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Rational evaluation subgroups. (English) Zbl 0855.55009

We prove that the rationalizations of Gottlieb’s evaluation subgroups of \(Y\) are, under suitable restrictions, rational homotopy invariants of maps \(f : X \to Y\). We then determine conditions on the rational homotopy of a map \(f : X \to Y\) which imply the vanishing or nonvanishing of the corresponding rational evaluation subgroup. Our results precisely establish a relationship between the vanishing of rational evaluation subgroups and the “freeness” of the image of the rational homotopy of \(X\) under \(f\) in \(Y\). As applications, we extend known examples of vanishing rational Gottlieb groups to other evaluation subgroups, make explicit calculations of rational homotopy groups of function spaces, and show that, for certain classes of spaces, the rational Gottlieb groups faithfully detect the fundamental division in rational homotopy theory between elliptic and hyperbolic spaces.

MSC:

55P62 Rational homotopy theory
55P15 Classification of homotopy type
58D99 Spaces and manifolds of mappings (including nonlinear versions of 46Exx)
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