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Sectional category of fibrations of fibre \(K(\mathbb Q,2k)\). (English) Zbl 1078.55012

Let \(P: E\to B\) be a fibration. The sectional category of \(p\), \(\text{secat}(p)\), is the least integer \(n\) such that \(B\) can be covered by \(n+ 1\) open subsets, over each of which \(p\) has a section. The genus of \(p\) is defined is a similar way, as the least integer \(n\) such that \(B\) can be covered by \(n+ 1\) open subsets, over each of which \(p\) is trivial. Clearly \(\text{secat}(p)\geq\text{genus}(p)\). It is also well known that \(\text{secat}(p)\) is the least integer \(n\) such that the \((n+ 1)\)-fold fibre join \(p*\cdots * p\) admits a homotopy section.
In this paper the author considers non-trivial rational fibrations with fibre an Eilenberg-MacLane space \(K(\mathbb{Q}, 2n)\). He proves that in this case \(\text{secat}(p)= 1\) and that the iterated joint fibrations \(p*\cdots * p\) are never trivial.

MSC:

55P62 Rational homotopy theory
55M30 Lyusternik-Shnirel’man category of a space, topological complexity à la Farber, topological robotics (topological aspects)
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Full Text: Euclid