Loop near-rings and unique decompositions of H-spaces

TL;DR

This paper develops an algebraic framework for loop near-rings associated with H-spaces, leading to new unique decomposition theorems for products of H-spaces, including infinite products and finite p-local cases.

Contribution

It introduces the algebraic theory of local loop near-rings and characterizes indecomposable H-spaces, enabling new decomposition results and simplified proofs of classical theorems.

Findings

Established a Krull--Schmidt type decomposition for H-spaces.

Proved that certain infinite products of H-spaces have unique decompositions.

Showed finite p-local H-spaces are strongly indecomposable, simplifying existing proofs.

Abstract

For every H-space $X$ the set of homotopy classes $[X,X]$ possesses a natural algebraic structure of a loop near-ring. Albeit one cannot say much about general loop near-rings, it turns out that those that arise from H-spaces are sufficiently close to rings to have a viable Krull--Schmidt type decomposition theory, which is then reflected into decomposition results of H-spaces. In the paper we develop the algebraic theory of local loop near-rings and derive an algebraic characterization of indecomposable and strongly indecomposable H-spaces. As a consequence, we obtain unique decomposition theorems for products of H-spaces. In particular, we are able to treat certain infinite products of H-spaces, thanks to a recent breakthrough in the Krull--Schmidt theory for infinite products. Finally, we show that indecomposable finite $p$-local H-spaces are automatically strongly indecomposable, which leads to an easy alternative proof of classical unique decomposition theorems of Wilkerson and Gray.