Alexandroff Manifolds and Homogeneous Continua

TL;DR

This paper proves that homogeneous, metric ANR-continua with nontrivial cohomology are Alexandroff manifolds and cannot be separated by certain lower-dimensional compacta, advancing understanding of their topological structure.

Contribution

It establishes that such continua are V^n_G-continua and cannot be separated by specific compacta, providing partial answers to longstanding topological questions.

Findings

Homogeneous metric ANR-continua with nontrivial cohomology are V^n_G-continua.

Such continua cannot be separated by compacta with trivial lower-dimensional cohomology.

Partial resolution to whether 2D homogeneous Peano continua can be separated by arcs.

Abstract

We prove the following result announced in Todorov and Valov: Any homogeneous, metric $ANR$-continuum is a $V^n_G$-continuum provided $\dim_GX=n\geq 1$ and $\check{H}^n(X;G)\neq 0$, where $G$ is a principal ideal domain. This implies that any homogeneous $n$-dimensional metric $ANR$-continuum with $\check{H}^n(X;G)\neq 0$ is a $V^n$-continuum in the sense of Alexandroff (1957). We also prove that any finite-dimensional homogeneous metric continuum $X$, satisfying $\check{H}^n(X;G)\neq 0$ for some group $G$ and $n\geq 1$, cannot be separated by a compactum $K$ with $\check{H}^{n-1}(K;G)=0$ and $\dim_G K\leq n-1$. This provides a partial answer to a question of Kallipoliti-Papasoglu (2007) whether any two-dimensional homogeneous Peano continuum cannot be separated by arcs.