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Spaces realized and non-realized as Dold-Lashof classifying spaces. (English) Zbl 1423.55021

Summary: Let \(X\) be a simply connected CW-complex of finite type. Denote by \({\mathrm{Baut}}_{1}(X)\) the Dold-Lashof classifying space of fibrations with fiber \(X\). This paper is a survey about the problem of realizing Dold-Lashof classifying spaces. We will also present some new results: we show that not all rank-two rational \(H\)-spaces can be realized as \({\mathrm{Baut}}_{1}(X)\) for simply connected, rational elliptic space \(X\). Moreover, we construct an infinite family of rational spaces \(X,\) such that \({\mathrm{Baut}}_{1}(X)\) is rationally a finite \(H\)-space of rank-two (up to rational homotopy type).

MSC:

55R05 Fiber spaces in algebraic topology
55R35 Classifying spaces of groups and \(H\)-spaces in algebraic topology
55P62 Rational homotopy theory
55-02 Research exposition (monographs, survey articles) pertaining to algebraic topology
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