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Loop near-rings and unique decompositions of H-spaces

Damir Franetič and Petar Pavešić

Algebraic & Geometric Topology 16 (2016) 3563–3580

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.

H-space, near-ring, algebraic loop, idempotent, strongly indecomposable space, Krull-Schmidt-Remak-Azumaya theorem
Mathematical Subject Classification 2010
Primary: 55P45
Secondary: 16Y30
Received: 25 November 2015
Revised: 28 February 2016
Accepted: 20 May 2016
Published: 15 December 2016
Damir Franetič
Faculty of Computer and Information Science
University of Ljubljana
Večna pot 113
1000 Ljubljana
Petar Pavešić
Faculty of Mathematics and Physics
University of Ljubljana
Jadranska 19
1111 Ljubljana