2005/09/27 by Yves Félix, Yves Felix, Felix, Yves +2
Mathematics · #Advanced Topics in Algebra #Algebraic Topology (math.AT) #Algebraic structures and combinatorial models #FOS: Mathematics #Homotopy and Cohomology in Algebraic Topology #math.AT
paper · pdf · doi:10.48550/arxiv.math/0509632
25 pages, AMSLaTeX2e
arxiv created 2005/09/27 · openalex publication_date 2005/09/27 · arxiv updated 2009/12/01 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
Let E be an H-space acting on a based space X. Then we refer to ev: E -> X, the map obtained by acting on the base point of X, as a ``generalized evaluation map." We establish several fundamental results about the rational homotopy behaviour of a generalized evaluation map, all of which apply to the usual evaluation map Map(X,X;1)->X. With mild hypotheses on X, we show that a generalized evaluation map ev factors, up to rational homotopy, through a map Gammaev: Sev -> X where Sev is a (relatively small) finite product of odd-dimensional spheres and the map induced by Gammaev on rational homotopy groups is injective. This result has strong consequences: if the image in rational homotopy groups of ev is trivial, then the generalized evaluation map is null-homotopic after rationalization; unless X satisfies a very strong splitting condition, any generalized evaluation map induces the trivial homomorphism in rational cohomology; the map Gammaev is rationally a homotopy monomorphism and a generalized evaluation map may be written as a composition of a homotopy epimorphism and this homotopy monomorphism. We include illustrative examples and prove numerous subsidiary results of interest.