On the evaluation map

Murillo, Aniceto

doi:10.1090/S0002-9947-1993-1112376-1

On the evaluation map
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Trans. Amer. Math. Soc. 339 (1993), 611-622 Request permission

Abstract:

The evaluation map of a differential graded algebra or of a space is described under two different approaches. This concept turns out to have geometric implications: (i) A $1$-connected topological space, with finite-dimensional rational homotopy, has finite-dimensional rational cohomology if and only if it has nontrivial evaluation map. (ii) Let $E\xrightarrow {\rho }B$ be a fibration of simplyconnected spaces. If the rational cohomology of the fibre is finite dimensional and the evaluation map of the base is different from zero, then the evaluation map of the total space is nonzero. Also, if $\rho$ is surjective in rational homotopy and the evaluation map of $E$ is nontrivial, then the evaluation map of the fibre is different from zero.

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