Universal Abelian H-spaces

TL;DR

This paper investigates the existence and properties of universal Abelian H-spaces, which are spaces that facilitate unique H-maps from any given space, and discusses conditions affecting their universality.

Contribution

It provides examples, discusses limitations, and explores conditions under which universal Abelian H-spaces can or cannot exist, including analysis of Anick spaces.

Findings

Anick spaces are not universal Abelian H-spaces for Moore spaces.

Conditions for universality depend on the target space properties.

Examples of (X,T) pairs illustrating the concept.

Abstract

The question of the existence of Universal homotopy commutative and homotopy associative H-spaces (called Abelian H-spaces) is studied. Such a space T(X) would prolong a map from X into an Abelian H-space to a unique H-map from T into X. Examples of such pairs (X,T) are given and conditions are discussed which limit the possible spaces X for which such a T can exist. The Anick spaces are shown not to be universal Abelian H-spaces for the corresponding Moore spaces, but conditions are discussed which could lead to a universal property with respect to a more limited range of targets.