[Paper Review] Generalized Cantor manifolds and indecomposable continua

Vladimir Todorov, Vesko Valov|arXiv (Cornell University)|Apr 13, 2012

Advanced Topology and Set Theory14 references3 citations

TL;DR

This paper establishes that indecomposable and hereditarily indecomposable continua are generalized Cantor manifolds with respect to the class of all continua, extending classical notions of Cantor manifolds and Alexandroff, Mazurkiewicz, and strong Cantor manifolds. It provides new proofs of Bing’s theorems on separating metric compacta by hereditarily indecomposable continua, using dimension theory and inverse limits of continua.

ABSTRACT

We review results concerning homogeneous compacta and discuss some open questions. It is established that indecomposable continua are Alexandroff (resp., Mazurkiewicz, or strong Cantor) manifolds with respect to the class of all continua. We also provide some new proofs of Bing's theorems about separating metric compacta by hereditarily indecomposable compacta.

Motivation & Objective

• To generalize the concepts of Cantor manifolds, strong Cantor manifolds, and Mazurkiewicz manifolds with respect to arbitrary classes of topological spaces.
• To investigate the topological properties of indecomposable and hereditarily indecomposable continua in the context of generalized dimension theories.
• To provide new, dimension-theoretically grounded proofs of Bing’s theorems on separation by hereditarily indecomposable compacta.
• To establish that every continuum in the class of indecomposable or hereditarily indecomposable compacta is the inverse limit of a system of such compacta.
• To explore the relationship between cohomological dimension, extension dimension, and the structure of homogeneous continua.

Proposed method

• Introduces generalized Cantor manifold concepts via a class of spaces 𝒞, defining Alexandroff, Mazurkiewicz, and strong Cantor 𝒞-manifolds using open covers and maps to spaces in 𝒞.
• Applies the dimension function $ D_{\mathcal{K}} $ associated with a stratum of CW-complexes to unify covering, cohomological, and extension dimensions.
• Uses the monotone-light factorization theorem and Hurewicz’s dimension inequality to bound $ D_{\mathcal{K}}(X) $ in terms of fiber dimensions.
• Employs inverse limits of inverse systems of compacta in the class $ \mathfrak{In} $ (indecomposable or hereditarily indecomposable continua) to construct limit spaces with desired properties.
• Applies Hart-Pol’s factorization theorem for hereditarily indecomposable continua and extends it to general dimension theories using elementary substructures and Löwenheim-Skolem arguments.
• Utilizes the characterization of indecomposable continua via open set separation to construct factorizations preserving indecomposability.

Experimental results

Research questions

• RQ1 Are indecomposable and hereditarily indecomposable continua generalized Cantor manifolds with respect to the class of all continua?
• RQ2 Can Bing’s theorems on separation by hereditarily indecomposable compacta be reproven using dimension-theoretic and inverse limit techniques?
• RQ3 What is the relationship between the extension dimension $ D_{\mathcal{K}} $ and the structure of continua in $ \mathfrak{In} $?
• RQ4 Is every continuum in $ \mathfrak{In} $ representable as the inverse limit of an inverse system of continua from $ \mathfrak{In} $?
• RQ5 How do the generalized notions of Cantor manifolds (Alexandroff, Mazurkiewicz, strong Cantor) relate when restricted to the class of continua?

Key findings

• Indecomposable continua are Alexandroff, Mazurkiewicz, and strong Cantor manifolds with respect to the class of all continua.
• Hereditarily indecomposable continua are also generalized Cantor manifolds in the same sense, extending classical separation theorems.
• Every strongly infinite-dimensional metric compactum contains a strongly infinite-dimensional hereditarily indecomposable continuum.
• There exists a hereditarily indecomposable continuum with $ \dim_{\mathbb{Z}}(X) \in \{2,3\} $, demonstrating that such continua can have finite cohomological dimension.
• Every continuum in $ \mathfrak{In} $ is the inverse limit of an inverse system of compacta from $ \mathfrak{In} $, proving a structural characterization of these spaces.
• For any perfect map $ f: X \to Y $ from a connected metric space $ X $, there exists a closed partition $ H $ between sets $ A $ and $ B $ such that fibers $ f^{-1}(y) \cap H $ are hereditarily indecomposable.