 
Summary: MAPS BETWEEN CLASSIFYING SPACES OF KACMOODY
GROUPS
JAUME AGUAD´E AND ALBERT RU´IZ
1. Introduction
The study of the maps between classifying spaces of compact Lie groups from a ho
motopy point of view has been one of the highlights of algebraic topology in the final
quarter of the xxth century. The project was started by Sullivan's construction of
unstable Adams operations in his deeply influential manuscript Geometric Topology,
part I. Along the seventies, Hubbuck, AdamsMahmud, Wilkerson and Friedlander
developed a deep analysis of the maps between classifying spaces using localization
theory, Steenrod and Ktheory operations and ´etale homotopy. In that same manu
script of Sullivan, he pointed out that the first obstruction to understand the maps
between classifying spaces consists in understanding the maps from BZ/p to a com
pact space. This problem was called the Sullivan conjecture and the way to its solution
(Miller, Carlsson) crossed some of the most beautiful landscapes of homotopy theory
at the end of the century: the unstable Adams spectral sequence and the structure
of injective objects in the category of unstable modules over the Steenrod algebra.
Once this very strong weapon was available, DwyerZabrodsky and Notbohm were
able to understand the mod p homotopy type of the space of maps from the classify
ing space of a ptoral group to the classifying space of a compact Lie group. Then,
