2003/09/26 by Gregory Lupton, Lupton, Gregory, Samuel Bruce Smith +1
Mathematics · Medicine · #55D23 (Primary) #55P62 #Advanced Topics in Algebra #Algebraic Topology (math.AT) #FOS: Mathematics #Homotopy and Cohomology in Algebraic Topology #Ophthalmology and Eye Disorders #math.AT #msc:55D23 #msc:55P62
paper · pdf · doi:10.48550/arxiv.math/0309432
34 pages
arxiv created 2003/09/26 · openalex publication_date 2003/09/26 · arxiv updated 2009/12/01 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
Let f: X -> Y be a based map of simply connected spaces. The corresponding evaluation map w: map(X,Y;f) -> Y induces a homomorphism of homotopy groups whose image in pin(Y) is called the nth evaluation subgroup of f. The nth Gottlieb group of X occurs as the special case in which Y = X and f = 1X. We identify the homomorphism induced on rational homotopy groups by this evaluation map, in terms of a map of complexes of derivations constructed using Sullivan minimal models. Our identification allows for the characterization of the rationalization of the nth evaluation subgroup of f. It also allows for the identification of several long exact sequences of rational homotopy groups, including the long exact sequence induced on rational homotopy groups by the evaluation fibration. As a consequence, we obtain an identification of the rationalization of the so-called G-sequence of the map f. This is a sequence--in general not exact--of groups and homomorphisms that includes the Gottlieb groups of X and the evaluation subgroups of f. We use these results to study the G-sequence in the context of rational homotopy theory. We give new examples of non-exact G-sequences and uncover a relationship between the homology of the rational G-sequence and negative derivations of rational cohomology. We also relate the splitting of the rational G-sequence of a fibre inclusion to a well-known conjecture in rational homotopy theory.