Rationalized evaluation subgroups of a map. I: Sullivan models, derivations and \(G\)-sequences. (English) Zbl 1112.55012
Let \(f : X \to Y\) be a based map of simply connected CW complexes. Denote by \(map(X,Y:f)\) the path component of the space of maps \(X \to Y\) homotopic to \(f\). Evaluation at the base point of \(X\) gives the evaluation map \(\omega : map (X,Y:f) \to Y\).
Denote by \(\mathcal M_f : \mathcal M_Y \to \mathcal M_X\) the Sullivan minimal model of \(f\), and by \(\varepsilon : \mathcal M_X \to \mathbb{Q}\) the natural augmentation. Let now \(Der_n (\mathcal M_Y, \mathcal M_X)\) denote the vector space of \(\mathcal M_f\)-derivations of degree \(n\), \(n>0\) from \(\mathcal M_Y\) into \(\mathcal M_X\). With the differential on \(\mathcal M_X\) and \(\mathcal M_Y\), \(Der_*(\mathcal M_Y, \mathcal M_X)\) is a complex, and the composition with \(\varepsilon\),
\[ \varepsilon_* : Der_*(\mathcal M_Y, \mathcal M_X) \to Der_*(\mathcal M_Y, \mathbb{Q}) \]
is a morphism of complexes.
The authors prove that the map induced in homology is in a natural way isomorphic to the map induced by \(\omega\) on the rational homotopy groups, \[ \pi_*(map(X,Y:f))\otimes \mathbb{Q} \to \pi_*(Y)\otimes \mathbb{Q}\,. \] They deduce informations on the rational relative Gottlieb groups and the rationalized \(G\)-sequence associated to the evaluation map.
Denote by \(\mathcal M_f : \mathcal M_Y \to \mathcal M_X\) the Sullivan minimal model of \(f\), and by \(\varepsilon : \mathcal M_X \to \mathbb{Q}\) the natural augmentation. Let now \(Der_n (\mathcal M_Y, \mathcal M_X)\) denote the vector space of \(\mathcal M_f\)-derivations of degree \(n\), \(n>0\) from \(\mathcal M_Y\) into \(\mathcal M_X\). With the differential on \(\mathcal M_X\) and \(\mathcal M_Y\), \(Der_*(\mathcal M_Y, \mathcal M_X)\) is a complex, and the composition with \(\varepsilon\),
\[ \varepsilon_* : Der_*(\mathcal M_Y, \mathcal M_X) \to Der_*(\mathcal M_Y, \mathbb{Q}) \]
is a morphism of complexes.
The authors prove that the map induced in homology is in a natural way isomorphic to the map induced by \(\omega\) on the rational homotopy groups, \[ \pi_*(map(X,Y:f))\otimes \mathbb{Q} \to \pi_*(Y)\otimes \mathbb{Q}\,. \] They deduce informations on the rational relative Gottlieb groups and the rationalized \(G\)-sequence associated to the evaluation map.
Reviewer: Yves Félix (Louvain-La-Neuve)
Keywords:
rational homotopy; Gottlieb groups; mapping spaces; evaluation map; derivations; evaluation subgroup; minimal models; \(\omega\)-homology; \(G\)-sequenceReferences:
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