  For primes p>=3, Cohen, Moore, and Neisendorfer showed that the exponent of
the p-torsion in the homotopy groups of S^2n+1 is p^n. This was obtained as a
consequence of a thorough analysis of the homotopy theory of Moore spaces.
Anick further developed this for p>=5 by constructing a homotopy fibration
S^2n-1 --> T^2n+1(p^r) --> Loop S^2n+1 whose connecting map is degree p^r on
the bottom cell. A much simpler construction of such a fibration for p>=3 was
given by Gray and the author using new methods. In this paper the new methods
are used to start over, first constructing Anick's fibration for p>=3, and then
using it to obtain the exponent result for spheres.
