Tiling spaces and inverse limits

An inverse limit of finite groups has been called in the literature a pro-finite group and we have extensive studies of profinite groups from the cohomological point of view by J. P. Serre. The general theory of non-abelian modules has not yet been developed and therefore we consider a generalization of profinite abelian groups. We study inverse systems of discrete finite length R-modules. Profinite modules are inverse limits of discrete finite length R-modules with the inverse limit topology.

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Approximate inverse limits and (m,n)-dimensions

Glasnik Matematicki ◽

10.3336/gm.56.1.11 ◽

2021 ◽

Vol 56 (1) ◽

pp. 175-194

Author(s):

James F. Peters ◽

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Keyword(s):

Fixed Point ◽

Abelian Group ◽

Free Abelian Group ◽

Betti Numbers ◽

Boundary Region ◽

Inverse Limits ◽

Jordan Curve Theorem ◽

Group Representations ◽

Approximate Inverse ◽

Fixed Cell

This paper introduces shape boundary regions in descriptive proximity forms of CW (Closure-finite Weak) spaces as a source of amiable fixed subsets as well as almost amiable fixed subsets of descriptive proximally continuous (dpc) maps. A dpc map is an extension of an Efremovič-Smirnov proximally continuous (pc) map introduced during the early-1950s by V.A. Efremovič and Yu.M. Smirnov. Amiable fixed sets and the Betti numbers of their free Abelian group representations are derived from dpc's relative to the description of the boundary region of the sets. Almost amiable fixed sets are derived from dpc's by relaxing the matching description requirement for the descriptive closeness of the sets. This relaxed form of amiable fixed sets works well for applications in which closeness of fixed sets is approximate rather than exact. A number of examples of amiable fixed sets are given in terms of wide ribbons. A bi-product of this work is a variation of the Jordan curve theorem and a fixed cell complex theorem, which is an extension of the Brouwer fixed point theorem.

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