On the realization and classification of symmetric algebras as cohomology rings

Abstract:

In this note, we prove that every algebra of the form $E[\sigma , \ldots ,{\sigma _n}] \otimes P[{\rho _1}, \ldots ,{\rho _n}]$ that is an unstable algebra over the $\mod p$ Steenrod algebra, $p$ an odd prime, that satisfies \[ \beta {\sigma _i} = {\rho _i},\quad \deg \rho _i \not \equiv 0(p),\quad i = 1, \ldots ,n,\] arises as the cohomology of at least one topological space. Moreover, we show that the classification of such algebras is implicit in the work of Adams and Wilkerson and Clark and Ewing.