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. Contrary to published assertions, the Anick spaces are shown not to be universal Abelian H-spaces for the corresponding Moore spaces; however conditions are discussed which could lead to a universal property with respect to a more limited range of targets, and a restricted universal property is proven.
Highlights
By an Abelian H-space we mean a connected CW complex with an Hspace structure that is homotopy associative and homotopy commutative
The goal of this work is to study the conditions under which an Abelianization exists as well as to seek a more restricted universal property which is appropriate for the Anick spaces
Suppose T is an Abelianization of X and k = Fp or Q
Summary
By an Abelian H-space we mean a connected CW complex with an Hspace structure that is homotopy associative and homotopy commutative. Every known example of an Abelianization T of a space X satisfies the condition that the H-map ΩSX −→h T extending the inclusion i : X → T has a right homotopy inverse. In this case we construct a fibration sequence:. We will develop some general properties of an Abelianization T of X and conclude that there are some severe limitations on the spaces X that are available. This will imply that prξ = 0 by 2.1, completing the proof To accomplish this we examine the cohomology group Hm(T ∧ T ) using a Kunneth Theorem available for all spaces of finite type [HW60, 5.7.26].
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