Realizing Spaces as Classifying Spaces

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Date
2015-04-10MFO Scientific Program
Research in Pairs 2014Series
Oberwolfach Preprints;2015,01Author
Lupton, Gregory
Smith, Samuel Bruce
Metadata
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Abstract
Which spaces occur as a classifying space for fibrations with a given fibre? We address this question in the context of rational homotopy theory. We construct an infinite family of finite complexes realized (up to rational homotopy) as classifying spaces. We also give several non-realization results, including the following: the rational homotopy types of $\mathbb{C}P^2$ and $S^4$ are not realized as the classifying space of any simply connected, rational space with finite-dimensional homotopy groups.