Loop near-rings and unique decompositions of H-spaces

Franetič, Damir¹ ; Pavešić, Petar²

¹Faculty of Computer and Information Science, University of Ljubljana, Večna pot 113, 1000 Ljubljana, Slovenia
²Faculty of Mathematics and Physics, University of Ljubljana, Jadranska 19, 1111 Ljubljana, Slovenia

Algebraic and Geometric Topology, Tome 16 (2016) no. 6, pp. 3563-3580

Résumé

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.

Détail

DOI : 10.2140/agt.2016.16.3563

Classification : 55P45, 16Y30
Keywords: H-space, near-ring, algebraic loop, idempotent, strongly indecomposable space, Krull-Schmidt-Remak-Azumaya theorem

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