Ordered group invariants for one-dimensional spaces

Abstract

Abstract. We show that the Bruschlinsky group with the winding order is a homeomorphism invariant for a class of one-dimensional inverse limit spaces. In particular we show that if a presentation of an inverse limit space satisfies the Simplicity Condition, then the Bruschlinsky group with the winding order of the inverse limit space is a dimension group and is a quotient of the dimension group with the standard order of the adjacency matrices associated with the presentation. Introduction.Ordered groups have been useful invariants for the classification of many different categories. A class of ordered groups, dimension groups, was used in the study of C * -algebras to classify AF-algebras ([6]), and Giordano, Herman, Putnam and Skau ([8,9]) defined (simple) dimension groups in terms of dynamical concepts to give complete information about the orbit structure of zero-dimensional minimal dynamical systems. Swanson and Volkmer ([15]) showed that the dimension group of a primitive matrix is a complete invariant for weak equivalence, which is called C * -equivalence by Bratteli, Jørgensen, Kim and Roush ([5]). And Barge, Jacklitch and Vago ([3]) showed that, for a certain class of one-dimensional inverse limit spaces, two spaces are homeomorphic if and only if their associated substitutions are weak equivalent, and that if two inverse limit spaces are homeomorphic and the squares of their connection maps are orientation preserving, then the dimension groups of the adjacency matrices associated with the substitutions are order isomorphic.A recent development ([2, 3, 4, 7, 8, 15]) is the refinement ofȞ 1 (X) as a topological invariant for certain one-dimensional spaces X, by making this group an ordered group. HereȞ 1 (X) is the direct limit of first cohomology groups on graphs approximating the space X. There is a natural order on the first cohomology of a graph (a coset is positive if it contains a nonnegative

Ordered group invariants for nonorientable one-dimensional generalized solenoids

“…In recent developments, ordered group invariants in C * -algebras and topological dynamics have been used to provide invariants of homeomorphisms of certain one-dimensional compact metrizable spaces ( [1], [2], [3], [6], [9], [10], [11], [16]). We can interpret these results as follows: the firstČech cohomology group of the space is given a preorder, and the isomorphism class of the resulting preordered group becomes an invariant of the homeomorphism class of the space.…”

“…Every 1-solenoid has an elementary presentation ( [14], [15]). If (X, f ) is an elementary presentation of a 1-solenoid X and M = M X,f is the adjacency matrix of f : X → X, theň H 1 (X) is isomorphic to ∆ M ( [16]). This invariant of the homeomorphism class of X was refined by Jacklitch ( [3]), who showed that if (X, f ) and (Y, g) are elementary presentations of oriented 1-solenoids X and Y , then the dimension groups…”

“…We can define a positive seť H 1 ⊕ for the winding order onȞ 1 for a large class of one-dimensional spaces, and explain the Jacklitch result as the computation of Ȟ 1 ,Ȟ 1 ⊕ for 1-solenoids ( [16]). This can also be done by analyzing the direct limit order onȞ 1 , and we can recast the arguments of [3] and [9] in this framework.…”

“…The assumption of an elementary presentation is necessary and natural. If X is orientable and elementary presented, then ∆ M X,f is order isomorphic toȞ 1 (X); but this may fail if the presentation is not elementary ( [16]). Likewise, if X is not orientable and (X, f ) is not an elementary presentation, then ∆ M X,f need not be isomorphic tǒ…”

Zeta Functions for One-Dimensional Generalized Solenoids

“…Apart from the matter of canonical symbolic dynamics, we mention renewed interest in Williams' systems and related systems on account of connections with ordered group invariants ( [3,12,18]) and substitutions and tilings ( [2,6]).…”