Paper:       Retractive transfers and p-local finite groups

Author:      Kári Ragnarsson

Institution: Department of Mathematical Sciences,
             University of Aberdeen
             Aberdeen AB243UE
             United Kingdom

Status:      To appear in Proc. Ed. Math. Soc

Abstract:  

In this paper we explore the possibility of defining $p$-local
finite groups (\cite{BLO2}) in terms of transfer properties of their
classifying spaces. More precisely, we consider the question posed
by Haynes Miller, whether an equivalent theory can be obtained by
studying triples $(f,t,X)$, where $X$ is a $p$-complete, nilpotent
space with finite fundamental group, \mbox{$f\negmedspace: BS \to
X$} is a map from the classifying space of a finite $p$-group, and
$t$ is a stable retraction of $f$ satisfying Frobenius reciprocity
at the level of stable homotopy. We refer to $t$ as a
\emph{retractive transfer} of $f$ and to $(f,t,X)$ as a
\emph{retractive transfer triple over $S$}.

In the case where $S$ is elementary abelian, we answer this question
in the affirmative by showing that a retractive transfer triple
$(f,t,X)$ over $S$ does indeed induce a $p$-local finite group over
$S$ with $X$ as its classifying space.

Using results from \cite{KR:ClSpec} we show that the converse is
true for general finite $p$-groups. That is, for a $p$-local finite
group $\plfg$, the natural inclusion \mbox{$\theta \negmedspace: BS
\to X$} has a retractive transfer $t$, making $(\theta,t,\ClSp)$ a
retractive transfer triple over $S$. This also requires a proof,
obtained jointly with Ran Levi, that $\ClSp$ is a nilpotent space,
which is of independent interest.