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Compact Maltsev spaces that are not retracts of compact groups. (Espaces de Maltsev compacts qui ne sont pas des rétractes de groupes compacts.) (French) Zbl 0893.22002

Let \(X\) be a topological space. A Maltsev operation on \(X\) is, by definition, a mapping \(m:X^3\to X\) satisfying \(m(x,y,y) =m(y,y,x) =x\) identically. The following known results are quoted: every retract of a topological group is a Maltsev space; every compact Maltsev space is a retract of its free topological group [see P. M. Gartside, E. A. Reznichenko and D. V. Sipacheva, Topology Appl. 80, 115-129 (1997)]. The following question posed there is answered in the negative, in the paper under review: is every compact Maltsev space a retract of some compact group? Specifically, Prop. 2 states that if a finite, simply connected CW-complex has \(p\)-torsion, for some prime \(p\geq 7\), then it is not a retract of any compact group. Furthermore, it is shown that for every prime \(p\geq 7\), there exist CW-complexes which are also Maltsev spaces, satisfying the assumptions of Prop. 2. The proofs are extremely sketchy; they draw heavily upon homotopy and Lie group theory, and the theory of \(H\)-spaces.
Reviewer: P.Flor (Graz)

MSC:

22C05 Compact groups
54C15 Retraction
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