Realizing spaces as classifying spaces

TL;DR

This paper investigates which spaces can serve as classifying spaces for fibrations with a fixed fiber within rational homotopy theory, constructing examples and proving non-realization results for certain spaces.

Contribution

It constructs an infinite family of rational classifying spaces and establishes non-realization results for specific rational homotopy types.

Findings

Constructed an infinite family of rational classifying spaces.

Proved that $\\mathbb{C}P^2$ and $S^4$ are not realizable as classifying spaces.

Identified limitations in realizing certain rational homotopy types as classifying spaces.

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.