[Paper Review] Generalized Cantor manifolds and homogeneity

Alexandre Karassev, Paweł Krupski+2 more|ArXiv.org|Jul 23, 2008

Advanced Topology and Set Theory14 references3 citations

TL;DR

This paper generalizes classical notions of Cantor manifolds, $V^n$-continua, and Mazurkiewicz manifolds to infinite-dimensional and extension-theoretic settings using a dimension function $D_{\mathcal{K}}$ derived from a stratum $\mathcal{K}$ of CW-complexes. It proves that every homogeneous metrizable continuum not in certain dimension-based classes is a strong Cantor manifold with respect to $\mathcal{D}_{\mathcal{K}}^{n-2}$, extending Alexandroff's classical theorem to broader topological contexts.

ABSTRACT

A classical theorem of Alexandroff states that every $n$-dimensional compactum $X$ contains an $n$-dimensional Cantor manifold. This theorem has a number of generalizations obtained by various authors. We consider extension-dimensional and infinite dimensional analogs of strong Cantor manifolds, Mazurkiewicz manifolds, and $V^n$-continua, and prove corresponding versions of the above theorem. We apply our results to show that each homogeneous metrizable continuum which is not in a given class $\mathcal C$ is a strong Cantor manifold (or at least a Cantor manifold) with respect to $\mathcal C$. Here, the class $\mathcal C$ is one of four classes that are defined in terms of dimension-like invariants. A class of spaces having bases of neighborhoods satisfying certain special conditions is also considered.

Motivation & Objective

• To extend Alexandroff's classical theorem on $n$-dimensional compacta containing $n$-dimensional Cantor manifolds to infinite-dimensional and extension-theoretic settings.
• To define generalized notions of strong Cantor manifolds, $V^n$-continua, and Mazurkiewicz manifolds using a general dimension function $D_{\mathcal{K}}$ based on a stratum $\mathcal{K}$ of CW-complexes.
• To establish conditions under which homogeneous metrizable continua are strong Cantor manifolds with respect to $\mathcal{D}_{\mathcal{K}}^{n-2}$, particularly when not belonging to specific dimension-based classes.
• To introduce and analyze property $(H)$ as a sufficient condition for the extension property $(\alpha)$ in compact $n$-dimensional spaces, aiding in the verification of homogeneity-related topological properties.

Proposed method

• Define a dimension function $D_{\mathcal{K}}(X)$ using a stratum $\mathcal{K} = \{K_0, K_1, \dots\}$ of CW-complexes, where $D_{\mathcal{K}}(X) \leq n$ iff $K_n \in AE(X)$, generalizing covering, cohomological, and extraordinary dimensions.
• Introduce generalized Cantor manifolds, strong Cantor manifolds, and $V^n$-continua with respect to $D_{\mathcal{K}}$, extending classical definitions to abstract dimension theories.
• Define Mazurkiewicz manifolds with respect to an admissible class $\mathcal{C}$ as spaces where any two disjoint closed sets with non-empty interior are joined by a continuum in the complement of any $F_\sigma$-set in $\mathcal{C}$.
• Establish that local homogeneity and property $(\alpha)$ imply that a space has a basis of compact, $n$-dimensional strong Cantor manifolds with respect to $\mathcal{D}_{\mathcal{K}}^{n-2}$, using a contradiction argument based on connectedness.
• Introduce property $(H)$: for every open set $U$ in a basis, $D_{\mathcal{K}}(\mathrm{bd}\,U) \leq n-1$ and maps from $\mathrm{bd}\,U$ to $K_{n-1}$ extend over $\mathrm{cl}U \setminus V$ for any non-empty open $V \subset U$, which implies property $(\alpha)$.
• Prove that if a compact $n$-dimensional space satisfies $(H)$, then it satisfies $(\alpha)$, using a minimal non-extendable set argument and boundary extension lemmas.

Experimental results

Research questions

• RQ1 Under what conditions does a homogeneous metrizable continuum become a strong Cantor manifold with respect to $\mathcal{D}_{\mathcal{K}}^{n-2}$?
• RQ2 Can the classical Alexandroff theorem on $n$-dimensional compacta containing $n$-dimensional Cantor manifolds be extended to infinite-dimensional or extension-theoretic settings using $D_{\mathcal{K}}$?
• RQ3 What is the relationship between property $(H)$ and property $(\alpha)$ in the context of compact $n$-dimensional spaces?
• RQ4 How do generalized notions of $V^n$-continua and Mazurkiewicz manifolds behave under $D_{\mathcal{K}}$-dimension theories?
• RQ5 Is property $(H)$ equivalent to property $(\alpha)$ for finite-dimensional, locally homogeneous compact spaces?

Key findings

• Every homogeneous metrizable continuum that is not in a given class $\mathcal{C}$ (defined via dimension invariants) is a strong Cantor manifold with respect to $\mathcal{D}_{\mathcal{K}}^{n-2}$, generalizing Alexandroff’s theorem.
• The space $X$ is a locally connected strong Cantor manifold with respect to $\mathcal{D}_{\mathcal{K}}^{n-2}$ under the hypotheses of Proposition 4.12, due to a basis of small $n$-dimensional strong Cantor manifolds and local homogeneity.
• Property $(H)$ implies property $(\alpha)$ for compact $n$-dimensional spaces, providing a practical criterion for verifying $(\alpha)$ in extension-theoretic contexts.
• The proof of Theorem 5.3 shows that if a compact $n$-dimensional space satisfies $(H)$, then it satisfies $(\alpha)$, using a minimal non-extendable set argument and boundary extension lemmas.
• The class of spaces with property $(H)$ includes manifolds, manifolds with removed open $n$-cells, and certain finite polyhedra, but not all spaces with property $(\alpha)$, such as the simple triod.
• The paper leaves open whether $(\alpha)$ and $(H)$ are equivalent for finite-dimensional, locally homogeneous compact spaces, posing a key question for future research.