Homological characterizations of $Q$-manifolds and $l_2$-manifolds

TL;DR

This paper explores how homological properties can replace certain density conditions in characterizing $Q$-manifolds and $l_2$-manifolds, leading to new homological characterizations.

Contribution

It introduces homological versions of existing density conditions, providing new characterizations of $Q$-manifolds and $l_2$-manifolds.

Findings

Homological $Z_n$-maps characterize $Q$-manifolds.

Homological $Z$-maps characterize $l_2$-manifolds.

Homological conditions can replace density assumptions in manifold characterizations.

Abstract

We investigate to what extend the density of $Z_n$-maps in the characterization of $Q$-manifolds, and the density of maps $f\in C(\mathbb N\times Q,X)$ having discrete images in the $l_2$-manifolds characterization can be weakened to the density of homological $Z_n$-maps and homological $Z$-maps, respectively. As a result, we obtain homological characterizations of $Q$-manifolds and $l_2$-manifolds.