2004/01/15 by Gregory Lupton, Lupton, Gregory, Samuel Bruce Smith +1
Biochemistry, Genetics and Molecular Biology · Mathematics · #55P62 (Primary) 55Q52 (Secondary) #Advanced Topics in Algebra #Algebraic Topology (math.AT) #FOS: Mathematics #Homotopy and Cohomology in Algebraic Topology #Sphingolipid Metabolism and Signaling #math.AT #msc:55P62 #msc:55Q52
paper · pdf · doi:10.48550/arxiv.math/0401178
33 pages
arxiv created 2004/01/15 · openalex publication_date 2004/01/15 · arxiv updated 2009/12/01 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
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.