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Homotopy Actions, Cyclic Maps and their Duals
#
Homotopy Actions, Cyclic Maps and their Duals

##
Martin Arkowitz and Gregory Lupton

An \emph{action of $A$ on $X$} is a map $F\colon A\times X \to X$
such that $F\vert_X = \id \colon X\to X$. The restriction $F\vert_A
\colon A \to X$ of an action is called a \emph{cyclic map}. Special
cases of these notions include group actions and the Gottlieb groups
of a space, each of which has been studied extensively. We prove
some general results about actions and their Eckmann-Hilton duals.
For instance, we classify the actions on an $H$-space that are
compatible with the $H$-structure. As a corollary, we prove that if
any two actions $F$ and $F'$ of $A$ on $X$ have cyclic maps $f$ and
$f'$ with $\Omega f = \Omega f'$, then $\Omega F$ and $\Omega F'$
give the same action of $\Omega A$ on $\Omega X$. We introduce a new
notion of the category of a map $g$ and prove that $g$ is cocyclic
if and only if the category is less than or equal to $1$. From this
we conclude that if $g$ is cocyclic, then the Berstein-Ganea
category of $g$ is $\le 1$. We also briefly discuss the
relationship between a map being cyclic and its cocategory being
$\le 1$.

Homology, Homotopy and Applications, Vol. 7(2005), No. 1, pp. 169-184
http://www.rmi.acnet.ge/hha/volumes/2005/n1a9/v7n1a9.dvi (ps, dvi.gz, ps.gz, pdf)
ftp://ftp.rmi.acnet.ge/pub/hha/volumes/2005/n1a9/v7n1a9.dvi (ps, dvi.gz, ps.gz, pdf)