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Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)

 

 

Formal spaces with finite-dimensional rational homotopy


Authors: Yves Félix and Stephen Halperin
Journal: Trans. Amer. Math. Soc. 270 (1982), 575-588
MSC: Primary 55P62; Secondary 18G99
DOI: https://doi.org/10.1090/S0002-9947-1982-0645331-9
MathSciNet review: 645331
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Abstract: Let $ S$ be a simply connected space. There is a certain principal fibration $ {K_1} \to E\mathop \to \limits^\pi {K_0}$ in which $ {K_1}$ and $ {K_0}$ are products of rational Eilenberg-Mac Lane spaces and a continuous map $ \phi :S \to E$ such that in particular $ {\phi _0} = \pi \circ \phi $ maps the primitive rational homology of $ S$ isomorphically to that of $ {K_0}$. A main result of this paper is the

Theorem. If $ \dim \pi {}_{\ast}(S) \otimes {\mathbf{Q}} < \infty $ then $ \phi $ is a rational homotopy equivalence if and only if all the primitive homology in $ H{}_{\ast}(S;\,{\mathbf{Q}})$ and $ H{}_{\ast}({K_0},\,S;\,{\mathbf{Q}})$ can (up to integral multiples) be represented by spheres and disk-sphere pairs.

Corollary. If $ S$ is formal, $ \phi $ is a rational homotopy equivalence.


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Additional Information

DOI: https://doi.org/10.1090/S0002-9947-1982-0645331-9
Keywords: Rational homotopy, minimal model, formality, Eilenberg-Moore spectral sequence
Article copyright: © Copyright 1982 American Mathematical Society