Abstract
Let a simply-connected homogeneous space \({X}\) satisfy the condition of \({{\rm dim} \pi_{\rm even}(X)\otimes {\mathbb{Q}}=2}\) and \({{\rm dim} \pi_{\rm odd}(X)\otimes {\mathbb{Q}}=3}\) (then, we say it is of (2, 3) type), which is the smallest rank in non-formal pure spaces. Then, we compute the Sullivan minimal model of the Dold–Lashof classifying space \({{\rm Baut}_1 X}\) according to Nishinobu and Yamaguchi (Topol Appl 196:290–307, 2015) and observe whether or not its rational cohomology is a polynomial algebra, which is a necessary condition for the Serre spectral sequence of any fibration over a sphere with fibre \({X}\) to degenerate at term \({E_2}\).
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Nishinobu, H., Yamaguchi, T. Rational cohomologies of classifying spaces for homogeneous spaces of small rank. Arab. J. Math. 5, 225–237 (2016). https://doi.org/10.1007/s40065-016-0156-y
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DOI: https://doi.org/10.1007/s40065-016-0156-y