Rational fibrations in differential homological algebra

Murillo, Aniceto

doi:10.1090/S0002-9947-1992-1079055-X

Rational fibrations in differential homological algebra
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by Aniceto Murillo

Trans. Amer. Math. Soc. 332 (1992), 79-91

DOI: https://doi.org/10.1090/S0002-9947-1992-1079055-X

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Abstract:

In this paper, a result of [6] is generalized as follows: Given a fibration $F \to E\xrightarrow {p}B$ of simply connected spaces in which either, the fibre has finite dimensional rational cohomology, or, it has finite dimensional rational homotopy and $\rho$ induces a surjection in rational homotopy, we construct an explicit isomorphism, \[ \begin {array}{*{20}{c}} {\varphi :\operatorname {Ext}_{{C^\ast }(B,{\mathbf {Q}})}({\mathbf {Q}},{C^\ast }(B;{\mathbf {Q}}))\hat \otimes \operatorname {Ext}_{{C^\ast }(F;{\mathbf {Q}})}({\mathbf {Q}},{C^\ast }(F,{\mathbf {Q}}))} \\ {\xrightarrow { \cong }\operatorname {Ext}_{{C^\ast }(E;{\mathbf {Q}})}(Q,{C^\ast }(E;{\mathbf {Q}})).} \\ \end {array} \] This is deduced from its "algebraic translation," a more general result in the environment of graded differential homological algebra.

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