Winding numbers, due to Sol Schwartzman, are introduced and are shown to have an equivalent interpretation in terms of the first cohomology of a compact dynamical system. In §3 we show there is a natural invariant measure of a time change system in terms of the invariant measure of the original compact dynamical system. We now prove that ergodicity and unique ergodicity are preserved under a positive continuous change of velocity. Finally we relate the winding numbers of a time change system to the winding numbers of the original system, and show that under certain conditions they are invariant. In §4, we show that a compact dynamical system admits a Global Cross-Section if and only there exists an Eigen function, with non-zero eigenvalue, of a time change system. Lastly we show that, under certain conditions, a non-zero winding number is an eigenvalue associated to an eigen function of a time change dynamical system. In §5 we show that it is possible to eliminate eigen functions with non-zero eigenvalue under a positive continuous change of velocity of a compact dynamical system, if there exists at least one orbit homeomorphic to the real numbers : if, in addition, the original dynamical is ergodic we prove that weak-mixing is not invariant under a change velocity

On general cohomology theory

Introduction. Due to the somewhat complicated character of the original definition of the Cech cohomology groups (in terms of open coverings and direct limit groups), several alternative approaches have been proposed in the literature for the purpose of achieving better conceptual insight or greater technical convenience, or preferably both. The purpose of this note is to propose still another approach to general cohomology theory. For the sake of economy in presentation, we first state our definition of the cohomology groups HPi(X, A), for a pair of topological spaces such that A CX, and defer comments on motivation and relationship to other approaches to the concluding sections.

Homology Groups of Relations

Any relation between the elements of a set X and the elements of a set Y is associated with two simplicial complexes K and L. A simplex of K is a finite set of elements of X related to a common element of Y; a simplex of L is a finite set of elements of Y related to a common element of X. In particular, the relation of being an element of a set of a covering is a relation between the points of a space and the sets of the covering the space. One of the complexes associated with this relation is the nerve of the covering; its homology and cohomology groups are used in defining the Cech homology and cohomology groups of the space. The other associated complex, which has points of the space as its vertices, is used in defining the Vietoris homology groups and the Alexander cohomology groups. The two complexes associated with a relation will be shown (Theorem 1) to have isomorphic homology and cohomology groups; if the complexes are geometrically realized, they even have the same homotopy type. In particular, the nerve and the Vietoris complex of any covering have isomorphic homology and cohomology groups. It follows that, when the Cech [5] and Vietoris [12] homology groups are based on the same family of coverings, these groups are isomorphic for arbitrary spaces.' It also follows that, when the Alexander [2] and Cech cohomology groups are based on the same family of coverings, they are isomorphic for arbitrary spaces.2 The Alexander cohomology theory based on all open coverings is found to satisfy the seven Eilenberg-Steenrod axioms.3 The proof consists of showing that this cohomology theory is isomorphic with the Cech cohomology theory which is known [6] to satisfy the axioms.

A Theorem on Periodic Transformations of Homology Spheres

In his theory of periodic transformations in a topological space' Smith proved the following theorem: THEOREM A2. Let X be a compact Hausdorff space of finite dimension which is a homology n-sphere over the cyclic group Jp of prime order p and let T be a periodic transformation on X of period p. Then the fixed point set is a homology r-sphere over Jp with -1 < r ? n. If X is an n-sphere and T acts analytically in X, it is well known that T preserves or reverses orientation according as the integer n r is even or odd. The question naturally arises [FP, p. 373]2 whether this remains true without this restriction that T is analytic. A partial solution of this problem has been given by Smith. He showed that,3 in the above theorem if p is an odd prime, and X is a compact metric space which has the same integral (Cech) cohomology groups as an n-sphere S' and the open subsets of which possess certain homology properties of those of S', then n r is even. The main aim of this paper is to establish the following theorem: THEOREM B. In Theorem A, let all the integral cohomology groups of X be finitely generated groups. Then T preserves or reverses orientation according as n r is even or odd. The meaning of orientation-preserving or reversing is defined here in terms of

Continuous Connectivity Groups in Terms of Limit Groups

It is known that the concept of a group spectrum and its limit group yields a particularly elegant method for defining the Cech homology and cohomology groups of arbitrary spaces. The object of this paper is to show that these fundamental group-theoretic notions can also be applied to define the connectivity groups, in the sense of homology theory, based on continuous mappings of n-cells into the space. With this approach, the homology groups appear as direct limits and the cohomology groups are inverse limits, in exact contrast with the Cech theory. This approach avoids the cumbersome use of singular chains and has the additional advantage that in many cases theorems established for polytopes can be immediately extended to arbitrary spaces. It will be remarked that the Vietoris homology theory can also be developed within the framework of the limit group concept, and it turns out that the spectrum which has the Vietoris group as its inverse limit group yields, for its direct limit group, the comology group of a space as originally defined by Alexander in terms of antisymmetric functions of sets of points. It will moreover be shown that the Alexander and Cech cohomology groups are isomorphic for a wide class of spaces.