Loop Spaces and Finiteness

Clarence W. Wilkerson
Purdue University
cwilkers@purdue.edu

This expository note began as comments on a shorter note of
F.\,R. Cohen \cite{cohen}.
Cohen's paper is an elegant application of powerful recent results in
unstable homotopy theory to a problem of interest to analysts. \\
{\sl {\bf Theorem :} (F.\,R. Cohen,\cite{cohen}) Let $X$ be a simply connected 
finite complex which is not contractible and let $\Omega^j_0X$ be 
the component of the constant map in the $j$-th pointed loop space of 
$X$. If $j \geq 2$, then the  Lusternik-Schnirlman category of 
$\Omega^j_0X$ is not finite. }\\

\indent This note includes a  rederivation of the above theorem
using H-space methods of W. Browder from the 60's, \cite{Browder-loop}, \cite{Browder-Torsion}.
The aim is to reduce the prerequisites for Cohen's theorem  to those 
available after a
second course in algebraic topology. We end with a discussion of 
recent work of  Lannes-Schwartz on various notions 
of finiteness properties and the behavior under looping.
The common theme is extensive use of the action of the Steenrod 
algebra on the cohomology of a topological space.