Rationalized Evaluation Subgroups of a Map II: Quillen Models and Adjoint Maps

Details Rationalized Evaluation Subgroups of a Map II: Quillen Models and Adjoint Maps Rationalized Evaluation Subgroups of a Map II: Quillen Models and Adjoint Maps

2004-01-15

arxiv.org/abs/math/0401178v1

Mathematics

Algebraic Topology

33 pages

Scientific paper

Let w: Map(X,Y;f) -> Y denote a general evaluation fibration. Working in the setting of rational homotopy theory via differential graded Lie algebras, we identify the long exact sequence induced on rational homotopy groups by w in terms of (generalized) derivation spaces and adjoint maps. As a consequence, we obtain a unified description of the rational homotopy theory of function spaces, at the level of rational homotopy groups, in terms of derivations of Quillen models and adjoints. In particular, as a natural extension of a result of Tanre, we identify the rationalization of the evaluation subgroups of a map f: X -> Y in this setting. As applications, we consider a generalization of a question of Gottlieb, within the context of rational homotopy theory. We also identify the rationalization of the G-sequence of f and make explicit computations of the homology of this sequence. In a separate result of independent interest, we give an explicit Quillen minimal model of a product AxX, in the case in which A is a rational co-H-space.

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