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This book contains a collection of articles summarizing together
the state of knowl- edge in a broad portion of modern homotopy
theory. These articles were assembled during 1998 and 1999, on the
occasion of an emphasis semester organized by the Centre de Recerca
Matematica (CRM) and its highlight, the 1998 Barcelona Con- ference
on Algebraic Topology (BCAT). First of all, we are indebted to all
the authors for submitting their work, and to the referees for
their help in the selec- tion and for their generous contribution
to the content of the articles. Many talks given during the CRM
semester or at the conference focused on aspects of the following
topics: abstract stable homotopy, model categories, homotopical
localizations and cellular approximations, p-compact groups, mod-
ules over the Steenrod algebra, classifying spaces for proper
actions of discrete groups, K-theory and other generalized
cohomology theories, cohomology of fi- nite and profinite groups,
Hochschild homology, configuration spaces, Lusternik- Schnirelmann
category, stable and unstable splittings. Other talks treated
multi- disciplinary subjects related to quantum field theory,
differential geometry, homo- topical dynamics, tilings, and various
aspects of group theory. In addition, an advanced course on
Classifying Spaces and Cohomology of Groups was organized by the
CRM in the days preceding the conference. Lecture notes from this
course will be published by Birkhauser Verlag as the first volume
of a newly created CRM Advanced Course series.
Central to this collection of papers are new developments in the
general theory of localization of spaces. This field has undergone
tremendous change of late and is yielding new insight into the
mysteries of classical homotopy theory. The present volume
comprises the refereed articles submitted at the Conference on
Algebraic Topology held in Sant Feliu de Guixols, Spain, in June
1994. Several comprehensive articles on general localization
clarify the basic tools and give a report on the state of the art
in the subject matter. The text is therefore accessible not only to
the professional mathematician but also to the advanced student.
This book contains a collection of articles summarizing together
the state of knowl- edge in a broad portion of modern homotopy
theory. These articles were assembled during 1998 and 1999, on the
occasion of an emphasis semester organized by the Centre de Recerca
Matematica (CRM) and its highlight, the 1998 Barcelona Con- ference
on Algebraic Topology (BCAT). First of all, we are indebted to all
the authors for submitting their work, and to the referees for
their help in the selec- tion and for their generous contribution
to the content of the articles. Many talks given during the CRM
semester or at the conference focused on aspects of the following
topics: abstract stable homotopy, model categories, homotopical
localizations and cellular approximations, p-compact groups, mod-
ules over the Steenrod algebra, classifying spaces for proper
actions of discrete groups, K-theory and other generalized
cohomology theories, cohomology of fi- nite and profinite groups,
Hochschild homology, configuration spaces, Lusternik- Schnirelmann
category, stable and unstable splittings. Other talks treated
multi- disciplinary subjects related to quantum field theory,
differential geometry, homo- topical dynamics, tilings, and various
aspects of group theory. In addition, an advanced course on
Classifying Spaces and Cohomology of Groups was organized by the
CRM in the days preceding the conference. Lecture notes from this
course will be published by Birkhauser Verlag as the first volume
of a newly created CRM Advanced Course series.
For a finite group $G$ of Lie type and a prime $p$, the authors
compare the automorphism groups of the fusion and linking systems
of $G$ at $p$ with the automorphism group of $G$ itself. When $p$
is the defining characteristic of $G$, they are all isomorphic,
with a very short list of exceptions. When $p$ is different from
the defining characteristic, the situation is much more complex but
can always be reduced to a case where the natural map from
$\mathrm{Out}(G)$ to outer automorphisms of the fusion or linking
system is split surjective. This work is motivated in part by
questions involving extending the local structure of a group by a
group of automorphisms, and in part by wanting to describe self
homotopy equivalences of $BG^\wedge _p$ in terms of
$\mathrm{Out}(G)$.
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