The Goodwillie Tower of the identity functor and the unstable 
                     periodic homotopy of spheres

        	AMS Classifiaction: 55P47, 55Q40, 55S12

			Greg Arone
                        arone@math.uchicago.edu

			Mark Mahowald
                        mark@math.nwu.edu


We investigate Goodwillie's ``Taylor tower'' of the identity functor 
from spaces to spaces. More specifically, we reformulate Johnson's 
description of the Goodwillie derivatives of the identity, and prove
that when evaluated at an odd-dimensional sphere, the only layers in 
the tower that are not contractible are those indexed by a prime power. 
Furthermore, in the case of a sphere the tower is finite in $v_k$-pe-
riodic homotopy. It has $k+1$ stages if the sphere is odd dimensional,
and $2(k+1)$ stages if the sphere is even-dimensional.


This is a revised version of a previously uploaded preprint. The paper
has been accepted for publication, and is now in its final form.