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

Authors: Gregory Lupton and Samuel Bruce Smith

Authors' e-mail addresses: G.Lupton@csuohio.edu and smith@sju.edu

Authors' mailing addresses: Department of Mathematics, Cleveland
State University, 2121 Euclid Ave., Cleveland OH 44115 and
Department of Mathematics, Saint Joseph's University,
Philadelphia, PA 19131

AMS classification number: 55P62, 55Q52

Other useful information: 33 pages;  http://arXiv.org/abs/math.AT/0401178

Abstract: 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.