Decomposition theorems in rational homotopy theory

Oprea, John

doi:10.1090/S0002-9939-1986-0822450-4

Decomposition theorems in rational homotopy theory
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by John Oprea

Proc. Amer. Math. Soc. 96 (1986), 505-512

DOI: https://doi.org/10.1090/S0002-9939-1986-0822450-4

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Abstract:

Let $F \to E \to B$ be a fibration of simply connected rational spaces with finite rational betti numbers. Denote the connecting homomorphism of the fibration by ${\partial _\# }$ and the Hurewicz map of the fibre $F$ by $h$. Then, it is shown that there is a decomposition $F \simeq \mathcal {F} \times K$ where $K$ is the product of rational Eilenberg-Mac Lane spaces contained in $\Omega B$ maximal with respect to the conditions: ${\pi _ * }\left ( K \right ) \cap \operatorname {Ker}{\partial _\# } = 0$ and ${\partial _\# }\left ( {{\pi _ * }\left ( K \right )} \right ) \cap \operatorname {Ker}h = 0$. This decomposition is obtained using Sullivan’s theory of minimal models. Applications are given of the main theorem and a dual result is proved for rational cofibrations.

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