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ISSN (electronic): 1472-2739
ISSN (print): 1472-2747
Loop near-rings and unique decompositions of H-spaces

Damir Franetič and Petar Pavešić

Algebraic & Geometric Topology 16 (2016) 3563–3580
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.

Keywords
H-space, near-ring, algebraic loop, idempotent, strongly indecomposable space, Krull-Schmidt-Remak-Azumaya theorem
Mathematical Subject Classification 2010
Primary: 55P45
Secondary: 16Y30
References
Publication
Received: 25 November 2015
Revised: 28 February 2016
Accepted: 20 May 2016
Published: 15 December 2016
Authors
Damir Franetič
Faculty of Computer and Information Science
University of Ljubljana
Večna pot 113
1000 Ljubljana
Slovenia
http://www.fri.uni-lj.si/si/damir-franetic
Petar Pavešić
Faculty of Mathematics and Physics
University of Ljubljana
Jadranska 19
1111 Ljubljana
Slovenia
http://www.fmf.uni-lj.si/~pavesic