Abelian properties of Anick spaces

TL;DR

This paper proves that Anick spaces are homotopy associative and commutative H-spaces, establishing their universal mapping property and developing an unstable composition theory using advanced algebraic topology techniques.

Contribution

It demonstrates the homotopy associativity and commutativity of Anick spaces and introduces a universal mapping property, advancing the understanding of their algebraic structure.

Findings

Anick spaces are homotopy associative and commutative H-spaces.

Established a universal mapping property for Anick spaces.

Developed an unstable composition theory using generalized Whitehead products.

Abstract

The Anick spaces play a key role in an unstable filtration of the stable homotopy of V(0) to produce secondary EHP sequences. This work establishes that the Anick spaces are homotopy associative and homotopy commutative H-spaces, and that they have a universal mapping property for maps into a homotopy commutative and homotopy associative H-space with a (necessary) condition on the growth of the p torsion. This is utilized to establish an unstable composition theory. The techniques involve generalized Whitehead products based on co-H spaces, calculations in a congruence category that lies between the unstable category and the stable category, and a controlled version of the extension theorem for principal fibrations.