Associativity in two-cell complexes
Brayton Gray
55D99
Dept. of Mathematics, Statistics, and Computer Science
University of Illinois at Chicago
851 South Morgan Street
Chicago, Il 60607-7045
brayton@uic.edu
Let P be the mapping cone of an element in an even stem in the homotopy
groups of spheres localized at an odd prime. Generalizing the case of a
mod p^r Moore space, we show that the smash square of P splits as a wedge
of two iterated suspensions of P. Furthermore, this can be done in a
unique way satisfying certain identities, and if p>3, one of these
identities is an associativity condition. There are two consequences:
1) If E is a (commutative) associative ring spectrum, then E^P is as well
when localized at p>3.
2) A Samelson product can be defined in homotopy with coefficients in P
which will satisfy all the usual identities including the Jacobi identity
if p>3.