Symmetric Products and Q-manifolds

TL;DR

This paper explores properties of symmetric products and Q-manifolds, providing new examples, a factor theorem, and a simplified proof of symmetric products preserving Hilbert cube manifold properties.

Contribution

It introduces a novel example of a compact absolute retract with specific symmetric product properties and generalizes the preservation theorem for Hilbert cube manifolds.

Findings

An example of a compact absolute retract not a Hilbert cube manifold but with a Hilbert cube symmetric product.

A factor theorem for nth symmetric product of ARs with the Hilbert cube.

A short proof that symmetric products preserve Hilbert cube manifold properties.

Abstract

An example is given of a compact absolute retract that is not a Hilbert cube manifold but whose second symmetric porduct is the Hilbert cube. A factor theorem is given for nth symmetric product of the cartesian product of any absolute neighborhood retract with the Hilbert cube. A short proof is included of the known fact that symmetric products preserve the property of being a compact Hilbert cube manifold (the theorem is proved here for all Hilbert cube manifolds).