Title: Evaluation Maps in Rational Homotopy

Authors: Yves Felix and Gregory Lupton


e-mail: felix@math.ucl.ac.be and G.Lupton@csuohio.edu

Mailing addresses:

Institut Mathematique, Universite Catholique de Louvain,
B-1348 Louvain-la-Neuve Belgique

Department of Mathematics,
Cleveland State University,
2121 Euclid Avenue,
Cleveland OH 44115 U.S.A.

AMS MSC2000: 55P62, 55Q05


arXiv: math.AT/0509632


Abstract:  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 Gamma_ev: S_ev -> X where S_ev is a 
(relatively small) finite product of odd-dimensional 
spheres and the map induced by Gamma_ev 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 Gamma_ev 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.